Finite Group Databasebeta

← #29595928

Subgroups — #29595928

Order 15616.

730 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21447508C212yesmaximal, complemented
2C2×#16839561C211yesmaximal, characteristic, complemented
3#21446357C212yesmaximal, complemented
4#21446213C211yesmaximal, characteristic, complemented
5C61×#3544C211yesmaximal, characteristic
6#16840240C2211yescharacteristic
7#16839779—22no—
8#16839771—21no—
9C2×#14519696C2211yescharacteristic
10C2×#14519696C2212yes—
11#16839561—21no—
12#16839561—21no—
13#16839561—22no—
14#16839559—22no—
15#16839522—21no—
16#16839518—22no—
17C61×#563—21no—
18C61×#563C412yescomplemented
19Q82×C61C411yescharacteristic, complemented
20C61×#555C411yescharacteristic, complemented
21C61×#554—21no—
22C61×#552—22no—
23C61×#547—22no—
24C61×#544—22no—
25C61×#542—22no—
26C61×#536—21no—
27C61×#536C2212yes—
28C2×C4⋊Q8×C61C2211yescharacteristic
29C61×#529—21no—
30C2×#11758194—21no—
31C2×#11758194—22no—
32Q8×Dic61—42no—
33#14519736—22no—
34#14519736—21no—
35#14519704—44no—
36#14519701—42no—
37#14519696—22no—
38#14519696—22no—
39#14519696—22no—
40C2×#41×C61—21no—
41C2×#41×C61D412yes—
42C22×Q8×C61D411yescharacteristic
43C22×Q8×C61C2311yescharacteristic
44C4⋊Q8×C61—21no—
45C4⋊Q8×C61—21no—
46C4⋊Q8×C61—22no—
47C4⋊Q8×C61—22no—
48C4⋊Q8×C61C2×C411yescharacteristic
49C4⋊Q8×C61C2×C411yescharacteristic
50C4⋊Q8×C61C2×C412yes—
51C4⋊Q8×C61D411yescharacteristic
52C4⋊Q8×C61D411yescharacteristic
53C4⋊Q8×C61D412yes—
54#126×C61—22no—
55#126×C61—22no—
56#126×C61—22no—
57#125×C61—21no—
58#125×C61—22no—
59#125×C61—22no—
60#125×C61—22no—
61#124×C61—21no—
62#124×C61—21no—
63#124×C61—21no—
64#124×C61C2×C412yes—
65#124×C61D412yes—
66#123×C61—22no—
67#123×C61—22no—
68#123×C61—22no—
69#123×C61—22no—
70#123×C61—22no—
71C22⋊Q8×C61—22no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—21no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61—22no—
77C22⋊Q8×C61—22no—
78C22⋊Q8×C61—22no—
79#121×C61—21no—
80#121×C61—22no—
81C4×Q8×C61—22no—
82C4×Q8×C61—21no—
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61—21no—
86#117×C61—21no—
87#117×C61—22no—
88#117×C61—22no—
89#117×C61D412yes—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C42×C61D411yescharacteristic
93#41×C61—42no—
94#41×C61—42no—
95#41×C61—81no—
96#41×C61—81no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—41no—
101C2×Q8×C61—41no—
102C2×Q8×C61—41no—
103C2×Q8×C61—41no—
104C2×Q8×C61C22⋊C411yescharacteristic
105C2×Q8×C61C22⋊C412yes—
106C2×Q8×C61C22⋊C411yescharacteristic
107C2×Q8×C61C22×C411yescharacteristic
108C2×Q8×C61C2×D411yescharacteristic
109C2×Q8×C61C2×D412yes—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61C22⋊C412yes—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—22no—
146C42×C61—21no—
147C42×C61—22no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61C22⋊C411yescharacteristic
151C42×C61C22⋊C411yescharacteristic
152C42×C61C22⋊C412yes—
153C42×C61C22⋊C412yes—
154C22⋊Dic61—41no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157C4×Dic61—24no—
158C4×Dic61—41no—
159#11758194—42no—
160#11758194—41no—
161#11758194—41no—
162#11758194—42no—
163C2×#92418—41no—
164C2×#92418—42no—
165C23×C61—22no—
166C23×C61#12011yescharacteristic
167Q8×C61—41no—
168Q8×C61—41no—
169Q8×C61—42no—
170Q8×C61—42no—
171Q8×C61—41no—
172Q8×C61—41no—
173Q8×C61—42no—
174Q8×C61—81no—
175Q8×C61—81no—
176D4×C61—42no—
177D4×C61—42no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—81no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—82no—
197C2×C4×C61—82no—
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×C4×C61C2×C22⋊C412yes—
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12012yes—
202C2×Dic61—42no—
203#92418—82no—
204#92418—81no—
205#12782—611nomaximal, Sylow
206C22×C61—41no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61#37911yescharacteristic
211C22×C61B(4,𝔽2)11yescharacteristic
212C22×C61#45811yescharacteristic
213C4×C61—22no—
214C4×C61—22no—
215C4×C61—21no—
216C4×C61—22no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—41no—
220C4×C61—42no—
221C4×C61—41no—
222C4×C61—41no—
223C4×C61—42no—
224Dic61—82no—
225#3544D6111yescharacteristic
226#2133—612no—
227C2×#356—611no—
228#1912—612no—
