Finite Group Databasebeta

← #30598997

Subgroups — #30598997

Order 15616.

550 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 106 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#21461696C212yesmaximal, complemented
2#21461679C212yesmaximal
3#21460682C212yesmaximal, complemented
4C2×#16839825C211yesmaximal, characteristic
5C2×#14519846—22no—
6C2×#14519835—22no—
7#16840704C2214yes—
8#16840640—22no—
9#16840636—22no—
10C23×#92418C411yescharacteristic, complemented
11C22×C8×D61C2211yescharacteristic
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839825—22no—
14#16839825—22no—
15C2×#14519706C411yescharacteristic, complemented
16C2×#14519689C2211yescharacteristic
17C2×C8×Dic61C411yescharacteristic, complemented
18C2×C22⋊C8×C61C411yescharacteristic, complemented
19C2×C4⋊#6923—24no—
20C2×C22⋊#6923—24no—
21#14519846—44no—
22#14519835—44no—
23(C2×#6923)⋊C4—24no—
24C22×C22⋊D61D411yescharacteristic
25C22×C4×D61C2311yescharacteristic
26C22×#92418—21no—
27C22×#92418—21no—
28C22×#92418—42no—
29C22×#92418C2×C411yescharacteristic
30C22×#92418C2×C411yescharacteristic
31C2×C8×D61—42no—
32C2×C8×D61D412yes—
33C2×C8×D61D412yes—
34C4×C22⋊D61—22no—
35C4×C22⋊D61—22no—
36C2×#11758196Q811yescharacteristic
37C2×C22⋊Dic61Q811yescharacteristic
38C2×#11758198D411yescharacteristic
39C2×C4×Dic61C2×C411yescharacteristic
40#14519706—22no—
41#14519689—22no—
42C8×Dic61—21no—
43C8×Dic61—21no—
44C23×C4×C61C2×C411yescharacteristic
45C22×C8×C61C2×C411yescharacteristic
46C22×C8×C61C2×C411yescharacteristic
47C22⋊C8×C61—22no—
48C24×C61C4⋊C411yescharacteristic
49C23×D61C2×D411yescharacteristic
50C22×#6923—44no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61C4⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C2×C22⋊D61—22no—
58C2×C22⋊D61—22no—
59C2×C22⋊D61D812yes—
60C2×C22⋊D61Q1612yes—
61C22×Dic61C4⋊C411yescharacteristic
62C22×Dic61C4⋊C411yescharacteristic
63C22×Dic61C2×Q811yescharacteristic
64C2×C4×D61—41no—
65C2×C4×D61—41no—
66C2×C4×D61C2×D412yes—
67C2×C4×D61#4112yes—
68C22⋊#6923—44no—
69C22⋊#6923—44no—
70C4⋊#6923—44no—
71C4⋊#6923—44no—
72C2×C8×C61—21no—
73C2×C8×C61—41no—
74C2×C8×C61—41no—
75C2×C8×C61C22⋊C412yes—
76C22⋊Dic61—22no—
77#11758198—22no—
78#11758196—22no—
79C4×Dic61—21no—
80C4×Dic61—21no—
81C2×#92418—22no—
82C2×#92418—21no—
83C2×#92418—42no—
84C2×#92418—42no—
85C2×#92418—41no—
86C2×#92418C22⋊C411yescharacteristic
87C2×#92418C22⋊C411yescharacteristic
88C8×D61—24no—
89C8×D61—42no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61#10712yescomplemented
95C23×C61C2×C4⋊C411yescharacteristic
96C22×D61—41no—
97C22×D61—41no—
98C22×D61#12112yes—
99C22×D61C2×D811yescharacteristic
100C22×D61C2×Q1611yescharacteristic
101C22×D61#13512yes—
102C2×#6923—84no—
103C2×#6923—84no—
104C2×#6923—84no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61C2×C22⋊C411yescharacteristic
113C2×C4×C61#11711yescharacteristic
114C22⋊D61—24no—
115C22⋊D61—24no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—41no—
120C2×Dic61—41no—
121C2×Dic61#10712yescomplemented
122C2×Dic61C22⋊Q812yes—
123C4×D61—22no—
124C4×D61—22no—
125C4×D61—42no—
126C8×C61—22no—
127C8×C61—41no—
128#92418—21no—
129#92418—41no—
130#92418#9912yescomplemented
131#8609—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—22no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61#38011yescharacteristic
140C22×C61C2×#10711yescharacteristic
141C22×C61#42711yescharacteristic
142C2×D61—22no—
143C2×D61—22no—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—42no—
147C2×D61#46612yes—
148C2×D61#46712yes—
149#6923—164no—
150C4×C61—21no—
151C4×C61—41no—
152C4×C61C2×#9911yescharacteristic
153C4×C61#41011yescharacteristic
154Dic61—22no—
155Dic61—22no—
156Dic61—41no—
157C2×C8×D4—611no—
158#1808—612no—
159#1799—612no—
160#1289—612no—
161C2×C61—21no—
