Finite Group Databasebeta

← #30599133

Subgroups — #30599133

Order 15616.

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

normal subgroups only

This group has 408 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×#14519847C211yesmaximal, characteristic, complemented
2#21461378C211yesmaximal, characteristic, complemented
3#21461155C212yesmaximal
4#21460705C212yesmaximal, complemented
5C8×#11758224C211yesmaximal, characteristic
6C2×#14519847—21no—
7#16840766C2212yes—
8C4×#11758213C2212yes—
9#16840638—21no—
10#16840626—21no—
11#16840626—21no—
12#16840612—22no—
13D4⋊#92420—22no—
14C4×#11758224C2211yescharacteristic
15C8×Dic122C411yescharacteristic
16#16839886C411yescharacteristic
17D4×#92418C411yescharacteristic
18C8×C22⋊D61—21no—
19#16839825—21no—
20#16839814C2211yescharacteristic
21#16839812—21no—
22C2×C8×Dic61—21no—
23C4×C8×D61C2211yescharacteristic
24C8×D4×C61C411yescharacteristic
25C2×#11758213—22no—
26C2×#11758213—22no—
27#14519847—24no—
28C4⋊#92420—22no—
29C4⋊#92420—22no—
30#14519832—22no—
31C4×#92420—22no—
32C2×#11758224Q811yescharacteristic
33C22×#92418—21no—
34C2×C8×D61D411yescharacteristic
35C2×C8×D61D411yescharacteristic
36C4×Dic122C2×C411yescharacteristic
37#14519742D411yescharacteristic
38D4×Dic61D411yescharacteristic
39C4×C22⋊D61—21no—
40#14519737—21no—
41#14519735Q811yescharacteristic
42#14519733—21no—
43C2×C4×Dic61—21no—
44C42×D61C2311yescharacteristic
45C4⋊#92418C2×C411yescharacteristic
46#14519706—21no—
47#14519706—21no—
48C22⋊#92418—21no—
49#14519689—21no—
50C4×#92418C2×C411yescharacteristic
51C8×Dic61—21no—
52C8×Dic61—21no—
53C8×Dic61—21no—
54C8×Dic61D411yescharacteristic
55C8×Dic61D411yescharacteristic
56C22×C8×C61—21no—
57C4×D4×C61C2×C411yescharacteristic
58C4⋊C8×C61C2×C411yescharacteristic
59C22⋊C8×C61—21no—
60C4×C8×C61C2×C411yescharacteristic
61C2×D4×C61C4⋊C411yescharacteristic
62C22×C4×C61—21no—
63C2×C22⋊D61—21no—
64C22×Dic61—21no—
65C22×Dic61—21no—
66#11758224—22no—
67#11758224—22no—
68C2×C4×D61C2×D411yescharacteristic
69C2×C4×D61C2×Q811yescharacteristic
70C2×C4×D61#4111yescharacteristic
71C2×Dic122C4⋊C411yescharacteristic
72C2×#92420—42no—
73C2×#92420—42no—
74#11758213—24no—
75#11758213—42no—
76#11758213—82no—
77C2×C8×C61—21no—
78C2×C8×C61—42no—
79C2×C8×C61C22⋊C411yescharacteristic
80C2×C8×C61C22⋊C411yescharacteristic
81C4⋊C4×C61C4⋊C411yescharacteristic
82C22⋊C4×C61—21no—
83C42×C61C22×C411yescharacteristic
84C22⋊Dic61—21no—
85#11758198—21no—
86C4⋊Dic61C4⋊C411yescharacteristic
87#11758196—21no—
88#11758196—21no—
89C4×Dic61—21no—
90C4×Dic61—21no—
91C4×Dic61—21no—
92C4×Dic61C2×D411yescharacteristic
93C4×Dic61C2×D411yescharacteristic
94C4×Dic61#4111yescharacteristic
95C2×#92418—21no—
96C2×#92418—42no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C22⋊C411yescharacteristic
99C8×D61—22no—
100C8×D61—41no—
101C23×C61—21no—
102C22×D61C22⋊Q811yescharacteristic
103D4×C61—21no—
104D4×C61#10812yescomplemented
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61C2×C4⋊C411yescharacteristic
110C2×C4×C61#11711yescharacteristic
111C22⋊D61—24no—
112C2×Dic61—24no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61#12111yescharacteristic
118C2×Dic61#12111yescharacteristic
119C2×Dic61C22⋊Q811yescharacteristic
120C4×D61—21no—
121C4×D61—21no—
122C4×D61—21no—
123C4×D61—21no—
124C4×D61—21no—
125C4×D61—21no—
126Dic122—21no—
127Dic122#10812yescomplemented
128#92420—44no—
129#92420—82no—
130C8×C61—21no—
131C8×C61—21no—
132C8×C61—41no—
133#92418—21no—
134#92418—41no—
135#92418#10412yes—
136#8671—611nomaximal, Sylow
137C22×C61—22no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61#38011yescharacteristic
141C2×D61—21no—
142C2×D61—21no—
143C2×D61—21no—
144C4×C61—21no—
145C4×C61—41no—
146C4×C61C2×#10411yescharacteristic
147C4×C61#42111yescharacteristic
