Finite Group Databasebeta

← #30606580

Subgroups — #30606580

Order 15616.

452 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 92 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#21462546C211yesmaximal, characteristic
2#21462093C211yesmaximal, characteristic
3#21461578C212yesmaximal, complemented
4#21460921C212yesmaximal, complemented
5#21447503C211yesmaximal, characteristic
6C2×#14519865C411yescharacteristic, complemented
7C2×#14519865C2211yescharacteristic
8#16840792—22no—
9#16840792—22no—
10#16840767C411yescharacteristic, complemented
11#16840743C2212yes—
12#16840737C411yescharacteristic, complemented
13#16840737C2211yescharacteristic
14#16840728—22no—
15#16840721C411yescharacteristic, complemented
16#16840668C412yes—
17#16840667C412yes—
18#16840645—22no—
19#16840313C2211yescharacteristic
20#16839745C2212yes—
21#16839561C411yescharacteristic, complemented
22#16839558C412yescomplemented
23C61×#356C411yescharacteristic, complemented
24#14519865—42no—
25#14519865—42no—
26C2×#11758213—21no—
27C2×#11758213C2×C411yescharacteristic
28C2×#11758213C2×C411yescharacteristic
29C2×#11758213D412yes—
30C2×D61⋊C8—21no—
31C2×D61⋊C8C2×C411yescharacteristic
32C2×D61⋊C8C2×C411yescharacteristic
33C2×D61⋊C8D412yes—
34#14519856—22no—
35#14519844—22no—
36#14519836—22no—
37#14519836C2×C414yes—
38#14519834—22no—
39#14519834—22no—
40#14519832—21no—
41#14519832—21no—
42#14519831—42no—
43C8×#6923—42no—
44C2×#11758226C2×C411yescharacteristic
45C2×#11758226C2311yescharacteristic
46M16×D61—22no—
47#14519774C2×C411yescharacteristic
48#14519766C2×C411yescharacteristic
49#14519766C2×C411yescharacteristic
50#14519752—22no—
51#14519739—22no—
52C42×D61C2×C411yescharacteristic
53#14519696C2×C412yes—
54#14519693—22no—
55C4⋊Q8×C61C2×C411yescharacteristic
56#101×C61C2×C412yes—
57C2×Q8×C61C4211yescharacteristic
58C2×Q8×C61C22×C411yescharacteristic
59#11758226—41no—
60#11758226—41no—
61#11758226—41no—
62#11758226—41no—
63C2×D244—21no—
64C2×D244—22no—
65C2×D244C4211yescharacteristic
66C2×D244C22×C411yescharacteristic
67C2×D244#4112yes—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61C22×C411yescharacteristic
71C2×C4×D61C2×D412yes—
72C22⋊#6923—24no—
73C2×#92420—21no—
74C2×#92420—21no—
75C2×#92420C22⋊C411yescharacteristic
76C2×#92420C22⋊C412yes—
77C2×#92420C22⋊C411yescharacteristic
78C4⋊#6923—42no—
79C4×#6923—42no—
80#11758213—41no—
81#11758213—42no—
82#11758213—42no—
83#11758213—41no—
84D61⋊C8—22no—
85D61⋊C8—22no—
86D61⋊C8—22no—
87D61⋊C8—42no—
88D61⋊C8—41no—
89M16×C61—22no—
90C4⋊C4×C61—22no—
91C42×C61C4211yescharacteristic
92#11758198—22no—
93#11758198—22no—
94C4×Dic61—21no—
95C4×Dic61C4211yescharacteristic
96#11758194—22no—
97#11758189—42no—
98C8×D61—42no—
99C22×D61—21no—
100C22×D61#11711yescharacteristic
101C22×D61C4×D412yes—
102C2×#6923—44no—
103Q8×C61—41no—
104Q8×C61—41no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61C2×C4211yescharacteristic
108C2×C4×C61C4×D412yes—
109C2×Dic61—21no—
110C2×Dic61C2×C22⋊C411yescharacteristic
111D244—41no—
112D244—42no—
113D244—41no—
114D244—42no—
115D244—82no—
116C4×D61—22no—
117C4×D61—21no—
118C4×D61—22no—
119C4×D61—21no—
120C4×D61—41no—
121C4×D61—41no—
122C4×D61—81no—
123#92420—21no—
124#92420—22no—
125#92420—41no—
126#92420—41no—
127#92420#9912yes—
128C8×C61—42no—
129#92418—42no—
130#12634—611nomaximal, Sylow
131C22×C61C4×C22⋊C411yescharacteristic
132C2×D61—21no—
133C2×D61—22no—
134C2×D61—41no—
135C2×D61—42no—
136C2×D61—42no—
137C2×D61—41no—
138#6923—84no—
139C4×C61—22no—
