Finite Group Databasebeta

← #30620426

Subgroups — #30620426

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462620C212yesmaximal, complemented
2#21462013C212yesmaximal, complemented
3#21462013C212yesmaximal, complemented
4#21462011C212yesmaximal, complemented
5#21461684C211yesmaximal, characteristic
6#21461684C211yesmaximal, characteristic
7#21461683C211yesmaximal, characteristic
8#21461569C211yesmaximal, characteristic
9#21461569C211yesmaximal, characteristic
10C4×#14519837C211yesmaximal, characteristic
11C2×#16840466C211yesmaximal, characteristic
12#16840814—22no—
13#16840814—22no—
14#16840812—22no—
15#16840811—22no—
16#16840765C2212yes—
17#16840765C2212yes—
18#16840765C2212yes—
19C2×#14519837C2211yescharacteristic
20C2×#14519837C2211yescharacteristic
21C2×#14519837C2211yescharacteristic
22C2×#14519837C2211yescharacteristic
23C4×#11758217C2212yes—
24C4×#11758217C2212yes—
25C4×#11758217C2212yes—
26#16840642C2212yes—
27#16840642C2212yes—
28#16840642C2212yes—
29#16840642C2212yes—
30#16840642C2212yes—
31#16840642C2212yes—
32#16840466—22no—
33#16840466—22no—
34#16840466—22no—
35#16840466—22no—
36C2×#14519775C2211yescharacteristic
37C2×#14519767C2211yescharacteristic
38C2×#14519767C2211yescharacteristic
39C2×#14519755C411yescharacteristic
40C2×#14519755C411yescharacteristic
41C2×#14519754C411yescharacteristic
42C2×#14519754C411yescharacteristic
43C2×#14519753C411yescharacteristic
44C2×#14519742C2211yescharacteristic
45C2×C4×C22⋊D61C2211yescharacteristic
46C2×#14519737C2211yescharacteristic
47C2×#14519737C2211yescharacteristic
48C22×#11758196C411yescharacteristic
49C2×C4⋊C4×D61C411yescharacteristic
50C2×C22⋊Q8×C61C411yescharacteristic
51C2×#11758217C2312yes—
52C2×#11758217C2312yes—
53C2×#11758217C2312yes—
54C2×#11758217C2312yes—
55#14519844—22no—
56#14519844—22no—
57#14519844—22no—
58C4⋊#92420—22no—
59C4⋊#92420—22no—
60C4⋊#92420—22no—
61#14519837—22no—
62#14519837—22no—
63#14519837—22no—
64#14519837Q814yescomplemented
65#14519832—22no—
66#14519832—22no—
67#14519832—22no—
68C4×#92420—22no—
69C4×#92420—22no—
70C4×#92420—22no—
71C22×Dic122C2×C411yescharacteristic
72C22×Dic122C2×C411yescharacteristic
73C22×C22⋊D61C2311yescharacteristic
74C23×Dic61C2×C411yescharacteristic
75C23×Dic61C2×C411yescharacteristic
76C22×C4×D61C2×C411yescharacteristic
77C22×C4×D61C2×C411yescharacteristic
78C22×C4×D61C2×C411yescharacteristic
79#14519775—22no—
80#14519775—22no—
81#14519767—22no—
82#14519767—22no—
83#14519767—22no—
84#14519767—22no—
85#14519755—22no—
86#14519755—22no—
87#14519755—22no—
88#14519755—22no—
89#14519754—22no—
90#14519754—22no—
91#14519754—22no—
92#14519754—22no—
93#14519753—22no—
94#14519753—22no—
95#14519742D412yes—
96#14519742D412yes—
97#14519742D412yes—
98#14519742D412yes—
99C4×C22⋊D61—22no—
100C4×C22⋊D61—22no—
101#14519737—22no—
102#14519737—22no—
103#14519737—22no—
104#14519737—22no—
105C2×C4⋊Dic61C2×C411yescharacteristic
106C2×C4⋊Dic61C2×C411yescharacteristic
107C2×#11758196—21no—
108C2×#11758196—21no—
109C2×#11758196—21no—
110C2×#11758196—21no—
111C2×#11758196C2×C411yescharacteristic
112C2×#11758196C2×C411yescharacteristic
113C2×#11758196C2×C411yescharacteristic
114C2×#11758196C2×C411yescharacteristic
115C2×#11758196C2×C411yescharacteristic
116C2×#11758196C2×C411yescharacteristic
117C2×#11758196C2×C411yescharacteristic
