Finite Group Databasebeta

← #30604841

Subgroups — #30604841

Order 15616.

432 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461525C211yesmaximal, characteristic
2#21461051C211yesmaximal, characteristic
3#21460818C212yesmaximal, complemented
4#21460817C212yesmaximal, complemented
5#21451645C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840779—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608C2212yes—
11#16840605—21no—
12C2×#14519776C411yescharacteristic
13C2×#14519733C2211yescharacteristic
14#16839702C2211yescharacteristic
15#16839700C2211yescharacteristic
16#16839698C411yescharacteristic
17#16839682—21no—
18#16839674—21no—
19#16839657—21no—
20#16839626C411yescharacteristic
21#16839620—21no—
22C61×#399C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—22no—
25C2×#11758217—41no—
26C2×#11758217—42no—
27C22×#92420—41no—
28#14519844—41no—
29C22⋊#92420—22no—
30C4×#92420—41no—
31C22×Dic122—21no—
32C23×Dic61D411yescharacteristic
33#14519776—41no—
34#14519776—41no—
35#14519733—41no—
36#14519733—41no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×#11758196D411yescharacteristic
42C2×#11758196C2311yescharacteristic
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61D411yescharacteristic
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684C2×C411yescharacteristic
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61—21no—
56C2×C22⋊C4×C61C2×C411yescharacteristic
57#95×C61C2×C411yescharacteristic
58C24×C61C22⋊C411yescharacteristic
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61—21no—
64C22×Dic61—21no—
65C22×Dic61—22no—
66C22×Dic61—21no—
67C22×Dic61—21no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61SD1612yes—
71C22×Dic61Q1612yes—
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×Dic122—42no—
75C2×Dic122—81no—
76C2×Dic122—81no—
77#11758217—24no—
78#11758217—44no—
79#11758217—42no—
80C2×#92420—42no—
81C2×#92420—44no—
82C2×#92420—42no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—81no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—81no—
87C22⋊Dic61—41no—
88C22⋊Dic61—81no—
89C4⋊Dic61—41no—
90#11758196—41no—
91#11758196—41no—
92#11758196—41no—
93#11758196—41no—
94#11758196—42no—
95#11758196—81no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61#10312yes—
105C23×C61C2×C22⋊C411yescharacteristic
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—22no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61#9911yescharacteristic
133C2×Dic61#9911yescharacteristic
134C2×Dic61#10312yes—
135C2×Dic61#10412yes—
136C2×Dic61C2×SD1611yescharacteristic
137C2×Dic61C2×Q1611yescharacteristic
138Dic122—84no—
139#92420—44no—
140#92420—82no—
141#11723—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—22no—
145C22×C61—42no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—81no—
149C22×C61C2×#9911yescharacteristic
150C22×C61C2×#10311yescharacteristic
151C22×C61C2×#10411yescharacteristic
152C4×C61—41no—
153C4×C61—41no—
154C4×C61—81no—
155C4×C61—81no—
156Dic61—21no—
157Dic61—21no—
158Dic61—41no—
159Dic61—41no—
160Dic61—81no—
161Dic61—81no—
162Dic61#32812yes—
163Dic61#32812yes—
164#2628—611no—
165#1742—611no—
166#1458—611no—
167#1353—612no—
168#1353—612no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—42no—
172C2×C61C2×#32811yescharacteristic
173C2×C61#212311yescharacteristic
174C2×C61#213411yescharacteristic
175C22×M16—1221no—
176C2×C4⋊Q8—611no—
177C2×#117—611no—
178#431—1221no—
179#400—611no—
180#400—611no—
181#399—611no—
182#399#692311yescharacteristic
183#395—1221no—
184#393—1221no—
185#389—1221no—
186#382—611no—
187#380—1221no—
188#343—612no—
189#343—1221no—
190#324—612no—
191#323—1221no—
192C61#1172311yescharacteristic, complemented, Sylow
193C22×Q8—1221no—
194C23×C4—611no—
195C2×M16—1222no—
196C2×M16—1224no—
197C2×M16—2442no—
198C2×M16—2441no—
199C22×C8—2441no—
200C4⋊Q8—2441no—
201C4⋊Q8—2441no—
202#117—2441no—
203#117—2441no—
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—
218C4⋊C8—2441no—
219C22⋊C8—1222no—
220C4×C8—2441no—
221#95—611no—
222#95—1221no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226#95—1221no—
227#95C2×#692311yescharacteristic
228C24C22⋊#692311yescharacteristic
229C2×Q8—4881no—
230C2×Q8—4881no—
231C2×Q8—2442no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—611no—
235C22×C4—612no—
236C22×C4—612no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1222no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4C22⋊#692311yescharacteristic
246C22×C4C22×#692311yescharacteristic
247M16—1224no—
248M16—2442no—
249M16—2444no—
250C2×C8—2442no—
251C2×C8—2444no—
252C2×C8—2442no—
253C4⋊C4—41no—
254C4⋊C4—81no—
255C4⋊C4—4881no—
256C4⋊C4—2441no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C4⋊C4—2442no—
260C4⋊C4—2441no—
261C4⋊C4—2441no—
262C22⋊C4—41no—
263C22⋊C4—81no—
264C22⋊C4—4881no—
265C22⋊C4—2441no—
266C42—1221no—
267C42—1221no—
268C42—1221no—
269C42—1221no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23#1451984012yes—
275C23C2×C22⋊#692311yescharacteristic
276Q8—4884no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
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—611no—
289C2×C4—612no—
290C2×C4—612no—
291C2×C4—611no—
292C2×C4—611no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C8—4882no—
310C8—2444no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22D61⋊#9911yescharacteristic
319C22C2×#1451984011yescharacteristic
320C22D61⋊#10411yescharacteristic
321C4—41no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—612no—
326C4—612no—
327C4—1221no—
328C4—1221no—
329C4—4881no—
330C4—4881no—
331C4—2441no—
332C4—2441no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2D61⋊#32811yescharacteristic
337C2#2146208211yescharacteristic
338C2#2146211911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#399C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×#11758196C23
hypercenter#399#6923
hypocenterC61#11723
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #399

Complements

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

NH ≅ G/Ntype
3. #21460818C2semidirect
4. #21460817C2semidirect
192. C61#11723semidirect

Maximal subgroups

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

subgroupindex
#214615252
#214610512
#214608182
#214608172
#214516452
#1172361