Finite Group Databasebeta

← #29577765

Subgroups — #29577765

Order 15616.

726 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 107 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#21449473C211yesmaximal, characteristic
2#21448405C211yesmaximal, characteristic, complemented
3#21444349C212yesmaximal
4#21443673C212yesmaximal, complemented
5C61×#1784C211yesmaximal, characteristic, complemented
6C2×#14519766C412yescomplemented
7C2×#14519751C2211yescharacteristic
8C2×#14519736—21no—
9#16839673—22no—
10#16839669—24no—
11#16839669C412yescomplemented
12#16839641—21no—
13#16839641—21no—
14#16839625—21no—
15#16839625C2212yes—
16C4×C22⋊Dic61C2211yescharacteristic
17#16839497—24no—
18C23×Q8×C61C2211yescharacteristic
19C2×#101×C61—22no—
20C61×#407C2212yes—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23#14519766—44no—
24#14519751—22no—
25#14519751—22no—
26#14519736—44no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×C22⋊Dic61—22no—
30C2×C22⋊Dic61—21no—
31C2×#11758198—22no—
32C2×#11758198—22no—
33C2×#11758198—21no—
34C2×C4×Dic61—22no—
35C2×C4×Dic61C2×C412yes—
36#14519684—24no—
37#14519684—22no—
38#14519684—22no—
39#14519684—21no—
40#14519684—22no—
41#14519684D414yes—
42C22×Q8×C61—22no—
43C22×Q8×C61—22no—
44C22×Q8×C61—44no—
45C22×Q8×C61C2×C412yes—
46C22×Q8×C61D412yes—
47C23×C4×C61D411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C23×C4×C61C2311yescharacteristic
50C2×M16×C61—22no—
51#101×C61—44no—
52C22⋊C8×C61—22no—
53C22⋊C8×C61—22no—
54C24×C61C2×D411yescharacteristic
55C23×D61—21no—
56C2×Q8×C61—24no—
57C2×Q8×C61—24no—
58C2×Q8×C61—24no—
59C2×Q8×C61—24no—
60C2×Q8×C61—42no—
61C2×Q8×C61—42no—
62C2×Q8×C61—44no—
63C2×Q8×C61—44no—
64C2×Q8×C61—48no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—24no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61C22⋊C412yes—
73C22×C4×C61C22⋊C412yes—
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D412yes—
77C22×C4×C61#4112yes—
78C2×C22⋊D61—44no—
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C22×Dic61—21no—
82C22×Dic61—22no—
83C22×Dic61—24no—
84C2×D244—22no—
85C2×D244—42no—
86M16×C61—44no—
87C2×C8×C61—42no—
88C22⋊Dic61—42no—
89C22⋊Dic61—44no—
90#11758198—44no—
91#11758198—44no—
92#11758198—42no—
93C4⋊Dic61—22no—
94C4⋊Dic61—44no—
95C4×Dic61—44no—
96C4×Dic61—42no—
97C23×C61—22no—
98C23×C61—22no—
99C23×C61—41no—
100C23×C61C2×C22⋊C411yescharacteristic
101C23×C61#12011yescharacteristic
102C23×C61#12311yescharacteristic
103C22×D61—42no—
104C22×D61—42no—
105C22×D61—41no—
106C22×D61—42no—
107Q8×C61—48no—
108Q8×C61—48no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61#9914yes—
124C2×C4×C61C4×D412yes—
125C2×C4×C61#12112yes—
126C22⋊D61—82no—
127C22⋊D61—82no—
128C2×Dic61—44no—
129C2×Dic61—42no—
130C2×Dic61—44no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—42no—
135C2×Dic61—44no—
136D244—82no—
137C8×C61—82no—
138#9466—611nomaximal, Sylow
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—44no—
147C22×C61#38611yescharacteristic
148C22×C61C2×#9912yes—
149C22×C61#45412yes—
150C2×D61—84no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C4×C61—22no—
155C4×C61—22no—
156C4×C61—24no—
157C4×C61—42no—
158Dic61—84no—
159Dic61—82no—
160Dic61—81no—
161#2436—611no—
162#2310—611no—
163#1784D6111yescharacteristic, complemented
164#1510—612no—
165#1284—612no—
