Finite Group Databasebeta

← #29702041

Subgroups — #29702041

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448909C211yesmaximal, characteristic, complemented
2#21448729C211yesmaximal, characteristic
3#21446793C211yesmaximal, characteristic, complemented
4#21446589C211yesmaximal, characteristic, complemented
5#21446558C211yesmaximal, characteristic, complemented
6#21446553C211yesmaximal, characteristic, complemented
7#21446413C211yesmaximal, characteristic, complemented
8#21446392C211yesmaximal, characteristic, complemented
9#21446082C211yesmaximal, characteristic
10#21446068C211yesmaximal, characteristic
11#21445696C211yesmaximal, characteristic, complemented
12#21445646C211yesmaximal, characteristic
13#21445644C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C2×C61×#474C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519688C2211yescharacteristic
23C2×#14519687C2211yescharacteristic
24C2×C8×Dic61C2211yescharacteristic
25#16839669—21no—
26#16839669—21no—
27#16839661—21no—
28#16839661C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839661C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839637—21no—
33#16839637—21no—
34#16839628C2211yescharacteristic
35#16839622—21no—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C2×#14519684—21no—
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C61×#474C418yescomplemented
55C2×#107×C61C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—41no—
67D4×Dic61—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519711—41no—
86#14519709—41no—
87#14519698—22no—
88#14519698Q814yescomplemented
89#14519688—22no—
90#14519687—41no—
91C8×Dic61—41no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×Q8×C61—21no—
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4⋊Q8×C61—22no—
109C4⋊Q8×C61C2×C414yescomplemented
110C4×D4×C61—22no—
111C4×D4×C61C2×C414yes—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#107×C61C2×C414yescomplemented
119#107×C61C2×C414yescomplemented
120C4⋊C8×C61C2×C414yescomplemented
121#102×C61C2×C414yes—
122#102×C61C2×C414yes—
123C24×C61—21no—
124C2×Q8×C61—24no—
125C2×Q8×C61—81no—
126C2×Q8×C61—81no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C22×C412yes—
131C2×D4×C61C2×Q812yes—
132C2×D4×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—24no—
137C22×C4×C61—81no—
138C22×C4×C61—81no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—81no—
151C22×Dic61—81no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C22×C412yes—
161C2×C8×C61C22×C412yes—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22×C412yes—
167C4⋊C4×C61C22×C412yes—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—24no—
170C42×C61C22×C412yes—
171C42×C61#4112yes—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C4⋊Dic61—21no—
175C4⋊Dic61—22no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—41no—
183C4⋊Dic61—81no—
184C4⋊Dic61—81no—
185C4⋊Dic61—81no—
186C4⋊Dic61—81no—
187C4⋊Dic61C2×Q812yes—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—42no—
193C4×Dic61—81no—
194C4×Dic61—81no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C2×Q812yes—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61C22×D411yescharacteristic
209Q8×C61—84no—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61C4×Q814yes—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—84no—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—84no—
260C2×Dic61—84no—
261C2×Dic61#13712yes—
262C8×C61—22no—
263C8×C61—22no—
264#92418—22no—
265#92418—41no—
266#22979—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61C2×C4×D411yescharacteristic
275C22×C61#54611yescharacteristic
276C22×C61D4×Q811yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57611yescharacteristic
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61C2×C4×Q811yescharacteristic
287C4×C61#52011yescharacteristic
288C4×C61#55211yescharacteristic
289C4×C61#55411yescharacteristic
290Dic61—22no—
291Dic61—42no—
292Dic61—41no—
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298C2×#474D6111yescharacteristic
299#2369—611no—
300#2349—611no—
301#1993—611no—
302#1949—611no—
303#1943—611no—
304#1942—611no—
305#1921—611no—
306#1918—611no—
307#1864—611no—
308#1862—611no—
309#1771—611no—
310#1762—611no—
311#1762—611no—
312C4×#102—611no—
313C2×C61—44no—
314C2×C61#236011yescharacteristic
315C2×C61C4×#13711yescharacteristic
316C2×C61#294811yescharacteristic
317C2×C61#331411yescharacteristic
318C2×C61#331911yescharacteristic
319C2×C61#338511yescharacteristic
320C2×C61#338911yescharacteristic
321C2×C4⋊Q8C2×D6111yescharacteristic
322C2×C4×D4—611no—
323C2×C4×D4—611no—
