Finite Group Databasebeta

← #30625916

Subgroups — #30625916

Order 15616.

1748 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462728C214yesmaximal, complemented
2#21462723C214yesmaximal, complemented
3#21462719C214yesmaximal, complemented
4#21462717C211yesmaximal, characteristic, complemented
5#21462715C211yesmaximal, characteristic, complemented
6#21462714C211yesmaximal, characteristic, complemented
7#21462712C211yesmaximal, characteristic, complemented
8#21462614C214yesmaximal, complemented
9#21462607C214yesmaximal, complemented
10#21462508C214yesmaximal
11#21462495C211yesmaximal, characteristic
12C2×#16840778C211yesmaximal, characteristic
13C2×#16840314C211yesmaximal, characteristic
14#16840814C2218yescomplemented
15#16840814C2218yescomplemented
16#16840813C2214yescomplemented
17#16840813C2214yescomplemented
18#16840812C2214yescomplemented
19#16840812C2214yescomplemented
20#16840812C2218yescomplemented
21#16840811C2214yescomplemented
22#16840811C2218yescomplemented
23#16840811C2214yescomplemented
24Q8×#92420C2214yescomplemented
25Q8×#92420C2214yescomplemented
26#16840786C2218yes—
27#16840786C2218yes—
28#16840783C2214yes—
29#16840783C2212yes—
30#16840782C2214yes—
31#16840782C2212yes—
32#16840779C2214yes—
33#16840779C2212yes—
34#16840778C2211yescharacteristic
35#16840778C2211yescharacteristic
36#16840778C2214yes—
37#16840778C2211yescharacteristic
38#16840778C2211yescharacteristic
39#16840778C2212yes—
40#16840778C2212yes—
41#16840778C2212yes—
42#16840768C2218yes—
43#16840765C2212yes—
44C2×C4⋊#92420C2212yes—
45#16840732C2212yes—
46C2×C22⋊#92420C2212yes—
47#16840726C2218yes—
48C4×#11758217C2212yes—
49C2×C4×#92420C2212yes—
50#16840314C412yes—
51#16840314C412yes—
52#16840314C412yes—
53#16840314C412yes—
54#16840314C2212yes—
55#16840314C2212yes—
56#16840314C2212yes—
57#16840314C2212yes—
58C2×#14519776C411yescharacteristic
59C2×#14519775C411yescharacteristic
60C2×#14519767C412yes—
61C2×#14519756C411yescharacteristic
62C2×#14519754C412yes—
63C2×Q8×Dic61C2211yescharacteristic
64C2×#14519742C2211yescharacteristic
65C2×#14519742C2212yes—
66C2×#14519733C2212yes—
67C22×C4×Dic61C2211yescharacteristic
68C2×C22⋊Q8×C61C411yescharacteristic
69C2×#11758217C2314yes—
70C22×#92420C2314yes—
71#14519844C2314yes—
72#14519844C2318yes—
73#14519844C2318yes—
74C4⋊#92420C2312yes—
75C4⋊#92420C2312yes—
76C4⋊#92420C2314yes—
77C4⋊#92420C2312yes—
78C4⋊#92420C2312yes—
79C4⋊#92420C2318yes—
80C4⋊#92420C2318yes—
81C22⋊#92420C2312yes—
82C22⋊#92420C2312yes—
83C22⋊#92420C2314yes—
84C22⋊#92420C2312yes—
85C22⋊#92420C2312yes—
86C22⋊#92420C2314yes—
87#14519832C2314yes—
88#14519832C2318yes—
89#14519832C2318yes—
90C4×#92420C2312yes—
91C4×#92420C2314yes—
92C4×#92420C2312yes—
93C4×#92420C2312yes—
94C4×#92420C2312yes—
95C4×#92420C2318yes—
96C4×#92420C2318yes—
97C22×Dic122C2×C411yescharacteristic
98C22×Dic122C2×C412yes—
99C23×Dic61C2×C411yescharacteristic
100C23×Dic61C2311yescharacteristic
101#14519776C2×C412yes—
102#14519776C2×C412yes—
103#14519776C2×C412yes—
104#14519776C2×C412yes—
105#14519775C2×C412yes—
106#14519775C2×C412yes—
107#14519775C2×C412yes—
108#14519775C2×C412yes—
109#14519767C2×C414yes—
