Finite Group Databasebeta

← #29833117

Subgroups — #29833117

Order 15616.

600 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#405×D61C211yesmaximal, characteristic
2#21447627C212yesmaximal, complemented
3#21447624C212yesmaximal, complemented
4#21447620C212yesmaximal, complemented
5#21447616C212yesmaximal, complemented
6#21447579C211yesmaximal, characteristic, complemented
7#21447577C211yesmaximal, characteristic, complemented
8#21447429C211yesmaximal, characteristic
9C2×#16839541C211yesmaximal, characteristic
10#21447416C211yesmaximal, characteristic, complemented
11C61×#2108C211yesmaximal, characteristic, complemented
12C2×M16×D61C411yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#117×D61C411yescharacteristic
15C2×#14519717C2211yescharacteristic
16C22×#11758185C2211yescharacteristic
17#16839720C411yescharacteristic
18#16839719C412yes—
19#16839718C411yescharacteristic
20#16839717C2211yescharacteristic
21#16839716C2212yes—
22#97×D61C412yes—
23C4×C8×D61C2212yes—
24#16839580C2212yescomplemented
25#16839580C2212yescomplemented
26#16839580C2212yescomplemented
27#16839580C2212yescomplemented
28#16839541C2211yescharacteristic
29#16839541C2211yescharacteristic
30#16839541C2211yescharacteristic
31#16839541C2211yescharacteristic
32#16839541C2212yes—
33#16839541C2212yes—
34#16839538C2212yescomplemented
35C4×#11758185C2212yescomplemented
36C2×M32×C61C2211yescharacteristic
37C22×C16×C61C2211yescharacteristic
38C61×#405C2211yescharacteristic
39C61×#346C2212yescomplemented
40C4×C16×C61C2212yescomplemented
41C22×C4×D61C2×C411yescharacteristic
42C2×#11758194C2×C411yescharacteristic
43C2×#11758189C2×C412yes—
44M16×D61—24no—
45C22×#92418C2×C411yescharacteristic
46C2×C8×D61—22no—
47C2×C8×D61—22no—
48C2×C8×D61C2×C412yes—
49C2×C8×D61C2×C411yescharacteristic
50C2×C8×D61C2×C411yescharacteristic
51C2×C8×D61C2311yescharacteristic
52C2×C8×D61C2311yescharacteristic
53#14519736C2×C411yescharacteristic
54#14519735C811yescharacteristic
55#14519735C811yescharacteristic
56#14519734C2×C412yes—
57#14519733C812yes—
58C4⋊C4×D61C811yescharacteristic
59C4⋊C4×D61C811yescharacteristic
60C22⋊C4×D61C812yes—
61C42×D61C2×C412yes—
62#14519717—22no—
63C2×#11758185—22no—
64C2×#11758185C2311yescharacteristic
65C2×#11758185C2312yes—
66C2×#11758185C2311yescharacteristic
67#14519688C2×C412yes—
68#14519687C2×C412yes—
69#14519687C2×C412yes—
70C4×#92418C2×C412yes—
71C8×Dic61C2×C412yes—
72C8×Dic61C2312yes—
73C2×M16×C61C2×C411yescharacteristic
74C22×C8×C61C2311yescharacteristic
75#117×C61C2×C411yescharacteristic
76M32×C61—22no—
77C2×C16×C61—22no—
78C2×C16×C61C2311yescharacteristic
79C2×C16×C61C2312yes—
80C2×C16×C61C2311yescharacteristic
81#97×C61C2×C412yes—
82C4×C8×C61C2312yes—
83C23×D61C2×C811yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×Dic61C2×C811yescharacteristic
86C2×C4×D61—22no—
87C2×C4×D61—22no—
88C2×C4×D61C2×C811yescharacteristic
89C2×C4×D61C2×C811yescharacteristic
90C2×C4×D61C2×C812yes—
91C2×C4×D61C2×C812yes—
92C2×C4×D61C22×C411yescharacteristic
93C2×C4×D61C22×C411yescharacteristic
94M16×C61M1614yes—
95C2×C8×C61C22×C411yescharacteristic
96C2×C8×C61C22×C412yes—
97C2×C8×C61#4112yes—
98C2×C8×C61#4112yes—
99C2×C8×C61C2411yescharacteristic
100C4⋊C4×C61C2×C811yescharacteristic
101C4⋊C4×C61C2×C811yescharacteristic
102C22⋊C4×C61C2×C812yes—
103C42×C61C22×C412yes—
104C22⋊Dic61C2×C812yes—
105#11758198C2×C812yes—
106#11758198C2×C812yes—
107C4⋊Dic61C2×C811yescharacteristic
108C4⋊Dic61C2×C811yescharacteristic
109#11758196C2×C812yes—
110#11758196C2×C812yes—
111C4×Dic61C2×C812yes—
112C4×Dic61C2×C812yes—
113C4×Dic61C22×C412yes—
114#11758194—22no—
115C2×#92418—22no—
116C2×#92418C22×C411yescharacteristic
117C2×#92418C22×C412yes—
118C2×#92418C22×C411yescharacteristic
119#11758189—24no—
120C8×D61—22no—
