Finite Group Databasebeta

← #30601106

Subgroups — #30601106

Order 15616.

474 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 76 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#21462501C211yesmaximal, characteristic
2#21461717C211yesmaximal, characteristic
3#21461229C212yesmaximal
4#21460667C212yesmaximal
5#21450274C211yesmaximal, characteristic
6#16840787—24no—
7#16840765—21no—
8#16840734C2212yes—
9C4×#11758217—21no—
10#16840705C2212yes—
11#16840640—21no—
12#16840640—21no—
13#16840616—24no—
14C2×#14519753C2211yescharacteristic
15C2×#14519735C2211yescharacteristic
16#16839689—21no—
17#16839689—21no—
18#16839669C411yescharacteristic, complemented
19#16839645—21no—
20#16839645—21no—
21#16839632C411yescharacteristic, complemented
22#16839627C411yescharacteristic, complemented
23C4×#11758198C2211yescharacteristic
24C61×#391C411yescharacteristic, complemented
25C2×#11758217—22no—
26C2×C22⋊#6923—22no—
27#14519844—22no—
28C4⋊#92420—22no—
29#14519834—22no—
30#14519834D414yes—
31#14519832—22no—
32C4×#92420—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—22no—
35C22×C4×D61C2311yescharacteristic
36#14519753—22no—
37#14519753—22no—
38#14519735—22no—
39#14519735—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198D411yescharacteristic
48C2×#11758198D411yescharacteristic
49C2×C4×Dic61C2×C411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62C23×D61C2×D411yescharacteristic
63C22×#6923—42no—
64C2×Q8×C61—22no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C2×D411yescharacteristic
77C2×C4×D61—41no—
78C2×C4×D61—41no—
79C2×C4×D61C2×D412yes—
80C2×C4×D61#4112yes—
81C22⋊#6923—42no—
82C22⋊#6923—42no—
83#11758217—42no—
84#11758217—42no—
85C2×#92420—24no—
86C4⋊C4×C61—22no—
87C42×C61—41no—
88#11758198—22no—
89#11758198—22no—
90#11758198—42no—
91#11758198—42no—
92C4⋊Dic61—22no—
93C4⋊Dic61—22no—
94C4⋊Dic61—42no—
95C4⋊Dic61—42no—
96#11758196—41no—
97#11758196—41no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—41no—
103C23×C61C2×C22⋊C411yescharacteristic
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61#12112yes—
109C2×#6923—82no—
110C2×#6923—82no—
111C2×#6923—82no—
112Q8×C61—82no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61C4×D412yes—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61#12011yescharacteristic
139C2×Dic61#12311yescharacteristic
140C4×D61—42no—
141C4×D61—42no—
142#92420—44no—
143#9444—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61#38611yescharacteristic
147C22×C61#41011yescharacteristic
148C22×C61#41011yescharacteristic
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—22no—
152C2×D61—81no—
153C2×D61—81no—
154#6923—162no—
155C4×C61—22no—
156C4×C61—42no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—81no—
160Dic61—21no—
161Dic61—21no—
162Dic61—41no—
163Dic61—41no—
164Dic61—81no—
165Dic61—81no—
166#2921—611no—
167#2499—611no—
168#1820—611no—
169#1511—612no—
170#1284—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61#188411yescharacteristic
174C2×C61#189911yescharacteristic
175C2×C61#190911yescharacteristic
176D61—42no—
177C2×C22⋊Q8—611no—
178C2×#117—611no—
179#433—1224no—
180#423—1221no—
181#407—612no—
182C4×M16—1221no—
183#397—1221no—
184#397—1221no—
185#391—611no—
186#391#692311yescharacteristic, complemented
187#387—1221no—
188#387—1221no—
189#384—611no—
190#382—611no—
191#381—612no—
192C4×C22⋊C4—611no—
193#342—1221no—
194#342—1221no—
195#328—1224no—
196C61#944411yescharacteristic, complemented, Sylow
197C22×Q8C2×#692311yescharacteristic
198C23×C4—611no—
199C2×M16—1222no—
200C22⋊Q8—1222no—
201C22⋊Q8—1222no—
202#117—1222no—
203#117—1222no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4C2×#692311yescharacteristic
210C2×C22⋊C4—611no—
211C2×C22⋊C4—611no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1222no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42C2×#692311yescharacteristic
219C4⋊C8—1222no—
220C4⋊C8—1222no—
221C22⋊C8—614no—
222C22⋊C8—1222no—
223#97—1222no—
224C4×C8—1222no—
225#95—21no—
226#95—21no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1222no—
231#95—1222no—
232#95—1221no—
233#95—1221no—
234C24—611no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—611no—
241C22×C4—612no—
242C22×C4—612no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—2441no—
248C22×C4—2441no—
249C22×C4—2442no—
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22×#692311yescharacteristic
253M16—2442no—
254M16—2442no—
255C2×C8—1224no—
256C4⋊C4—22no—
257C4⋊C4—1222no—
258C4⋊C4—1222no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2441no—
262C4⋊C4—2441no—
263C22⋊C4—1222no—
264C22⋊C4—1222no—
265C22⋊C4—2442no—
266C22⋊C4—2442no—
267C22⋊C4—2442no—
268C22⋊C4—2442no—
269C42—41no—
270C42—1221no—
271C42—1221no—
272C42—1221no—
273C42—1221no—
274C42—2441no—
275C23—612no—
276C23—1221no—
277C23—1221no—
278C23—2441no—
279C23—2441no—
280C23C2×C22⋊#692311yescharacteristic
281Q8—82no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—4882no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4D4×#692312yes—
314C8—2444no—
315C22—21no—
316C22—21no—
317C22—1222no—
318C22—1222no—
319C22—1222no—
320C22—4881no—
321C22—4881no—
322C22#1684071011yescharacteristic
323C22#1684074111yescharacteristic
324C22#1684074211yescharacteristic
325C4—22no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—1221no—
331C4—1221no—
332C4—9762no—
333C4—4881no—
334C4—4881no—
335C4—2441no—
336C4—2441no—
337C2—21no—
338C2—21no—
339C2—2442no—
340C2#2146182111yescharacteristic
341C2#2146184211yescharacteristic
342C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#9444
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
18. #16839669C4semidirect
21. #16839632C4semidirect
22. #16839627C4semidirect
24. C61×#391C4semidirect
186. #391#6923semidirect
196. C61#9444semidirect

Maximal subgroups

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

subgroupindex
#214625012
#214617172
#214612292
#214606672
#214502742
#944461