110#14519767C2×C414yes—
111#14519767C2×C414yes—
112#14519767C2×C414yes—
113#14519756C2×C412yes—
114#14519756C2×C412yes—
115#14519756C2×C412yes—
116#14519756C2×C412yes—
117#14519754C2×C414yes—
118#14519754C2×C414yes—
119#14519754C2×C414yes—
120#14519754C2×C414yes—
121Q8×Dic61C2×C412yes—
122Q8×Dic61C2×C412yes—
123Q8×Dic61C2312yes—
124Q8×Dic61C2312yes—
125#14519742C2×C414yes—
126#14519742C2×C414yes—
127#14519742C2×C412yes—
128#14519742C2×C412yes—
129#14519742C2312yes—
130#14519742C2312yes—
131#14519742C2314yes—
132#14519742C2314yes—
133#14519733C2×C414yes—
134#14519733C2×C414yes—
135#14519733C2314yes—
136#14519733C2314yes—
137C2×C4⋊Dic61C2×C412yes—
138C2×C4⋊Dic61C2×C411yescharacteristic
139C2×#11758196C2×C412yes—
140C2×#11758196C2×C412yes—
141C2×#11758196C2×C412yes—
142C2×#11758196C2×C412yes—
143C2×#11758196C2×C412yes—
144C2×C22⋊Dic61C2×C412yes—
145C2×C4×Dic61C2×C411yescharacteristic
146C2×C4×Dic61C2×C411yescharacteristic
147C2×C4×Dic61C2×C411yescharacteristic
148C2×C4×Dic61C2×C411yescharacteristic
149C2×C4×Dic61C2×C411yescharacteristic
150C2×C4×Dic61C2×C411yescharacteristic
151C2×C4×Dic61C2311yescharacteristic
152C2×C4×Dic61C2311yescharacteristic
153C2×C4×Dic61C2311yescharacteristic
154C2×C4×Dic61C2311yescharacteristic
155C2×C4×Dic61C2311yescharacteristic
156C2×C4×Dic61C2311yescharacteristic
157C2×C4×Dic61C2312yes—
158C2×C4×Dic61C2312yes—
159C22×Q8×C61C2×C411yescharacteristic
160C23×C4×C61C2×C411yescharacteristic
161C22⋊Q8×C61C2×C412yes—
162C22⋊Q8×C61C2×C412yes—
163C22⋊Q8×C61C2×C412yes—
164C22⋊Q8×C61C2×C412yes—
165C2×C4⋊C4×C61C2×C412yes—
166C2×C4⋊C4×C61C2×C411yescharacteristic
167C2×C22⋊C4×C61C2×C412yes—
168C24×C61C22×C411yescharacteristic
169C2×Q8×C61—24no—
170C2×Q8×C61C22×C412yes—
171C2×Q8×C61C22×C412yes—
172C22×C4×C61—24no—
173C22×C4×C61C22×C412yes—
174C22×C4×C61C22×C411yescharacteristic
175C22×C4×C61C22×C411yescharacteristic
176C22×C4×C61C22×C411yescharacteristic
177C22×C4×C61C22×C411yescharacteristic
178C22×C4×C61C22×C411yescharacteristic
179C22×C4×C61C22×C412yes—
180C22×C4×C61C22×C411yescharacteristic
181C22×Dic61—24no—
182C22×Dic61—42no—
183C22×Dic61C22×C411yescharacteristic
184C22×Dic61C22×C411yescharacteristic
185C22×Dic61C22×C411yescharacteristic
186C22×Dic61C22×C411yescharacteristic
187C22×Dic61C22×C411yescharacteristic
188C22×Dic61C22×C411yescharacteristic
189C22×Dic61C22×C412yes—
190C22×Dic61C22×C411yescharacteristic
191C22×Dic61C22×C411yescharacteristic
192C22×Dic61C22×C411yescharacteristic
193C22×Dic61C22×C412yes—
194C22×Dic61C2411yescharacteristic
195C22×Dic61C2411yescharacteristic
196C22×Dic61C2411yescharacteristic
197C2×Dic122—24no—
198C2×Dic122—44no—
199C2×Dic122C22×C414yes—
200C2×Dic122C22×C412yes—
201C2×Dic122C22×C412yes—
202C2×Dic122C22×C414yes—
203#11758217—28no—
204C2×#92420—28no—
205C2×#92420C2414yes—
206C2×#92420C2414yes—
207C2×#92420C2414yes—
208C2×#92420C2414yes—
209C4⋊C4×C61C22×C414yes—
210C4⋊C4×C61C22×C414yes—
211C4⋊C4×C61C22×C412yes—
212C4⋊C4×C61C22×C412yes—
213C22⋊C4×C61C22×C414yes—
214C22⋊C4×C61C22×C414yes—
215C22⋊Dic61C22×C414yes—
216C22⋊Dic61C22×C414yes—
217C4⋊Dic61C22×C414yes—
218C4⋊Dic61C22×C414yes—
219C4⋊Dic61C22×C412yes—
