Finite Group Databasebeta

← #29571067

Subgroups — #29571067

Order 15616.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454795C211yesmaximal, characteristic
2#21445506C211yesmaximal, characteristic, complemented
3#21444407C211yesmaximal, characteristic
4#21444109C211yesmaximal, characteristic, complemented
5#21443719C211yesmaximal, characteristic, complemented
6#21443712C211yesmaximal, characteristic, complemented
7C61×#1500C211yesmaximal, characteristic
8#16839893C411yescharacteristic, complemented
9Q8×#92418C411yescharacteristic, complemented
10#16839840C411yescharacteristic, complemented
11#16839834—21no—
12D4×#92418—21no—
13D4×#92418C411yescharacteristic, complemented
14#16839823—21no—
15#16839819C2211yescharacteristic
16#16839718—21no—
17C4×#11758194C2211yescharacteristic
18C2×#14519688—21no—
19C2×#14519688C2211yescharacteristic
20C2×C4×#92418—21no—
21C2×C4×#92418C2211yescharacteristic
22#16839504C412yescomplemented
23#16839502C412yescomplemented
24#16839484—21no—
25#16839483—21no—
26C61×#529C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C61×#328C412yes—
30C61×#327C412yes—
31C2×#11758194—21no—
32C2×#11758194—21no—
33C22×#92418—21no—
34C22×#92418—21no—
35C22×#92418D411yescharacteristic
36C22×#92418D411yescharacteristic
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418C2×C411yescharacteristic
40C4⋊#92418C2×C411yescharacteristic
41C22⋊#92418—21no—
42C22⋊#92418—21no—
43C22⋊#92418—21no—
44#14519688—24no—
45#14519688—21no—
46#14519688—21no—
47#14519688—21no—
48#14519688C2×C412yes—
49#14519688C2×C412yes—
50C4×#92418—24no—
51C4×#92418—21no—
52C4×#92418—21no—
53C4×#92418C2×C411yescharacteristic
54C4×#92418C2×C411yescharacteristic
55C4×#92418D411yescharacteristic
56C4×#92418D411yescharacteristic
57#126×C61—21no—
58#125×C61C2×C411yescharacteristic
59#124×C61C2×C411yescharacteristic
60#123×C61—21no—
61C22⋊Q8×C61C2×C411yescharacteristic
62#121×C61C2×C411yescharacteristic
63C4×Q8×C61C2×C411yescharacteristic
64C4×D4×C61—21no—
65C4×D4×C61C2×C411yescharacteristic
66#117×C61—21no—
67C2×C42×C61—21no—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C22⋊C8×C61C2×C412yes—
71C4×C8×C61—21no—
72#95×C61—21no—
73#95×C61C2×C412yes—
74C2×Q8×C61C4211yescharacteristic
75C2×D4×C61—21no—
76C2×D4×C61C4211yescharacteristic
77C22×C4×C61—22no—
78C22×C4×C61—21no—
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4111yescharacteristic
82C2×C8×C61—22no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61C4211yescharacteristic
88C4⋊C4×C61C4211yescharacteristic
89C22⋊C4×C61—21no—
90C22⋊C4×C61—21no—
91C22⋊C4×C61—21no—
92C22⋊C4×C61—21no—
93C22⋊C4×C61—21no—
94C42×C61—22no—
95C42×C61—22no—
96C42×C61—21no—
97C42×C61C22×C411yescharacteristic
98C42×C61C22×C411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61#4111yescharacteristic
101#11758194—42no—
102#11758194—42no—
103C2×#92418—22no—
104C2×#92418—22no—
105C2×#92418—22no—
106C2×#92418—22no—
107C2×#92418—22no—
108C2×#92418—21no—
109C2×#92418—21no—
110C2×#92418—22no—
111C2×#92418—42no—
112C2×#92418C22⋊C411yescharacteristic
113C2×#92418C22⋊C411yescharacteristic
114C2×#92418C22⋊C411yescharacteristic
115C2×#92418C22⋊C411yescharacteristic
116C23×C61—21no—
117C23×C61C4×D411yescharacteristic
118Q8×C61—41no—
119D4×C61—41no—
120D4×C61—42no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—24no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—41no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61C2×C4211yescharacteristic
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61#11711yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C8×C61—42no—
140#92418—24no—
141#92418—21no—
142#92418—21no—
143#92418—42no—
144#92418—41no—
145#92418—41no—
146#92418#10412yes—
147#92418#10412yes—
148#8249—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61C4×C22⋊C411yescharacteristic
155C22×C61#44412yes—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—44no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61—41no—
165C4×C61C2×#10411yescharacteristic
166C4×C61C2×#10411yescharacteristic
167#2966—611no—
168#1726—611no—
