Finite Group Databasebeta

← #16808822

Subgroups — #16808822

Order 3328.

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#13131745C214yesmaximal, complemented
2#13131740C214yesmaximal, complemented
3#13131736C214yesmaximal, complemented
4#13131734C211yesmaximal, characteristic, complemented
5#13131732C211yesmaximal, characteristic, complemented
6#13131731C211yesmaximal, characteristic, complemented
7#13131729C211yesmaximal, characteristic, complemented
8#13131631C214yesmaximal, complemented
9#13131624C214yesmaximal, complemented
10#13131525C214yesmaximal
11#13131512C211yesmaximal, characteristic
12C2×#11719119C211yesmaximal, characteristic
13C2×#11719327C211yesmaximal, characteristic
14C2×C13×C22⋊Q8C411yescharacteristic
15C2×Q8×Dic13C2211yescharacteristic
16C2×#88198C411yescharacteristic
17C2×#88178C411yescharacteristic
18C22×C4×Dic13C2211yescharacteristic
19#11719327C412yes—
20#11719327C412yes—
21#11719327C412yes—
22#11719327C412yes—
23#11719327C2212yes—
24#11719327C2212yes—
25#11719327C2212yes—
26#11719327C2212yes—
27C2×#88143C412yes—
28C2×#88142C411yescharacteristic
29C2×#88141C2211yescharacteristic
30C2×#88141C2212yes—
31C2×#88132C412yes—
32C2×#88131C2212yes—
33#11719169C2212yes—
34C2×C22⋊#1057C2212yes—
35#11719158C2214yescomplemented
36#11719158C2214yescomplemented
37Q8×#1057C2214yescomplemented
38Q8×#1057C2214yescomplemented
39#11719130C2214yes—
40#11719130C2212yes—
41#11719129C2212yes—
42#11719128C2214yes—
43#11719128C2212yes—
44#11719125C2214yes—
45#11719125C2212yes—
46C4×#6122C2212yes—
47#11719119C2214yes—
48#11719119C2211yescharacteristic
49#11719119C2211yescharacteristic
50#11719119C2211yescharacteristic
51#11719119C2211yescharacteristic
52#11719119C2212yes—
53#11719119C2212yes—
54#11719119C2212yes—
55C2×C4⋊#1057C2212yes—
56C2×C4×#1057C2212yes—
57#11719080C2214yescomplemented
58#11719080C2214yescomplemented
59#11719080C2218yescomplemented
60#11719078C2218yescomplemented
61#11719078C2218yescomplemented
62#11719074C2214yescomplemented
63#11719074C2218yescomplemented
64#11719074C2214yescomplemented
65#11719068C2218yes—
66#11719068C2218yes—
67#11719067C2218yes—
68#11719062C2218yes—
69C22×Q8×C13C2×C411yescharacteristic
70C23×C4×C13C2×C411yescharacteristic
71C23×Dic13C2×C411yescharacteristic
72C23×Dic13C2311yescharacteristic
73C22×Dic26C2×C411yescharacteristic
74C22×Dic26C2×C412yes—
75C2×#6122C2314yes—
76C22×#1057C2314yes—
77C13×C22⋊Q8C2×C412yes—
78C13×C22⋊Q8C2×C412yes—
79C13×C22⋊Q8C2×C412yes—
80C13×C22⋊Q8C2×C412yes—
81C2×C13×C4⋊C4C2×C411yescharacteristic
82C2×C13×C4⋊C4C2×C412yes—
83C2×C13×C22⋊C4C2×C412yes—
84C2×C22⋊Dic13C2×C412yes—
85Q8×Dic13C2×C412yes—
86Q8×Dic13C2×C412yes—
