Finite Group Databasebeta

← #29689158

Subgroups — #29689158

Order 15616.

1362 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454264C211yesmaximal, characteristic, complemented
2#21448310C211yesmaximal, characteristic, complemented
3#21446268C212yesmaximal, complemented
4#21446263C212yesmaximal, complemented
5#21446122C214yesmaximal, complemented
6#21445918C212yesmaximal, complemented
7#21445763C212yesmaximal, complemented
8C61×#3449C211yesmaximal, characteristic
9C22×#11758194C2212yescomplemented
10C2×#14519736C2211yescharacteristic, complemented
11C22×C4×Dic61C2211yescharacteristic, complemented
12C2×#14519704C2214yes—
13C2×#14519704C2214yes—
14#16839729—22no—
15#16839729—22no—
16#16839726C2212yes—
17#16839726C2212yescomplemented
18#16839669—21no—
19#16839665—21no—
20#16839642—21no—
21#16839642—21no—
22C4×C22⋊Dic61C2212yes—
23C4×C22⋊Dic61C2212yescomplemented
24#16839521C2214yescomplemented
25#16839521C2214yescomplemented
26C22×#41×C61C2211yescharacteristic
27C22×#41×C61C2211yescharacteristic
28C61×#537C414yescomplemented
29C61×#536—21no—
30C61×#536C412yescomplemented
31C61×#535—21no—
32C61×#535C412yescomplemented
33C61×#534—21no—
34C61×#534—21no—
35C61×#532—22no—
36C61×#525—22no—
37C61×#525C2212yes—
38C61×#525C2212yes—
39C2×#117×C61C2211yescharacteristic
40C23×Dic61—21no—
41C2×#11758194—22no—
42C2×#11758194—42no—
43C2×#11758194—42no—
44C2×#11758194C2314yes—
45C22×#92418—22no—
46#14519736—42no—
47C2×C4⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—22no—
50C2×C22⋊Dic61—22no—
51C2×C4×Dic61—21no—
52C2×C4×Dic61—42no—
53C2×C4×Dic61C2312yes—
54C2×C4×Dic61C2312yes—
55#14519704—44no—
56#14519704—44no—
57C22⋊#92418—22no—
58C22⋊#92418—22no—
59C22⋊#92418—24no—
60#14519684—22no—
61#14519684—22no—
62C2×#41×C61—21no—
63C2×#41×C61—21no—
64C2×#41×C61—42no—
65C2×#41×C61—42no—
66C2×#41×C61—42no—
67C2×#41×C61—42no—
68C2×#41×C61C2×C412yes—
69C2×#41×C61C2×C412yes—
70C2×#41×C61C2×C412yescomplemented
71C2×#41×C61D412yes—
72C2×#41×C61D412yes—
73C2×#41×C61C2312yes—
74C22×Q8×C61C2×C411yescharacteristic, complemented
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61—21no—
77C22×D4×C61C2×C411yescharacteristic, complemented
78C22×D4×C61D411yescharacteristic
79C22×D4×C61D411yescharacteristic
80C22×D4×C61D411yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C4⋊Q8×C61—22no—
86#127×C61—22no—
87#124×C61—22no—
88#124×C61—22no—
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93C22⋊Q8×C61—24no—
94C22⋊Q8×C61C2×C414yes—
95C22⋊Q8×C61C2×C414yescomplemented
96#121×C61—24no—
97#121×C61—24no—
98#121×C61—24no—
99#121×C61C2×C414yes—
100#121×C61C2×C414yescomplemented
101#120×C61—22no—
102#120×C61—22no—
103#120×C61—22no—
104#120×C61—22no—
105C4×D4×C61—24no—
106C4×D4×C61—24no—
107#117×C61—21no—
108#117×C61—21no—
109#117×C61C2×C412yescomplemented
110#117×C61C2312yes—
111C2×C4⋊C4×C61C2×C412yescomplemented
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114C24×C61—21no—
115C24×C61C2×D411yescharacteristic
116C24×C61C2×D411yescharacteristic
117C24×C61C2×D411yescharacteristic
118#41×C61—24no—
119#41×C61—24no—
120#41×C61—44no—
121#41×C61—44no—
122#41×C61—42no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61—44no—
127#41×C61—42no—
128#41×C61—42no—
129#41×C61—44no—