118C2×#11758196C2×C411yescharacteristic
119C2×#11758196C2311yescharacteristic
120C2×#11758196C2311yescharacteristic
121C2×#11758196C2311yescharacteristic
122C4⋊C4×D61—21no—
123C4⋊C4×D61—21no—
124C4⋊C4×D61—21no—
125C4⋊C4×D61—21no—
126C2×C22⋊Dic61C2×C411yescharacteristic
127C2×#11758198C2×C411yescharacteristic
128C2×#11758198C2×C411yescharacteristic
129C2×#11758198C2×C411yescharacteristic
130C2×C4×Dic61C2311yescharacteristic
131C2×C4×Dic61C2311yescharacteristic
132C2×C4×Dic61C2311yescharacteristic
133C22×Q8×C61C2×C411yescharacteristic
134C23×C4×C61C2×C411yescharacteristic
135C22⋊Q8×C61—22no—
136C22⋊Q8×C61—22no—
137C2×C4⋊C4×C61C2×C411yescharacteristic
138C2×C4⋊C4×C61C2×C411yescharacteristic
139C2×C4⋊C4×C61C2×C411yescharacteristic
140C2×C22⋊C4×C61C2×C411yescharacteristic
141C2×C22⋊C4×C61C2×C411yescharacteristic
142C24×C61C22×C411yescharacteristic
143C23×D61C22×C411yescharacteristic
144C2×Q8×C61—22no—
145C2×Q8×C61—81no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—81no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C22×C411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61C2×Q812yes—
158C2×C22⋊D61#4112yes—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—42no—
164C22×Dic61—81no—
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167C22×Dic61C22×C411yescharacteristic
168C22×Dic61C22×C411yescharacteristic
169C22×Dic61C22×C411yescharacteristic
170C22×Dic61C22×C411yescharacteristic
171C22×Dic61C2411yescharacteristic
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—42no—
179C2×C4×D61—81no—
180C2×C4×D61—81no—
181C2×Dic122—22no—
182C2×Dic122—42no—
183C2×Dic122—81no—
184C2×Dic122C22⋊C412yes—
185C2×Dic122C22⋊C412yes—
186#11758217—22no—
187#11758217—22no—
188#11758217—22no—
189#11758217—22no—
190#11758217—22no—
191#11758217—22no—
192#11758217—22no—
193#11758217C2×Q814yes—
194C2×#92420—22no—
195C2×#92420—22no—
196C2×#92420—22no—
197C2×#92420—22no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61C22⋊C412yes—
202C4⋊C4×C61C22⋊C412yes—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊Dic61—22no—
206#11758198—22no—
207#11758198—22no—
208#11758198—22no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—22no—
212#11758196—21no—
213#11758196—21no—
214#11758196—22no—
215#11758196—21no—
216#11758196—21no—
217#11758196—21no—
218#11758196—21no—
219#11758196—21no—
220#11758196—21no—
221#11758196—22no—
222#11758196—22no—
223#11758196—22no—
224#11758196—22no—
225#11758196C22⋊C411yescharacteristic
226#11758196C22⋊C411yescharacteristic
227#11758196C22⋊C411yescharacteristic
228#11758196C22⋊C411yescharacteristic
229#11758196C22⋊C412yes—
230#11758196C22⋊C412yes—
231C4×Dic61C2×D412yes—
232C4×Dic61C2×D412yes—
233C4×Dic61C2×D412yes—
234C4×Dic61C2×D412yes—
235C4×Dic61C2×D412yes—
236C4×Dic61C2×D412yes—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61C4×Q812yes—
242C23×C61C23×C411yescharacteristic
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—41no—
246C22×D61—41no—
247C22×D61C4×Q812yes—
248Q8×C61—82no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—81no—
259C2×C4×C61—81no—
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C412yes—
263C2×C4×C61C2×C22⋊C412yes—
264C22⋊D61—24no—
265C22⋊D61—24no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—81no—
278C2×Dic61—81no—
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C412yes—
