Finite Group Databasebeta

← #30600615

Subgroups — #30600615

Order 15616.

566 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C4×Q8×#6923C211yesmaximal, characteristic
2#21462004C211yesmaximal, characteristic
3#21461203C211yesmaximal, characteristic
4#21461148C211yesmaximal, characteristic
5#21460716C211yesmaximal, characteristic
6#21460646C211yesmaximal, characteristic
7#21455860C211yesmaximal, characteristic
8C2×Q8×#6923—21no—
9#16840781C2211yescharacteristic
10#16840766C2211yescharacteristic
11C4⋊C4×#6923—21no—
12C4⋊C4×#6923—22no—
13C4×C4⋊#6923—21no—
14C4×C4⋊#6923C2211yescharacteristic
15C42×#6923—21no—
16C42×#6923C2211yescharacteristic
17#16840632—21no—
18#16840632—21no—
19#16840616—22no—
20#16840612—21no—
21Q8⋊#92420—21no—
22#16840317C2211yescharacteristic
23C4×Q8×D61C2211yescharacteristic
24#16840099—21no—
25#16840054C411yescharacteristic, complemented
26#16839973C411yescharacteristic, complemented
27#16839972C411yescharacteristic, complemented
28#16839919—21no—
29#16839856—21no—
30#16839854—21no—
31#16839814C2211yescharacteristic
32C61×#477C411yescharacteristic, complemented
33C2×#11758213—21no—
34Q8×#6923—24no—
35Q8×#6923—24no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923—21no—
38C2×C4×#6923—22no—
39C2×C4×#6923—21no—
40C2×C4×#6923—22no—
41C2×C4×#6923—21no—
42C2×C4×#6923—21no—
43C2×C4×#6923D411yescharacteristic
44C2×C4×#6923D411yescharacteristic
45C4⋊#92420—21no—
46C4⋊#92420—21no—
47C4⋊#92420—21no—
48#14519834—21no—
49C4×#92420—21no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4D412yes—
52C2×Q8×D61D411yescharacteristic
53C2×Dic244—21no—
54C2×#11758202—21no—
55#14519777C2×C411yescharacteristic
56#14519768—21no—
57#14519757—21no—
58#14519753—21no—
59Q8×Dic61—21no—
60Q8×Dic61D411yescharacteristic
61C4×Dic122—21no—
62C4×Dic122C2×C411yescharacteristic
63#14519742—21no—
64#14519742—21no—
65#14519742D411yescharacteristic
66#14519735—21no—
67C4⋊C4×D61—21no—
68C4⋊C4×D61D411yescharacteristic
69C42×D61—21no—
70C42×D61C2311yescharacteristic
71#14519710—21no—
72C4⋊#92418C2×C411yescharacteristic
73#14519702—21no—
74#14519700—21no—
75#14519700—21no—
76#14519689—21no—
77C8×Dic61—21no—
78C4⋊Q8×C61C2×C411yescharacteristic
79C4×Q8×C61C2×C411yescharacteristic
80#107×C61—21no—
81C4⋊C8×C61C2×C411yescharacteristic
82#103×C61—21no—
83C22×#6923—21no—
84C22×#6923—21no—
85C22×#6923—22no—
86C2×Q8×C61—21no—
87C2×Q8×C61C22⋊C411yescharacteristic
88Q8×D61—22no—
89Q8×D61—22no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61C2×D411yescharacteristic
93C2×C4×D61C2×D411yescharacteristic
94C2×C4×D61C2×D411yescharacteristic
95C2×Dic122—21no—
96C2×Dic122—21no—
97C2×Dic122C22⋊C411yescharacteristic
98C2×#92420—41no—
99C2×#92420—41no—
100C4⋊#6923—28no—
101C4⋊#6923—22no—
102C4⋊#6923—22no—
103C4×#6923—28no—
104C4×#6923—22no—
105C4×#6923—22no—
106C4×#6923—21no—
107C4×#6923—21no—
108C4×#6923—21no—
109C4×#6923—42no—
110C4×#6923—44no—
111C4×#6923—41no—
112C4×#6923—41no—
113C4×#6923Q1612yescomplemented
114#11758213—41no—
115#11758213—41no—
116C2×C8×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61C22⋊C411yescharacteristic
121C42×C61—21no—
