Finite Group Databasebeta

← #30620449

Subgroups — #30620449

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758213C211yesmaximal, characteristic, complemented
2#21462278C211yesmaximal, characteristic, complemented
3#21462006C212yesmaximal, complemented
4#21462004C211yesmaximal, characteristic, complemented
5#21461996C212yesmaximal, complemented
6#21461995C211yesmaximal, characteristic, complemented
7#21461692C211yesmaximal, characteristic
8#21461692C212yesmaximal
9#21461583C212yesmaximal
10C4×#14519842C211yesmaximal, characteristic
11#21459401C211yesmaximal, characteristic
12#16840812—21no—
13#16840812—22no—
14Q8×#92420—21no—
15#16840766C2211yescharacteristic
16#16840766C2212yes—
17C2×#14519842C2211yescharacteristic
18C2×#14519842C2212yes—
19C2×#14519842C2211yescharacteristic
20C4×#11758213C2211yescharacteristic
21C4×#11758213C2212yes—
22(C2×#6923)⋊Q8—21no—
23#16840713—21no—
24#16840711—22no—
25C4×C4⋊#6923C2211yescharacteristic
26C4×C4⋊#6923C2212yes—
27#16840698C2212yes—
28C42×#6923C2211yescharacteristic
29#16840632C2211yescharacteristic
30#16840632C2211yescharacteristic
31#16840632C2212yes—
32#16840632C2212yes—
33#16840632C2212yes—
34#16840632C2212yes—
35#16840632C2212yes—
36Q8×Dic122C411yescharacteristic, complemented
37#16840488C412yescomplemented
38#16840472—21no—
39#16840472C411yescharacteristic, complemented
40#16840468—22no—
41#16840468C412yescomplemented
42#16840427—21no—
43#16840427C411yescharacteristic, complemented
44#16840318—22no—
45#16840318C2212yes—
46#16840317—21no—
47#16840317C2211yescharacteristic
48C4×#11758226C2211yescharacteristic
49C4×#11758226C2212yes—
50C4×Q8×D61—21no—
51C4×Q8×D61C2211yescharacteristic
52C61×#557C411yescharacteristic, complemented
53C2×#11758213C2311yescharacteristic
54C2×#11758213C2312yes—
55C2×#11758213C2311yescharacteristic
56C2×C4⋊#6923—22no—
57C2×C4⋊#6923—21no—
58C2×C4×#6923—21no—
59C2×C4×#6923—21no—
60C2×C4×#6923C2311yescharacteristic
61C2×C4×#6923C2312yes—
62C2×C4×#6923C2311yescharacteristic
63C4⋊#92420—21no—
64C4⋊#92420—22no—
65C4⋊#92420—22no—
66C4⋊#92420—21no—
67#14519842—22no—
68#14519842—24no—
69#14519842Q814yescomplemented
70#14519832—22no—
71#14519832—21no—
72C4×#92420—21no—
73C4×#92420—22no—
74(C2×#6923)⋊C4—21no—
75(C2×#6923)⋊C4—22no—
76(C2×#6923)⋊C4—22no—
77(C2×#6923)⋊C4—22no—
78C2×#11758226C2311yescharacteristic
79C2×Q8×D61—21no—
80C2×Q8×D61D411yescharacteristic
81#14519777—21no—
82#14519777C2×C411yescharacteristic
83#14519776—22no—
84#14519776—22no—
85#14519776C2×C412yes—
86#14519776C2×C412yes—
87#14519769—22no—
88#14519769C2×C412yes—
89#14519768—22no—
90#14519768C2×C412yes—
91#14519757—21no—
92#14519757—22no—
93#14519755—22no—
94#14519755—22no—
95#14519755—22no—
96#14519755—22no—
97#14519755—22no—
98#14519755—22no—
99#14519753—21no—
100#14519753—21no—
101#14519753—21no—
102Q8×Dic61—21no—
103Q8×Dic61C2×C411yescharacteristic
104Q8×Dic61C2×C412yes—
105Q8×Dic61D411yescharacteristic
106C4×Dic122—21no—
107C4×Dic122—22no—
108C4×Dic122C2×C412yes—
109C4×Dic122C2×C411yescharacteristic
110#14519742—22no—
111#14519742—22no—
112#14519742—21no—
113#14519742D411yescharacteristic
114#14519742D412yes—
115C4×D244—22no—
116C4×D244—21no—
117C4×D244C2×C412yes—
118C4×D244C2×C411yescharacteristic
119#14519739—21no—
120#14519739—22no—
121#14519739C2×C411yescharacteristic
122#14519739C2×C412yes—
123#14519735—22no—
124#14519735—21no—
125#14519735C2312yes—