169#1524—611no—
170#1500D6111yescharacteristic
171#1451—611no—
172#1302—611no—
173#1300—611no—
174C2×C61—22no—
175C2×C61—81no—
176C2×C61#175711yescharacteristic
177C2×C61C4×#10411yescharacteristic
178C2×C61#176511yescharacteristic
179#529C2×D6111yescharacteristic
180#446—611no—
181C8×Q8—611no—
182#436—611no—
183#435—1221no—
184C8×D4—611no—
185C8×D4—1221no—
186#433—1221no—
187#432—611no—
188#431C2×D6111yescharacteristic
189#405—1221no—
190C4×M16—611no—
191C2×#97—611no—
192C2×#97—1221no—
193C2×C4×C8—611no—
194C2×C4×C8—1221no—
195#376C2×D6111yescharacteristic
196#330—612no—
197#329—612no—
198#328Dic6112yes—
199#327Dic6112yes—
200#322—1221no—
201#322—1221no—
202C61#824911yescharacteristic, complemented, Sylow
203C2×M16—1221no—
204C2×M16—1221no—
205C22×C8—611no—
206C22×C8—611no—
207C22×C8—1221no—
208C22×C8—1221no—
209#126—21no—
210#125C4×D6111yescharacteristic
211#124C4×D6111yescharacteristic
212#123—21no—
213C22⋊Q8C2×Dic6111yescharacteristic
214#121C2×Dic6111yescharacteristic
215C4×Q8C4×D6111yescharacteristic
216C4×D4—21no—
217C4×D4C4×D6111yescharacteristic
218#117—21no—
219C2×C42—21no—
220C2×C42C22×D6111yescharacteristic
221C4⋊C8—21no—
222C4⋊C8—611no—
223C4⋊C8—611no—
224C4⋊C8—1221no—
225C4⋊C8—1221no—
226C22⋊C8—1221no—
227C22⋊C8—1221no—
228C22⋊C8—1221no—
229C22⋊C8C2×Dic6112yes—
230#97—612no—
231#97—612no—
232#97—1221no—
233#97—1224no—
234#97—1221no—
235#97—1221no—
236C4×C8—21no—
237C4×C8—611no—
238C4×C8—611no—
239C4×C8—611no—
240C4×C8—611no—
241C4×C8—1221no—
242C4×C8—1224no—
243C4×C8—1221no—
244#95—21no—
245#95C2×Dic6112yes—
246C2×Q8C4×Dic6111yescharacteristic
247C2×D4—21no—
248C2×D4C4×Dic6111yescharacteristic
249C22×C4—22no—
250C22×C4—21no—
251C22×C4D4×D6111yescharacteristic
252C22×C4#1175822411yescharacteristic
253C22×C4C22×Dic6111yescharacteristic
254M16—2442no—
255M16—2442no—
256C2×C8—22no—
257C2×C8—611no—
258C2×C8—611no—
259C2×C8—611no—
260C2×C8—611no—
261C2×C8—1222no—
262C2×C8—1222no—
263C2×C8—1222no—
264C2×C8—1222no—
265C2×C8—1222no—
266C2×C8—1221no—
267C2×C8—1221no—
268C2×C8—1222no—
269C2×C8—2442no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4C4×Dic6111yescharacteristic
275C4⋊C4C4×Dic6111yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C42—22no—
282C42—22no—
283C42—21no—
284C42C2×C4×D6111yescharacteristic
285C42C2×C4×D6111yescharacteristic
286C42D4×D6111yescharacteristic
287C42#1175822411yescharacteristic
288C23—21no—
289C23D4×Dic6111yescharacteristic
290Q8—41no—
291D4—41no—
292D4—42no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4C2×C4×Dic6111yescharacteristic
306C2×C4C22⋊C4×D6111yescharacteristic
307C2×C4#1451973311yescharacteristic
308C2×C4#1451973711yescharacteristic
309C2×C4#1451973711yescharacteristic
310C2×C4D4×Dic6111yescharacteristic
311C8—42no—
312C8—612no—
313C8—612no—
314C8—1224no—
315C8—1221no—
316C8—1221no—
317C8—2442no—
318C8—2441no—
319C8—2441no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22C22⋊C4×Dic6111yescharacteristic
326C22#1683988412yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—44no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4#104×D6111yescharacteristic
337C4#104×D6111yescharacteristic
338C2—22no—
339C2—81no—
340C2#2144562111yescharacteristic
341C2#104×Dic6111yescharacteristic
342C2#2144566611yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C2×C42
FrattiniC2×C42C22×D61
FittingC61×#1500C2
socleC22×C61C4×C22⋊C4
prime radical#121×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#1500D61
hypocenterC61#8249
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1500

Complements

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

NH ≅ G/Ntype
2. #21445506C2semidirect
4. #21444109C2semidirect
5. #21443719C2semidirect
6. #21443712C2semidirect
8. #16839893C4semidirect
9. Q8×#92418C4semidirect
10. #16839840C4semidirect
13. D4×#92418C4semidirect
22. #16839504C4semidirect
23. #16839502C4semidirect
202. C61#8249semidirect

Maximal subgroups

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

subgroupindex
#214547952
#214455062
#214444072
#214441092
#214437192
#214437122
C61×#15002
#824961