220C4⋊Dic61C22×C412yes—
221#11758196C22×C414yes—
222#11758196C22×C414yes—
223#11758196C22×C414yes—
224#11758196C22×C414yes—
225#11758196C22×C414yes—
226#11758196C22×C414yes—
227#11758196C22×C414yes—
228#11758196C22×C414yes—
229#11758196C22×C414yes—
230#11758196C22×C414yes—
231C4×Dic61C22×C411yescharacteristic
232C4×Dic61C22×C411yescharacteristic
233C4×Dic61C22×C411yescharacteristic
234C4×Dic61C22×C411yescharacteristic
235C4×Dic61C22×C411yescharacteristic
236C4×Dic61C22×C411yescharacteristic
237C4×Dic61C22×C411yescharacteristic
238C4×Dic61C22×C411yescharacteristic
239C4×Dic61C22×C411yescharacteristic
240C4×Dic61C22×C411yescharacteristic
241C4×Dic61C22×C411yescharacteristic
242C4×Dic61C22×C411yescharacteristic
243C4×Dic61C22×C414yes—
244C4×Dic61C22×C414yes—
245C4×Dic61C2411yescharacteristic
246C4×Dic61C2411yescharacteristic
247C4×Dic61C2411yescharacteristic
248C4×Dic61C2414yes—
249C4×Dic61C2414yes—
250C4×Dic61C2411yescharacteristic
251C23×C61—22no—
252C23×C61—22no—
253C23×C61—41no—
254C23×C61C23×C411yescharacteristic
255C23×C61C23×C411yescharacteristic
256C23×C61C23×C411yescharacteristic
257Q8×C61—24no—
258Q8×C61—24no—
259C2×C4×C61—22no—
260C2×C4×C61—24no—
261C2×C4×C61—22no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—42no—
266C2×C4×C61#13114yes—
267C2×C4×C61C23×C411yescharacteristic
268C2×C4×C61C23×C411yescharacteristic
269C2×C4×C61C23×C414yes—
270C2×C4×C61C23×C414yes—
271C2×C4×C61C23×C411yescharacteristic
272C2×C4×C61C23×C411yescharacteristic
273C2×Dic61—22no—
274C2×Dic61—24no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—42no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61C23×C411yescharacteristic
287C2×Dic61C23×C411yescharacteristic
288C2×Dic61C23×C411yescharacteristic
289C2×Dic61C23×C411yescharacteristic
290C2×Dic61C23×C411yescharacteristic
291C2×Dic61C23×C414yes—
292C2×Dic61C23×C414yes—
293C2×Dic61C23×C411yescharacteristic
294C2×Dic61C23×C411yescharacteristic
295C2×Dic612−1+414yes—
296C2×Dic61C2511yescharacteristic
297Dic122—24no—
298Dic122—24no—
299Dic122—28no—
300Dic122—44no—
301#92420—24no—
302#92420—24no—
303#92420—24no—
304#92420—24no—
305#60250—611nomaximal, Sylow
306C22×C61—21no—
307C22×C61—21no—
308C22×C61—21no—
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—44no—
314C22×C61#51912yes—
315C22×C61C2×#13112yes—
316C22×C61C24×C411yescharacteristic
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—24no—
320C4×C61—24no—
321C4×C61C2×#13112yes—
322C4×C61C2×#13112yes—
323Dic61—21no—
324Dic61—21no—
325Dic61—24no—
326Dic61—24no—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61#51912yes—
331Dic61#56812yes—
332Dic61C2×2−1+412yes—
333C2×#533—611no—
334#2986—614no—
335#2985—614no—
336#2984—614no—
337#2983—611no—
338#2983—611no—
339#2982—611no—
340#2982—611no—
341#2962—614no—
342#2960—614no—
343#2924—614no—
344#2920—611no—
345C2×#431—611no—
346C2×C61—22no—
347C2×C61—22no—
348C2×C61—41no—
349C2×C61C2×#51911yescharacteristic
350C2×C61C22×#13111yescharacteristic
351C2×C61C2×#56811yescharacteristic
352#533—612no—
353#533—612no—
354#533—612no—
355#533—612no—
356#533—612no—
357#533—612no—
358#533—612no—
359#533—612no—
360C2×C4⋊Q8—611no—