122C42×C61C22×C411yescharacteristic
123#11758202—42no—
124#11758198—41no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—41no—
129C4⋊Dic61C22⋊C411yescharacteristic
130#11758196—21no—
131#11758196—21no—
132#11758196—21no—
133#11758196—41no—
134C4×Dic61—21no—
135C4×Dic61—21no—
136C4×Dic61—21no—
137C4×Dic61—41no—
138C4×Dic61C2×D411yescharacteristic
139C4×Dic61#4111yescharacteristic
140C4×Dic61#4111yescharacteristic
141C2×#92418—21no—
142Dic244—42no—
143C22×D61#12011yescharacteristic
144C2×#6923—22no—
145C2×#6923—22no—
146C2×#6923—22no—
147C2×#6923—22no—
148C2×#6923—24no—
149C2×#6923—41no—
150C2×#6923—42no—
151C2×#6923—41no—
152C2×#6923—42no—
153Q8×C61—21no—
154Q8×C61—81no—
155Q8×C61#10412yes—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—41no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61#12111yescharacteristic
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12311yescharacteristic
168C4×D61—24no—
169C4×D61—21no—
170C4×D61—21no—
171C4×D61—41no—
172C4×D61—42no—
173C4×D61C2×Q1611yescharacteristic
174C4×D61#13511yescharacteristic
175C4×D61#13711yescharacteristic
176C4×D61#13711yescharacteristic
177Dic122—24no—
178Dic122—21no—
179Dic122—41no—
180Dic122—41no—
181Dic122#10412yes—
182#92420—81no—
183#92420—81no—
184C8×C61—41no—
185#92418—41no—
186#9232—611nomaximal, Sylow
187C22×C61#38611yescharacteristic
188C2×D61—21no—
189C2×D61—21no—
190C2×D61#45111yescharacteristic
191C2×D61#45211yescharacteristic
192#6923—24no—
193#6923—42no—
194#6923—44no—
195C4×C61—22no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—81no—
199C4×C61C2×#10411yescharacteristic
200C4×C61C2×#10411yescharacteristic
201C4×C61C4×Q1611yescharacteristic
202C4×C61#44111yescharacteristic
203Dic61—21no—
204Dic61—22no—
205Dic61—21no—
206Dic61—41no—
207Dic61—81no—
208Dic61#46211yescharacteristic
209Dic61#46411yescharacteristic
210#3104—611no—
211C42×Q8—611no—
212#1994—611no—
213#1503—611no—
214#1487—611no—
215#1303—611no—
216#1276—611no—
217C2×C61#187811yescharacteristic
218C2×C61#189611yescharacteristic
219C2×C61#190611yescharacteristic
220D61—21no—
221D61—21no—
222#533—611no—
223C2×C4×Q8—611no—
224C2×C4×Q8—1221no—
225#484—1221no—
226#477—611no—
227#477#692311yescharacteristic, complemented
228#462—611no—
229#462—611no—
230#451—1221no—
231C4×Q16—1221no—
232C4×Q16—1221no—
233#431—611no—
234#431—611no—
235#423—611no—
236C4×C4⋊C4—611no—
237C4×C4⋊C4—1222no—
238C4×C4⋊C4—1221no—
239C4×C4⋊C4—1221no—
240C43—611no—
241C43—1221no—
242#339—1221no—
243#339—1221no—
244#328—1222no—
245Q8⋊C8—1221no—
246Q8⋊C8—1221no—
247C61#923211yescharacteristic, complemented, Sylow
248C22×Q8—611no—
249C2×Q16—1221no—
250C2×Q16—1221no—
251C2×M16—1221no—
252C4⋊Q8—611no—
253C4⋊Q8C2×#692311yescharacteristic
254#125—1221no—
255C22⋊Q8—1221no—
256C22⋊Q8—1221no—
257C4×Q8—611no—
258C4×Q8—611no—
259C4×Q8—611no—
260C4×Q8—1221no—
261C4×Q8—1221no—
262C4×Q8—1221no—
263C4×Q8—1221no—
264C4×Q8—1224no—
265C4×Q8—1224no—
266C4×Q8C2×#692311yescharacteristic
267#117—1221no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1222no—
272C2×C42—611no—
273C2×C42—611no—
274C2×C42—611no—
275C2×C42—1221no—
