Finite Group Databasebeta

← #30602075

Subgroups — #30602075

Order 15616.

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

normal subgroups only

This group has 422 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
1#21462004C211yesmaximal, characteristic
2#21462004C211yesmaximal, characteristic
3#21461532C211yesmaximal, characteristic
4#21461377C211yesmaximal, characteristic
5#21461169C211yesmaximal, characteristic
6#21461168C211yesmaximal, characteristic
7#21456404C211yesmaximal, characteristic
8#16840766C2211yescharacteristic
9#16840766C2211yescharacteristic
10C4×D61⋊C8C2211yescharacteristic
11C42×#6923C2211yescharacteristic
12#16840632—22no—
13#16840632—22no—
14(C2×#6923)⋊C8—22no—
15#16840625—22no—
16#16840612—22no—
17Q8⋊#92420—22no—
18#16840317C2211yescharacteristic
19#16840317C2211yescharacteristic
20#16840165C411yescharacteristic, complemented
21#16840141C411yescharacteristic, complemented
22#16840113C411yescharacteristic, complemented
23#16840002—22no—
24#16839854—22no—
25C4×C8×D61C2211yescharacteristic
26C61×#487C411yescharacteristic, complemented
27C2×#11758213—21no—
28C2×#11758213—21no—
29C2×D61⋊C8—21no—
30C2×C4×#6923—21no—
31C2×C4×#6923—21no—
32C2×C4×#6923D412yes—
33C4⋊#92420—21no—
34C4⋊#92420—21no—
35C4⋊#92420—21no—
36C4⋊#92420—21no—
37C4×#92420—21no—
38C4×#92420D412yes—
39C2×#11758202—22no—
40C2×C8×D61D412yes—
41#14519777C2×C411yescharacteristic
42#14519777C2×C411yescharacteristic
43#14519768—21no—
44#14519768—21no—
45#14519757—21no—
46#14519757—21no—
47#14519753—21no—
48#14519753—21no—
49Q8×Dic61—21no—
50Q8×Dic61—21no—
51#14519742—21no—
52#14519742—21no—
53#14519735—21no—
54#14519735—21no—
55C42×D61C2311yescharacteristic
56#14519710—22no—
57#14519702—22no—
58C4×#92418C2×C411yescharacteristic
59C8×Dic61D412yes—
60C4⋊Q8×C61C2×C411yescharacteristic
61C4⋊Q8×C61C2×C411yescharacteristic
62#103×C61—22no—
63C4×C8×C61C2×C411yescharacteristic
64C22×#6923—22no—
65C2×Q8×C61—21no—
66C2×Q8×C61—21no—
67C2×C4×D61C2×D411yescharacteristic
68C2×C4×D61C2×D411yescharacteristic
69C2×C4×D61#4111yescharacteristic
70C2×Dic122—21no—
71C2×Dic122—21no—
72C2×#92420—22no—
73C2×#92420—41no—
74C2×#92420—41no—
75C4×#6923—22no—
76C4×#6923—42no—
77C4×#6923—42no—
78C4×#6923Q1614yescomplemented
79#11758213—42no—
80#11758213—42no—
81D61⋊C8—42no—
82C2×C8×C61C22⋊C412yes—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—21no—
87C42×C61C22×C411yescharacteristic
88#11758202—44no—
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—21no—
92C4⋊Dic61—21no—
93C4⋊Dic61—21no—
94C4⋊Dic61—21no—
95C4⋊Dic61—41no—
96C4⋊Dic61—41no—
97#11758196—41no—
98#11758196—41no—
99C4×Dic61—41no—
100C4×Dic61—41no—
101C4×Dic61C2×D411yescharacteristic
102C4×Dic61C2×D411yescharacteristic
103C4×Dic61#4111yescharacteristic
104C2×#92418C22⋊C412yes—
105C8×D61—42no—
106C22×D61#12111yescharacteristic
107C2×#6923—24no—
108C2×#6923—42no—
109C2×#6923—42no—
110Q8×C61—42no—
111Q8×C61—42no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61C2×C22⋊C411yescharacteristic
115C2×C4×C61C4×D411yescharacteristic
116C2×C4×C61C4×D411yescharacteristic
117C2×Dic61—41no—
118C2×Dic61—41no—
119C2×Dic61#12111yescharacteristic
120C2×Dic61#12411yescharacteristic
121C2×Dic61#12711yescharacteristic
122C4×D61—22no—
123C4×D61—22no—
124C4×D61C2×Q1612yes—
125C4×D61#13512yes—
126Dic122—42no—
127Dic122—42no—
128#92420—42no—
129#92420—81no—
130#92420—81no—
131C8×C61—22no—
132#92418—22no—
133#9848—611nomaximal, Sylow
134C22×C61#39011yescharacteristic
135C2×D61—21no—
136C2×D61—21no—
137C2×D61#46712yes—
138#6923—44no—
139C4×C61—81no—
140C4×C61—81no—
