Finite Group Databasebeta

← #30599156

Subgroups — #30599156

Order 15616.

418 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461741C211yesmaximal, characteristic, complemented
2#21461367C211yesmaximal, characteristic, complemented
3#21461249C212yesmaximal
4#21460674C212yesmaximal, complemented
5#21454528C211yesmaximal, characteristic
6#16840781C2212yes—
7#16840706C2212yes—
8#16840636—21no—
9#16840636—21no—
10#16840626—21no—
11#16840623—21no—
12#16840614—22no—
13#16840613—22no—
14C2×#14519785—21no—
15#16840309C2211yescharacteristic
16#16839841C411yescharacteristic, complemented
17#16839838—21no—
18D4×#92418C411yescharacteristic, complemented
19#16839827C411yescharacteristic, complemented
20#16839825—21no—
21#16839784C2211yescharacteristic
22#16839727—21no—
23C4×C8×D61C2211yescharacteristic
24C61×#436C411yescharacteristic, complemented
25C2×C4⋊#6923—22no—
26C4⋊#92420—22no—
27#14519834—22no—
28C4×#92420—22no—
29(C2×#6923)⋊C4—22no—
30(C2×#6923)⋊C4—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#14519776Q811yescharacteristic
40#14519773Q811yescharacteristic
41#14519763—21no—
42#14519752D411yescharacteristic
43#14519750—21no—
44#14519748D411yescharacteristic
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
61C22×#6923—42no—
62C2×D4×C61C4⋊C411yescharacteristic
63C22×C4×C61—21no—
64C2×C22⋊D61—21no—
65C2×C22⋊D61—21no—
66C2×C22⋊D61—21no—
67#11758222—42no—
68#11758222—42no—
69C2×D244—21no—
70C2×D244C4⋊C411yescharacteristic
71C2×C4×D61—21no—
72C2×C4×D61C2×D411yescharacteristic
73C2×C4×D61C2×D411yescharacteristic
74C2×C4×D61#4111yescharacteristic
75C2×Dic122—21no—
76C2×#92420—42no—
77C4⋊#6923—42no—
78C4⋊#6923—42no—
79M16×C61—42no—
80C2×C8×C61—21no—
81C2×C8×C61C22⋊C411yescharacteristic
82C2×C8×C61C22⋊C411yescharacteristic
83C4⋊C4×C61C4⋊C411yescharacteristic
84C22⋊C4×C61—21no—
85C42×C61C22×C411yescharacteristic
86C22⋊Dic61—21no—
87#11758198—21no—
88#11758198—21no—
89C4⋊Dic61C4⋊C411yescharacteristic
90#11758196—21no—
91C4×Dic61C2×D411yescharacteristic
92C4×Dic61C2×Q811yescharacteristic
93C4×Dic61#4111yescharacteristic
94C2×#92418—21no—
95C2×#92418—42no—
96C2×#92418C22⋊C411yescharacteristic
97C2×#92418C22⋊C411yescharacteristic
98#11758189—42no—
99#11758189—42no—
100C8×D61—22no—
101C8×D61—22no—
102C8×D61—42no—
103C23×C61—21no—
104C22×D61—21no—
105C22×D61#12111yescharacteristic
106C2×#6923—82no—
107C2×#6923—82no—
108C2×#6923—82no—
109D4×C61—41no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61C2×C22⋊C411yescharacteristic
114C2×C4×C61C2×C4⋊C411yescharacteristic
115C2×C4×C61#11711yescharacteristic
116C22⋊D61—42no—
117C22⋊D61—42no—
118C22⋊D61—42no—
119C2×Dic61—21no—
120C2×Dic61#12111yescharacteristic
121C2×Dic61C22⋊Q811yescharacteristic
122C2×Dic61C22⋊Q811yescharacteristic
123D244—41no—
124D244—42no—
125C4×D61—41no—
126C4×D61—41no—
127C4×D61—41no—
128C4×D61—41no—
129C4×D61#13512yes—
130C4×D61AGL(1,ℤ/8)12yes—
131Dic122—42no—
132#92420—82no—
133C8×C61—21no—
134C8×C61—21no—
135C8×C61—41no—
136#92418—21no—
137#92418—21no—
138#92418—41no—
139#8680—611nomaximal, Sylow
140C22×C61—41no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61#38011yescharacteristic
144C2×D61—21no—
145C2×D61—21no—
146C2×D61—41no—
147C2×D61—41no—
148C2×D61—41no—
149C2×D61#46312yes—
150#6923—162no—
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—41no—
154C4×C61#41611yescharacteristic
155C4×C61#41811yescharacteristic
156Dic61—21no—
157Dic61—21no—
158Dic61—41no—
159Dic61#47812yes—
160#2931—611no—
161#1835—611no—
162#1564—611no—
163#1517—612no—
164#1286—612no—
165C2×C61—81no—
166C2×C61#180411yescharacteristic