87Q8×Dic13C2312yes—
88Q8×Dic13C2312yes—
89#88198C2×C412yes—
90#88198C2×C412yes—
91#88198C2×C412yes—
92#88198C2×C412yes—
93C2×C4⋊Dic13C2×C412yes—
94C2×C4⋊Dic13C2×C411yescharacteristic
95#88178C2×C412yes—
96#88178C2×C412yes—
97#88178C2×C412yes—
98#88178C2×C412yes—
99C2×#6101C2×C412yes—
100C2×#6101C2×C412yes—
101C2×#6101C2×C412yes—
102C2×#6101C2×C412yes—
103C2×#6101C2×C412yes—
104C2×C4×Dic13C2×C411yescharacteristic
105C2×C4×Dic13C2×C411yescharacteristic
106C2×C4×Dic13C2×C411yescharacteristic
107C2×C4×Dic13C2×C411yescharacteristic
108C2×C4×Dic13C2×C411yescharacteristic
109C2×C4×Dic13C2×C411yescharacteristic
110C2×C4×Dic13C2312yes—
111C2×C4×Dic13C2312yes—
112C2×C4×Dic13C2311yescharacteristic
113C2×C4×Dic13C2311yescharacteristic
114C2×C4×Dic13C2311yescharacteristic
115C2×C4×Dic13C2311yescharacteristic
116C2×C4×Dic13C2311yescharacteristic
117C2×C4×Dic13C2311yescharacteristic
118#88143C2×C414yes—
119#88143C2×C414yes—
120#88143C2×C414yes—
121#88143C2×C414yes—
122#88142C2×C412yes—
123#88142C2×C412yes—
124#88142C2×C412yes—
125#88142C2×C412yes—
126#88141C2×C412yes—
127#88141C2×C412yes—
128#88141C2×C414yes—
129#88141C2×C414yes—
130#88141C2314yes—
131#88141C2312yes—
132#88141C2312yes—
133#88141C2314yes—
134#88132C2×C414yes—
135#88132C2×C414yes—
136#88132C2×C414yes—
137#88132C2×C414yes—
138#88131C2×C414yes—
139#88131C2×C414yes—
140#88131C2314yes—
141#88131C2314yes—
142C22⋊#1057C2314yes—
143C22⋊#1057C2312yes—
144C22⋊#1057C2312yes—
145C22⋊#1057C2314yes—
146C22⋊#1057C2312yes—
147C22⋊#1057C2312yes—
148#88112C2318yes—
149#88112C2318yes—
150#88112C2314yes—
151#88110C2318yes—
152#88110C2318yes—
153#88110C2314yes—
154C4⋊#1057C2318yes—
155C4⋊#1057C2318yes—
156C4⋊#1057C2312yes—
157C4⋊#1057C2312yes—
158C4⋊#1057C2314yes—
159C4⋊#1057C2312yes—
160C4⋊#1057C2312yes—
161C4×#1057C2318yes—
162C4×#1057C2318yes—
163C4×#1057C2314yes—
164C4×#1057C2312yes—
165C4×#1057C2312yes—
166C4×#1057C2312yes—
167C4×#1057C2312yes—
168#60250—131nomaximal, Sylow
169C24×C13C22×C411yescharacteristic
170C2×Q8×C13—24no—
171C2×Q8×C13C22×C412yes—
172C2×Q8×C13C22×C412yes—
173C22×C4×C13—24no—
174C22×C4×C13C22×C411yescharacteristic
175C22×C4×C13C22×C411yescharacteristic
176C22×C4×C13C22×C411yescharacteristic
177C22×C4×C13C22×C411yescharacteristic
178C22×C4×C13C22×C411yescharacteristic
179C22×C4×C13C22×C412yes—
180C22×C4×C13C22×C412yes—
181C22×C4×C13C22×C411yescharacteristic
182C22×Dic13—24no—
183C22×Dic13—42no—
184C22×Dic13C22×C411yescharacteristic
185C22×Dic13C22×C411yescharacteristic
186C22×Dic13C22×C411yescharacteristic
187C22×Dic13C22×C411yescharacteristic
188C22×Dic13C22×C411yescharacteristic
189C22×Dic13C22×C411yescharacteristic
