Finite Group Databasebeta

← #30604814

Subgroups — #30604814

Order 15616.

468 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461518C211yesmaximal, characteristic
2#21461052C211yesmaximal, characteristic
3#21460816C212yesmaximal
4#21460814C212yesmaximal
5#21449089C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840778—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×#14519734C2211yescharacteristic
13#16839700C411yescharacteristic, complemented
14#16839697C2211yescharacteristic
15#16839695—21no—
16#16839684—21no—
17#16839681C411yescharacteristic, complemented
18#16839673C2211yescharacteristic
19C4×#11758196—21no—
20C4×C4⋊Dic61C411yescharacteristic, complemented
21C4×#11758198—21no—
22C2×C4×Q8×C61C411yescharacteristic, complemented
23#14519856—24no—
24#14519856—42no—
25C2×C4⋊#6923—41no—
26C2×C22⋊#6923—22no—
27C2×C4×#6923—41no—
28C4⋊#92420—41no—
29C22⋊#92420—22no—
30C4×#92420—41no—
31C22×C4×D61D411yescharacteristic
32#14519734—41no—
33#14519734—41no—
34C2×C4⋊Dic61—21no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×#11758196C2×C411yescharacteristic
38C2×#11758196D411yescharacteristic
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198D411yescharacteristic
42C2×#11758198C2311yescharacteristic
43C2×C4×Dic61—21no—
44C2×C4×Dic61D411yescharacteristic
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic
50#14519684C2×C411yescharacteristic
51C22×Q8×C61C2×C411yescharacteristic
52C4×Q8×C61—41no—
53C4×Q8×C61—41no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C42×C61—21no—
57C2×C42×C61C2×C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—42no—
61C2×Q8×C61—41no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—22no—
75C2×C4×D61SD1612yes—
76C2×C4×D61Q1612yes—
77C22⋊#6923—24no—
78C22⋊#6923—42no—
79C2×#92420—42no—
80C4⋊#6923—44no—
81C4×#6923—44no—
82C4⋊C4×C61—42no—
83C4⋊C4×C61—41no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—41no—
87C42×C61—41no—
88#11758198—41no—
89#11758198—41no—
90#11758198—42no—
91#11758198—42no—
92C4⋊Dic61—42no—
93#11758196—41no—
94#11758196—41no—
95#11758196—42no—
96#11758196—42no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61—42no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61C2×SD1611yescharacteristic
106C22×D61C2×Q1611yescharacteristic
107C2×#6923—44no—
108C2×#6923—82no—
109C2×#6923—82no—
110Q8×C61—42no—
111Q8×C61—42no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61#9911yescharacteristic
123C2×C4×C61#9911yescharacteristic
124C2×C4×C61#10312yes—
125C2×C4×C61#10412yes—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—22no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61#10312yes—
139C4×D61—22no—
140C4×D61—22no—
141C4×D61—22no—
142C4×D61—22no—
143#92420—82no—
144#11715—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61C2×#9911yescharacteristic
148C22×C61C2×#10311yescharacteristic
149C22×C61C2×#10411yescharacteristic
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—41no—
153C2×D61—41no—
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—81no—
157#6923—84no—
158C4×C61—42no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—81no—
162C4×C61#32812yes—
163C4×C61#32812yes—
164Dic61—22no—
165Dic61—42no—
166Dic61—41no—
167Dic61—41no—
168Dic61—81no—
169#2396—611no—
170#1740—611no—
171#1458—611no—
172#1352—612no—
173#1351—612no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61C2×#32811yescharacteristic
177C2×C61#212011yescharacteristic
178C2×C61#212611yescharacteristic
179D61—41no—
180D61—41no—
181C2×C4×Q8#692311yescharacteristic, complemented
182C2×#117—611no—
183C2×#117—1221no—
184#431—1221no—
185#400—611no—
186#399—611no—
187#398—1221no—
188#396—1221no—
189#395—611no—
190#393—611no—
191C4×C4⋊C4—611no—
192C4×C4⋊C4—1221no—
193C4×C22⋊C4—1221no—
194#342—612no—
195#342—1221no—
196#324—1221no—
197#323—612no—
198C61#1171511yescharacteristic, complemented, Sylow
199C22×Q8C2×#692311yescharacteristic
200C23×C4—611no—
201C4×Q8—41no—
202C4×Q8—41no—
203#117—1224no—
204#117—2441no—
205#117—2441no—
206#117—2442no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—2441no—
214C2×C4⋊C4C2×#692311yescharacteristic
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C42—21no—
221C2×C42—611no—
222C2×C42—1221no—
223C2×C42—2441no—
224C2×C42C2×#692311yescharacteristic
225C4⋊C8—2441no—
226C22⋊C8—1222no—
227C4×C8—2441no—
228#95—611no—
229#95—611no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234C24—611no—
235C2×Q8—42no—
236C2×Q8—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—611no—
240C22×C4—612no—
241C22×C4—612no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1222no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—2442no—
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22×#692311yescharacteristic
253C2×C8—2442no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—2441no—
257C4⋊C4—2441no—
258C4⋊C4—2442no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2444no—
262C22⋊C4—1224no—
263C22⋊C4—2441no—
264C22⋊C4—2441no—
265C22⋊C4—2442no—
266C22⋊C4—2442no—
267C22⋊C4—2442no—
268C42—21no—
269C42—21no—
270C42—41no—
271C42—41no—
272C42—1221no—
273C42—1221no—
274C42—2444no—
275C42—2442no—
276C23—611no—
277C23—611no—
278C23—1221no—
279C23—1221no—
280C23—2441no—
281C23—2441no—
282C23C2×C22⋊#692311yescharacteristic
283Q8—42no—
284Q8—42no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—612no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1222no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4—1222no—
302C2×C4—1222no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—4882no—
306C2×C4—4882no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2444no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4#1451983511yescharacteristic
316C2×C4#1451983511yescharacteristic
317C2×C4#1451984012yes—
318C2×C4#1451984212yes—
319C8—4882no—
320C22—21no—
321C22—21no—
322C22—1221no—
323C22—1221no—
324C22—4881no—
325C22—2441no—
326C22—2441no—
327C22—2441no—
328C22—2441no—
329C22C2×#1451983511yescharacteristic
330C22C2×#1451984011yescharacteristic
331C22C2×#1451984211yescharacteristic
332C4—42no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—1222no—
337C4—4881no—
338C4—4884no—
339C4—2442no—
340C4—2441no—
341C4—2441no—
342C4#1684061612yes—
343C4#1684061612yes—
344C2—21no—
345C2—21no—
346C2—2441no—
347C2—2441no—
348C2C2×#1684061611yescharacteristic
349C2#2146207011yescharacteristic
350C2#2146209011yescharacteristic
subgroupG/H
centerC2×C4#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenterC2×C4×Q8#6923
hypocenterC61#11715
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
13. #16839700C4semidirect
17. #16839681C4semidirect
20. C4×C4⋊Dic61C4semidirect
22. C2×C4×Q8×C61C4semidirect
181. C2×C4×Q8#6923semidirect
198. C61#11715semidirect

Maximal subgroups

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

subgroupindex
#214615182
#214610522
#214608162
#214608142
#214490892
#1171561