Finite Group Databasebeta

← #29590403

Subgroups — #29590403

Order 15616.

400 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 83 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#21451679C211yesmaximal, characteristic
2C4×#14519696C211yesmaximal, characteristic, complemented
3#21443850C212yesmaximal, complemented
4#21443823C212yesmaximal
5C61×#1461C211yesmaximal, characteristic, complemented
6C2×#14519765C411yescharacteristic
7C2×#14519696—21no—
8C4×#11758194—21no—
9#16839697—21no—
10#16839682C411yescharacteristic
11#16839682C2211yescharacteristic
12#16839679C411yescharacteristic
13#16839679C2211yescharacteristic
14#16839673—21no—
15#16839653—21no—
16#16839653—21no—
17#16839630C411yescharacteristic
18#16839531C2212yes—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#431—21no—
21C61×#324—21no—
22C61×#324C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C22×D244C2×C411yescharacteristic
25C2×#11758194—22no—
26#14519765—41no—
27#14519765—41no—
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×#11758198—21no—
31C2×#11758198—21no—
32C2×#11758198—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36#14519696—24no—
37#14519688—41no—
38C4×#92418—41no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684C2×C411yescharacteristic
44#14519684C2×C411yescharacteristic
45C22×Q8×C61D411yescharacteristic
46C4×Q8×C61—41no—
47C4×Q8×C61—41no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61C2311yescharacteristic
51C2×C42×C61C2×C411yescharacteristic
52C2×C42×C61D411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C4⋊C8×C61—41no—
55C22⋊C8×C61—22no—
56C4×C8×C61—41no—
57C23×D61—21no—
58C2×Q8×C61—22no—
59C2×Q8×C61—22no—
60C2×Q8×C61—41no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C2×D244—41no—
72C2×D244—81no—
73C2×Dic122—41no—
74C2×Dic122—81no—
75C2×C8×C61—42no—
76C4⋊C4×C61—41no—
77C4⋊C4×C61—41no—
78C4⋊C4×C61—41no—
79C42×C61—21no—
80C42×C61—21no—
81C42×C61—21no—
82C42×C61—21no—
83C42×C61—21no—
84C42×C61—21no—
85#11758198—42no—
86#11758198—41no—
87#11758198—41no—
88#11758198—42no—
89#11758198—42no—
90#11758198—81no—
91#11758198—81no—
92#11758196—81no—
93#11758196—81no—
94#11758194—24no—
95#11758194—42no—
96C2×#92418—42no—
97C23×C61C2×C22⋊C411yescharacteristic
98C22×D61—41no—
99C22×D61—41no—
100C22×D61—41no—
101C22×D61—41no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—41no—
105Q8×C61—44no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61#9911yescharacteristic
121C2×C4×C61#9911yescharacteristic
122C2×C4×C61#10412yes—
123C2×C4×C61#13512yes—
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—41no—
133D244—82no—
134Dic122—82no—
135C8×C61—82no—
136#92418—44no—
137#11738—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61C2×#9911yescharacteristic
141C22×C61#41611yescharacteristic
142C22×C61C2×#10411yescharacteristic
143C2×D61—81no—
144C2×D61—81no—
145C2×D61—81no—
146C2×D61—81no—
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—81no—
150C4×C61—21no—
151C4×C61—21no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61—41no—
158Dic61—81no—
159Dic61—81no—
160Dic61—81no—
161#2632—611no—
162C4×#101—611no—
163#1461D6111yescharacteristic, complemented
164#1354—612no—
165#1345—612no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61#150111yescharacteristic
169C2×C61#212211yescharacteristic
170C2×C61#213311yescharacteristic
171D61—161no—
172C2×#124—611no—
173C2×C4×Q8C2×D6111yescharacteristic
174#431—21no—
175C2×#101—1221no—
176C4×M16—1221no—
177#399—1221no—
178#395—611no—
179#395—611no—
180#394—611no—
181#394—611no—
182#393—1221no—
183#388—1221no—
184#388—1221no—
185#383—611no—
186#343—612no—
187#324—21no—
188#324C2×D6112yes—
189C61#1173811yescharacteristic, complemented, Sylow
190C22×Q8—611no—
191C22×Q8D24411yescharacteristic
192C22×D4—611no—
193C2×M16—1222no—
194#124—2441no—
195#124—2441no—
196C4×Q8—41no—
197C4×Q8—41no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4C4×D6111yescharacteristic
201C2×C4⋊C4C22⋊D6111yescharacteristic
202C2×C4⋊C4C22×D6111yescharacteristic
203C2×C22⋊C4—1221no—
204C2×C22⋊C4—1221no—
205C2×C22⋊C4—1221no—
206C2×C22⋊C4—1221no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1221no—
209C2×C42C4×D6111yescharacteristic
210C2×C42D24411yescharacteristic
211C2×C42C22⋊D6111yescharacteristic
212C4⋊C8—41no—
213#101—1224no—
214C22⋊C8—22no—
215#97—2441no—
216C4×C8—41no—
217C4×C8—2441no—
218#95—611no—
219#95—611no—
220#95—1221no—
221#95—1221no—
222#95—1221no—
223#95—1221no—
224C24—1221no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—4881no—
229C2×Q8—2441no—
230C2×D4—4881no—
231C2×D4—2441no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4#1175819811yescharacteristic
236C22×C4#1175819811yescharacteristic
237C22×C4#1175819811yescharacteristic
238C22×C4#1175819811yescharacteristic
239C22×C4C2×C4×D6111yescharacteristic
240C22×C4C2×D24411yescharacteristic
241C22×C4C2×C22⋊D6111yescharacteristic
242M16—1224no—
243M16—2442no—
244C2×C8—42no—
245C2×C8—2442no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—4881no—
250C4⋊C4—4881no—
251C22⋊C4—4881no—
252C22⋊C4—4881no—
253C22⋊C4—2442no—
254C22⋊C4—2441no—
255C22⋊C4—2441no—
256C22⋊C4—2442no—
257C22⋊C4—2442no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—2441no—
265C23—2441no—
266C23—2441no—
267C23—2441no—
268C23—2441no—
269C23—2441no—
270C23—2441no—
271C23C2×#1175819811yescharacteristic
272Q8—44no—
273Q8—4882no—
274D4—4882no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—2441no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4#1451969111yescharacteristic
299C2×C4#1451969111yescharacteristic
300C2×C4#1451970312yes—
301C2×C4#1451979912yes—
302C8—82no—
303C8—2444no—
304C22—21no—
305C22—21no—
306C22—4881no—
307C22—4881no—
308C22—4881no—
309C22—4881no—
310C22—4881no—
311C22—4881no—
312C22—4881no—
313C22C2×#1451969111yescharacteristic
314C22#1683975911yescharacteristic
315C22C2×#1451970311yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4881no—
325C4—4881no—
326C4—4881no—
327C2—21no—
328C2—21no—
329C2—9761no—
330C2#2144431111yescharacteristic
331C2#2144746811yescharacteristic
332C2#2144750611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1461C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1461D61
hypocenterC61#11738
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. C4×#14519696C2semidirect
3. #21443850C2semidirect
5. C61×#1461C2semidirect
163. #1461D61semidirect
189. C61#11738semidirect

Maximal subgroups

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

subgroupindex
#214516792
C4×#145196962
#214438502
#214438232
C61×#14612
#1173861