Finite Group Databasebeta

← #29590621

Subgroups — #29590621

Order 15616.

590 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 87 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#21449184C211yesmaximal, characteristic
2#21444160C211yesmaximal, characteristic
3#21443839C214yesmaximal, complemented
4C4×#99×C61C211yesmaximal, characteristic
5C23×Dic122C411yescharacteristic, complemented
6C2×#14519756—21no—
7#16839819—21no—
8#16839682C412yescomplemented
9#16839674C2212yes—
10#16839668—21no—
11#16839642—21no—
12#16839634—21no—
13#16839622C411yescharacteristic, complemented
14#16839486—21no—
15#16839486C2212yes—
16C2×C4×D4×C61C2211yescharacteristic
17C2×#99×C61—21no—
18C4×C22⋊C4×C61—21no—
19C61×#342C2212yes—
20C22×Dic122—21no—
21C22×Dic122—44no—
22C22×Dic122—41no—
23C22×Dic122—41no—
24C22×Dic122C2×C412yes—
25C23×Dic61—21no—
26#14519756—44no—
27C2×C4⋊Dic61—21no—
28C2×#11758196—22no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—22no—
31C4⋊#92418—41no—
32C22⋊#92418—22no—
33#14519688—41no—
34#14519684—22no—
35#14519684—22no—
36#14519684C2×C412yes—
37C22×D4×C61D411yescharacteristic
38C23×C4×C61C2×C411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C4×D4×C61—41no—
41C4×D4×C61—41no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61D411yescharacteristic
45C2×C22⋊C4×C61C2311yescharacteristic
46C2×C42×C61—41no—
47C2×C42×C61D411yescharacteristic
48#99×C61—24no—
49#95×C61—41no—
50C24×C61C22⋊C411yescharacteristic
51C24×C61C2×D411yescharacteristic
52C2×D4×C61—22no—
53C2×D4×C61—22no—
54C2×D4×C61—41no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61—22no—
69C22×Dic61—44no—
70C22×Dic61—42no—
71C2×Dic122—24no—
72C2×Dic122—42no—
73C2×Dic122—48no—
74C2×Dic122—44no—
75C2×Dic122—41no—
76C2×Dic122—41no—
77C2×Dic122—42no—
78C2×Dic122—44no—
79C2×Dic122—44no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—24no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C22⋊C4×C61—42no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—82no—
88C22⋊Dic61—44no—
89C22⋊Dic61—42no—
90C4⋊Dic61—42no—
91#11758196—44no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61#10412yes—
102C23×C61C2×C22⋊C411yescharacteristic
103C23×C61#13712yes—
104D4×C61—44no—
105C2×C4×C61—22no—
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—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—44no—
119C2×C4×C61—81no—
120C2×C4×C61—82no—
121C2×C4×C61—82no—
122C2×C4×C61#9911yescharacteristic
123C2×C4×C61#9911yescharacteristic
124C2×Dic61—24no—
125C2×Dic61—42no—
126C2×Dic61—48no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131Dic122—44no—
132Dic122—44no—
133Dic122—44no—
134Dic122—44no—
135#92418—82no—
136#11785—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—42no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—81no—
149C22×C61C2×#9911yescharacteristic
150C22×C61#41911yescharacteristic
151C22×C61C2×#10411yescharacteristic
152C4×C61—21no—
153C4×C61—41no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61—84no—
158C4×C61AΣL(1,𝔽16)12yes—
159Dic61—44no—
160Dic61—82no—
161#2409—611no—
162C4×#99D6111yescharacteristic
163#1466—611no—
164#1351—614no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61—42no—
168C2×C61—42no—
169C2×C61#150811yescharacteristic
170C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
171C2×C61#212911yescharacteristic
172C23×Q8—611no—
173C2×C22⋊Q8—1221no—
174C2×C4×D4C2×D6111yescharacteristic
175#432—1221no—
176C2×#99—21no—
177#395—612no—
178#393—612no—
179#391—1221no—
180#386—1221no—
181#385—1221no—
182#380—611no—
183C4×C22⋊C4—21no—
184#342C2×D6112yes—
185#323—612no—
186#323—1221no—
187C61#1178511yescharacteristic, complemented, Sylow
188C22×Q8—612no—
189C22×Q8—1221no—
190C22×Q8—2441no—
191C22×Q8—2441no—
192C22×Q8—2444no—
193C22×D4C22⋊D6111yescharacteristic
194C23×C4—1221no—
195C23×C4C4×D6111yescharacteristic
196C23×C4C22⋊D6111yescharacteristic
197C22⋊Q8—2444no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4C4×D6111yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1221no—
205C2×C22⋊C4—1222no—
206C2×C22⋊C4D24411yescharacteristic
207C2×C22⋊C4C22×D6111yescharacteristic
208C2×C42—41no—
209C2×C42D24411yescharacteristic
210C4⋊C8—2441no—
211#99—24no—
212C22⋊C8—1222no—
213#97—2441no—
214#95—41no—
215#95—612no—
216#95—1222no—
217#95—1222no—
218C24#1175819811yescharacteristic
219C24C2×C22⋊D6111yescharacteristic
220C2×Q8—1224no—
221C2×Q8—2442no—
222C2×Q8—2442no—
223C2×Q8—2444no—
224C2×Q8—2444no—
225C2×Q8—2448no—
226C2×Q8—2444no—
227C2×Q8—2441no—
228C2×Q8—2441no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—1222no—
240C22×C4—1222no—
241C22×C4—2444no—
242C22×C4—2442no—
243C22×C4#1175819811yescharacteristic
244C22×C4#1175819811yescharacteristic
245C22×C4#1175819811yescharacteristic
246C22×C4C2×C4×D6111yescharacteristic
247C22×C4C2×D24411yescharacteristic
248C2×C8—2442no—
249C4⋊C4—41no—
250C4⋊C4—2442no—
251C4⋊C4—2444no—
252C22⋊C4—24no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—2444no—
257C22⋊C4—2442no—
258C42—21no—
259C42—21no—
260C42—82no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1451970512yes—
270C23C2×#1175819811yescharacteristic
271C23#1451980712yes—
272Q8—2444no—
273Q8—2444no—
274Q8—2444no—
275Q8—2444no—
276D4—44no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—1224no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2448no—
300C2×C4—2442no—
301C2×C4#1451969111yescharacteristic
302C2×C4#1451969111yescharacteristic
303C8—4882no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#1451969111yescharacteristic
317C22#1683977011yescharacteristic
318C22C2×#1451970511yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—84no—
325C4—4882no—
326C4—2444no—
327C4#1683954812yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#2144433811yescharacteristic
333C2C2×#1683954811yescharacteristic
334C2#2144749311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC4×#99×C61C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C22⋊C4×C61C23
hypercenterC4×#99D61
hypocenterC61#11785
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21443839C2semidirect
5. C23×Dic122C4semidirect
8. #16839682C4semidirect
13. #16839622C4semidirect
187. C61#11785semidirect

Maximal subgroups

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

subgroupindex
#214491842
#214441602
#214438392
C4×#99×C612
#1178561