Finite Group Databasebeta

← #30606563

Subgroups — #30606563

Order 15616.

464 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462310C211yesmaximal, characteristic
2D61⋊#356C211yesmaximal, characteristic
3#101×#6923C212yesmaximal
4#21460915C212yesmaximal
5#356×D61C211yesmaximal, characteristic
6C2×Q8×#6923C411yescharacteristic
7C2×Q8×#6923C2211yescharacteristic
8D61⋊#101C2212yes—
9M16×#6923—22no—
10(C2×#6923)⋊Q8C411yescharacteristic
11(C2×#6923)⋊Q8C2211yescharacteristic
12#16840711—22no—
13#16840711—22no—
14#16840706C411yescharacteristic
15#16840698C411yescharacteristic
16#16840673C412yescomplemented
17#16840672C412yescomplemented
18#16840641—22no—
19C4⋊Q8×D61C2211yescharacteristic
20#101×D61C2212yes—
21#16839561C411yescharacteristic, complemented
22#16839560C412yescomplemented
23C61×#356C411yescharacteristic, complemented
24D61⋊M16—22no—
25Q8×#6923—42no—
26Q8×#6923—42no—
27C2×C4⋊#6923—21no—
28C2×C4⋊#6923C2×C411yescharacteristic
29C2×C4⋊#6923C2×C411yescharacteristic
30C2×C4⋊#6923D412yes—
31C2×C4×#6923—21no—
32C2×C4×#6923C2×C411yescharacteristic
33C2×C4×#6923C2×C411yescharacteristic
34C2×C4×#6923D412yes—
35#14519838—22no—
36#14519838C2×C414yes—
37#14519831—42no—
38C8×#6923—42no—
39(C2×#6923)⋊C4—22no—
40(C2×#6923)⋊C4—22no—
41(C2×#6923)⋊C4—21no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—22no—
44C2×Q8×D61C2×C411yescharacteristic
45C2×Q8×D61C2311yescharacteristic
46M16×D61—22no—
47#14519777C2×C411yescharacteristic
48#14519776—22no—
49#14519775C2×C411yescharacteristic
50#14519775C2×C411yescharacteristic
51C4⋊C4×D61—22no—
52C42×D61C2×C411yescharacteristic
53#14519696C2×C412yes—
54#14519695—22no—
55C4⋊Q8×C61C2×C411yescharacteristic
56#101×C61C2×C412yes—
57C22×#6923—21no—
58C22×#6923—21no—
59C22×#6923C22⋊C412yes—
60C22×#6923C22⋊C411yescharacteristic
61C22×#6923C22⋊C411yescharacteristic
62C2×Q8×C61C4211yescharacteristic
63C2×Q8×C61C22×C411yescharacteristic
64Q8×D61—41no—
65Q8×D61—41no—
66Q8×D61—41no—
67Q8×D61—41no—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61C22×C411yescharacteristic
71C2×C4×D61C2×D412yes—
72C2×Dic122—21no—
73C2×Dic122C4211yescharacteristic
74C2×Dic122C22×C411yescharacteristic
75C2×Dic122#4112yes—
76#11758217—24no—
77C4⋊#6923—41no—
78C4⋊#6923—42no—
79C4⋊#6923—41no—
80C4⋊#6923—42no—
81C4×#6923—22no—
82C4×#6923—22no—
83C4×#6923—22no—
84C4×#6923—41no—
85C4×#6923—42no—
86#11758213—42no—
87D61⋊C8—42no—
88M16×C61—22no—
89C4⋊C4×C61—22no—
90C42×C61C4211yescharacteristic
91C4⋊Dic61—22no—
92#11758196—22no—
93#11758196—22no—
94C4×Dic61—21no—
95C4×Dic61C4211yescharacteristic
96#11758194—22no—
97#11758189—42no—
98C8×D61—42no—
99C22×D61C2×C22⋊C411yescharacteristic
100C2×#6923—21no—
101C2×#6923—22no—
102C2×#6923—41no—
103C2×#6923—42no—
104C2×#6923—41no—
105C2×#6923—41no—
106C2×#6923—41no—
107C2×#6923—42no—
108C2×#6923—41no—
109C2×#6923—41no—
110C2×#6923#9912yes—
111Q8×C61—41no—
112Q8×C61—41no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61C2×C4211yescharacteristic
116C2×C4×C61C4×D412yes—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61#11711yescharacteristic
120C2×Dic61C4×D412yes—
121C4×D61—22no—
122C4×D61—21no—
123C4×D61—22no—
124C4×D61—21no—
125C4×D61—41no—
126C4×D61—41no—
127C4×D61—41no—
128C4×D61—41no—
129Dic122—41no—
130Dic122—42no—
131Dic122—42no—
132Dic122—41no—
133#92420—44no—
134C8×C61—42no—
135#92418—42no—
136#12631—611nomaximal, Sylow
137C22×C61C4×C22⋊C411yescharacteristic
138C2×D61—21no—
139C2×D61—21no—
140C2×D61C2×#9911yescharacteristic
141C2×D61#41011yescharacteristic
142#6923—22no—
143#6923—41no—