130C2×Q8×C61—21no—
131C2×Q8×C61—41no—
132C2×Q8×C61—41no—
133C2×Q8×C61—42no—
134C2×Q8×C61C22⋊C412yes—
135C2×Q8×C61C22⋊C412yes—
136C2×Q8×C61C22×C412yes—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—24no—
141C2×D4×C61—21no—
142C2×D4×C61—42no—
143C2×D4×C61—41no—
144C2×D4×C61—41no—
145C2×D4×C61—41no—
146C2×D4×C61—41no—
147C2×D4×C61—41no—
148C2×D4×C61—41no—
149C2×D4×C61—42no—
150C2×D4×C61—42no—
151C2×D4×C61C22⋊C412yes—
152C2×D4×C61C22⋊C412yes—
153C2×D4×C61C22⋊C412yes—
154C2×D4×C61C22⋊C412yes—
155C2×D4×C61C22⋊C412yes—
156C2×D4×C61C22⋊C412yes—
157C2×D4×C61C22×C412yes—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—22no—
162C22×C4×C61—22no—
163C22×C4×C61—22no—
164C22×C4×C61—21no—
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—22no—
168C22×C4×C61—44no—
169C22×C4×C61C22×C411yescharacteristic
170C22×C4×C61C22×C411yescharacteristic
171C22×C4×C61C22×C412yes—
172C22×C4×C61C22×C412yes—
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2411yescharacteristic
181C22×Dic61—22no—
182C22×Dic61—22no—
183C22×Dic61—44no—
184C22×Dic61—42no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—24no—
187C4⋊C4×C61—24no—
188C4⋊C4×C61C22×C414yes—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—24no—
191C22⋊C4×C61—24no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—24no—
194C22⋊C4×C61—24no—
195C42×C61—22no—
196C42×C61—22no—
197C22⋊Dic61—44no—
198C22⋊Dic61—44no—
199C22⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4×Dic61—24no—
202C4×Dic61—24no—
203C4×Dic61—42no—
204#11758194—24no—
205#11758194—24no—
206#11758194—44no—
207C2×#92418—24no—
208C2×#92418—44no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—22no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—82no—
221C23×C61C2×C22⋊C411yescharacteristic
222C23×C61C2×C22⋊C412yes—
223C23×C61C2×C22⋊C412yes—
224C23×C61C2×C22⋊C411yescharacteristic
225C23×C61C22×D411yescharacteristic
226Q8×C61—24no—
227Q8×C61—42no—
228Q8×C61—42no—
229Q8×C61—42no—
230D4×C61—24no—
231D4×C61—42no—
232D4×C61—42no—
233D4×C61—42no—
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61—42no—
238D4×C61—44no—
239D4×C61—42no—
240D4×C61—42no—
241D4×C61—84no—
242D4×C61—84no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—24no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—44no—
262C2×C4×C61—42no—
263C2×C4×C61—44no—
264C2×C4×C61—82no—
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C2×C22⋊C412yes—
267C2×C4×C61C2×C22⋊C412yes—
268C2×C4×C61C2×C22⋊C411yescharacteristic
269C2×C4×C61C23×C411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61H(5,𝔽2)12yes—
272C2×C4×C61H(5,𝔽2)12yes—
273C2×Dic61—24no—
274C2×Dic61—42no—
275C2×Dic61—48no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280#92418—44no—
281#21775—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—24no—
289C22×C61—24no—
290C22×C61—22no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—44no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—84no—
300C22×C61C22×C22⋊C411yescharacteristic
301C22×C61#53411yescharacteristic
302C22×C61#53411yescharacteristic
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—44no—
309C4×C61#51812yes—
310C4×C61#53412yes—
311Dic61—44no—
312Dic61—82no—
313#3449D6111yescharacteristic
314#2879—611no—
315#2297—611no—