166C2×C61—22no—
167C2×C61—22no—
168C2×C61—41no—
169C2×C61#188811yescharacteristic
170C2×C61#189811yescharacteristic
171C2×C61#191211yescharacteristic
172D61—161no—
173C23×Q8C2×D6111yescharacteristic
174C2×#124—612no—
175C2×#121—611no—
176C2×#117—1221no—
177C2×#101—22no—
178#407C2×D6112yes—
179#393—1222no—
180#391—612no—
181#391—1224no—
182#386—1221no—
183#386—1221no—
184#381—612no—
185#381—1221no—
186C4×C22⋊C4—611no—
187#327—1224no—
188C61#946611yescharacteristic, complemented, Sylow
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—44no—
192C22×Q8C4×D6112yes—
193C22×Q8C22⋊D6112yes—
194C22×D4—611no—
195C22×D4—1221no—
196C23×C4D24411yescharacteristic
197C23×C4D24411yescharacteristic
198C23×C4C22×D6111yescharacteristic
199C2×M16—22no—
200#124—2444no—
201#121—1222no—
202#121—1222no—
203#117—2444no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—1222no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4—1221no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1221no—
211C2×C42—612no—
212C2×C42—1222no—
213#101—44no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—614no—
217#95—1224no—
218#95—1222no—
219#95—1222no—
220#95—1222no—
221#95—1221no—
222C24—1221no—
223C24C2×D24411yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—48no—
233C2×D4—1222no—
234C2×D4—2442no—
235C2×D4—2444no—
236C2×D4—2441no—
237C2×D4—2441no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1221no—
246C22×C4—1222no—
247C22×C4—1224no—
248C22×C4#1175819812yes—
249C22×C4#1175819812yes—
250C22×C4C2×C4×D6111yescharacteristic
251C22×C4D4×D6112yes—
252C22×C4#1175822412yes—
253C22×C4C2×C22⋊D6111yescharacteristic
254M16—44no—
255C2×C8—42no—
256C4⋊C4—1222no—
257C4⋊C4—2444no—
258C22⋊C4—2444no—
259C22⋊C4—2444no—
260C22⋊C4—2442no—
261C22⋊C4—2444no—
262C22⋊C4—2442no—
263C42—2442no—
264C42—2444no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—2442no—
269C23—2442no—
270C23—2442no—
271C23—2441no—
272C23C2×#1175819811yescharacteristic
273C23#1451974711yescharacteristic
274C23#1451976111yescharacteristic
275Q8—48no—
276Q8—48no—
277D4—4882no—
278D4—4882no—
279D4—4882no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—2444no—
295C2×C4—2442no—
296C2×C4—2444no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2444no—
300C2×C4—2442no—
301C2×C4—2441no—
302C2×C4#1451969114yes—
303C2×C4#1451973712yes—
304C2×C4#1451974912yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—4884no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22#1683963911yescharacteristic
319C22C2×#1451969112yes—
320C22#1683993712yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—4884no—
326C4—4882no—
327C4—4881no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—9761no—
332C2#2144622011yescharacteristic
333C2#2144627011yescharacteristic
334C2#2144635411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1784C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1784D61
hypocenterC61#9466
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #21448405C2semidirect
4. #21443673C2semidirect
5. C61×#1784C2semidirect
6. C2×#14519766C4semidirect
11. #16839669C4semidirect
163. #1784D61semidirect
188. C61#9466semidirect

Maximal subgroups

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

subgroupindex
#214494732
#214484052
#214443492
#214436732
C61×#17842
#946661