167C2×C61#181211yescharacteristic
168C2×C61#181411yescharacteristic
169D61—22no—
170D61—81no—
171C2×#131—1221no—
172#532—611no—
173#436—611no—
174#436—1221no—
175#436#692311yescharacteristic, complemented
176C8×D4—611no—
177C8×D4—611no—
178C8×D4—1221no—
179#431—612no—
180#423—611no—
181#408—1221no—
182C2×C4×C8—611no—
183#382—612no—
184#340—1221no—
185#340—1221no—
186#335—1221no—
187#334—1221no—
188#327—1222no—
189#327—1222no—
190C61#868011yescharacteristic, complemented, Sylow
191C2×#41—1221no—
192#131—2444no—
193C2×M16—21no—
194C2×M16—1221no—
195C2×M16—1221no—
196C22×C8—611no—
197C22×C8—611no—
198C22×C8—1221no—
199C22×C8—1221no—
200C4⋊Q8—611no—
201#127—611no—
202#124—1221no—
203#121—611no—
204#121—611no—
205#121—1221no—
206C4×D4—611no—
207C4×D4—1221no—
208C4×D4C2×#692311yescharacteristic
209C2×C4⋊C4—1222no—
210C2×C42—611no—
211C4⋊C8—611no—
212C4⋊C8—1221no—
213C4⋊C8—1222no—
214C4⋊C8C2×#692311yescharacteristic
215C22⋊C8—21no—
216C22⋊C8—1221no—
217C22⋊C8—1221no—
218C22⋊C8—1221no—
219C22⋊C8—1222no—
220C4×C8—611no—
221C4×C8—611no—
222C4×C8—611no—
223C4×C8—1222no—
224C4×C8C2×#692311yescharacteristic
225#95—1222no—
226#95—1222no—
227#41—2442no—
228#41—2442no—
229C2×Q8—1221no—
230C2×D4—611no—
231C2×D4—1221no—
232C2×D4—1221no—
233C2×D4—1221no—
234C2×D4—1221no—
235C2×D4C4⋊#692311yescharacteristic
236C22×C4—21no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—1221no—
241C22×C4—2442no—
242M16—42no—
243M16—2442no—
244M16—2442no—
245C2×C8—21no—
246C2×C8—611no—
247C2×C8—611no—
248C2×C8—1222no—
249C2×C8—1222no—
250C2×C8—1221no—
251C2×C8—2442no—
252C2×C8—2442no—
253C2×C8—2442no—
254C2×C8C22⋊#692311yescharacteristic
255C2×C8C22⋊#692311yescharacteristic
256C4⋊C4—611no—
257C4⋊C4—1221no—
258C4⋊C4—2442no—
259C4⋊C4—2442no—
260C4⋊C4C4⋊#692311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—1221no—
263C22⋊C4—1221no—
264C22⋊C4—1221no—
265C42—611no—
266C42—611no—
267C42—611no—
268C42C22×#692311yescharacteristic
269C23—21no—
270C23—611no—
271C23—1221no—
272Q8—2442no—
273D4—41no—
274D4—2441no—
275D4—2442no—
276D4—2442no—
277D4—2442no—
278D4—2442no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—611no—
283C2×C4—612no—
284C2×C4—612no—
285C2×C4—611no—
286C2×C4—611no—
287C2×C4—1221no—
288C2×C4—4882no—
289C2×C4—4882no—
290C2×C4—4882no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4C2×C22⋊#692311yescharacteristic
296C2×C4C2×C4⋊#692311yescharacteristic
297C2×C4#1451985611yescharacteristic
298C8—21no—
299C8—21no—
300C8—41no—
301C8—1221no—
302C8—1221no—
303C8—4882no—
304C8—2441no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—612no—
309C22—1221no—
310C22—1221no—
311C22—2441no—
312C22—2441no—
313C22—2441no—
314C22#1684070411yescharacteristic
315C4—21no—
316C4—21no—
317C4—41no—
318C4—612no—
319C4—1221no—
320C4—1221no—
321C4—9762no—
322C4—2441no—
323C4#1684075211yescharacteristic
324C4#1684075411yescharacteristic
325C2—81no—
326C2—1222no—
327C2—4881no—
328C2#2146168911yescharacteristic
329C2#2146170211yescharacteristic
330C2#2146170511yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#436C4
socleC22×C61#380
prime radical#16840309C22
cosocleC42×D61C23
hypercenter#436#6923
hypocenterC61#8680
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. #21461741C2semidirect
2. #21461367C2semidirect
4. #21460674C2semidirect
16. #16839841C4semidirect
18. D4×#92418C4semidirect
19. #16839827C4semidirect
24. C61×#436C4semidirect
175. #436#6923semidirect
190. C61#8680semidirect

Maximal subgroups

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

subgroupindex
#214617412
#214613672
#214612492
#214606742
#214545282
#868061