Finite Group Databasebeta

← #30599130

Subgroups — #30599130

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#21461157C212yesmaximal
4#21460705C212yesmaximal, complemented
5C8×#11758226C211yesmaximal, characteristic
6C2×#14519848—21no—
7#16840766C2212yes—
8C4×#11758213C2212yes—
9#16840638—21no—
10#16840624—21no—
11#16840624—21no—
12Q8⋊#92420—22no—
13#16840610—22no—
14C4×#11758226C2211yescharacteristic
15Q8×#92418C411yescharacteristic
16C8×D244—21no—
17C8×D244C411yescharacteristic
18#16839827—21no—
19#16839827C411yescharacteristic
20#16839814—21no—
21#16839814C2211yescharacteristic
22C4×C8×D61—21no—
23C4×C8×D61C2211yescharacteristic
24C8×Q8×C61C411yescharacteristic
25C2×#11758213—22no—
26C2×#11758213—22no—
27#14519848—24no—
28C4⋊#92420—22no—
29C4⋊#92420—22no—
30#14519832—22no—
31C4×#92420—22no—
32C2×#11758226Q811yescharacteristic
33C2×C8×D61—21no—
34C2×C8×D61—21no—
35C2×C8×D61D411yescharacteristic
36C2×C8×D61D411yescharacteristic
37Q8×Dic61D411yescharacteristic
38C4×D244—21no—
39C4×D244C2×C411yescharacteristic
40#14519739—21no—
41#14519739D411yescharacteristic
42#14519735—21no—
43#14519735Q811yescharacteristic
44C42×D61—21no—
45C42×D61C2311yescharacteristic
46C4⋊#92418—21no—
47C4⋊#92418C2×C411yescharacteristic
48#14519689—21no—
49#14519689—21no—
50#14519689—21no—
51C4×#92418—21no—
52C4×#92418C2×C411yescharacteristic
53C8×Dic61—21no—
54C8×Dic61D411yescharacteristic
55C8×Dic61D411yescharacteristic
56C4×Q8×C61C2×C411yescharacteristic
57C4⋊C8×C61—21no—
58C4⋊C8×C61C2×C411yescharacteristic
59C4×C8×C61—21no—
60C4×C8×C61C2×C411yescharacteristic
61C2×Q8×C61C4⋊C411yescharacteristic
62#11758226—22no—
63#11758226—22no—
64C2×D244—21no—
65C2×D244C4⋊C411yescharacteristic
66C2×C4×D61—21no—
67C2×C4×D61—21no—
68C2×C4×D61—21no—
69C2×C4×D61C2×D411yescharacteristic
70C2×C4×D61C2×Q811yescharacteristic
71C2×C4×D61#4111yescharacteristic
72C2×#92420—42no—
73C2×#92420—42no—
74#11758213—24no—
75#11758213—42no—
76#11758213—82no—
77C2×C8×C61—21no—
78C2×C8×C61C22⋊C411yescharacteristic
79C2×C8×C61C22⋊C411yescharacteristic
80C4⋊C4×C61—21no—
81C4⋊C4×C61C4⋊C411yescharacteristic
82C42×C61—21no—
83C42×C61C22×C411yescharacteristic
84#11758198—21no—
85#11758198—21no—
86#11758198—21no—
87C4⋊Dic61—21no—
88C4⋊Dic61C4⋊C411yescharacteristic
89C4×Dic61—21no—
90C4×Dic61—21no—
91C4×Dic61C2×D411yescharacteristic
92C4×Dic61C2×D411yescharacteristic
93C4×Dic61#4111yescharacteristic
94C2×#92418—21no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C22⋊C411yescharacteristic
97C8×D61—22no—
98C8×D61—41no—
99C8×D61—42no—
100C8×D61—42no—
101C22×D61—21no—
102C22×D61C22⋊Q811yescharacteristic
103Q8×C61—21no—
104Q8×C61#10812yescomplemented
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61C2×C22⋊C411yescharacteristic
108C2×C4×C61C2×C4⋊C411yescharacteristic
109C2×C4×C61#11711yescharacteristic
110C2×Dic61—21no—
111C2×Dic61#12111yescharacteristic
112C2×Dic61#12111yescharacteristic
113C2×Dic61C22⋊Q811yescharacteristic
114D244—24no—
115D244—21no—
116D244#10812yescomplemented
117C4×D61—24no—
118C4×D61—21no—
119C4×D61—21no—
120C4×D61—21no—
121C4×D61—21no—
122C4×D61—22no—
123C4×D61—41no—
124C4×D61—41no—
125C4×D61—42no—
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#38011yescharacteristic
136C2×D61—21no—
137C2×D61—21no—
138C2×D61—22no—
139C2×D61—21no—
140C2×D61—41no—
141C2×D61—41no—
142C4×C61—22no—
143C4×C61—21no—
144C4×C61—41no—