324C2×C4×D4C2×D6111yescharacteristic
325#474Dic6118yescomplemented
326C2×#107—611no—
327C2×#107C2×D6111yescharacteristic
328C2×#107C2×D6111yescharacteristic
329C2×#106—611no—
330C2×C4⋊C8C2×D6111yescharacteristic
331C2×#102—611no—
332C2×#102—611no—
333C2×#102C2×D6111yescharacteristic
334C2×#102C2×D6111yescharacteristic
335C2×#97—611no—
336C2×#97—611no—
337C2×C4×C8—611no—
338#391—1221no—
339#391—1221no—
340#389—611no—
341#389—611no—
342#389—611no—
343#389—611no—
344#389—1221no—
345#385—1221no—
346#385—1221no—
347#382—611no—
348#380—1221no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—1221no—
354C2×#95—1221no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#336—611no—
364#336—611no—
365C61#2297911yescharacteristic, complemented, Sylow
366C22×Q8—21no—
367C22×D4C22×D6111yescharacteristic
368C23×C4—21no—
369C23×C4—1221no—
370C23×C4—1221no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4⋊Q8—22no—
376C4⋊Q8C2×Dic6114yescomplemented
377C4×D4—22no—
378C4×D4—1222no—
379C4×D4—1222no—
380C4×D4—2441no—
381C4×D4—2441no—
382C4×D4C2×Dic6114yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42C22×D6111yescharacteristic
406#107—2441no—
407#107C2×Dic6114yescomplemented
408#107C2×Dic6114yescomplemented
409#106—2441no—
410C4⋊C8C2×Dic6114yescomplemented
411#102—614no—
412#102—1222no—
413#102C2×Dic6114yes—
414#102C2×Dic6114yes—
415#97—1222no—
416#97—2441no—
417C4×C8—2441no—
418#95—611no—
419#95—611no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429C24—21no—
430C2×Q8—24no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D6112yes—
437C2×D4#1175822612yes—
438C2×D4C22×Dic6112yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—4881no—
454C22×C4—4881no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—612no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8C22×Dic6112yes—
473C2×C8C22×Dic6112yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—612no—
479C4⋊C4—1222no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—2441no—
492C4⋊C4C22×Dic6112yes—
493C4⋊C4C22×Dic6112yes—
494C4⋊C4C22×Dic6112yes—
495C22⋊C4—24no—
496C22⋊C4—2442no—
497C22⋊C4—2442no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C42—1221no—
502C42—4881no—
503C42—4881no—
504C42—2442no—
505C42#1175822412yes—
506C42C22×Dic6112yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D6111yescharacteristic
515Q8—84no—
516D4—22no—
517D4—22no—
518D4—22no—
519D4Q8×Dic6114yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
534C2×C4—612no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—4884no—
548C2×C4—4884no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4D4×Dic6112yes—
561C2×C4D4×Dic6112yes—
562C2×C4C23×Dic6111yescharacteristic
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4C2×#1175822611yescharacteristic
566C2×C4#1451982311yescharacteristic
567C2×C4#1451982411yescharacteristic
568C8—22no—
569C8—22no—
570C8—1222no—
571C8—2441no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×D4×Dic6111yescharacteristic
580C22#1684040311yescharacteristic
581C22#1684042511yescharacteristic
582C22#137×D6111yescharacteristic
583C22#1684056111yescharacteristic
584C22#1684056311yescharacteristic
585C22#1684056311yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—1222no—
592C4—4881no—
593C4—4881no—
594C4—4881no—
595C4—2441no—
596C4—2441no—
597C4—2442no—
598C4—2441no—
599C4C2×Q8×Dic6111yescharacteristic
600C4#1684024511yescharacteristic
601C4#1684045411yescharacteristic
602C4#1684047011yescharacteristic
603C2—44no—
604C2#2144882811yescharacteristic
605C2#137×Dic6111yescharacteristic
606C2#2145467111yescharacteristic
607C2#2145789711yescharacteristic
608C2#2145797211yescharacteristic
609C2#2145872411yescharacteristic
610C2#2145877211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#474D61
hypocenterC61#22979
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #21448909C2semidirect
3. #21446793C2semidirect
4. #21446589C2semidirect
5. #21446558C2semidirect
6. #21446553C2semidirect
7. #21446413C2semidirect
8. #21446392C2semidirect
11. #21445696C2semidirect
54. C61×#474C4semidirect
88. #14519698Q8semidirect
109. C4⋊Q8×C61C2×C4semidirect
118. #107×C61C2×C4semidirect
119. #107×C61C2×C4semidirect
120. C4⋊C8×C61C2×C4semidirect
325. #474Dic61semidirect
365. C61#22979semidirect
376. C4⋊Q8C2×Dic61semidirect
407. #107C2×Dic61semidirect
408. #107C2×Dic61semidirect
410. C4⋊C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214489092
#214487292
#214467932
#214465892
#214465582
#214465532
#214464132
#214463922
#214460822
#214460682
#214456962
#214456462
#214456442
#102×Dic612
C2×C61×#4742
#2297961