141C4×C61#42112yes—
142C4×C61C4×Q1612yes—
143C4×C61#44412yes—
144Dic61—21no—
145Dic61—21no—
146Dic61—81no—
147Dic61—81no—
148Dic61#49411yescharacteristic
149Dic61#49511yescharacteristic
150#3170—611no—
151#1994—611no—
152#1994—611no—
153#1744—611no—
154#1567—611no—
155#1493—611no—
156#1493—611no—
157C2×C61#196311yescharacteristic
158C2×C61#196911yescharacteristic
159C2×C61#197511yescharacteristic
160D61—22no—
161#533—611no—
162#533—611no—
163#500—611no—
164#494—611no—
165#487—611no—
166#487#692311yescharacteristic, complemented
167#467—1222no—
168C4×Q16—1222no—
169#423—611no—
170#423—611no—
171C2×C4×C8—611no—
172C2×C4×C8—611no—
173C43—611no—
174#339—1222no—
175#339—1222no—
176#336—1222no—
177#335—1222no—
178Q8⋊C8—1222no—
179Q8⋊C8—1222no—
180C61#984811yescharacteristic, complemented, Sylow
181C2×Q16—1222no—
182C2×M16—1221no—
183C2×M16—1221no—
184C22×C8—612no—
185C22×C8—1221no—
186C4⋊Q8—611no—
187C4⋊Q8—611no—
188C4⋊Q8C2×#692311yescharacteristic
189C4⋊Q8C2×#692311yescharacteristic
190#125—1221no—
191#125—1221no—
192C22⋊Q8—1221no—
193C22⋊Q8—1221no—
194C22⋊Q8—1221no—
195C22⋊Q8—1221no—
196C4×Q8—1221no—
197C4×Q8—1221no—
198C4×Q8—1221no—
199C4×Q8—1221no—
200#117—1221no—
201#117—1221no—
202C2×C42—611no—
203C2×C42—612no—
204C2×C42—1221no—
205C2×C42—1221no—
206#107—1222no—
207C4⋊C8—1221no—
208C4⋊C8—1221no—
209C4⋊C8—1221no—
210C4⋊C8—1221no—
211#103—22no—
212#103—1222no—
213C4×C8—612no—
214C4×C8—612no—
215C4×C8—611no—
216C4×C8—1221no—
217C4×C8C2×#692311yescharacteristic
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8—1221no—
221C2×Q8—1221no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—1222no—
226Q16—2444no—
227M16—2442no—
228M16—2442no—
229C2×C8—612no—
230C2×C8—1222no—
231C2×C8—2442no—
232C2×C8—2441no—
233C2×C8—2441no—
234C2×C8—2442no—
235C2×C8C22⋊#692312yes—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—1221no—
241C4⋊C4—1221no—
242C4⋊C4—1221no—
243C4⋊C4—1221no—
244C4⋊C4—2441no—
245C4⋊C4—2441no—
246C4⋊C4—2441no—
247C4⋊C4—2441no—
248C22⋊C4—2441no—
249C22⋊C4—2441no—
250C42—611no—
251C42—611no—
252C42—611no—
253C42—614no—
254C42—1222no—
255C42—2441no—
256C42—2441no—
257C42—2442no—
258C42—2442no—
259C42C22×#692311yescharacteristic
260C23—611no—
261Q8—42no—
262Q8—42no—
263Q8—2442no—
264Q8—2442no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—611no—
268C2×C4—612no—
269C2×C4—612no—
270C2×C4—611no—
271C2×C4—611no—
272C2×C4—1222no—
273C2×C4—1222no—
274C2×C4—1224no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4—2442no—
278C2×C4—2442no—
279C2×C4C2×C22⋊#692311yescharacteristic
280C2×C4D4×#692311yescharacteristic
281C2×C4D4×#692311yescharacteristic
282C8—22no—
283C8—1222no—
284C8—4881no—
285C8—4881no—
286C8—2442no—
287C22—612no—
288C22—1221no—
289C22—1221no—
290C22#1684071411yescharacteristic
291C4—81no—
292C4—81no—
293C4—611no—
294C4—611no—
295C4—1221no—
296C4—1221no—
297C4—4881no—
298C4—4881no—
299C4—2444no—
300C4#1684075912yes—
301C4Q16×#692312yes—
302C4#1684080712yes—
303C2—1222no—
304C2#2146195611yescharacteristic
305C2#2146196511yescharacteristic
306C2#2146197511yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#390
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9848
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
20. #16840165C4semidirect
21. #16840141C4semidirect
22. #16840113C4semidirect
26. C61×#487C4semidirect
78. C4×#6923Q16semidirect
166. #487#6923semidirect
180. C61#9848semidirect

Maximal subgroups

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

subgroupindex
#214620042
#214615322
#214613772
#214611692
#214611682
#214564042
#984861