Finite Group Databasebeta

← #29738867

Subgroups — #29738867

Order 15616.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840168C211yesmaximal, characteristic, complemented
2C2×#16839802C211yesmaximal, characteristic
3#21449532C211yesmaximal, characteristic
4#21447295C211yesmaximal, characteristic, complemented
5#21447291C211yesmaximal, characteristic, complemented
6#21447157C211yesmaximal, characteristic, complemented
7#21447149C211yesmaximal, characteristic, complemented
8#21446554C211yesmaximal, characteristic, complemented
9#21446549C211yesmaximal, characteristic, complemented
10#21446016C211yesmaximal, characteristic
11#21446012C211yesmaximal, characteristic
12#21445975C211yesmaximal, characteristic, complemented
13#21445850C211yesmaximal, characteristic
14#21445603C211yesmaximal, characteristic
15C61×#2440C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16840168—24no—
20#16839802—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839698—21no—
35#16839685—21no—
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839659—21no—
39#16839645—21no—
40#16839632C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C22⋊Q8×C61—21no—
52C22×C4⋊C4×C61C2211yescharacteristic
53C61×#398—21no—
54C61×#395—21no—
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#385—21no—
58C61×#384C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C4×C22⋊C4×C61C2211yescharacteristic
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194Q814yescomplemented
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519769—22no—
69#14519769—41no—
70#14519736—22no—
71#14519736—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85#14519710—22no—
86#14519710—22no—
87#14519710—22no—
88#14519710D414yes—
89#14519708—22no—
90#14519708—22no—
91#14519708—22no—
92#14519708—22no—
93#14519708—41no—
94#14519708—41no—
95C22⋊#92418Q814yescomplemented
96#14519688—22no—
97C4×#92418—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101C22×Q8×C61—21no—
102C23×C4×C61—21no—
103C23×C4×C61C2311yescharacteristic
104C4⋊Q8×C61—22no—
105C4⋊Q8×C61—41no—
106C22⋊Q8×C61—44no—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—41no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61D411yescharacteristic
121C2×C22⋊C4×C61D411yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61D411yescharacteristic
128#95×C61D411yescharacteristic
129C24×C61C2×D411yescharacteristic
130C2×Q8×C61—42no—
131C2×Q8×C61—42no—
132C2×Q8×C61—81no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—81no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×Q812yes—
150C22×C4×C61C2×Q812yes—
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61—81no—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—42no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C22⋊Dic61—42no—
178C22⋊Dic61—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—42no—
187C4⋊Dic61—81no—
188C4⋊Dic61C2×D412yes—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—81no—
193C4×Dic61—42no—
194#11758194—28no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C2×D412yes—
202C2×#92418C2×Q812yes—
203C2×#92418C2×Q812yes—
204C2×#92418C2×Q812yes—
205C2×#92418C2×Q812yes—
206C2×#92418#4112yes—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61C4⋊Q812yes—
214C23×C61C22×D411yescharacteristic
215Q8×C61—84no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—84no—
245C2×C4×C61C4⋊Q812yes—
246C2×C4×C61AGL(1,ℤ/8)12yes—
247C2×C4×C61#13712yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×Q811yescharacteristic
250C2×C4×C61C22×Q811yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266#92418—22no—
267#92418—22no—
268#92418—22no—
269#92418C22⋊Q814yescomplemented
270#26078—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4⋊Q811yescharacteristic
279C22×C61D4211yescharacteristic
280C22×C61#54711yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57711yescharacteristic
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—42no—
290C4×C61—41no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61C2×C22⋊Q811yescharacteristic
295C4×C61#54411yescharacteristic
296C4×C61D4×Q811yescharacteristic
297C4×C61#55411yescharacteristic
298Dic61—42no—
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302C2×#501—611no—
303C2×#428—611no—
304#2441—611no—
305#2440D6111yescharacteristic
306#2084—611no—
307#2083—611no—
308#2059—611no—
309#2058—611no—
310#1943—611no—
311#1942—611no—
312#1854—611no—
313#1853—611no—
314#1844—611no—
315#1809—611no—
316#1753—611no—
317C2×C61—42no—
318C2×C61#244011yescharacteristic
319C2×C61#316311yescharacteristic
320C2×C61#316511yescharacteristic
321C2×C61#327211yescharacteristic
