Finite Group Databasebeta

← #30599129

Subgroups — #30599129

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×#14519848C211yesmaximal, characteristic, complemented
2#21461375C211yesmaximal, characteristic, complemented
3#21461155C212yesmaximal
4#21460705C212yesmaximal, complemented
5C8×#11758224C211yesmaximal, characteristic
6C2×#14519848—21no—
7#16840766C2212yes—
8C4×#11758213C2212yes—
9#16840638—21no—
10#16840624—21no—
11#16840624—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#14519848—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—42no—
116C2×Dic61#12111yescharacteristic
117C2×Dic61#12111yescharacteristic
118C2×Dic61C22⋊Q811yescharacteristic
119C4×D61—21no—
120C4×D61—21no—
121C4×D61—21no—
122C4×D61—21no—
123C4×D61—22no—
124Dic122—21no—
125Dic122#10812yescomplemented
126#92420—44no—
127#92420—82no—
128C8×C61—21no—
129C8×C61—21no—
130C8×C61—41no—
131#92418—21no—
132#92418—41no—
133#92418#10412yes—
134#8670—611nomaximal, Sylow
135C22×C61—22no—
136C22×C61—41no—
137C22×C61—41no—
138C22×C61#38011yescharacteristic
139C2×D61—21no—
140C2×D61—21no—
141C2×D61—21no—
142C4×C61—21no—
143C4×C61—41no—
144C4×C61C2×#10411yescharacteristic
145C4×C61#42111yescharacteristic
146C4×C61C2×#10811yescharacteristic
147C4×C61C2×#10811yescharacteristic
148Dic61—21no—
149Dic61—22no—
150Dic61—21no—
151Dic61—41no—
152Dic61#47112yes—
153C8×#41—611no—
154C4×#108—611no—
155#1566—611no—
156#1489—612no—
157#1299—612no—
158C2×C61—42no—
159C2×C61#180611yescharacteristic
160C2×C61#181011yescharacteristic
161C2×C61#181511yescharacteristic
162D61—41no—
163C4×#41—611no—
164C8×Q8—611no—
165C8×Q8—611no—
166C8×D4—611no—
167C8×D4—1221no—
168C8×D4—1221no—
169C8×D4#692311yescharacteristic
170#431—611no—
171#431—1221no—
172C2×#108—1221no—
173#423—612no—
174C4×M16—612no—
175C2×C4×C8—611no—
176C2×C4×C8—1221no—
177#341—1221no—
178#334—1221no—
179#334—1221no—
180Q8⋊C8—1222no—
181D4⋊C8—1222no—
182C61#867011yescharacteristic, complemented, Sylow
183C2×#41—611no—
184C2×M16—1222no—
185C2×M16—1222no—
186C22×C8—21no—
187C22×C8—611no—
188C22×C8—611no—
189C22×C8—1221no—
190C4×Q8—611no—
191C4×Q8—611no—
192C4×D4—611no—
193C4×D4—1221no—
194C4×D4—1221no—
195C4×D4C2×#692311yescharacteristic
196#117—611no—
197#117—1221no—
198C2×C42—611no—
199C2×C42—1221no—
200#108—1224no—
201C4⋊C8—611no—
202C4⋊C8—1222no—
203C4⋊C8—1222no—
204C4⋊C8—1221no—
205C4⋊C8—1221no—
206C4⋊C8C2×#692311yescharacteristic
207C22⋊C8—21no—
208C22⋊C8—1221no—
209C22⋊C8—1221no—
210#97—1222no—
211C4×C8—611no—
212C4×C8—611no—
213C4×C8—611no—
214C4×C8—1221no—
215C4×C8—1222no—
216C4×C8—1221no—
217C4×C8—1221no—
218C4×C8C2×#692311yescharacteristic
219#41—1222no—
220#41—1222no—
221C2×Q8—611no—
222C2×D4—1221no—
223C2×D4C4⋊#692311yescharacteristic
224C22×C4—21no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—1221no—
229C22×C4—1221no—
230M16—1224no—
231M16—4882no—
232M16—2442no—
233C2×C8—21no—
234C2×C8—42no—
235C2×C8—611no—
236C2×C8—611no—
237C2×C8—1222no—
238C2×C8—1221no—
239C2×C8—2441no—
240C2×C8—2442no—
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8C22⋊#692311yescharacteristic
244C2×C8C22⋊#692311yescharacteristic
245C4⋊C4—611no—
246C4⋊C4—1221no—
247C4⋊C4—1221no—
248C4⋊C4C4⋊#692311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—1221no—
251C22⋊C4—1221no—
252C42—611no—
253C42—611no—
254C42—611no—
255C42—1221no—
256C42—1221no—
257C42—1221no—
258C42C22×#692311yescharacteristic
259C23—21no—
260C23—611no—
261Q8—612no—
262Q8—1221no—
263D4—21no—
264D4—1224no—
265D4#1451984812yescomplemented
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—611no—
270C2×C4—611no—
271C2×C4—611no—
272C2×C4—1222no—
273C2×C4—1224no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—2442no—
281C2×C4C2×C22⋊#692311yescharacteristic
282C2×C4C2×C4⋊#692311yescharacteristic
283C2×C4#1451985611yescharacteristic
284C8—21no—
285C8—21no—
286C8—41no—
287C8—612no—
288C8—1221no—
289C8—4882no—
290C8—2444no—
291C8—2441no—
292C22—22no—
293C22—41no—
294C22—41no—
295C22—1221no—
296C22—1221no—
297C22—1221no—
298C22#1684070411yescharacteristic
299C4—21no—
300C4—41no—
301C4—612no—
302C4—1221no—
303C4—1221no—
304C4—1222no—
305C4—2441no—
306C4D61⋊#10411yescharacteristic
307C4#1684076011yescharacteristic
308C4D61⋊#10811yescharacteristic
309C4C2×#1451984811yescharacteristic
310C2—42no—
311C2—2441no—
312C2#2146169311yescharacteristic
313C2#2146169911yescharacteristic
314C2#2146170711yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC8×D4×C61C4
socleC22×C61#380
prime radicalC2×#11758224Q8
cosocleC42×D61C23
hypercenterC8×D4#6923
hypocenterC61#8670
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×#14519848C2semidirect
2. #21461375C2semidirect
4. #21460705C2semidirect
104. D4×C61#108semidirect
125. Dic122#108semidirect
182. C61#8670semidirect
265. D4#14519848semidirect

Maximal subgroups

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

subgroupindex
C4×#145198482
#214613752
#214611552
#214607052
C8×#117582242
#867061