148C4×C61C2×#10811yescharacteristic
149C4×C61C2×#10811yescharacteristic
150Dic61—21no—
151Dic61—22no—
152Dic61—21no—
153Dic61—41no—
154Dic61#47211yescharacteristic
155Dic61#47311yescharacteristic
156C8×#41—611no—
157C4×#108—611no—
158#1567—611no—
159#1489—612no—
160#1299—612no—
161C2×C61—42no—
162C2×C61#180611yescharacteristic
163C2×C61#181011yescharacteristic
164C2×C61#181611yescharacteristic
165D61—41no—
166C4×#41—611no—
167C8×Q8—611no—
168C8×Q8—611no—
169C8×D4—611no—
170C8×D4—1221no—
171C8×D4—1221no—
172C8×D4#692311yescharacteristic
173#431—611no—
174#431—1221no—
175C2×#108—1221no—
176#423—612no—
177C4×M16—612no—
178C2×C4×C8—611no—
179C2×C4×C8—1221no—
180#341—1221no—
181#335—1221no—
182#335—1221no—
183Q8⋊C8—1222no—
184D4⋊C8—1222no—
185C61#867111yescharacteristic, complemented, Sylow
186C2×#41—611no—
187C2×M16—1222no—
188C2×M16—1222no—
189C22×C8—21no—
190C22×C8—611no—
191C22×C8—611no—
192C22×C8—1221no—
193C4×Q8—611no—
194C4×Q8—611no—
195C4×D4—611no—
196C4×D4—1221no—
197C4×D4—1221no—
198C4×D4C2×#692311yescharacteristic
199#117—611no—
200#117—1221no—
201C2×C42—611no—
202C2×C42—1221no—
203#108—1224no—
204C4⋊C8—611no—
205C4⋊C8—1222no—
206C4⋊C8—1222no—
207C4⋊C8—1221no—
208C4⋊C8—1221no—
209C4⋊C8C2×#692311yescharacteristic
210C22⋊C8—21no—
211C22⋊C8—1221no—
212C22⋊C8—1221no—
213#97—1222no—
214C4×C8—611no—
215C4×C8—611no—
216C4×C8—611no—
217C4×C8—1221no—
218C4×C8—1222no—
219C4×C8—1221no—
220C4×C8—1221no—
221C4×C8C2×#692311yescharacteristic
222#41—1222no—
223#41—1222no—
224C2×Q8—611no—
225C2×D4—1221no—
226C2×D4C4⋊#692311yescharacteristic
227C22×C4—21no—
228C22×C4—611no—
229C22×C4—611no—
230C22×C4—611no—
231C22×C4—1221no—
232C22×C4—1221no—
233M16—1224no—
234M16—4882no—
235M16—2442no—
236C2×C8—21no—
237C2×C8—42no—
238C2×C8—611no—
239C2×C8—611no—
240C2×C8—1222no—
241C2×C8—1221no—
242C2×C8—2441no—
243C2×C8—2442no—
244C2×C8—2442no—
245C2×C8—2442no—
246C2×C8C22⋊#692311yescharacteristic
247C2×C8C22⋊#692311yescharacteristic
248C4⋊C4—611no—
249C4⋊C4—1221no—
250C4⋊C4—1221no—
251C4⋊C4C4⋊#692311yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—1221no—
254C22⋊C4—1221no—
255C42—611no—
256C42—611no—
257C42—611no—
258C42—1221no—
259C42—1221no—
260C42—1221no—
261C42C22×#692311yescharacteristic
262C23—21no—
263C23—611no—
264Q8—612no—
265Q8—1221no—
266D4—21no—
267D4—1224no—
268D4#1451984712yescomplemented
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—611no—
273C2×C4—611no—
274C2×C4—611no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1224no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4C2×C22⋊#692311yescharacteristic
287C2×C4C2×C4⋊#692311yescharacteristic
288C2×C4#1451985611yescharacteristic
289C8—21no—
290C8—21no—
291C8—41no—
292C8—612no—
293C8—1221no—
294C8—4882no—
295C8—2444no—
296C8—2441no—
297C22—22no—
298C22—41no—
299C22—41no—
300C22—1221no—
301C22—1221no—
302C22—1221no—
303C22#1684070411yescharacteristic
304C4—21no—
305C4—41no—
306C4—611no—
307C4—611no—
308C4—1221no—
309C4—1221no—
310C4—1222no—
311C4—2441no—
312C4D61⋊#10411yescharacteristic
313C4#1684076011yescharacteristic
314C4C2×#1451984711yescharacteristic
315C4D61⋊#10811yescharacteristic
316C2—42no—
317C2—2441no—
318C2#2146169311yescharacteristic
319C2#2146169811yescharacteristic
320C2#2146170811yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC8×D4×C61C4
socleC22×C61#380
prime radicalC2×#11758224Q8
cosocleC42×D61C23
hypercenterC8×D4#6923
hypocenterC61#8671
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C4×#14519847C2semidirect
2. #21461378C2semidirect
4. #21460705C2semidirect
104. D4×C61#108semidirect
127. Dic122#108semidirect
185. C61#8671semidirect
268. D4#14519847semidirect

Maximal subgroups

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

subgroupindex
C4×#145198472
#214613782
#214611552
#214607052
C8×#117582242
#867161