Finite Group Databasebeta

← #29595800

Subgroups — #29595800

Order 15616.

736 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21447514C211yesmaximal, characteristic, complemented
2#21447473C211yesmaximal, characteristic, complemented
3#21447460C211yesmaximal, characteristic
4#21446372C211yesmaximal, characteristic, complemented
5#21446342C211yesmaximal, characteristic, complemented
6#21446238C211yesmaximal, characteristic
7C61×#3538C211yesmaximal, characteristic
8#16840529C2211yescharacteristic
9#16839771—21no—
10#16839746C2211yescharacteristic
11#16839746C2211yescharacteristic
12C2×#14519696—21no—
13#16839736C2211yescharacteristic
14#16839726—21no—
15#16839561—22no—
16#16839552—21no—
17#16839552—22no—
18#16839549—21no—
19#16839532—21no—
20#16839532—21no—
21#16839518—21no—
22C61×#563—21no—
23C61×#563C412yescomplemented
24C61×#552—21no—
25C61×#552—21no—
26C61×#551—21no—
27C61×#551C412yescomplemented
28D4×Q8×C61—21no—
29D4×Q8×C61—21no—
30C61×#544—22no—
31C61×#541—21no—
32C61×#539—22no—
33C61×#536C2211yescharacteristic
34C61×#536C2211yescharacteristic
35C61×#525C2211yescharacteristic
36C2×C22⋊Q8×C61—21no—
37#14519784—42no—
38C2×#11758194—21no—
39C2×#11758194—21no—
40C2×#11758194—21no—
41C2×#11758194—21no—
42C2×#11758194—21no—
43#14519736—21no—
44#14519701—42no—
45#14519696—21no—
46#14519696—21no—
47#14519696—42no—
48#14519694—21no—
49#14519694—21no—
50#14519692—21no—
51#14519692—21no—
52C22⋊#92418—42no—
53C2×#41×C61C2311yescharacteristic
54C22×Q8×C61—21no—
55C22×Q8×C61D411yescharacteristic
56C22×Q8×C61D411yescharacteristic
57C23×C4×C61D411yescharacteristic
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61D412yes—
63#126×C61—22no—
64#126×C61—22no—
65#126×C61—22no—
66#126×C61—22no—
67#125×C61—22no—
68#125×C61—22no—
69#125×C61—21no—
70#125×C61—22no—
71#125×C61—22no—
72#124×C61—21no—
73#124×C61C2×C412yes—
74#123×C61—22no—
75#123×C61—21no—
76#123×C61—21no—
77#123×C61—22no—
78#123×C61C2×C412yes—
79#123×C61C2×C411yescharacteristic
80#123×C61D412yes—
81#123×C61D411yescharacteristic
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95#121×C61—21no—
96#120×C61C2×C411yescharacteristic
97#120×C61D411yescharacteristic
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104#117×C61—22no—
105#117×C61—22no—
106#117×C61D411yescharacteristic
107#117×C61D411yescharacteristic
108#117×C61D411yescharacteristic
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61—21no—
111C2×C22⋊C4×C61—22no—
112C24×C61C22⋊C411yescharacteristic
113#41×C61—41no—
114#41×C61—41no—
115C2×Q8×C61—22no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—41no—
120C2×Q8×C61—41no—
121C2×Q8×C61—41no—
122C2×Q8×C61—41no—
123C2×Q8×C61—81no—
124C2×Q8×C61—81no—
125C2×Q8×C61C22⋊C412yes—
126C2×Q8×C61C2×D411yescharacteristic
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61C22×C411yescharacteristic
130C2×D4×C61C2×D411yescharacteristic
131C2×D4×C61C2×D411yescharacteristic
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—41no—
138C22×C4×C61—41no—
139C22×C4×C61C22⋊C412yes—
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61C22⋊C411yescharacteristic
174C22⋊C4×C61C22⋊C411yescharacteristic
175C22⋊C4×C61C22⋊C412yes—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61C22⋊C412yes—
181C22⋊Dic61—41no—
182C4⋊Dic61—41no—
183C4×Dic61—41no—
184#11758194—22no—
185#11758194—41no—
186#11758194—41no—
187#11758194—41no—
188#11758194—41no—
189#11758194—41no—
190#11758194—41no—
191#11758194—41no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—41no—
195C2×#92418—41no—
196C2×#92418—41no—
197C23×C61—21no—
198C23×C61—41no—
199C23×C61—42no—
200C23×C61C2×C22⋊C411yescharacteristic
201C23×C61C2×C22⋊C411yescharacteristic
202C23×C61#12011yescharacteristic
203Q8×C61—41no—
204Q8×C61—42no—
205Q8×C61—41no—
206Q8×C61—41no—
207Q8×C61—41no—
208Q8×C61—41no—
209Q8×C61—81no—
210Q8×C61—82no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—42no—
215D4×C61—42no—
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—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—44no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61—81no—
235C2×C4×C61—81no—
236C2×C4×C61—82no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×Dic61—81no—
242#92418—42no—
243#92418—81no—
244#92418—81no—
245#12757—611nomaximal, Sylow
246C22×C61—22no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—42no—
255C22×C61—41no—
256C22×C61—81no—
257C22×C61#37911yescharacteristic
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—42no—
262C4×C61—42no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61B(4,𝔽2)11yescharacteristic
269C4×C61#45811yescharacteristic
270Dic61—161no—
271#3538D6111yescharacteristic
272#2135—611no—
273#2123—611no—
274#2120—611no—
275#1915—611no—
276#1909—611no—
277#1891—611no—
278C2×C61—21no—