140C4×C61—21no—
141C4×C61—41no—
142C4×C61—41no—
143Dic61—41no—
144Dic61C2×#9911yescharacteristic
145Dic61#41011yescharacteristic
146#2932—611no—
147#2132—611no—
148#2126—611no—
149#1757—612no—
150#1395—612no—
151C2×C61—21no—
152C2×C61C4×#9911yescharacteristic
153D61—42no—
154D61—41no—
155D61—81no—
156C2×#131—611no—
157C2×#131—611no—
158#532—611no—
159#435—1222no—
160#435—1222no—
161#423—611no—
162#413—612no—
163#410—612no—
164#408—611no—
165#408—611no—
166#405—1222no—
167C2×#97—611no—
168#356—611no—
169#356#692311yescharacteristic, complemented
170#355—612no—
171#354—612no—
172#353—612no—
173#344—1222no—
174C61#1263411yescharacteristic, complemented, Sylow
175C2×#41—611no—
176C2×#41—611no—
177#131—2442no—
178#131—2442no—
179C2×M16—611no—
180C2×M16—612no—
181C2×M16—611no—
182C2×M16—1221no—
183C2×M16—1222no—
184C22×C8—611no—
185C22×C8—612no—
186C22×C8—611no—
187C22×C8—1221no—
188C4⋊Q8C2×#692311yescharacteristic
189#127—611no—
190#124—611no—
191#124—611no—
192#121—1222no—
193C4×D4—1222no—
194#117—1222no—
195C2×C42—611no—
196C4⋊C8—1222no—
197#101—612no—
198#101C2×#692312yes—
199#100—1222no—
200#99—614no—
201#99—1222no—
202C22⋊C8—1222no—
203C22⋊C8—1222no—
204#97—1221no—
205#97—1221no—
206#97—2442no—
207C4×C8—2442no—
208#41—2441no—
209#41—2441no—
210#41—2441no—
211#41—2441no—
212C2×Q8C4×#692311yescharacteristic
213C2×Q8C22×#692311yescharacteristic
214C2×D4—611no—
215C2×D4—611no—
216C2×D4—612no—
217C2×D4—1222no—
218C2×D4—1221no—
219C22×C4—611no—
220C22×C4—612no—
221C22×C4—1221no—
222C22×C4—1221no—
223M16—22no—
224M16—1222no—
225M16—2442no—
226M16—2442no—
227M16—2442no—
228M16—2441no—
229M16—2441no—
230C2×C8—611no—
231C2×C8—611no—
232C2×C8—612no—
233C2×C8—1222no—
234C2×C8—1222no—
235C2×C8—1222no—
236C2×C8—1221no—
237C2×C8—1221no—
238C2×C8—2442no—
239C2×C8—2442no—
240C2×C8—2441no—
241C4⋊C4—22no—
242C4⋊C4—2442no—
243C22⋊C4—1222no—
244C22⋊C4—1222no—
245C22⋊C4—1224no—
246C42—611no—
247C42—1221no—
248C42—2442no—
249C42C4×#692311yescharacteristic
250C23—611no—
251C23—612no—
252C23—1221no—
253Q8—41no—
254Q8—41no—
255D4—4882no—
256D4—2442no—
257D4—2441no—
258D4—2441no—
259D4—2442no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—611no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1221no—
266C2×C4—1221no—
267C2×C4—1221no—
268C2×C4—4881no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2444no—
272C2×C4C2×C4×#692311yescharacteristic
273C2×C4D4×#692312yes—
274C8—42no—
275C8—612no—
276C8—1221no—
277C8—1222no—
278C8—2442no—
279C8—2441no—
280C8—2441no—
281C22—1222no—
282C22—1221no—
283C22—2442no—
284C22—2441no—
285C22—2442no—
286C22—2441no—
287C22C22⋊C4×#692311yescharacteristic
288C4—22no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4—611no—
293C4—611no—
294C4—4884no—
295C4—2441no—
296C2—21no—
297C2—4881no—
298C2—2442no—
299C2—2441no—
300C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×Q8C22×#6923
FittingC61×#356C4
socleC2×C61C4×#99
prime radical#16840313C22
cosocleC2×#11758226C23
hypercenter#356#6923
hypocenterC61#12634
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #21461578C2semidirect
4. #21460921C2semidirect
6. C2×#14519865C4semidirect
10. #16840767C4semidirect
12. #16840737C4semidirect
15. #16840721C4semidirect
21. #16839561C4semidirect
22. #16839558C4semidirect
23. C61×#356C4semidirect
169. #356#6923semidirect
174. C61#12634semidirect

Maximal subgroups

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

subgroupindex
#214625462
#214620932
#214615782
#214609212
#214475032
#1263461