190C22×Dic13C22×C412yes—
191C22×Dic13C22×C411yescharacteristic
192C22×Dic13C22×C411yescharacteristic
193C22×Dic13C22×C411yescharacteristic
194C22×Dic13C22×C412yes—
195C22×Dic13C2411yescharacteristic
196C22×Dic13C2411yescharacteristic
197C22×Dic13C2411yescharacteristic
198C2×Dic26—24no—
199C2×Dic26—44no—
200C2×Dic26C22×C414yes—
201C2×Dic26C22×C412yes—
202C2×Dic26C22×C412yes—
203C2×Dic26C22×C414yes—
204#6122—28no—
205C2×#1057—28no—
206C2×#1057C2414yes—
207C2×#1057C2414yes—
208C2×#1057C2414yes—
209C2×#1057C2414yes—
210C13×C4⋊C4C22×C412yes—
211C13×C4⋊C4C22×C412yes—
212C13×C4⋊C4C22×C414yes—
213C13×C4⋊C4C22×C414yes—
214C13×C22⋊C4C22×C414yes—
215C13×C22⋊C4C22×C414yes—
216C22⋊Dic13C22×C414yes—
217C22⋊Dic13C22×C414yes—
218C4⋊Dic13C22×C414yes—
219C4⋊Dic13C22×C414yes—
220C4⋊Dic13C22×C412yes—
221C4⋊Dic13C22×C412yes—
222#6101C22×C414yes—
223#6101C22×C414yes—
224#6101C22×C414yes—
225#6101C22×C414yes—
226#6101C22×C414yes—
227#6101C22×C414yes—
228#6101C22×C414yes—
229#6101C22×C414yes—
230#6101C22×C414yes—
231#6101C22×C414yes—
232C4×Dic13C22×C414yes—
233C4×Dic13C22×C414yes—
234C4×Dic13C22×C411yescharacteristic
235C4×Dic13C22×C411yescharacteristic
236C4×Dic13C22×C411yescharacteristic
237C4×Dic13C22×C411yescharacteristic
238C4×Dic13C22×C411yescharacteristic
239C4×Dic13C22×C411yescharacteristic
240C4×Dic13C22×C411yescharacteristic
241C4×Dic13C22×C411yescharacteristic
242C4×Dic13C22×C411yescharacteristic
243C4×Dic13C22×C411yescharacteristic
244C4×Dic13C22×C411yescharacteristic
245C4×Dic13C22×C411yescharacteristic
246C4×Dic13C2411yescharacteristic
247C4×Dic13C2411yescharacteristic
248C4×Dic13C2411yescharacteristic
249C4×Dic13C2414yes—
250C4×Dic13C2414yes—
251C4×Dic13C2411yescharacteristic
252C2×#533—131no—
253#2986—134no—
254#2985—134no—
255#2984—134no—
256#2983—131no—
257#2983—131no—
258#2982—131no—
259#2982—131no—
260#2962—134no—
261#2960—134no—
262#2924—134no—
263#2920—131no—
264C2×#431—131no—
265C23×C13—22no—
266C23×C13—22no—
267C23×C13—41no—
268C23×C13C23×C411yescharacteristic
269C23×C13C23×C411yescharacteristic
270C23×C13C23×C411yescharacteristic
271Q8×C13—24no—
272Q8×C13—24no—
273C2×C4×C13—22no—
274C2×C4×C13—24no—
275C2×C4×C13—22no—
276C2×C4×C13—22no—
277C2×C4×C13—22no—
278C2×C4×C13—22no—
279C2×C4×C13—42no—
280C2×C4×C13#13114yes—
281C2×C4×C13C23×C411yescharacteristic
282C2×C4×C13C23×C411yescharacteristic
283C2×C4×C13C23×C414yes—
284C2×C4×C13C23×C414yes—
285C2×C4×C13C23×C411yescharacteristic
286C2×C4×C13C23×C411yescharacteristic
287C2×Dic13—22no—
288C2×Dic13—24no—
289C2×Dic13—22no—
290C2×Dic13—22no—
291C2×Dic13—22no—
292C2×Dic13—22no—
293C2×Dic13—22no—
294C2×Dic13—22no—
295C2×Dic13—22no—
296C2×Dic13—42no—
