Finite Group Databasebeta

← #30600141

Subgroups — #30600141

Order 15616.

404 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462730C211yesmaximal, characteristic
2#21461277C211yesmaximal, characteristic
3#21461143C211yesmaximal, characteristic
4#21461142C211yesmaximal, characteristic
5#21460651C211yesmaximal, characteristic
6#21460651C211yesmaximal, characteristic
7#21456330C211yesmaximal, characteristic
8#16840813—21no—
9#16840812—21no—
10#16840811—21no—
11Q8×#92420—21no—
12#16840787—22no—
13#16840781C2211yescharacteristic
14#16840781C2211yescharacteristic
15C4×#11758213C2211yescharacteristic
16C4×#11758213C2211yescharacteristic
17#16840619—22no—
18#16840612—21no—
19#16840612—21no—
20Q8⋊#92420—21no—
21Q8⋊#92420—21no—
22#16840317C2211yescharacteristic
23#16840317C2211yescharacteristic
24#16840167C411yescharacteristic
25#16840163—22no—
26#16840158C411yescharacteristic
27#16840158C411yescharacteristic
28#16839799—22no—
29C4×C8×D61C2211yescharacteristic
30C61×#500C411yescharacteristic
31C2×#11758213—21no—
32C2×#11758213—21no—
33#14519844—22no—
34C4⋊#92420—22no—
35C4⋊#92420D411yescharacteristic
36C4⋊#92420D411yescharacteristic
37C4⋊#92420D411yescharacteristic
38C4⋊#92420D411yescharacteristic
39#14519834—21no—
40#14519834—21no—
41#14519832—22no—
42#14519832—21no—
43#14519832—21no—
44C4×#92420—22no—
45C4×#92420—21no—
46C4×#92420—21no—
47C4×#92420—21no—
48C4×#92420—21no—
49C2×C8×D61—21no—
50#14519777C2×C411yescharacteristic
51#14519777C2×C411yescharacteristic
52#14519768—21no—
53#14519768D412yes—
54#14519757—21no—
55#14519757—21no—
56#14519753—21no—
57#14519753—21no—
58Q8×Dic61—21no—
59Q8×Dic61—21no—
60#14519742—21no—
61#14519742—21no—
62#14519735—21no—
63#14519735D412yes—
64C42×D61C2311yescharacteristic
65#14519711—22no—
66#14519708—22no—
67#14519708—22no—
68C4×#92418C2×C411yescharacteristic
69C8×Dic61—21no—
70C4⋊Q8×C61C2×C411yescharacteristic
71C4⋊Q8×C61C2×C411yescharacteristic
72#107×C61—22no—
73C4×C8×C61C2×C411yescharacteristic
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76C2×C4×D61C2×D411yescharacteristic
77C2×C4×D61#4111yescharacteristic
78C2×C4×D61#4111yescharacteristic
79C2×Dic122—21no—
80C2×Dic122—21no—
81C2×#92420—21no—
82C2×#92420—21no—
83C2×#92420—21no—
84C2×#92420—21no—
85C2×#92420—41no—
86C2×#92420—41no—
87#11758213—22no—
88#11758213—22no—
89#11758213—41no—
90#11758213—41no—
91C2×C8×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61C22⋊C412yes—
96C42×C61C22×C411yescharacteristic
97#11758198—22no—
98#11758198—41no—
99C4⋊Dic61—21no—
100C4⋊Dic61—22no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—41no—
104C4⋊Dic61C22⋊C412yes—
105#11758196—22no—
106#11758196—41no—
107C4×Dic61—22no—
108C4×Dic61—41no—
109C4×Dic61C2×D411yescharacteristic
110C4×Dic61C2×D411yescharacteristic
111C4×Dic61C2×D411yescharacteristic
112C2×#92418—21no—
113C8×D61—22no—
114C8×D61—22no—
115C22×D61#12311yescharacteristic
116Q8×C61—41no—
117Q8×C61—41no—
118Q8×C61—41no—
119Q8×C61—41no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61C2×C22⋊C411yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×Dic61—22no—
126C2×Dic61—41no—
127C2×Dic61#12011yescharacteristic
128C2×Dic61#12111yescharacteristic
129C2×Dic61#12111yescharacteristic
130C4×D61—21no—
131C4×D61—21no—
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—21no—
135C4×D61—21no—
136Dic122—41no—
137Dic122—41no—
138Dic122—41no—
139Dic122—41no—
140#92420—42no—
141#92420—42no—
142#92420—41no—
143#92420—41no—
144#92420—41no—
145#92420—41no—
146C8×C61—22no—
147#92418—22no—
148#9068—611nomaximal, Sylow
149C22×C61#38611yescharacteristic
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—21no—
153C4×C61—21no—
154C4×C61—42no—
155C4×C61—81no—
156C4×C61#44311yescharacteristic