126#14519735C2311yescharacteristic
127C4⋊C4×D61—21no—
128C4⋊C4×D61—22no—
129C4⋊C4×D61D412yes—
130C4⋊C4×D61D411yescharacteristic
131C42×D61—21no—
132C42×D61—22no—
133C42×D61C2311yescharacteristic
134C42×D61C2312yes—
135C4⋊Q8×C61—21no—
136C4⋊Q8×C61C2×C411yescharacteristic
137#125×C61—22no—
138#125×C61C2×C412yes—
139C4×Q8×C61—21no—
140C4×Q8×C61C2×C412yes—
141C4×Q8×C61C2×C411yescharacteristic
142C4×Q8×C61C2×C411yescharacteristic
143C22×#6923—21no—
144C22×#6923—21no—
145C22×#6923—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22⋊C411yescharacteristic
148C2×Q8×C61C22×C411yescharacteristic
149#11758226—21no—
150#11758226—21no—
151#11758226C2×Q812yes—
152#11758226#4112yes—
153Q8×D61—42no—
154Q8×D61—44no—
155C2×D244—21no—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—21no—
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2411yescharacteristic
167C2×Dic122—22no—
168C2×Dic122—21no—
169C2×Dic122—22no—
170C2×Dic122C22⋊C411yescharacteristic
171C2×Dic122C22⋊C412yes—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—22no—
175C4⋊#6923—42no—
176C4⋊#6923—44no—
177C4×#6923—21no—
178C4×#6923—22no—
179C4×#6923—21no—
180C4×#6923—21no—
181C4×#6923—22no—
182C4×#6923—42no—
183C4×#6923—42no—
184C4×#6923C2×Q812yescomplemented
185#11758213—21no—
186#11758213—21no—
187#11758213—22no—
188#11758213—21no—
189#11758213—22no—
190#11758213C2×Q812yescomplemented
191C4⋊C4×C61—22no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61C22⋊C411yescharacteristic
197C4⋊C4×C61C22⋊C412yes—
198C4⋊C4×C61C22×C411yescharacteristic
199C4⋊C4×C61C22×C412yes—
200C42×C61—22no—
201C42×C61—21no—
202C42×C61C22×C412yes—
203C42×C61C22×C411yescharacteristic
204#11758198—22no—
205#11758198—21no—
206#11758198—22no—
207#11758198—21no—
208#11758198—22no—
209#11758198—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—22no—
212C4⋊Dic61—22no—
213C4⋊Dic61—21no—
214C4⋊Dic61C22⋊C411yescharacteristic
215C4⋊Dic61C22⋊C412yes—
216C4⋊Dic61C22×C412yes—
217C4⋊Dic61C22×C411yescharacteristic
218#11758196—22no—
219#11758196—22no—
220#11758196—22no—
221#11758196—22no—
222#11758196—22no—
223#11758196C22⋊C412yes—
224#11758196C22⋊C412yes—
225#11758196C22⋊C412yes—
226C4×Dic61—21no—
227C4×Dic61—22no—
228C4×Dic61—21no—
229C4×Dic61—22no—
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C22×C412yes—
232C4×Dic61C2×D411yescharacteristic
233C4×Dic61C2×D412yes—
234C4×Dic61C2×D412yes—
235C4×Dic61C2×D411yescharacteristic
236C22×D61—21no—
237C22×D61C22×D411yescharacteristic
238C2×#6923—22no—
239C2×#6923—22no—
240C2×#6923—41no—
241C2×#6923—41no—
242C2×#6923—41no—
243C2×#6923—41no—
244C2×#6923—42no—
245C2×#6923—42no—
246C2×#6923—42no—
247Q8×C61—21no—
248Q8×C61—22no—
249Q8×C61—42no—
250Q8×C61C4×Q812yes—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C23×C411yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C22×D411yescharacteristic
266D244—22no—
267D244—22no—
268D244—21no—
269D244C4×Q812yes—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—41no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—44no—
277C4×D61—42no—
278C4×D61—81no—
279C4×D61—82no—
280C4×D61C22×Q811yescharacteristic
281C4×D61C2×#4111yescharacteristic
282C4×D612−1+411yescharacteristic
283C4×D612−1+411yescharacteristic
284Dic122—22no—
285Dic122—42no—
286Dic122—44no—
287Dic122—82no—
288Dic122—81no—
289#92420—22no—
290#92420—41no—
291#92420—42no—
292#22529—611nomaximal, Sylow