279C2×C61—41no—
280C2×C61—41no—
281C2×C61—42no—
282C2×C61—41no—
283C2×C61#179011yescharacteristic
284#568—611no—
285#563—21no—
286#563Dic6112yescomplemented
287#552—21no—
288#552—21no—
289#551—21no—
290#551Dic6112yescomplemented
291D4×Q8—21no—
292D4×Q8—21no—
293#544—22no—
294#541—21no—
295#539—22no—
296#536C2×D6111yescharacteristic
297#536C2×D6111yescharacteristic
298#525C2×D6111yescharacteristic
299C2×C22⋊Q8—21no—
300#419—1221no—
301#413—611no—
302#413—611no—
303C2×#101—1221no—
304#410—611no—
305#407—1221no—
306#356—1222no—
307#352—1221no—
308#352—1222no—
309AΣL(1,𝔽16)—1221no—
310#344—1221no—
311#344—1221no—
312H(3,ℤ/4)—1221no—
313C61#1275711yescharacteristic, complemented, Sylow
314C2×#41C22×D6111yescharacteristic
315C22×Q8—21no—
316C22×Q8C22⋊D6111yescharacteristic
317C22×Q8C22⋊D6111yescharacteristic
318C23×C4C22⋊D6111yescharacteristic
319#131—2442no—
320C2×M16—1221no—
321C2×M16—1221no—
322C2×M16—1221no—
323C2×M16—1221no—
324C2×M16—1221no—
325C4⋊Q8—21no—
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—21no—
329C4⋊Q8C22⋊D6112yes—
330#126—22no—
331#126—22no—
332#126—22no—
333#126—22no—
334#125—22no—
335#125—22no—
336#125—21no—
337#125—22no—
338#125—22no—
339#124—21no—
340#124C2×Dic6112yes—
341#123—22no—
342#123—21no—
343#123—21no—
344#123—22no—
345#123C2×Dic6111yescharacteristic
346#123C2×Dic6112yes—
347#123C22⋊D6111yescharacteristic
348#123C22⋊D6112yes—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—21no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362#121—21no—
363#120C2×Dic6111yescharacteristic
364#120C22⋊D6111yescharacteristic
365C4×Q8—21no—
366C4×Q8—21no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371#117—22no—
372#117—22no—
373#117—1221no—
374#117C22⋊D6111yescharacteristic
375#117C22⋊D6111yescharacteristic
376#117C22⋊D6111yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C22⋊C4—22no—
380#103—2442no—
381#101—1221no—
382#101—1221no—
383#101—2442no—
384#100—1221no—
385#100—1221no—
386#99—1221no—
387#99—1221no—
388C22⋊C8—2442no—
389C24C22⋊Dic6111yescharacteristic
390#41—41no—
391#41—41no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8C22⋊Dic6112yes—
403C2×Q8C2×C22⋊D6111yescharacteristic
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C22×Dic6111yescharacteristic
407C2×D4C2×C22⋊D6111yescharacteristic
408C2×D4C2×C22⋊D6111yescharacteristic
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4C22⋊Dic6112yes—
417C22×C4C22⋊Dic6111yescharacteristic
418C22×C4C2×C22⋊D6111yescharacteristic
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421M16—1222no—
422M16—2441no—
423M16—2441no—
424M16—2441no—
425M16—2441no—
426M16—2441no—
427M16—2441no—
428M16—2441no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—2441no—
432C2×C8—2441no—
433C2×C8—2441no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—2441no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—2441no—
465C22⋊C4C22⋊Dic6111yescharacteristic
466C22⋊C4C22⋊Dic6111yescharacteristic
467C22⋊C4C22⋊Dic6112yes—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—2441no—
473C42C22⋊Dic6112yes—
474C23—21no—
475C23—41no—
476C23—42no—
477C23C2×C22⋊Dic6111yescharacteristic
478C23C2×C22⋊Dic6111yescharacteristic
479C23#1451974611yescharacteristic
480Q8—41no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—82no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—42no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—4881no—
515C2×C4C2×C22⋊Dic6111yescharacteristic
516C2×C4#1451974611yescharacteristic
517C2×C4#1451974611yescharacteristic
518C2×C4#1451974611yescharacteristic
519C8—4881no—
520C8—4881no—
521C8—2442no—
522C22—22no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—81no—
533C22#1683961811yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—42no—
538C4—42no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—42no—
543C4—42no—
544C4—9761no—
545C4#1683994911yescharacteristic
546C4#1683995211yescharacteristic
547C2—21no—
548C2—41no—
549C2—42no—
550C2—41no—
551C2—41no—
552C2#2144576711yescharacteristic
subgroupG/H
centerC2#21445767
commutatorC2×D4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#3538C2
socleC2×C61#1790
prime radicalC61×#525C22
cosocleC2×#41×C61C23
hypercenter#3538D61
hypocenterC61#12757
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3538

Complements

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

NH ≅ G/Ntype
1. #21447514C2semidirect
2. #21447473C2semidirect
4. #21446372C2semidirect
5. #21446342C2semidirect
23. C61×#563C4semidirect
27. C61×#551C4semidirect
286. #563Dic61semidirect
290. #551Dic61semidirect
313. C61#12757semidirect

Maximal subgroups

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

subgroupindex
#214475142
#214474732
#214474602
#214463722
#214463422
#214462382
C61×#35382
#1275761