Finite Group Databasebeta

← #30597504

Subgroups — #30597504

Order 15616.

582 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×#16840790C211yesmaximal, characteristic
2#21461564C211yesmaximal, characteristic
3#21461558C212yesmaximal, complemented
4#21460660C212yesmaximal, complemented
5#21444136C211yesmaximal, characteristic
6C22×D61⋊C8C411yescharacteristic, complemented
7C22×D61⋊C8C2211yescharacteristic
8#16840790—22no—
9#16840790—22no—
10C2×C4⋊#92420C411yescharacteristic, complemented
11C2×#14519836C412yes—
12C2×#14519834C411yescharacteristic, complemented
13C2×#14519834C2211yescharacteristic
14C2×#14519831—22no—
15C2×C4×#92420C411yescharacteristic, complemented
16C4×C22⋊#6923C2212yes—
17#16840640C412yes—
18(C2×#6923)⋊C8—22no—
19C2×#14519739C2211yescharacteristic
20C22⋊C8×D61C2212yes—
21#16839488C411yescharacteristic, complemented
22#16839485C412yescomplemented
23C61×#324C411yescharacteristic, complemented
24C2×D61⋊C8—21no—
25C2×D61⋊C8—21no—
26C2×D61⋊C8—42no—
27C2×D61⋊C8—42no—
28C2×D61⋊C8D412yes—
29C2×D61⋊C8D412yes—
30C22×#92420C2×C411yescharacteristic
31C22×#92420C2×C411yescharacteristic
32C22×#92420C2×C411yescharacteristic
33C22×#92420C2×C411yescharacteristic
34C2×C22⋊#6923C2×C414yes—
35C2×C4×#6923—22no—
36C4⋊#92420—22no—
37#14519836—24no—
38#14519834—22no—
39#14519834—22no—
40#14519831—44no—
41C4×#92420—21no—
42C4×#92420—21no—
43(C2×#6923)⋊C4—22no—
44C22×D244C2×C411yescharacteristic
45C22×C4×D61C2×C411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C2×C8×D61—22no—
48#14519739—22no—
49#14519739—22no—
50C2×#11758198C2×C411yescharacteristic
51C2×#11758198C2×C411yescharacteristic
52C2×C4×Dic61C2×C411yescharacteristic
53C22⋊#92418C2×C412yes—
54#14519689—22no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C22⋊C8×C61C2×C412yes—
57C23×D61C4211yescharacteristic
58C23×D61C22×C411yescharacteristic
59C22×#6923—24no—
60C22×C4×C61C4211yescharacteristic
61C22×C4×C61C4211yescharacteristic
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61C4211yescharacteristic
64C22×Dic61C22×C411yescharacteristic
65C2×D244—22no—
66C2×D244—22no—
67C2×D244M1614yes—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61—41no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C4×D61—41no—
74C2×C4×D61C2×D412yes—
75C2×C4×D61#4112yes—
76C22⋊#6923—24no—
77C22⋊#6923—24no—
78C2×#92420—22no—
79C2×#92420—21no—
80C2×#92420—21no—
81C2×#92420—41no—
82C2×#92420—41no—
83C2×#92420—41no—
84C2×#92420C22⋊C411yescharacteristic
85C2×#92420C22⋊C412yes—
86C2×#92420C22⋊C411yescharacteristic
87C4×#6923—42no—
88C4×#6923—42no—
89D61⋊C8—24no—
90D61⋊C8—42no—
91D61⋊C8—42no—
92D61⋊C8—42no—
93C2×C8×C61—22no—
94C4⋊C4×C61—22no—
95#11758198—22no—
96#11758198—22no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C2×#92418—22no—
100C8×D61—42no—
101C8×D61—42no—
102C23×C61C2×C4211yescharacteristic
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61#9712yescomplemented
110C22×D61C4×D412yes—
111C22×D61C2×M1612yes—
112C22×D61#13112yes—
113C2×#6923—44no—
114C2×#6923—44no—
115C2×#6923—44no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61#9712yescomplemented
122C2×C4×C61C4×D412yes—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—41no—
127C2×Dic61C2×C22⋊C411yescharacteristic
128C2×Dic61#11711yescharacteristic
129D244—28no—
130C4×D61—22no—
131C4×D61—22no—
132C4×D61—42no—
133C4×D61—42no—
134C4×D61—41no—
135C4×D61—41no—
136#92420—22no—
137#92420—21no—
138#92420—41no—
139#92420—41no—
140#92420#9912yescomplemented
141C8×C61—42no—
142#92418—42no—
143#8160—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61C4×C22⋊C411yescharacteristic
147C22×C61C2×#9711yescharacteristic
148C22×C61#40511yescharacteristic
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—22no—
152C2×D61—42no—
153C2×D61—42no—
154C2×D61—42no—
155C2×D61—41no—
156C2×D61—41no—
157C2×D61—42no—
158C2×D61#43514yes—
159#6923—84no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—41no—
163Dic61—21no—
164Dic61—41no—
165Dic61C2×#9911yescharacteristic
166Dic61#41011yescharacteristic