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—22no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61#54811yescharacteristic
300C2×D61D4×Q811yescharacteristic
301#6923—42no—
302#6923—42no—
303#6923—82no—
304C4×C61—22no—
305C4×C61—22no—
306C4×C61—21no—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—41no—
310C4×C61#42112yes—
311C4×C61C2×C4×Q811yescharacteristic
312C4×C61#51911yescharacteristic
313Dic61—22no—
314Dic61—21no—
315Dic61—22no—
316Dic61—21no—
317Dic61—41no—
318Dic61—42no—
319Dic61#54811yescharacteristic
320Dic61D4×Q811yescharacteristic
321Q8×#41—611no—
322Q8×M16—611no—
323#2295—611no—
324#1995—612no—
325#1994—611no—
326#1986—611no—
327#1986—612no—
328#1806—611no—
329#1806—612no—
330#1759—612no—
331C4×#104—611no—
332C2×C61Q8×C22⋊C411yescharacteristic
333C2×C61C2×#42111yescharacteristic
334C2×C61#290111yescharacteristic
335D61—21no—
336D61—21no—
337D61—42no—
338Q82—611no—
339#557—612no—
340#557#692311yescharacteristic, complemented
341#554—612no—
342#554—611no—
343#554—1221no—
344#554—1222no—
345D4×Q8—611no—
346D4×Q8—1221no—
347#533—611no—
348#533—612no—
349#533—1222no—
350#533—1221no—
351C4×#41—611no—
352C4×#41—612no—
353C2×C4×Q8—611no—
354C2×C4×Q8—1221no—
355#446—1221no—
356#446—1222no—
357C8×Q8—1221no—
358#423—611no—
359#423—612no—
360C2×#104—611no—
361C2×#104—611no—
362C2×#104—612no—
363C4×M16—611no—
364C4×M16—612no—
365#391—1221no—
366#389—1221no—
367#387—1222no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—612no—
370#376—612no—
371C43—611no—
372#339—612no—
373#339—612no—
374#339—612no—
375#339—611no—
376#339—612no—
377#339—612no—
378#339—611no—
379C61#2252911yescharacteristic, complemented, Sylow
380C2×#41—611no—
381C22×Q8—611no—
382C22×Q8—1221no—
383C2×M16—611no—
384C2×M16—612no—
385C2×M16—611no—
386C4⋊Q8—21no—
387C4⋊Q8—612no—
388C4⋊Q8—612no—
389C4⋊Q8—611no—
390C4⋊Q8—1221no—
391C4⋊Q8—1222no—
392C4⋊Q8—1222no—
393C4⋊Q8C2×#692311yescharacteristic
394#125—22no—
395#125—612no—
396#125—612no—
397#125—1222no—
398#125—1222no—
399#125C2×#692312yes—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1221no—
406C22⋊Q8—1222no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8—1222no—
411C4×Q8—21no—
412C4×Q8—611no—
413C4×Q8—611no—
414C4×Q8—612no—
415C4×Q8—611no—
416C4×Q8—612no—
417C4×Q8—611no—
418C4×Q8—612no—
419C4×Q8—1222no—
420C4×Q8—1221no—
421C4×Q8—1222no—
422C4×Q8—1222no—
423C4×Q8—1221no—
424C4×Q8—1221no—
425C4×Q8C2×#692312yes—
426C4×Q8C2×#692311yescharacteristic
427C4×Q8C2×#692311yescharacteristic
428C4×D4—612no—
429C4×D4—611no—
430C4×D4—611no—
431C4×D4—612no—
432C4×D4—1221no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1221no—
436#117—612no—
437#117—611no—
438#117—1222no—
439#117—1221no—
440C2×C4⋊C4—612no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1222no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1221no—
446C2×C42—611no—
447C2×C42—612no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—612no—
451C2×C42—1222no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C4⋊C8—1221no—
456C4⋊C8—1222no—
457C4⋊C8—1222no—
458C4⋊C8—1221no—
459#104—614no—
460#104—1222no—
461#104—1224no—
462#97—1222no—
463#97—1221no—
464C4×C8—1221no—
465C4×C8—1222no—
466#95—1222no—
467#95—1222no—
468#95—1221no—
469#95—1222no—
470#41—612no—
471#41—612no—
472#41—1221no—
473#41—1221no—
474C2×Q8—21no—
475C2×Q8—611no—
476C2×Q8—612no—
477C2×Q8—1222no—
478C2×Q8—1222no—
479C2×Q8—1221no—
480C2×Q8—2442no—
