Finite Group Databasebeta

← #30601003

Subgroups — #30601003

Order 15616.

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

normal subgroups only

This group has 532 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#21462383C211yesmaximal, characteristic
2#21461283C212yesmaximal, complemented
3#21461071C211yesmaximal, characteristic
4#21460707C212yesmaximal, complemented
5#21451510C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840783C2212yes—
8#16840726—24no—
9C4×#11758217—21no—
10C4×#11758217C2212yes—
11#16840619—24no—
12#16840608—21no—
13#16840605—21no—
14C2×#14519768C2211yescharacteristic
15C2×#14519733C2211yescharacteristic
16#16839700C411yescharacteristic
17#16839694C411yescharacteristic
18#16839687C411yescharacteristic
19#16839658—21no—
20#16839650—21no—
21#16839648—21no—
22#16839637—21no—
23#16839607C2211yescharacteristic
24C61×#399C411yescharacteristic
25C2×#11758217—22no—
26C2×#11758217—24no—
27C2×#11758217—24no—
28C2×#11758217—22no—
29C22×#92420—22no—
30C4⋊#92420—22no—
31C22⋊#92420—22no—
32#14519832—22no—
33#14519832—22no—
34#14519832D414yes—
35C4×#92420—22no—
36C4×#92420—22no—
37C23×Dic61D411yescharacteristic
38#14519768—22no—
39#14519768—22no—
40#14519733—22no—
41#14519733—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196C2×C411yescharacteristic
47C2×#11758196D411yescharacteristic
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61D411yescharacteristic
50C2×C4×Dic61—21no—
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61C2311yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684C2×C411yescharacteristic
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61C2×C411yescharacteristic
63#95×C61C2×C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79#11758217—28no—
80#11758217—24no—
81#11758217—24no—
82C2×#92420—28no—
83C2×#92420—24no—
84C2×#92420—42no—
85C4⋊C4×C61—22no—
86C4⋊C4×C61—42no—
87C22⋊C4×C61—22no—
88C22⋊C4×C61—42no—
89C22⋊Dic61—22no—
90C22⋊Dic61—42no—
91C4⋊Dic61—22no—
92C4⋊Dic61—42no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95#11758196—22no—
96#11758196—22no—
97#11758196—42no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C4×Dic61—42no—
101C4×Dic61C2×D412yes—
102C4×Dic61#4112yes—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61#10412yes—
108C23×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61—22no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61C4×D412yes—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—24no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61#10412yes—
136C2×Dic61#12011yescharacteristic
137C2×Dic61#12112yes—
138C2×Dic61#12311yescharacteristic
139#92420—28no—
140#92420—82no—
141#9388—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—42no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—81no—
149C22×C61#38611yescharacteristic
150C22×C61C2×#10411yescharacteristic
151C22×C61C2×#10411yescharacteristic
152C4×C61—22no—
153C4×C61—42no—
154C4×C61—41no—
155C4×C61—81no—
156Dic61—22no—
157Dic61—21no—
158Dic61—22no—
159Dic61—21no—
160Dic61—41no—
161Dic61—81no—
162Dic61—81no—
163#2873—611no—
164#2611—611no—
165#1528—612no—
166#1464—611no—
167#1300—612no—
168C2×C61—21no—
169C2×C61—21no—
170C2×C61—42no—
171C2×C61#188411yescharacteristic
172C2×C61#189611yescharacteristic
173C2×C61#191511yescharacteristic
174C22×M16—1221no—
175C2×#125—611no—
176C2×#117—611no—
177#432—612no—
178#405—1224no—
179C4×M16—612no—
180C4×M16—1221no—
181#400—611no—
182#399#692311yescharacteristic
183#398—611no—
184#397—611no—
185#389—1221no—
186#388—1221no—
187#387—1221no—
188#385—1221no—
189#376—611no—
190#330—1224no—
191#324—1221no—
192#323—1221no—
193C61#938811yescharacteristic, complemented, Sylow
194C23×C4—611no—
195C2×M16—1222no—
196C2×M16—1224no—
197C2×M16—1224no—
198C2×M16—1222no—
199C22×C8—1222no—
200#125—1222no—
201#125—1222no—
202#117—1222no—
203#117—1222no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4C2×#692311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—611no—
214C2×C22⋊C4—1221no—
215C2×C22⋊C4C2×#692311yescharacteristic
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—1221no—
219C4⋊C8—1222no—
220C22⋊C8—1222no—
221#97—614no—
222#97—1222no—
223#97—1222no—
224C4×C8—1222no—
225C4×C8—1222no—
226#95—611no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95C2×#692311yescharacteristic
233C24C22⋊#692311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—611no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4C22⋊#692311yescharacteristic
247C22×C4C22×#692311yescharacteristic
248M16—1228no—
249M16—1224no—
250M16—1224no—
251C2×C8—1224no—
252C2×C8—1228no—
253C2×C8—2442no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—1222no—
257C4⋊C4—1222no—
258C4⋊C4—1222no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2441no—
262C4⋊C4—2441no—
263C22⋊C4—22no—
264C22⋊C4—42no—
265C22⋊C4—1222no—
266C22⋊C4—2442no—
267C42—612no—
268C42—612no—
269C42—1221no—
270C42—1221no—
271C42—2442no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1451984212yes—
277C23C2×C22⋊#692311yescharacteristic
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—611no—
288C2×C4—612no—
289C2×C4—611no—
290C2×C4—612no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1224no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4D4×#692312yes—
308C8—1228no—
309C8—4882no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1684071011yescharacteristic
318C22D61⋊#10411yescharacteristic
319C22C2×#1451984211yescharacteristic
320C4—22no—
321C4—42no—
322C4—41no—
323C4—81no—
324C4—1222no—
325C4—1221no—
326C4—1222no—
327C4—1221no—
328C4—4881no—
329C4—4881no—
330C4—2441no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2146182211yescharacteristic
335C2#2146183611yescharacteristic
336C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#399C4
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#399#6923
hypocenterC61#9388
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #21461283C2semidirect
4. #21460707C2semidirect
193. C61#9388semidirect

Maximal subgroups

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

subgroupindex
#214623832
#214612832
#214610712
#214607072
#214515102
#938861