167C2×C8×D4—611no—
168C4×#99—611no—
169#1752—612no—
170#1459—611no—
171#1282—612no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#175211yescharacteristic
175C2×C61C4×#9911yescharacteristic
176C2×C61#175711yescharacteristic
177D61—24no—
178D61—42no—
179C23×C8—611no—
180C23×C8—611no—
181C2×C4×D4—611no—
182C8×D4—1222no—
183C8×D4—1222no—
184C2×C4⋊C8—611no—
185C2×#99—612no—
186C2×C22⋊C8—612no—
187C2×C22⋊C8—611no—
188C2×C22⋊C8—611no—
189C2×#97—1222no—
190C2×C4×C8—611no—
191C4×C22⋊C4—612no—
192#342—612no—
193#336—1222no—
194#324—611no—
195#324#692311yescharacteristic, complemented
196#323—612no—
197C61#816011yescharacteristic, complemented, Sylow
198C22×D4—611no—
199C23×C4—611no—
200C23×C4—611no—
201C22×C8—611no—
202C22×C8—612no—
203C22×C8—611no—
204C22×C8—612no—
205C22×C8—611no—
206C22×C8—611no—
207C22×C8—1222no—
208C22×C8—1221no—
209C22×C8—1221no—
210C22×C8—2442no—
211C22×C8—2442no—
212C4×D4—1222no—
213C4×D4—1222no—
214C2×C4⋊C4C2×#692311yescharacteristic
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—614no—
218C2×C42—611no—
219C2×C42—1222no—
220C4⋊C8—1222no—
221#99—1224no—
222C22⋊C8—612no—
223C22⋊C8—1222no—
224C22⋊C8—1222no—
225C22⋊C8—1222no—
226C22⋊C8C2×#692312yes—
227#97—2444no—
228C4×C8—1221no—
229C4×C8—1221no—
230#95—1222no—
231C24—611no—
232C24—611no—
233C2×D4—614no—
234C2×D4—1222no—
235C2×D4—1222no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—611no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1224no—
243C22×C4—2441no—
244C22×C4—2441no—
245C22×C4—2441no—
246C22×C4—2441no—
247C22×C4C4×#692311yescharacteristic
248C22×C4C4×#692311yescharacteristic
249C22×C4C22×#692311yescharacteristic
250C2×C8—22no—
251C2×C8—611no—
252C2×C8—611no—
253C2×C8—612no—
254C2×C8—1224no—
255C2×C8—1222no—
256C2×C8—1221no—
257C2×C8—1221no—
258C2×C8—1222no—
259C2×C8—2442no—
260C2×C8—2442no—
261C2×C8—2442no—
262C2×C8—2442no—
263C2×C8—2441no—
264C2×C8—2442no—
265C2×C8—2441no—
266C2×C8—2441no—
267C4⋊C4—22no—
268C22⋊C4—1222no—
269C22⋊C4—1222no—
270C22⋊C4—1224no—
271C22⋊C4—1224no—
272C42—1221no—
273C42—1221no—
274C42—2442no—
275C42—2442no—
276C23—612no—
277C23—612no—
278C23—612no—
279C23—612no—
280C23—1221no—
281C23—1221no—
282C23—2441no—
283C23—2441no—
284C23—2441no—
285C23—2441no—
286C23C2×C4×#692311yescharacteristic
287D4—1228no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—2442no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2442no—
304C2×C4—2444no—
305C2×C4—2444no—
306C2×C4—2444no—
307C2×C4—2441no—
308C2×C4#1451983112yescomplemented
309C2×C4D4×#692312yes—
310C8—42no—
311C8—612no—
312C8—1222no—
313C8—1221no—
314C8—2442no—
315C8—2441no—
316C8—2441no—
317C22—21no—
318C22—21no—
319C22—614no—
320C22—1222no—
321C22—1222no—
322C22—1222no—
323C22—2442no—
324C22—2442no—
325C22—2442no—
326C22—2442no—
327C22—2441no—
328C22—2441no—
329C22C22⋊C4×#692311yescharacteristic
330C22C2×#1451983111yescharacteristic
331C22#1684072811yescharacteristic
332C4—22no—
333C4—22no—
334C4—41no—
335C4—611no—
336C4—611no—
337C4—1221no—
338C4—4884no—
339C4—2441no—
340C2—21no—
341C2—21no—
342C2—1224no—
343C2—2442no—
344C2#2146155811yescharacteristic
345C2#99×#692311yescharacteristic
346C2#2146157811yescharacteristic
subgroupG/H
centerC22C22⋊C4×#6923
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#324C4
socleC22×C61C4×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC22×C4×D61C23
hypercenter#324#6923
hypocenterC61#8160
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. #21461558C2semidirect
4. #21460660C2semidirect
6. C22×D61⋊C8C4semidirect
10. C2×C4⋊#92420C4semidirect
12. C2×#14519834C4semidirect
15. C2×C4×#92420C4semidirect
21. #16839488C4semidirect
22. #16839485C4semidirect
23. C61×#324C4semidirect
109. C22×D61#97semidirect
121. C2×C4×C61#97semidirect
140. #92420#99semidirect
195. #324#6923semidirect
197. C61#8160semidirect
308. C2×C4#14519831semidirect

Maximal subgroups

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

subgroupindex
C2×#168407902
#214615642
#214615582
#214606602
#214441362
#816061