276C2×C42—1221no—
277C2×C42—1222no—
278C2×C42—1221no—
279C2×C42—1221no—
280C2×C42—1222no—
281#107—21no—
282#107—1221no—
283C4⋊C8—611no—
284C4⋊C8—1221no—
285C4⋊C8—1221no—
286C4⋊C8—1221no—
287C4⋊C8C2×#692311yescharacteristic
288#103—21no—
289#103—1221no—
290#103—1221no—
291#103—1221no—
292C22⋊C8—1221no—
293C22⋊C8—1221no—
294C4×C8—1221no—
295C4×C8—1221no—
296#95—612no—
297#95—1222no—
298C2×Q8—21no—
299C2×Q8—611no—
300C2×Q8—1221no—
301C2×Q8—1222no—
302C2×Q8—1222no—
303C2×Q8—1221no—
304C2×Q8C22⋊#692311yescharacteristic
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1222no—
312C22×C4—1221no—
313Q16—2442no—
314Q16—2442no—
315M16—2441no—
316M16—2441no—
317C2×C8—21no—
318C2×C8—1221no—
319C2×C8—2441no—
320C2×C8—2441no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—611no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—1221no—
328C4⋊C4—1228no—
329C4⋊C4—1222no—
330C4⋊C4—1222no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—2441no—
335C4⋊C4—2441no—
336C4⋊C4C22⋊#692311yescharacteristic
337C22⋊C4—2441no—
338C42—21no—
339C42—611no—
340C42—611no—
341C42—611no—
342C42—612no—
343C42—1221no—
344C42—1221no—
345C42—1221no—
346C42—1221no—
347C42—1221no—
348C42—1228no—
349C42—1222no—
350C42—1222no—
351C42—1221no—
352C42—2441no—
353C42—2441no—
354C42—2441no—
355C42—2442no—
356C42—2444no—
357C42C22×#692311yescharacteristic
358C23—611no—
359Q8—21no—
360Q8—81no—
361Q8—612no—
362Q8—1224no—
363Q8—1221no—
364Q8—2441no—
365Q8—2441no—
366Q8#1451984212yes—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—611no—
371C2×C4—611no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—611no—
375C2×C4—611no—
376C2×C4—611no—
377C2×C4—1224no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—1222no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—1224no—
385C2×C4—1222no—
386C2×C4—1221no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2441no—
394C2×C4C2×C22⋊#692311yescharacteristic
395C2×C4D4×#692311yescharacteristic
396C2×C4D4×#692311yescharacteristic
397C8—41no—
398C8—4881no—
399C8—4881no—
400C8—2441no—
401C22—611no—
402C22—611no—
403C22—1221no—
404C22—1221no—
405C22#1684071011yescharacteristic
406C4—22no—
407C4—21no—
408C4—41no—
409C4—81no—
410C4—611no—
411C4—611no—
412C4—1224no—
413C4—1221no—
414C4—1222no—
415C4—1221no—
416C4—4881no—
417C4—2442no—
418C4—2444no—
419C4—2441no—
420C4D61⋊#10411yescharacteristic
421C4C2×#1451984211yescharacteristic
422C4Q16×#692311yescharacteristic
423C4#1684080211yescharacteristic
424C2—1221no—
425C2—1221no—
426C2#2146181311yescharacteristic
427C2#2146183611yescharacteristic
428C2#2146185611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#477C4
socleC22×C61#386
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#477#6923
hypocenterC61#9232
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
25. #16840054C4semidirect
26. #16839973C4semidirect
27. #16839972C4semidirect
32. C61×#477C4semidirect
113. C4×#6923Q16semidirect
227. #477#6923semidirect
247. C61#9232semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×#69232
#214620042
#214612032
#214611482
#214607162
#214606462
#214558602
#923261