Finite Group Databasebeta

← #30599172

Subgroups — #30599172

Order 15616.

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

normal subgroups only

This group has 414 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#21461744C211yesmaximal, characteristic, complemented
2D61⋊#335C211yesmaximal, characteristic, complemented
3#21461330C212yesmaximal
4#21460688C212yesmaximal, complemented
5#21454528C211yesmaximal, characteristic
6#16840767C2212yes—
7D61⋊(C4⋊C8)C2212yes—
8#16840638—21no—
9#16840638—21no—
10#16840626—21no—
11#16840625—21no—
12#16840621—22no—
13#16840620—22no—
14C2×#14519785—21no—
15#16840309C2211yescharacteristic
16#16839841C411yescharacteristic
17#16839838—21no—
18D4×#92418C411yescharacteristic
19#16839827C411yescharacteristic
20#16839825—21no—
21#16839784C2211yescharacteristic
22#16839727—21no—
23C4×C8×D61C2211yescharacteristic
24C61×#436C411yescharacteristic
25C2×#11758213—22no—
26C2×D61⋊C8—22no—
27#14519844—22no—
28#14519844—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31C2×#11758222—21no—
32#14519785—44no—
33C2×#11758189—21no—
34C2×#11758189—21no—
35C22×#92418—21no—
36C2×C8×D61—21no—
37C2×C8×D61D411yescharacteristic
38C2×C8×D61D411yescharacteristic
39#14519776D411yescharacteristic
40#14519773D411yescharacteristic
41#14519763—21no—
42#14519752Q811yescharacteristic
43#14519750—21no—
44#14519748Q811yescharacteristic
45C4×D244C2×C411yescharacteristic
46C4×C22⋊D61—21no—
47C42×D61C2311yescharacteristic
48C4⋊#92418C2×C411yescharacteristic
49#14519706—21no—
50C22⋊#92418—21no—
51#14519689—21no—
52#14519689—21no—
53C4×#92418C2×C411yescharacteristic
54C8×Dic61D411yescharacteristic
55C8×Dic61D411yescharacteristic
56C2×M16×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—
64C2×C22⋊D61—21no—
65C2×C22⋊D61—21no—
66#11758222—42no—
67#11758222—42no—
68C2×D244—21no—
69C2×D244C4⋊C411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61C2×D411yescharacteristic
72C2×C4×D61C2×Q811yescharacteristic
73C2×C4×D61#4111yescharacteristic
74C2×Dic122—21no—
75C2×#92420—42no—
76C2×#92420—42no—
77#11758213—42no—
78#11758213—42no—
79D61⋊C8—82no—
80M16×C61—42no—
81C2×C8×C61—21no—
82C2×C8×C61C22⋊C411yescharacteristic
83C2×C8×C61C22⋊C411yescharacteristic
84C4⋊C4×C61C4⋊C411yescharacteristic
85C22⋊C4×C61—21no—
86C42×C61C22×C411yescharacteristic
87C22⋊Dic61—21no—
88#11758198—21no—
89#11758198—21no—
90C4⋊Dic61C4⋊C411yescharacteristic
91#11758196—21no—
92C4×Dic61C2×D411yescharacteristic
93C4×Dic61C2×D411yescharacteristic
94C4×Dic61#4111yescharacteristic
95C2×#92418—21no—
96C2×#92418—42no—
97C2×#92418C22⋊C411yescharacteristic
98C2×#92418C22⋊C411yescharacteristic
99#11758189—42no—
100#11758189—42no—
101C8×D61—22no—
102C8×D61—22no—
103C8×D61—42no—
104C23×C61—21no—
105C22×D61—21no—
106C22×D61C22⋊Q811yescharacteristic
107D4×C61—41no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61C2×C4⋊C411yescharacteristic
113C2×C4×C61#11711yescharacteristic
114C22⋊D61—42no—
115C22⋊D61—42no—
116C22⋊D61—41no—
117C22⋊D61—41no—
118C2×Dic61—21no—
119C2×Dic61#12111yescharacteristic
120C2×Dic61#12111yescharacteristic
121C2×Dic61C22⋊Q811yescharacteristic
122D244—41no—
123D244—42no—
124C4×D61—21no—
125C4×D61—21no—
126C4×D61—41no—
127C4×D61—41no—
128C4×D61—41no—
129C4×D61#13512yes—
130C4×D61AGL(1,ℤ/8)12yes—
131Dic122—42no—
132#92420—82no—
133#92420—82no—
134C8×C61—21no—
135C8×C61—21no—
136C8×C61—41no—
137#92418—21no—
138#92418—21no—
139#92418—41no—
140#8688—611nomaximal, Sylow
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61#38011yescharacteristic
145C2×D61—21no—
146C2×D61—21no—
147C2×D61—41no—
148C2×D61—41no—
149C2×D61—41no—
150C2×D61#47812yes—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—41no—
154C4×C61#41611yescharacteristic