361C2×C4⋊Q8—611no—
362C2×#125—612no—
363C2×C22⋊Q8—611no—
364C2×C22⋊Q8—612no—
365C2×C22⋊Q8#692311yescharacteristic
366C2×C4×Q8—611no—
367C2×C4×Q8—611no—
368C2×C4×Q8—612no—
369C2×#117—612no—
370C22×C42—611no—
371#446—614no—
372#446—614no—
373#446—618no—
374#446—614no—
375#446—614no—
376#446—618no—
377#446—618no—
378C8×Q8—614no—
379C8×Q8—614no—
380C8×Q8—614no—
381C8×Q8—618no—
382C8×Q8—614no—
383#433—618no—
384#433—618no—
385#432—612no—
386#432—614no—
387#432—612no—
388#432—614no—
389#431—611no—
390#431—611no—
391#431—612no—
392#431—612no—
393#431—612no—
394#431—611no—
395#431—611no—
396#431—614no—
397#431—612no—
398#431—614no—
399#424—618no—
400#423—612no—
401C2×C4⋊C8—612no—
402#407—612no—
403C2×C22⋊C8—612no—
404#405—618no—
405C4×M16—612no—
406C2×C4×C8—612no—
407C61#6025011yescharacteristic, complemented, Sylow
408C22×Q8—611no—
409C22×Q8—612no—
410C22×Q8C2×#692311yescharacteristic
411C23×C4—611no—
412C23×C4—611no—
413C23×C4C2×#692311yescharacteristic
414C2×M16—614no—
415C22×C8—614no—
416C4⋊Q8—612no—
417C4⋊Q8—612no—
418C4⋊Q8—612no—
419C4⋊Q8—612no—
420C4⋊Q8—612no—
421C4⋊Q8—612no—
422C4⋊Q8—612no—
423C4⋊Q8—612no—
424#125—614no—
425#125—614no—
426#125—614no—
427#125—614no—
428C22⋊Q8—612no—
429C22⋊Q8—612no—
430C22⋊Q8—612no—
431C22⋊Q8—612no—
432C22⋊Q8—614no—
433C22⋊Q8—614no—
434C22⋊Q8—614no—
435C22⋊Q8—614no—
436C22⋊Q8C2×#692312yes—
437C22⋊Q8C2×#692312yes—
438C22⋊Q8C2×#692312yes—
439C22⋊Q8C2×#692312yes—
440C4×Q8—612no—
441C4×Q8—612no—
442C4×Q8—612no—
443C4×Q8—612no—
444C4×Q8—612no—
445C4×Q8—612no—
446C4×Q8—614no—
447C4×Q8—614no—
448C4×Q8—612no—
449C4×Q8—612no—
450C4×Q8—614no—
451C4×Q8—614no—
452#117—614no—
453#117—614no—
454#117—614no—
455#117—614no—
456C2×C4⋊C4—612no—
457C2×C4⋊C4—612no—
458C2×C4⋊C4—612no—
459C2×C4⋊C4—612no—
460C2×C4⋊C4—612no—
461C2×C4⋊C4—612no—
462C2×C4⋊C4—611no—
463C2×C4⋊C4C2×#692312yes—
464C2×C4⋊C4C2×#692311yescharacteristic
465C2×C22⋊C4—612no—
466C2×C22⋊C4C2×#692312yes—
467C2×C42—612no—
468C2×C42—612no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42—611no—
477C2×C42—611no—
478C2×C42—611no—
479C2×C42—611no—
480C2×C42—611no—
481C4⋊C8—612no—
482C4⋊C8—612no—
483C4⋊C8—614no—
484C4⋊C8—612no—
485C4⋊C8—612no—
486C4⋊C8—618no—
487C4⋊C8—618no—
488C4⋊C8—614no—
489C4⋊C8—618no—
490C4⋊C8—618no—
491C22⋊C8—612no—
492C22⋊C8—612no—
493C22⋊C8—614no—
494C22⋊C8—612no—
495C22⋊C8—612no—
496C22⋊C8—614no—
497#97—614no—
498#97—618no—
499#97—618no—
500C4×C8—612no—
501C4×C8—614no—
502C4×C8—612no—
503C4×C8—612no—
504C4×C8—612no—
505C4×C8—618no—
506C4×C8—618no—
507C24C22×#692311yescharacteristic
508C2×Q8—24no—
509C2×Q8—614no—
510C2×Q8—612no—
511C2×Q8—612no—
512C2×Q8—614no—
513C2×Q8—1224no—
514C2×Q8—2444no—
515C2×Q8C22×#692312yes—
516C2×Q8C22×#692312yes—
517C22×C4—24no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—612no—
528C22×C4—611no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—612no—
532C22×C4—1224no—
533C22×C4—2442no—
534C22×C4C22×#692311yescharacteristic
535C22×C4C22×#692311yescharacteristic
536C22×C4C22×#692311yescharacteristic
537C22×C4C22×#692311yescharacteristic
538C22×C4C22×#692311yescharacteristic
539C22×C4C22×#692312yes—
540C22×C4C22×#692312yes—