162C2×C61—21no—
163C2×C61—42no—
164C2×C61#179911yescharacteristic
165C2×C61#180811yescharacteristic
166C2×C61#181211yescharacteristic
167D61—22no—
168D61—22no—
169C23×C8—611no—
170C23×C8—611no—
171C2×C4×D4—611no—
172C8×D4—1222no—
173C8×D4—1222no—
174C2×#107—1222no—
175C2×C4⋊C8—611no—
176C2×#99—1222no—
177C2×C22⋊C8—611no—
178C2×C22⋊C8#692311yescharacteristic, complemented
179C2×C4×C8—611no—
180#380—614no—
181#342—1222no—
182#340—1222no—
183C61#860911yescharacteristic, complemented, Sylow
184C22×D4—611no—
185C23×C4—611no—
186C23×C4C2×#692311yescharacteristic
187C22×C8—611no—
188C22×C8—612no—
189C22×C8—612no—
190C22×C8—611no—
191C22×C8—1221no—
192C22×C8—1221no—
193C22×C8—2442no—
194C22×C8—2442no—
195C22×C8C2×#692311yescharacteristic
196C22×C8C2×#692311yescharacteristic
197C4×D4—1222no—
198C4×D4—1222no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—1224no—
201C2×C22⋊C4—611no—
202C2×C22⋊C4—611no—
203C2×C22⋊C4—1224no—
204C2×C42—611no—
205#107—2444no—
206C4⋊C8—1222no—
207#99—2444no—
208C22⋊C8—22no—
209C22⋊C8—1222no—
210C4×C8—1221no—
211C4×C8—1221no—
212#95—1224no—
213C24—611no—
214C24C4⋊#692311yescharacteristic
215C2×D4—612no—
216C2×D4—612no—
217C2×D4—1222no—
218C2×D4—1222no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—612no—
226C22×C4—612no—
227C22×C4—611no—
228C22×C4—2441no—
229C22×C4—2441no—
230C22×C4—2444no—
231C22×C4C4⋊#692311yescharacteristic
232C22×C4C22×#692311yescharacteristic
233C2×C8—21no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—611no—
237C2×C8—611no—
238C2×C8—1224no—
239C2×C8—1222no—
240C2×C8—1221no—
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8—2442no—
244C2×C8—2441no—
245C2×C8C22⋊#692312yes—
246C4⋊C4—1222no—
247C4⋊C4—2444no—
248C4⋊C4—2444no—
249C22⋊C4—1222no—
250C22⋊C4—1222no—
251C22⋊C4—2444no—
252C22⋊C4—2444no—
253C42—1221no—
254C42—1221no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—612no—
260C23—611no—
261C23—611no—
262C23—612no—
263C23—2441no—
264C23—2441no—
265C23#1451984612yescomplemented
266C23C2×C4⋊#692311yescharacteristic
267D4—1224no—
268D4—1224no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—612no—
277C2×C4—612no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1222no—
281C2×C4—1222no—
282C2×C4—1221no—
283C2×C4—4884no—
284C2×C4—4884no—
285C2×C4—4884no—
286C2×C4—2442no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4C2×C22⋊#692311yescharacteristic
290C2×C4#1451985611yescharacteristic
291C8—22no—
292C8—41no—
293C8—612no—
294C8—1221no—
295C8—2441no—
296C22—21no—
297C22—21no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—612no—
304C22—612no—
305C22—1222no—
306C22—1222no—
307C22—2441no—
308C22—2441no—
309C22—2442no—
310C22#1684070411yescharacteristic
311C22C2×#1451984611yescharacteristic
312C22#1684077211yescharacteristic
313C4—21no—
314C4—41no—
315C4—1222no—
316C4—1222no—
317C4—9764no—
318C4—2441no—
319C4D61⋊#9911yescharacteristic
320C4#1684074211yescharacteristic
321C2—21no—
322C2—21no—
323C2—42no—
324C2—1222no—
325C2—1222no—
326C2#2146168111yescharacteristic
327C2#2146169611yescharacteristic
328C2#2146170211yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊C8×C61C4
socleC22×C61#380
prime radicalC22×C22⋊D61D4
cosocleC22×C4×D61C23
hypercenterC2×C22⋊C8#6923
hypocenterC61#8609
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #21461696C2semidirect
3. #21460682C2semidirect
10. C23×#92418C4semidirect
15. C2×#14519706C4semidirect
17. C2×C8×Dic61C4semidirect
18. C2×C22⋊C8×C61C4semidirect
94. C23×C61#107semidirect
121. C2×Dic61#107semidirect
130. #92418#99semidirect
178. C2×C22⋊C8#6923semidirect
183. C61#8609semidirect
265. C23#14519846semidirect

Maximal subgroups

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

subgroupindex
#214616962
#214616792
#214606822
C2×#168398252
#860961