229#1886—611no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—82no—
234C2×C61#179011yescharacteristic
235#563—21no—
236#563Dic6112yescomplemented
237Q82Dic6111yescharacteristic, complemented
238#555Dic6111yescharacteristic, complemented
239#554—21no—
240#552—22no—
241#547—22no—
242#544—22no—
243#542—22no—
244#536—21no—
245#536C2×D6112yes—
246C2×C4⋊Q8C2×D6111yescharacteristic
247#529—21no—
248#519—611no—
249#421—1222no—
250#419—1221no—
251C2×#101—612no—
252C2×#101—611no—
253#356—1221no—
254#356—1221no—
255#356—1222no—
256#355—1222no—
257#339—1221no—
258H(3,ℤ/4)—1222no—
259C61#1278211yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C2×#41C22⋊D6112yes—
262C22×Q8C22⋊D6111yescharacteristic
263C22×Q8C22×D6111yescharacteristic
264C2×M16—1222no—
265C2×M16—1221no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8C2×Dic6111yescharacteristic
271C4⋊Q8C2×Dic6111yescharacteristic
272C4⋊Q8C2×Dic6112yes—
273C4⋊Q8C22⋊D6111yescharacteristic
274C4⋊Q8C22⋊D6111yescharacteristic
275C4⋊Q8C22⋊D6112yes—
276#126—22no—
277#126—22no—
278#126—22no—
279#125—21no—
280#125—22no—
281#125—22no—
282#125—22no—
283#124—21no—
284#124—21no—
285#124—21no—
286#124C2×Dic6112yes—
287#124C22⋊D6112yes—
288#123—22no—
289#123—22no—
290#123—22no—
291#123—22no—
292#123—22no—
293C22⋊Q8—21no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301#121—21no—
302#121—22no—
303C4×Q8—21no—
304C4×Q8—22no—
305C4×Q8—2442no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—21no—
309#117—22no—
310#117—21no—
311#117—22no—
312#117—1222no—
313#117—1221no—
314#117C22⋊D6112yes—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C42C22⋊D6111yescharacteristic
318#104—2444no—
319#103—2442no—
320#101—1222no—
321#101—1222no—
322#101—1222no—
323#41—42no—
324#41—42no—
325#41—81no—
326#41—81no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8C22⋊Dic6111yescharacteristic
335C2×Q8C22⋊Dic6112yes—
336C2×Q8C22⋊Dic6111yescharacteristic
337C2×Q8C22×Dic6111yescharacteristic
338C2×Q8C2×C22⋊D6111yescharacteristic
339C2×Q8C2×C22⋊D6112yes—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4C22⋊Dic6112yes—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4C2×C22⋊D6111yescharacteristic
350C22×C4C2×C22⋊D6112yes—
351M16—2442no—
352M16—2441no—
353M16—2441no—
354M16—2442no—
355C2×C8—2441no—
356C2×C8—2442no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—1222no—
374C4⋊C4—2442no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—2441no—
385C42—21no—
386C42—22no—
387C42—21no—
388C42—21no—
389C42—1224no—
390C42—2441no—
391C42C22⋊Dic6111yescharacteristic
392C42C22⋊Dic6111yescharacteristic
393C42C22⋊Dic6112yes—
394C42C22⋊Dic6112yes—
395C23—22no—
396C23#1451974611yescharacteristic
397Q8—41no—
398Q8—41no—
399Q8—42no—
400Q8—41no—
401Q8—41no—
402Q8—42no—
403Q8—42no—
404Q8—81no—
405Q8—81no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—81no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—2442no—
429C2×C4C2×C22⋊Dic6111yescharacteristic
430C2×C4C2×C22⋊Dic6112yes—
431C2×C4#1451974611yescharacteristic
432C2×C4#1451974612yes—
433C8—4882no—
434C8—4881no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22#1683961811yescharacteristic
440C22#1683994911yescharacteristic
441C22#1683995211yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—22no—
446C4—21no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—41no—
451C4—41no—
452C4—42no—
453C4—4882no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—82no—
458C2#2144576711yescharacteristic
subgroupG/H
centerC2#21445767
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
Fitting, prime radicalC61×#3544C2
socleC2×C61#1790
cosocleC22×Q8×C61C23
hypercenter#3544D61
hypocenterC61#12782
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ #3544

Complements

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

NH ≅ G/Ntype
1. #21447508C2semidirect
2. C2×#16839561C2semidirect
3. #21446357C2semidirect
4. #21446213C2semidirect
18. C61×#563C4semidirect
19. Q82×C61C4semidirect
20. C61×#555C4semidirect
236. #563Dic61semidirect
237. Q82Dic61semidirect
238. #555Dic61semidirect
259. C61#12782semidirect

Maximal subgroups

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

subgroupindex
#214475082
C2×#168395612
#214463572
#214462132
C61×#35442
#1278261