541C22×C4C22×#692311yescharacteristic
542M16—1228no—
543C2×C8—614no—
544C2×C8—614no—
545C2×C8—614no—
546C2×C8—614no—
547C2×C8—1228no—
548C4⋊C4—614no—
549C4⋊C4—614no—
550C4⋊C4—614no—
551C4⋊C4—614no—
552C4⋊C4—614no—
553C4⋊C4—614no—
554C4⋊C4—614no—
555C4⋊C4—614no—
556C4⋊C4—614no—
557C4⋊C4—614no—
558C4⋊C4—614no—
559C4⋊C4—614no—
560C4⋊C4—612no—
561C4⋊C4—612no—
562C4⋊C4C22×#692314yes—
563C4⋊C4C22×#692314yes—
564C4⋊C4C22×#692312yes—
565C4⋊C4C22×#692312yes—
566C22⋊C4—614no—
567C22⋊C4—614no—
568C22⋊C4C22×#692314yes—
569C22⋊C4C22×#692314yes—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—614no—
574C42—614no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—611no—
579C42—611no—
580C42—611no—
581C42—611no—
582C42—611no—
583C42—611no—
584C42—611no—
585C42—611no—
586C42—611no—
587C42—611no—
588C42—614no—
589C42—614no—
590C23—22no—
591C23—22no—
592C23—41no—
593C23C23×#692311yescharacteristic
594C23C23×#692311yescharacteristic
595C23C23×#692311yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—1228no—
599Q8—1224no—
600Q8—1224no—
601Q8—2444no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—614no—
610C2×C4—611no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—611no—
617C2×C4—611no—
618C2×C4—614no—
619C2×C4—614no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1224no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2442no—
633C2×C4#1451986414yes—
634C2×C4C23×#692311yescharacteristic
635C2×C4C23×#692311yescharacteristic
636C2×C4C23×#692314yes—
637C2×C4C23×#692314yes—
638C2×C4C23×#692311yescharacteristic
639C2×C4C23×#692311yescharacteristic
640C8—1224no—
641C8—1224no—
642C8—1224no—
643C8—1224no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—44no—
652C22#1684083012yes—
653C22C2×#1451986412yes—
654C22C24×#692311yescharacteristic
655C4—21no—
656C4—21no—
657C4—24no—
658C4—24no—
659C4—612no—
660C4—612no—
661C4—612no—
662C4—1221no—
663C4—1221no—
664C4—1224no—
665C4—1221no—
666C4—1221no—
667C4—1221no—
668C4—1224no—
669C4C2×#1451986412yes—
670C4C2×#1451986412yes—
671C2—22no—
672C2—22no—
673C2—41no—
674C2C2×#1684083011yescharacteristic
675C2C22×#1451986411yescharacteristic
676C2D61⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC2×C22⋊Q8×C61C4
prime radicalC24×C61C22×C4
cosocleC2×Dic61C25
hypercenterC2×C22⋊Q8#6923
hypocenterC61#60250
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21462728C2semidirect
2. #21462723C2semidirect
3. #21462719C2semidirect
4. #21462717C2semidirect
5. #21462715C2semidirect
6. #21462714C2semidirect
7. #21462712C2semidirect
8. #21462614C2semidirect
9. #21462607C2semidirect
14. #16840814C22semidirect
15. #16840814C22semidirect
16. #16840813C22semidirect
17. #16840813C22semidirect
18. #16840812C22semidirect
19. #16840812C22semidirect
20. #16840812C22semidirect
21. #16840811C22semidirect
22. #16840811C22semidirect
23. #16840811C22semidirect
24. Q8×#92420C22semidirect
25. Q8×#92420C22semidirect
407. C61#60250semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214627282
#214627232
#214627192
#214627172
#214627152
#214627142
#214627122
#214626142
#214626072
#214625082
#214624952
C2×#168407782
C2×#168403142
#6025061