316#1897—612no—
317#1896—612no—
318#1871—614no—
319#1826—612no—
320#1789—612no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—42no—
324C2×C61—41no—
325C2×C61—42no—
326C2×C61—41no—
327C2×C61—82no—
328C2×C61#230811yescharacteristic
329C2×C61#290112yes—
330C22×#41C2×D6111yescharacteristic
331C22×#41C2×D6111yescharacteristic
332C22×M16—612no—
333#537Dic6114yescomplemented
334#536—21no—
335#536Dic6112yescomplemented
336#535—21no—
337#535Dic6112yescomplemented
338#534—21no—
339#534—21no—
340#532—22no—
341#525—22no—
342#525C2×D6112yes—
343#525C2×D6112yes—
344C2×#117—611no—
345C2×#117C2×D6111yescharacteristic
346C22×C42—611no—
347C2×#104—614no—
348C2×#104—614no—
349#408—1222no—
350#408—1222no—
351#407—612no—
352#407—612no—
353#391—1221no—
354#390—1221no—
355#386—1221no—
356#386—1221no—
357C4×C22⋊C4—612no—
358C4×C22⋊C4—612no—
359#339—614no—
360#339—614no—
361C61#2177511yescharacteristic, complemented, Sylow
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—42no—
365C2×#41—42no—
366C2×#41—42no—
367C2×#41—42no—
368C2×#41C2×Dic6112yes—
369C2×#41C2×Dic6112yes—
370C2×#41C2×Dic6112yescomplemented
371C2×#41C22⋊D6112yes—
372C2×#41C22⋊D6112yes—
373C2×#41C22×D6112yes—
374C22×Q8C2×Dic6111yescharacteristic, complemented
375C22×Q8C22⋊D6111yescharacteristic
376C22×D4—21no—
377C22×D4C2×Dic6111yescharacteristic, complemented
378C22×D4C22⋊D6111yescharacteristic
379C22×D4C22⋊D6111yescharacteristic
380C22×D4C22⋊D6111yescharacteristic
381C23×C4—21no—
382C23×C4—1221no—
383C23×C4C22×D6111yescharacteristic
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6111yescharacteristic
386C2×M16—614no—
387C2×M16—1222no—
388C2×M16—2442no—
389C2×M16—2442no—
390C22×C8—1222no—
391C4⋊Q8—22no—
392#127—22no—
393#124—22no—
394#124—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399C22⋊Q8—24no—
400C22⋊Q8C2×Dic6114yes—
401C22⋊Q8C2×Dic6114yescomplemented
402#121—24no—
403#121—24no—
404#121—24no—
405#121C2×Dic6114yes—
406#121C2×Dic6114yescomplemented
407#120—22no—
408#120—22no—
409#120—22no—
410#120—22no—
411C4×D4—24no—
412C4×D4—24no—
413#117—21no—
414#117—21no—
415#117—2442no—
416#117C2×Dic6112yescomplemented
417#117C22×D6112yes—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4C2×Dic6112yescomplemented
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C42—21no—
425C2×C42—612no—
426C2×C42—612no—
427C2×C42—1221no—
428C2×C42—2442no—
429#104—2444no—
430#104—2444no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1224no—
434#95—1222no—
435#95—1222no—
436C24—21no—
437C24C2×C22⋊D6111yescharacteristic
438C24C2×C22⋊D6111yescharacteristic
439C24C2×C22⋊D6111yescharacteristic
440#41—24no—
441#41—24no—
442#41—44no—
443#41—44no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—44no—
449#41—42no—
450#41—42no—
451#41—44no—
452C2×Q8—21no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—42no—
456C2×Q8C22⋊Dic6112yes—
457C2×Q8C22⋊Dic6112yes—
458C2×Q8C22×Dic6112yes—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4C22⋊Dic6112yes—
474C2×D4C22⋊Dic6112yes—
475C2×D4C22⋊Dic6112yes—
476C2×D4C22⋊Dic6112yes—
477C2×D4C22⋊Dic6112yes—
478C2×D4C22⋊Dic6112yes—
479C2×D4C22×Dic6112yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—2444no—
494C22×C4—2442no—
495C22×C4C22×Dic6111yescharacteristic