155C4×C61#41811yescharacteristic
156Dic61—21no—
157Dic61—41no—
158Dic61#46312yes—
159Dic61#47211yescharacteristic
160Dic61#47311yescharacteristic
161#2931—611no—
162#1837—611no—
163C2×#335—611no—
164#1547—612no—
165#1292—612no—
166C2×C61—81no—
167C2×C61#180411yescharacteristic
168C2×C61#181311yescharacteristic
169C2×C61#181611yescharacteristic
170D61—22no—
171D61—81no—
172C2×#131—1221no—
173#532—611no—
174#436—611no—
175#436—1221no—
176#436#692311yescharacteristic
177C8×D4—611no—
178C8×D4—611no—
179C8×D4—1221no—
180#423—611no—
181#423—612no—
182C2×C4⋊C8—612no—
183#408—1221no—
184C2×C4×C8—611no—
185#341—1221no—
186#341—1221no—
187#335—1221no—
188#335—1221no—
189#332—1222no—
190#331—1222no—
191C61#868811yescharacteristic, complemented, Sylow
192C2×#41—1221no—
193#131—2444no—
194C2×M16—21no—
195C2×M16—1221no—
196C2×M16—1221no—
197C2×M16—1222no—
198C22×C8—611no—
199C22×C8—611no—
200C22×C8—1221no—
201C22×C8—1221no—
202C22×C8—1222no—
203C4⋊Q8—611no—
204#127—611no—
205#124—1221no—
206#121—611no—
207#121—611no—
208#121—1221no—
209C4×D4—611no—
210C4×D4—1221no—
211C4×D4C2×#692311yescharacteristic
212C2×C42—611no—
213C4⋊C8—611no—
214C4⋊C8—1221no—
215C4⋊C8—1222no—
216C4⋊C8—1222no—
217C4⋊C8—1222no—
218C4⋊C8—1222no—
219C4⋊C8C2×#692311yescharacteristic
220C22⋊C8—21no—
221C22⋊C8—1221no—
222C22⋊C8—1221no—
223C22⋊C8—1221no—
224C4×C8—611no—
225C4×C8—611no—
226C4×C8—611no—
227C4×C8C2×#692311yescharacteristic
228#41—2442no—
229#41—2442no—
230C2×Q8—1221no—
231C2×D4—611no—
232C2×D4—1221no—
233C2×D4—1221no—
234C2×D4—1221no—
235C2×D4—1221no—
236C2×D4C4⋊#692311yescharacteristic
237C22×C4—21no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—1221no—
242M16—42no—
243M16—2442no—
244M16—2442no—
245M16—2442no—
246M16—2442no—
247C2×C8—21no—
248C2×C8—611no—
249C2×C8—611no—
250C2×C8—1222no—
251C2×C8—1222no—
252C2×C8—1221no—
253C2×C8—4882no—
254C2×C8—2442no—
255C2×C8—2442no—
256C2×C8—2442no—
257C2×C8—2442no—
258C2×C8C22⋊#692311yescharacteristic
259C2×C8C22⋊#692311yescharacteristic
260C4⋊C4—611no—
261C4⋊C4—1221no—
262C4⋊C4C4⋊#692311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—1221no—
265C22⋊C4—1221no—
266C22⋊C4—1221no—
267C42—611no—
268C42—611no—
269C42—611no—
270C42C22×#692311yescharacteristic
271C23—21no—
272C23—611no—
273C23—1221no—
274Q8—2442no—
275D4—41no—
276D4—2441no—
277D4—2442no—
278D4—2441no—
279D4—2441no—
280D4—2442no—
281D4—2442no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—611no—
286C2×C4—611no—
287C2×C4—612no—
288C2×C4—612no—
289C2×C4—611no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4C2×C22⋊#692311yescharacteristic
297C2×C4C2×C4⋊#692311yescharacteristic
298C2×C4#1451985611yescharacteristic
299C8—21no—
300C8—21no—
301C8—41no—
302C8—1221no—
303C8—1221no—
304C8—4882no—
305C8—4882no—
306C8—2441no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—612no—
311C22—1221no—
312C22—1221no—
313C22—2441no—
314C22—2441no—
315C22—2441no—
316C22#1684070411yescharacteristic
317C4—21no—
318C4—21no—
319C4—41no—
320C4—612no—
321C4—611no—
322C4—611no—
323C4—1221no—
324C4—2441no—
325C4#1684075211yescharacteristic
326C4#1684075411yescharacteristic
327C2—81no—
328C2—1222no—
329C2—4881no—
330C2#2146168911yescharacteristic
331C2#2146170411yescharacteristic
332C2#2146170811yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#436C4
socleC22×C61#380
prime radical#16840309C22
cosocleC42×D61C23
hypercenter#436#6923
hypocenterC61#8688
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #21461744C2semidirect
2. D61⋊#335C2semidirect
4. #21460688C2semidirect
191. C61#8688semidirect

Maximal subgroups

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

subgroupindex
#214617442
D61⋊#3352
#214613302
#214606882
#214545282
#868861