157C4×C61#44311yescharacteristic
158C4×C61#44411yescharacteristic
159C4×C61#44411yescharacteristic
160Dic61—21no—
161Dic61—21no—
162Dic61—42no—
163Dic61—81no—
164Dic61#45611yescharacteristic
165Dic61#45611yescharacteristic
166#3160—611no—
167#2986—611no—
168#1526—611no—
169#1485—611no—
170#1485—611no—
171#1278—611no—
172#1278—611no—
173C2×C61#187611yescharacteristic
174C2×C61#187611yescharacteristic
175C2×C61#188611yescharacteristic
176D61—41no—
177#533—611no—
178#533—611no—
179#500—611no—
180#500#692311yescharacteristic
181#499—1222no—
182#498—611no—
183#498—611no—
184#446—1221no—
185#446—1221no—
186C8×Q8—1221no—
187C8×Q8—1221no—
188#433—1222no—
189#431—611no—
190#431—611no—
191#427—1222no—
192C4×M16—611no—
193C4×M16—611no—
194C2×C4×C8—611no—
195#330—1222no—
196Q8⋊C8—1221no—
197Q8⋊C8—1221no—
198Q8⋊C8—1221no—
199Q8⋊C8—1221no—
200C61#906811yescharacteristic, complemented, Sylow
201C2×M16—1221no—
202C2×M16—1221no—
203C22×C8—1221no—
204C4⋊Q8—611no—
205C4⋊Q8—611no—
206C4⋊Q8C2×#692311yescharacteristic
207C4⋊Q8C2×#692311yescharacteristic
208#125—612no—
209#125—1221no—
210C22⋊Q8—1221no—
211C22⋊Q8—1221no—
212C22⋊Q8—1221no—
213C22⋊Q8—1221no—
214C4×Q8—1221no—
215C4×Q8—1221no—
216C4×Q8—1221no—
217C4×Q8—1221no—
218#117—612no—
219#117—1221no—
220C2×C42—611no—
221#107—22no—
222#107—1222no—
223#106—1222no—
224#106—1222no—
225C4⋊C8—611no—
226C4⋊C8—611no—
227C4⋊C8—611no—
228C4⋊C8—611no—
229C4⋊C8—1222no—
230C4⋊C8—1222no—
231C22⋊C8—1221no—
232C22⋊C8—1221no—
233#97—1221no—
234#97—1221no—
235#97—1222no—
236C4×C8—611no—
237C4×C8—1221no—
238C4×C8—1221no—
239C4×C8—1221no—
240C4×C8—1221no—
241C4×C8—1221no—
242C4×C8—1222no—
243C4×C8C2×#692311yescharacteristic
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—1221no—
247C2×Q8—1221no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251M16—1222no—
252M16—1222no—
253M16—2441no—
254M16—2441no—
255C2×C8—21no—
256C2×C8—1222no—
257C2×C8—1222no—
258C2×C8—1221no—
259C2×C8—1221no—
260C2×C8—1221no—
261C2×C8—1221no—
262C2×C8—1221no—
263C2×C8—2441no—
264C2×C8—2441no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—612no—
269C4⋊C4—1222no—
270C4⋊C4—1222no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4—1221no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C4⋊C4C22⋊#692312yes—
277C22⋊C4—1222no—
278C22⋊C4—2441no—
279C42—611no—
280C42—611no—
281C42—611no—
282C42—1222no—
283C42—2441no—
284C42C22×#692311yescharacteristic
285C23—611no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—2441no—
291Q8—2441no—
292Q8—2441no—
293Q8—2441no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—611no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1222no—
306C2×C4—2441no—
307C2×C4C2×C22⋊#692311yescharacteristic
308C2×C4D4×#692311yescharacteristic
309C2×C4D4×#692311yescharacteristic
310C8—22no—
311C8—1222no—
312C8—2441no—
313C8—2441no—
314C8—2441no—
315C8—2441no—
316C8—2442no—
317C8—2442no—
318C22—1221no—
319C22—1221no—
320C22—1221no—
321C22#1684071011yescharacteristic
322C4—21no—
323C4—42no—
324C4—81no—
325C4—611no—
326C4—611no—
327C4—1221no—
328C4—1221no—
329C4—4881no—
330C4—2442no—
331C4#1684080411yescharacteristic
332C4#1684080611yescharacteristic
333C4#1684080811yescharacteristic
334C4#1684080911yescharacteristic
335C2—2441no—
336C2#2146180911yescharacteristic
337C2#2146181011yescharacteristic
338C2#2146182511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#500C4
socleC22×C61#386
prime radicalC22×D61#123
cosocleC42×D61C23
hypercenter#500#6923
hypocenterC61#9068
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
200. C61#9068semidirect

Maximal subgroups

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

subgroupindex
#214627302
#214612772
#214611432
#214611422
#214606512
#214563302
#906861