297C2×Dic13—41no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300C2×Dic13C23×C411yescharacteristic
301C2×Dic13C23×C411yescharacteristic
302C2×Dic13C23×C411yescharacteristic
303C2×Dic13C23×C411yescharacteristic
304C2×Dic13C23×C411yescharacteristic
305C2×Dic13C23×C414yes—
306C2×Dic13C23×C414yes—
307C2×Dic13C23×C411yescharacteristic
308C2×Dic13C23×C411yescharacteristic
309C2×Dic132−1+414yes—
310C2×Dic13C2511yescharacteristic
311Dic26—24no—
312Dic26—24no—
313Dic26—28no—
314Dic26—44no—
315#1057—24no—
316#1057—24no—
317#1057—24no—
318#1057—24no—
319#533—132no—
320#533—132no—
321#533—132no—
322#533—132no—
323#533—132no—
324#533—132no—
325#533—132no—
326#533—132no—
327C2×C4⋊Q8—131no—
328C2×C4⋊Q8—131no—
329C2×#125—132no—
330C2×C22⋊Q8—131no—
331C2×C22⋊Q8—132no—
332C2×C22⋊Q8#26111yescharacteristic
333C2×C4×Q8—131no—
334C2×C4×Q8—131no—
335C2×C4×Q8—132no—
336C2×#117—132no—
337C22×C42—131no—
338#446—138no—
339#446—134no—
340#446—134no—
341#446—138no—
342#446—134no—
343#446—134no—
344#446—138no—
345C8×Q8—134no—
346C8×Q8—138no—
347C8×Q8—134no—
348C8×Q8—134no—
349C8×Q8—134no—
350#433—138no—
351#433—138no—
352#432—132no—
353#432—132no—
354#432—134no—
355#432—134no—
356#431—131no—
357#431—131no—
358#431—132no—
359#431—132no—
360#431—132no—
361#431—131no—
362#431—131no—
363#431—132no—
364#431—134no—
365#431—134no—
366#424—138no—
367#423—132no—
368C2×C4⋊C8—132no—
369#407—132no—
370C2×C22⋊C8—132no—
371#405—138no—
372C4×M16—132no—
373C2×C4×C8—132no—
374C22×C13—21no—
375C22×C13—21no—
376C22×C13—21no—
377C22×C13—21no—
378C22×C13—21no—
379C22×C13—21no—
380C22×C13—21no—
381C22×C13—44no—
382C22×C13#51912yes—
383C22×C13C2×#13112yes—
384C22×C13C24×C411yescharacteristic
385C4×C13—21no—
386C4×C13—21no—
387C4×C13—24no—
388C4×C13—24no—
389C4×C13C2×#13112yes—
390C4×C13C2×#13112yes—
391Dic13—21no—
392Dic13—21no—
393Dic13—24no—
394Dic13—24no—
395Dic13—21no—
396Dic13—21no—
397Dic13—21no—
398Dic13#51912yes—
399Dic13#56812yes—
400Dic13C2×2−1+412yes—
401C22×Q8—131no—
402C22×Q8—132no—
403C22×Q8C2×#26111yescharacteristic
404C23×C4—131no—
405C23×C4—131no—
406C23×C4C2×#26111yescharacteristic
407C2×M16—134no—
408C22×C8—134no—
409C4⋊Q8—132no—
410C4⋊Q8—132no—
411C4⋊Q8—132no—
412C4⋊Q8—132no—
413C4⋊Q8—132no—
414C4⋊Q8—132no—
415C4⋊Q8—132no—
416C4⋊Q8—132no—
417#125—134no—
418#125—134no—
419#125—134no—
420#125—134no—
421C22⋊Q8—132no—
422C22⋊Q8—132no—
423C22⋊Q8—132no—
424C22⋊Q8—132no—
425C22⋊Q8—134no—
426C22⋊Q8—134no—
427C22⋊Q8—134no—
428C22⋊Q8—134no—
429C22⋊Q8C2×#26112yes—
430C22⋊Q8C2×#26112yes—
431C22⋊Q8C2×#26112yes—
432C22⋊Q8C2×#26112yes—
433C4×Q8—132no—
434C4×Q8—132no—
435C4×Q8—132no—
436C4×Q8—132no—
437C4×Q8—132no—
438C4×Q8—132no—