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C22×D412yes—
285C2×Dic61C22×Q811yescharacteristic
286C2×Dic61C2×#4111yescharacteristic
287C2×Dic612−1+412yes—
288C4×D61—42no—
289C4×D61—42no—
290C4×D61—81no—
291C4×D61—81no—
292C4×D61—81no—
293C4×D61—81no—
294Dic122—24no—
295Dic122—42no—
296Dic122—82no—
297#92420—24no—
298#92420—42no—
299#92420—42no—
300#92420—42no—
301#22515—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—22no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61C22×C22⋊C411yescharacteristic
310C22×C61C2×C4×Q811yescharacteristic
311C22×C61#51911yescharacteristic
312C2×D61—22no—
313C2×D61—42no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—42no—
317C4×C61—22no—
318C4×C61—22no—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—41no—
324Dic61—22no—
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
331Dic61#41312yes—
332Dic61#54812yes—
333Dic61D4×Q812yes—
334C2×#554—611no—
335Q8×M16—612no—
336#1998—612no—
337#1998—612no—
338#1997—612no—
339#1800—611no—
340#1800—611no—
341#1800—611no—
342C4×#100—611no—
343C4×#100—611no—
344C4×#100—611no—
345C2×C61—21no—
346C2×C61—21no—
347C2×C61—42no—
348C2×C61Q8×C22⋊C411yescharacteristic
349C2×C61C2×#41311yescharacteristic
350C2×C61#288811yescharacteristic
351D61—42no—
352#554—1222no—
353#554—1222no—
354#554—1222no—
355#554—1222no—
356C2×C4⋊Q8—611no—
357C2×#125—611no—
358C2×#125—611no—
359C2×C22⋊Q8—611no—
360C2×C22⋊Q8—611no—
361C2×C22⋊Q8—611no—
362C2×C22⋊Q8—611no—
363C2×C22⋊Q8—611no—
364C2×C22⋊Q8#692311yescharacteristic
365C2×C4×Q8—611no—
366C2×C4×D4—611no—
367C2×C4×D4—611no—
368C2×C4×D4—611no—
369C22×C4⋊C4—611no—
370C22×C4⋊C4—611no—
371#446—1222no—
372#446—1222no—
373#446—1222no—
374C8×Q8—1222no—
375#423—612no—
376#423—612no—
377#423—612no—
378C2×#100—611no—
379C2×#100—611no—
380C2×#100—611no—
381C2×#100—611no—
382C4×M16—612no—
383C4×M16—612no—
384C4×M16—612no—
385#343—612no—
386#343—612no—
387#343—612no—
388#343—612no—
389#343—612no—
390#343—612no—
391C61#2251511yescharacteristic, complemented, Sylow
392C22×Q8—611no—
393C22×Q8—611no—
394C22×Q8C2×#692311yescharacteristic
395C22×D4—611no—
396C23×C4—611no—
397C23×C4—611no—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C23×C4C2×#692311yescharacteristic
402C2×M16—612no—
403C2×M16—612no—
404C2×M16—612no—
405C2×M16—612no—
406C4⋊Q8—1222no—
407C4⋊Q8—1222no—
408#125—1222no—
409#125—1222no—
410#125—1222no—
411#125—1222no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—1222no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417C22⋊Q8—1222no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C4×Q8—612no—
425C4×Q8—612no—
426C4×Q8—612no—
427C4×Q8—612no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4C2×#692311yescharacteristic
456C2×C4⋊C4C2×#692311yescharacteristic
457C2×C4⋊C4C2×#692311yescharacteristic
458C2×C22⋊C4—611no—
459C2×C22⋊C4—611no—
460C2×C22⋊C4—611no—
461C2×C22⋊C4—611no—
462C2×C22⋊C4C2×#692311yescharacteristic
463C2×C22⋊C4C2×#692311yescharacteristic
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C4⋊C8—1222no—
468C4⋊C8—1222no—
469C4⋊C8—1222no—
470C4⋊C8—1222no—
471C4⋊C8—1222no—
472C4⋊C8—1222no—
473#100—614no—
474#100—1222no—
475#100—1222no—
476#100—1222no—
477#97—1222no—
478#97—1222no—
479#97—1222no—
480C4×C8—1222no—
481C4×C8—1222no—
482C4×C8—1222no—
483C24—611no—