481C2×Q8—2444no—
482C2×Q8C22⋊#692311yescharacteristic
483C2×Q8C22×#692311yescharacteristic
484C2×D4—611no—
485C2×D4—1221no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—612no—
489C22×C4—611no—
490C22×C4—612no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1222no—
497C22×C4—1221no—
498C22×C4—1222no—
499M16—612no—
500M16—1221no—
501M16—1221no—
502M16—1222no—
503M16—1222no—
504M16—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1222no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—612no—
517C4⋊C4—611no—
518C4⋊C4—612no—
519C4⋊C4—611no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4—1222no—
529C4⋊C4—2442no—
530C4⋊C4—2444no—
531C4⋊C4C22⋊#692311yescharacteristic
532C4⋊C4C22⋊#692312yes—
533C4⋊C4C22×#692312yes—
534C4⋊C4C22×#692311yescharacteristic
535C22⋊C4—1221no—
536C22⋊C4—1222no—
537C22⋊C4—1221no—
538C22⋊C4—1222no—
539C22⋊C4—1221no—
540C22⋊C4—1222no—
541C42—22no—
542C42—21no—
543C42—611no—
544C42—612no—
545C42—612no—
546C42—611no—
547C42—611no—
548C42—612no—
549C42—612no—
550C42—1221no—
551C42—1222no—
552C42—1221no—
553C42—1222no—
554C42—1221no—
555C42—1222no—
556C42—1221no—
557C42—1222no—
558C42—1221no—
559C42—2442no—
560C42—2442no—
561C42C22×#692312yes—
562C42C22×#692311yescharacteristic
563C23—611no—
564C23—1221no—
565Q8—22no—
566Q8—21no—
567Q8—42no—
568Q8—1222no—
569Q8—4881no—
570Q8—4882no—
571Q8—2442no—
572Q8—2444no—
573Q8Q8×#692312yes—
574D4—612no—
575D4—1222no—
576D4—1222no—
577D4—1221no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—611no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1221no—
596C2×C4—1222no—
597C2×C4—4882no—
598C2×C4—4881no—
599C2×C4—2441no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2444no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2442no—
611C2×C4C2×C22⋊#692312yes—
612C2×C4C2×C22⋊#692311yescharacteristic
613C2×C4C2×C22⋊#692312yes—
614C2×C4C2×C22⋊#692311yescharacteristic
615C2×C4C23×#692311yescharacteristic
616C8—1222no—
617C8—2441no—
618C8—2442no—
619C22—611no—
620C22—611no—
621C22—1221no—
622C22—1221no—
623C22—1222no—
624C22—2441no—
625C22—2441no—
626C22C22×C22⋊#692311yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—21no—
631C4—42no—
632C4—41no—
633C4—611no—
634C4—611no—
635C4—1221no—
636C4—1222no—
637C4—1221no—
638C4—1222no—
639C4—4882no—
640C4—2442no—
641C4—2442no—
642C4—2441no—
643C4—2442no—
644C4#1684075912yes—
645C4C2×Q8×#692311yescharacteristic
646C4#1684083011yescharacteristic
647C2—1221no—
648C2—1221no—
649C2—2442no—
650C2Q8×C22⋊#692311yescharacteristic
651C2C2×#1684075911yescharacteristic
652C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#557C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758226C23
cosocleC2×C4×D61C24
hypercenter#557#6923
hypocenterC61#22529
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
1. Q8×#11758213C2semidirect
2. #21462278C2semidirect
3. #21462006C2semidirect
4. #21462004C2semidirect
5. #21461996C2semidirect
6. #21461995C2semidirect
36. Q8×Dic122C4semidirect
37. #16840488C4semidirect
39. #16840472C4semidirect
41. #16840468C4semidirect
43. #16840427C4semidirect
52. C61×#557C4semidirect
69. #14519842Q8semidirect
184. C4×#6923C2×Q8semidirect
190. #11758213C2×Q8semidirect
340. #557#6923semidirect
379. C61#22529semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582132
#214622782
#214620062
#214620042
#214619962
#214619952
#214616922
#214615832
C4×#145198422
#214594012
#2252961