439C4×Q8—134no—
440C4×Q8—134no—
441C4×Q8—132no—
442C4×Q8—132no—
443C4×Q8—134no—
444C4×Q8—134no—
445#117—134no—
446#117—134no—
447#117—134no—
448#117—134no—
449C2×C4⋊C4—132no—
450C2×C4⋊C4—132no—
451C2×C4⋊C4—132no—
452C2×C4⋊C4—132no—
453C2×C4⋊C4—132no—
454C2×C4⋊C4—132no—
455C2×C4⋊C4—131no—
456C2×C4⋊C4C2×#26112yes—
457C2×C4⋊C4C2×#26111yescharacteristic
458C2×C22⋊C4—132no—
459C2×C22⋊C4C2×#26112yes—
460C2×C42—132no—
461C2×C42—132no—
462C2×C42—131no—
463C2×C42—131no—
464C2×C42—131no—
465C2×C42—131no—
466C2×C42—131no—
467C2×C42—131no—
468C2×C42—131no—
469C2×C42—131no—
470C2×C42—131no—
471C2×C42—131no—
472C2×C42—131no—
473C2×C42—131no—
474C4⋊C8—132no—
475C4⋊C8—132no—
476C4⋊C8—134no—
477C4⋊C8—132no—
478C4⋊C8—132no—
479C4⋊C8—138no—
480C4⋊C8—138no—
481C4⋊C8—134no—
482C4⋊C8—138no—
483C4⋊C8—138no—
484C22⋊C8—132no—
485C22⋊C8—132no—
486C22⋊C8—134no—
487C22⋊C8—132no—
488C22⋊C8—132no—
489C22⋊C8—134no—
490#97—134no—
491#97—138no—
492#97—138no—
493C4×C8—132no—
494C4×C8—134no—
495C4×C8—132no—
496C4×C8—132no—
497C4×C8—132no—
498C4×C8—138no—
499C4×C8—138no—
500C2×C13—22no—
501C2×C13—22no—
502C2×C13—41no—
503C2×C13C2×#51911yescharacteristic
504C2×C13C22×#13111yescharacteristic
505C2×C13C2×#56811yescharacteristic
506C24C22×#26111yescharacteristic
507C2×Q8—24no—
508C2×Q8—134no—
509C2×Q8—132no—
510C2×Q8—132no—
511C2×Q8—134no—
512C2×Q8—264no—
513C2×Q8—524no—
514C2×Q8C22×#26112yes—
515C2×Q8C22×#26112yes—
516C22×C4—24no—
517C22×C4—131no—
518C22×C4—131no—
519C22×C4—131no—
520C22×C4—131no—
521C22×C4—131no—
522C22×C4—131no—
523C22×C4—131no—
524C22×C4—131no—
525C22×C4—131no—
526C22×C4—132no—
527C22×C4—131no—
528C22×C4—131no—
529C22×C4—131no—
530C22×C4—132no—
531C22×C4—264no—
532C22×C4—522no—
533C22×C4C22×#26111yescharacteristic
534C22×C4C22×#26111yescharacteristic
535C22×C4C22×#26111yescharacteristic
536C22×C4C22×#26111yescharacteristic
537C22×C4C22×#26111yescharacteristic
538C22×C4C22×#26112yes—
539C22×C4C22×#26112yes—
540C22×C4C22×#26111yescharacteristic
541M16—268no—
542C2×C8—134no—
543C2×C8—134no—
544C2×C8—134no—
545C2×C8—134no—
546C2×C8—268no—
547C4⋊C4—134no—
548C4⋊C4—134no—
549C4⋊C4—134no—
550C4⋊C4—134no—
551C4⋊C4—134no—
552C4⋊C4—134no—
553C4⋊C4—134no—
554C4⋊C4—134no—
555C4⋊C4—134no—
556C4⋊C4—134no—
557C4⋊C4—134no—
558C4⋊C4—134no—
559C4⋊C4—132no—
560C4⋊C4—132no—
561C4⋊C4C22×#26114yes—
562C4⋊C4C22×#26114yes—
563C4⋊C4C22×#26112yes—
564C4⋊C4C22×#26112yes—
565C22⋊C4—134no—
566C22⋊C4—134no—
567C22⋊C4C22×#26114yes—
568C22⋊C4C22×#26114yes—
569C42—131no—
570C42—131no—
571C42—131no—
572C42—134no—
573C42—134no—
574C42—131no—
575C42—131no—
576C42—131no—
577C42—131no—
578C42—131no—
579C42—131no—
580C42—131no—