322C2×C61#327811yescharacteristic
323C2×C61#334411yescharacteristic
324C2×C61#334511yescharacteristic
325C22×M16—611no—
326C2×C4⋊Q8C2×D6111yescharacteristic
327C2×#125—611no—
328C2×C22⋊Q8—21no—
329C2×#117—611no—
330C22×C4⋊C4C2×D6111yescharacteristic
331#501—1224no—
332#428—1224no—
333C2×#107—611no—
334C2×#107—611no—
335C2×#107—611no—
336C2×#107—611no—
337C2×#106—611no—
338C2×#106—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×C22⋊C8—611no—
344C2×#97—611no—
345C2×C4×C8—611no—
346#399—1221no—
347#398—21no—
348#396—1221no—
349#395—21no—
350#389—611no—
351#389—611no—
352#389—1221no—
353#389C2×D6111yescharacteristic
354#389C2×D6111yescharacteristic
355#387—1221no—
356#385—21no—
357#384—611no—
358#384C2×D6111yescharacteristic
359#380—611no—
360#380C2×D6111yescharacteristic
361C4×C22⋊C4C2×D6111yescharacteristic
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#336—611no—
369#336—611no—
370C61#2607811yescharacteristic, complemented, Sylow
371C22×Q8—21no—
372C23×C4—21no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—614no—
375C2×M16—1222no—
376C2×M16—1222no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—611no—
381C4⋊Q8—22no—
382C4⋊Q8—41no—
383#125—1222no—
384#125—2441no—
385C22⋊Q8—44no—
386#117—1222no—
387#117—2441no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—41no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4C22⋊D6111yescharacteristic
414C2×C22⋊C4C22⋊D6111yescharacteristic
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417C2×C42C22×D6111yescharacteristic
418#107—614no—
419#107—1222no—
420#107—1222no—
421#107—1222no—
422#106—1222no—
423#106—1222no—
424#106—1222no—
425#106—1222no—
426#106—2441no—
427#106—2441no—
428C22⋊C8—614no—
429#97—1222no—
430C4×C8—1222no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95C22⋊D6111yescharacteristic
438#95C22⋊D6111yescharacteristic
439C24C2×C22⋊D6111yescharacteristic
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—81no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—81no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4Q8×D6112yes—
459C22×C4Q8×D6112yes—
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—1228no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—612no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—4881no—
501C4⋊C4—4881no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4D4×D6112yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—4881no—
511C22⋊C4—2442no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—2442no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1451977512yes—
524C23C22×C22⋊D6111yescharacteristic
525Q8—84no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4—1222no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4#1451977512yes—
568C2×C4#1451980312yes—
569C2×C4#1451980812yes—
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×Q8×D6111yescharacteristic
572C2×C4C2×Q8×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#1451982211yescharacteristic
575C2×C4#1451982411yescharacteristic
576C8—614no—
577C8—1222no—
578C8—1222no—
579C8—1222no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1451977511yescharacteristic
588C22D4×C22⋊D6111yescharacteristic
589C22#1684041111yescharacteristic
590C22C2×#1451980311yescharacteristic
591C22C2×#1451980811yescharacteristic
592C22#1684057011yescharacteristic
593C22#1684057311yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—4881no—
604C4—4881no—
605C4—4881no—
606C4—2442no—
607C4C22⋊Q8×D6111yescharacteristic
608C4#1684039111yescharacteristic
609C4#1684042511yescharacteristic
610C4#1684046611yescharacteristic
611C2—42no—
612C2#2144951811yescharacteristic
613C2#2145634511yescharacteristic
614C2#2145636111yescharacteristic
615C2#2145741511yescharacteristic
616C2#2145748111yescharacteristic
617C2#2145826811yescharacteristic
618C2#2145828911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2440C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2440D61
hypocenterC61#26078
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C2×#16840168C2semidirect
4. #21447295C2semidirect
5. #21447291C2semidirect
6. #21447157C2semidirect
7. #21447149C2semidirect
8. #21446554C2semidirect
9. #21446549C2semidirect
12. #21445975C2semidirect
63. C2×#11758194Q8semidirect
95. C22⋊#92418Q8semidirect
269. #92418C22⋊Q8semidirect
370. C61#26078semidirect

Maximal subgroups

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

subgroupindex
C2×#168401682
C2×#168398022
#214495322
#214472952
#214472912
#214471572
#214471492
#214465542
#214465492
#214460162
#214460122
#214459752
#214458502
#214456032
C61×#24402
#2607861