484C24C22×#692311yescharacteristic
485C2×Q8—22no—
486C2×Q8—81no—
487C2×Q8—612no—
488C2×Q8—612no—
489C2×Q8—1222no—
490C2×Q8—4881no—
491C2×Q8—2442no—
492C2×D4—612no—
493C2×D4—612no—
494C2×D4—1222no—
495C2×D4—1222no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—81no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—4881no—
517C22×C4—4881no—
518C22×C4—4881no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4C22×#692311yescharacteristic
522C22×C4C22×#692311yescharacteristic
523C22×C4C22×#692311yescharacteristic
524C22×C4C22×#692311yescharacteristic
525C22×C4C22×#692311yescharacteristic
526C22×C4C22×#692311yescharacteristic
527M16—614no—
528M16—1222no—
529M16—1222no—
530M16—1222no—
531M16—1222no—
532M16—1222no—
533M16—1222no—
534M16—1222no—
535C2×C8—1222no—
536C2×C8—1222no—
537C2×C8—1222no—
538C2×C8—1222no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—611no—
543C4⋊C4—611no—
544C4⋊C4—611no—
545C4⋊C4—611no—
546C4⋊C4—612no—
547C4⋊C4—612no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1222no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C4⋊C4—1221no—
562C4⋊C4—1221no—
563C4⋊C4—1222no—
564C4⋊C4C22⋊#692312yes—
565C4⋊C4C22⋊#692312yes—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—1222no—
569C22⋊C4—1222no—
570C22⋊C4—1222no—
571C22⋊C4—1222no—
572C42—612no—
573C42—612no—
574C42—612no—
575C42—612no—
576C42—612no—
577C42—612no—
578C23—21no—
579C23—21no—
580C23—41no—
581C23—41no—
582C23—612no—
583C23—1221no—
584C23—1221no—
585C23—2441no—
586C23—2441no—
587C23Q8×#692312yes—
588C23C23×#692311yescharacteristic
589Q8—82no—
590Q8—1224no—
591Q8—4882no—
592Q8—2442no—
593D4—1224no—
594D4—1224no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—611no—
607C2×C4—612no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—4881no—
620C2×C4—4881no—
621C2×C4—4881no—
622C2×C4—4881no—
623C2×C4—4881no—
624C2×C4—4881no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2442no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4C2×C22⋊#692312yes—
635C2×C4C2×C22⋊#692312yes—
636C2×C4C2×C22⋊#692311yescharacteristic
637C2×C4C2×C22⋊#692311yescharacteristic
638C8—1224no—
639C8—2442no—
640C8—2442no—
641C8—2442no—
642C22—21no—
643C22—21no—
644C22—22no—
645C22—42no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—1222no—
650C22—2442no—
651C22—2442no—
652C22—2441no—
653C22—2441no—
654C22C22×C22⋊#692311yescharacteristic
655C22C2×Q8×#692311yescharacteristic
656C22#1684083011yescharacteristic
657C4—22no—
658C4—21no—
659C4—21no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—612no—
665C4—612no—
666C4—612no—
667C4—1221no—
668C4—1222no—
669C4—1221no—
670C4—1221no—
671C4—2441no—
672C4—2441no—
673C4—2441no—
674C2—21no—
675C2—21no—
676C2—42no—
677C2—2442no—
678C2Q8×C22⋊#692311yescharacteristic
679C2D61⋊#41311yescharacteristic
680C2#2146242911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C23
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#22515
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21462620C2semidirect
2. #21462013C2semidirect
3. #21462013C2semidirect
4. #21462011C2semidirect
64. #14519837Q8semidirect
391. C61#22515semidirect

Maximal subgroups

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

subgroupindex
#214626202
#214620132
#214620112
#214616842
#214616832
#214615692
C4×#145198372
C2×#168404662
#2251561