144#6923—42no—
145#6923—81no—
146#6923—81no—
147C4×C61—22no—
148C4×C61—21no—
149C4×C61—41no—
150C4×C61—41no—
151Dic61—22no—
152Dic61—21no—
153Dic61—41no—
154Dic61—41no—
155#2327—611no—
156C2×#356—611no—
157C2×#356—611no—
158C4×#101—612no—
159#1394—612no—
160C2×C61—21no—
161C2×C61C4×#9911yescharacteristic
162D61—21no—
163D61#213311yescharacteristic
164D61#213511yescharacteristic
165C2×C4⋊Q8—611no—
166C2×C4×Q8—611no—
167C2×C4×Q8—611no—
168C2×#101—612no—
169C2×#101—612no—
170C4×M16—1222no—
171#391—611no—
172#391—611no—
173#387—1222no—
174#387—1222no—
175#382—611no—
176#376—611no—
177#356—612no—
178#356—612no—
179#356—611no—
180#356—612no—
181#356#692311yescharacteristic, complemented
182#343—1222no—
183C61#1263111yescharacteristic, complemented, Sylow
184C22×Q8—611no—
185C22×Q8—611no—
186C2×M16—1222no—
187C2×M16—1222no—
188C4⋊Q8—611no—
189C4⋊Q8—611no—
190C4⋊Q8—611no—
191C4⋊Q8—1222no—
192C4⋊Q8C2×#692311yescharacteristic
193C4×Q8—2442no—
194C4×Q8—2442no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—612no—
197C2×C4⋊C4—611no—
198C2×C4⋊C4—1222no—
199C2×C4⋊C4—1221no—
200C2×C42—611no—
201C2×C42—611no—
202C2×C42—612no—
203C2×C42—611no—
204C2×C42—1221no—
205#101—614no—
206#101—612no—
207#101—1222no—
208#101—1222no—
209#101C2×#692312yes—
210#97—2442no—
211C4×C8—2442no—
212#95—1221no—
213#95—1221no—
214#95—1222no—
215#95—1222no—
216#95—1222no—
217C2×Q8—611no—
218C2×Q8—612no—
219C2×Q8—611no—
220C2×Q8—1221no—
221C2×Q8—2441no—
222C2×Q8—2441no—
223C2×Q8—2441no—
224C2×Q8—2441no—
225C2×Q8C4×#692311yescharacteristic
226C2×Q8C22×#692311yescharacteristic
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—612no—
230C22×C4—612no—
231C22×C4—611no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4—1221no—
236M16—22no—
237M16—1224no—
238M16—1222no—
239M16—2442no—
240M16—2442no—
241C2×C8—2442no—
242C2×C8—2442no—
243C4⋊C4—22no—
244C4⋊C4—1222no—
245C4⋊C4—1222no—
246C4⋊C4—1222no—
247C4⋊C4—2442no—
248C4⋊C4—2441no—
249C4⋊C4—2442no—
250C4⋊C4—2441no—
251C42—611no—
252C42—1221no—
253C42—1222no—
254C42—1222no—
255C42—1222no—
256C42—2441no—
257C42—2442no—
258C42C4×#692311yescharacteristic
259C23—611no—
260Q8—41no—
261Q8—41no—
262Q8—2442no—
263Q8—2441no—
264Q8—2441no—
265Q8—2442no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—612no—
269C2×C4—611no—
270C2×C4—612no—
271C2×C4—1221no—
272C2×C4—1222no—
273C2×C4—1221no—
274C2×C4—1222no—
275C2×C4—1221no—
276C2×C4—1222no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2442no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2442no—
290C2×C4—2441no—
291C2×C4C2×C4×#692311yescharacteristic
292C2×C4D4×#692312yes—
293C8—42no—
294C8—2442no—
295C8—2444no—
296C22—611no—
297C22—611no—
298C22—1221no—
299C22—1221no—
300C22C22⋊C4×#692311yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—1222no—
306C4—1222no—
307C4—1221no—
308C4—4881no—
309C4—4881no—
310C4—2441no—
311C4—2441no—
312C4—2442no—
313C4—2441no—
314C2—21no—
315C2—611no—
316C2—611no—
317C2—1221no—
318C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×Q8C22×#6923
FittingC61×#356C4
socleC2×C61C4×#99
prime radicalC22×D61C2×C22⋊C4
cosocleC2×Q8×D61C23
hypercenter#356#6923
hypocenterC61#12631
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
16. #16840673C4semidirect
17. #16840672C4semidirect
21. #16839561C4semidirect
22. #16839560C4semidirect
23. C61×#356C4semidirect
181. #356#6923semidirect
183. C61#12631semidirect

Maximal subgroups

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

subgroupindex
#214623102
D61⋊#3562
#101×#69232
#214609152
#356×D612
#1263161