496C22×C4C22×Dic6111yescharacteristic
497C22×C4C22×Dic6112yes—
498C22×C4C22×Dic6112yes—
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6112yes—
502C22×C4C2×C22⋊D6112yes—
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507M16—1224no—
508M16—1224no—
509M16—2444no—
510C2×C8—1224no—
511C2×C8—2444no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—2442no—
516C4⋊C4C22×Dic6114yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—2444no—
524C22⋊C4—2444no—
525C22⋊C4—2442no—
526C42—22no—
527C42—22no—
528C42—1224no—
529C42—1224no—
530C42—2442no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—82no—
543C23C2×C22⋊Dic6111yescharacteristic
544C23C2×C22⋊Dic6112yes—
545C23C2×C22⋊Dic6112yes—
546C23C2×C22⋊Dic6111yescharacteristic
547C23C22×C22⋊D6111yescharacteristic
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552D4—24no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—44no—
561D4—42no—
562D4—42no—
563D4—84no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—82no—
587C2×C4—1224no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2448no—
593C2×C4—2442no—
594C2×C4C2×C22⋊Dic6111yescharacteristic
595C2×C4C2×C22⋊Dic6112yes—
596C2×C4C2×C22⋊Dic6112yes—
597C2×C4C2×C22⋊Dic6111yescharacteristic
598C2×C4C23×Dic6111yescharacteristic
599C2×C4C22×C22⋊D6111yescharacteristic
600C2×C4#1451982212yes—
601C2×C4#1451982212yes—
602C8—2444no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—22no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—84no—
621C22C22×C22⋊Dic6111yescharacteristic
622C22#1684032111yescharacteristic
623C22#1684032111yescharacteristic
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—44no—
629C4—42no—
630C4—4882no—
631C4—2444no—
632C4#1684023712yes—
633C4#1684032112yes—
634C2—21no—
635C2—21no—
636C2—42no—
637C2—41no—
638C2—41no—
639C2—42no—
640C2—82no—
641C2#2144839711yescharacteristic
642C2#2145436612yes—
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3449C2
socleC22×C61C22×C22⋊C4
cosocleC22×C4×C61C24
hypercenter#3449D61
hypocenterC61#21775
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×#41 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21454264C2semidirect
2. #21448310C2semidirect
3. #21446268C2semidirect
4. #21446263C2semidirect
5. #21446122C2semidirect
6. #21445918C2semidirect
7. #21445763C2semidirect
9. C22×#11758194C22semidirect
10. C2×#14519736C22semidirect
11. C22×C4×Dic61C22semidirect
17. #16839726C22semidirect
23. C4×C22⋊Dic61C22semidirect
24. #16839521C22semidirect
25. #16839521C22semidirect
28. C61×#537C4semidirect
30. C61×#536C4semidirect
32. C61×#535C4semidirect
70. C2×#41×C61C2×C4semidirect
74. C22×Q8×C61C2×C4semidirect
77. C22×D4×C61C2×C4semidirect
95. C22⋊Q8×C61C2×C4semidirect
100. #121×C61C2×C4semidirect
109. #117×C61C2×C4semidirect
111. C2×C4⋊C4×C61C2×C4semidirect
333. #537Dic61semidirect
335. #536Dic61semidirect
337. #535Dic61semidirect
361. C61#21775semidirect
370. C2×#41C2×Dic61semidirect
374. C22×Q8C2×Dic61semidirect
377. C22×D4C2×Dic61semidirect
401. C22⋊Q8C2×Dic61semidirect
406. #121C2×Dic61semidirect
416. #117C2×Dic61semidirect
419. C2×C4⋊C4C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214542642
#214483102
#214462682
#214462632
#214461222
#214459182
#214457632
C61×#34492
#2177561