581C42—131no—
582C42—131no—
583C42—131no—
584C42—131no—
585C42—131no—
586C42—131no—
587C42—134no—
588C42—134no—
589C13#6025011yescharacteristic, complemented, Sylow
590C23—22no—
591C23—22no—
592C23—41no—
593C23C23×#26111yescharacteristic
594C23C23×#26111yescharacteristic
595C23C23×#26111yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—268no—
599Q8—264no—
600Q8—264no—
601Q8—524no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—42no—
609C2×C4—134no—
610C2×C4—131no—
611C2×C4—131no—
612C2×C4—131no—
613C2×C4—131no—
614C2×C4—131no—
615C2×C4—131no—
616C2×C4—131no—
617C2×C4—131no—
618C2×C4—134no—
619C2×C4—134no—
620C2×C4—262no—
621C2×C4—262no—
622C2×C4—262no—
623C2×C4—262no—
624C2×C4—262no—
625C2×C4—262no—
626C2×C4—262no—
627C2×C4—264no—
628C2×C4—262no—
629C2×C4—521no—
630C2×C4—521no—
631C2×C4—521no—
632C2×C4—522no—
633C2×C4#8823814yes—
634C2×C4C23×#26111yescharacteristic
635C2×C4C23×#26111yescharacteristic
636C2×C4C23×#26114yes—
637C2×C4C23×#26114yes—
638C2×C4C23×#26111yescharacteristic
639C2×C4C23×#26111yescharacteristic
640C8—264no—
641C8—264no—
642C8—264no—
643C8—264no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—44no—
652C22C2×#8823812yes—
653C22#1171962712yes—
654C22C24×#26111yescharacteristic
655C4—21no—
656C4—21no—
657C4—24no—
658C4—24no—
659C4—132no—
660C4—132no—
661C4—132no—
662C4—261no—
663C4—261no—
664C4—264no—
665C4—261no—
666C4—261no—
667C4—261no—
668C4—264no—
669C4C2×#8823812yes—
670C4C2×#8823812yes—
671C2—22no—
672C2—22no—
673C2—41no—
674C2C2×#1171962711yescharacteristic
675C2C22×#8823811yescharacteristic
676C2D13⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#261
commutator, socleC22×C13C24×C4
FittingC2×C13×C22⋊Q8C4
prime radicalC24×C13C22×C4
cosocleC2×Dic13C25
hypercenterC2×C22⋊Q8#261
hypocenterC13#60250
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #13131745C2semidirect
2. #13131740C2semidirect
3. #13131736C2semidirect
4. #13131734C2semidirect
5. #13131732C2semidirect
6. #13131731C2semidirect
7. #13131729C2semidirect
8. #13131631C2semidirect
9. #13131624C2semidirect
35. #11719158C22semidirect
36. #11719158C22semidirect
37. Q8×#1057C22semidirect
38. Q8×#1057C22semidirect
57. #11719080C22semidirect
58. #11719080C22semidirect
59. #11719080C22semidirect
60. #11719078C22semidirect
61. #11719078C22semidirect
62. #11719074C22semidirect
63. #11719074C22semidirect
64. #11719074C22semidirect
589. C13#60250semidirect

Maximal subgroups

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

subgroupindex
#131317452
#131317402
#131317362
#131317342
#131317322
#131317312
#131317292
#131316312
#131316242
#131315252
#131315122
C2×#117191192
C2×#117193272
#6025013