145C4×C61C2×#10411yescharacteristic
146C4×C61#42111yescharacteristic
147C4×C61C2×#10811yescharacteristic
148C4×C61C2×#10811yescharacteristic
149Dic61—21no—
150Dic61—21no—
151Dic61—41no—
152Dic61#47112yes—
153C8×#41—611no—
154C4×#108—611no—
155#1566—611no—
156#1489—612no—
157#1299—612no—
158C2×C61#180611yescharacteristic
159C2×C61#181011yescharacteristic
160C2×C61#181511yescharacteristic
161D61—41no—
162D61—42no—
163C4×#41—611no—
164C8×Q8—611no—
165C8×Q8#692311yescharacteristic
166C8×D4—611no—
167C8×D4—611no—
168C8×D4—1221no—
169C8×D4—1221no—
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—611no—
187C22×C8—611no—
188C22×C8—1221no—
189C22×C8—1221no—
190C4×Q8—611no—
191C4×Q8C2×#692311yescharacteristic
192C4×D4—611no—
193C4×D4—611no—
194C4×D4—1221no—
195C4×D4—1221no—
196#117—611no—
197#117—1221no—
198C2×C42—611no—
199C2×C42—1221no—
200#108—1224no—
201C4⋊C8—21no—
202C4⋊C8—611no—
203C4⋊C8—1221no—
204C4⋊C8—1222no—
205C4⋊C8—1222no—
206C4⋊C8C2×#692311yescharacteristic
207C22⋊C8—1221no—
208C22⋊C8—1221no—
209C22⋊C8—1221no—
210#97—1222no—
211C4×C8—21no—
212C4×C8—611no—
213C4×C8—611no—
214C4×C8—611no—
215C4×C8—1221no—
216C4×C8—1221no—
217C4×C8—1222no—
218C4×C8C2×#692311yescharacteristic
219#41—1222no—
220#41—1222no—
221C2×Q8C4⋊#692311yescharacteristic
222C2×D4—611no—
223C2×D4—1221no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—1221no—
228C22×C4—1221no—
229C22×C4—1221no—
230M16—1224no—
231M16—4882no—
232M16—2442no—
233C2×C8—21no—
234C2×C8—611no—
235C2×C8—611no—
236C2×C8—1222no—
237C2×C8—1221no—
238C2×C8—2441no—
239C2×C8—2442no—
240C2×C8—2442no—
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8C22⋊#692311yescharacteristic
244C2×C8C22⋊#692311yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—611no—
247C4⋊C4—1221no—
248C4⋊C4C4⋊#692311yescharacteristic
249C22⋊C4—1221no—
250C22⋊C4—1221no—
251C22⋊C4—1221no—
252C42—21no—
253C42—611no—
254C42—611no—
255C42—611no—
256C42—1221no—
257C42—1221no—
258C42C22×#692311yescharacteristic
259C23—611no—
260C23—1221no—
261Q8—21no—
262Q8#1451984812yescomplemented
263D4—612no—
264D4—1224no—
265D4—1221no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—611no—
269C2×C4—611no—
270C2×C4—611no—
271C2×C4—1222no—
272C2×C4—1224no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—2442no—
279C2×C4—2441no—
280C2×C4—2441no—
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—1221no—
293C22—1222no—
294C22—1221no—
295C22—1221no—
296C22—2441no—
297C22—2441no—
298C22#1684070411yescharacteristic
299C4—22no—
300C4—21no—
301C4—41no—
302C4—612no—
303C4—1221no—
304C4—1221no—
305C4—2441no—
306C4D61⋊#10411yescharacteristic
307C4#1684076111yescharacteristic
308C4D61⋊#10811yescharacteristic
309C4C2×#1451984811yescharacteristic
310C2—2441no—
311C2—2442no—
312C2#2146169411yescharacteristic
313C2#2146169911yescharacteristic
314C2#2146170711yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC8×Q8×C61C4
socleC22×C61#380
prime radicalC2×#11758226Q8
cosocleC42×D61C23
hypercenterC8×Q8#6923
hypocenterC61#8670
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
1. C4×#14519848C2semidirect
2. #21461375C2semidirect
4. #21460705C2semidirect
104. Q8×C61#108semidirect
116. D244#108semidirect
182. C61#8670semidirect
262. Q8#14519848semidirect

Maximal subgroups

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

subgroupindex
C4×#145198482
#214613752
#214611572
#214607052
C8×#117582262
#867061