121C8×D61—22no—
122C8×D61—24no—
123C8×D61—21no—
124C8×D61—21no—
125C8×D61—21no—
126C8×D61—21no—
127C16×C61—21no—
128C16×C61—21no—
129C16×C61—22no—
130#11758185—21no—
131#11758185—21no—
132#11758185—22no—
133C23×C61C22×C811yescharacteristic
134C22×D61—22no—
135C22×D61—22no—
136C22×D61—41no—
137C22×D61C22×C811yescharacteristic
138C22×D61C22×C811yescharacteristic
139C2×C4×C61#11712yes—
140C2×C4×C61C22×C812yes—
141C2×C4×C61C22×C812yes—
142C2×C4×C61C22×C811yescharacteristic
143C2×C4×C61C2×M1612yes—
144C2×C4×C61C23×C411yescharacteristic
145C2×Dic61—22no—
146C2×Dic61C22×C811yescharacteristic
147C2×Dic61C22×C812yes—
148C2×Dic61C22×C812yes—
149C2×Dic61C22×C811yescharacteristic
150C4×D61—24no—
151C4×D61—21no—
152C4×D61—21no—
153C4×D61—21no—
154C4×D61—21no—
155C4×D61—22no—
156C4×D61—22no—
157C4×D61—42no—
158C8×C61#11712yes—
159C8×C61C2×M1614yes—
160C8×C61C2×#4111yescharacteristic
161C8×C61C2×#4111yescharacteristic
162#92418—21no—
163#92418—21no—
164#92418—22no—
165#33422—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61#43112yes—
169C22×C61C23×C811yescharacteristic
170C2×D61—21no—
171C2×D61—21no—
172C2×D61—21no—
173C2×D61—21no—
174C2×D61—21no—
175C2×D61—44no—
176C2×D61#50412yes—
177C4×C61—22no—
178C4×C61—22no—
179C4×C61#43112yes—
180C4×C61C2×#11711yescharacteristic
181C4×C61C22×M1611yescharacteristic
182Dic61—22no—
183Dic61—22no—
184Dic61—21no—
185Dic61—21no—
186C2×#405—611no—
187#2164—612no—
188#2164—612no—
189#2163—612no—
190#2162—612no—
191#2151—611no—
192#2151—611no—
193#2112—611no—
194C2×C22⋊C16—611no—
195#2108—611no—
196#2108D6111yescharacteristic, complemented
197C2×C61—22no—
198C2×C61C2×#43111yescharacteristic
199C2×C61C2×#50411yescharacteristic
200C2×C61#340711yescharacteristic
201D61—22no—
202D61—41no—
203C22×M16—611no—
204C23×C8—611no—
205C2×#117—611no—
206C2×M32—611no—
207C2×M32C2×D6111yescharacteristic
208C22×C16—611no—
209C22×C16C2×D6111yescharacteristic
210#405—611no—
211#405—611no—
212#405—611no—
213#405—612no—
214#405—612no—
215#405C2×D6111yescharacteristic
216C2×#97—612no—
217C2×C4×C8—612no—
218C4⋊C16—612no—
219C4⋊C16—612no—
220C4⋊C16—612no—
221C4⋊C16—612no—
222C22⋊C16—611no—
223C22⋊C16—611no—
224C22⋊C16—611no—
225C22⋊C16—611no—
226C22⋊C16—612no—
227C22⋊C16—612no—
228#346—612no—
229#346C2×D6112yescomplemented
230C4×C16—612no—
231C4×C16C2×D6112yescomplemented
232C61#3342211yescharacteristic, complemented, Sylow
233C23×C4—611no—
234C2×M16—611no—
235C2×M16—612no—
236C2×M16—1224no—
237C2×M16C4×D6111yescharacteristic
238C22×C8—611no—
239C22×C8—611no—
240C22×C8—611no—
241C22×C8—612no—
242C22×C8—611no—
243C22×C8—611no—
244C22×C8—1222no—
245C22×C8—1222no—
246C22×C8C22×D6111yescharacteristic
247#117—611no—
248#117—611no—
249#117—611no—
250#117—612no—
251#117—612no—
252#117C4×D6111yescharacteristic
253C2×C4⋊C4—611no—
254C2×C4⋊C4—611no—
255C2×C22⋊C4—612no—
256C2×C42—612no—
257M32—22no—
258M32—1222no—
259C2×C16—22no—
260C2×C16—611no—
261C2×C16—612no—
262C2×C16—611no—
263C2×C16—1222no—
264C2×C16C22×D6111yescharacteristic
265C2×C16C22×D6112yes—
266C2×C16C22×D6111yescharacteristic
267#97—612no—
268#97—612no—
269#97—612no—
270#97C4×D6112yes—
271C4×C8—612no—
272C4×C8—612no—
273C4×C8—612no—
274C4×C8C22×D6112yes—
275C24—611no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—611no—
279C22×C4—612no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—612no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4C2×C4×D6111yescharacteristic
286M16—1222no—
287M16—1224no—
288M16#1175818914yes—
289C2×C8—611no—
290C2×C8—612no—
291C2×C8—611no—
292C2×C8—1222no—
293C2×C8—1221no—
294C2×C8—1221no—
295C2×C8—1221no—
296C2×C8—1221no—
297C2×C8—1222no—
298C2×C8—1222no—
299C2×C8—1224no—
300C2×C8C2×C4×D6111yescharacteristic
301C2×C8C2×C4×D6112yes—
302C2×C8#1175822212yes—
303C2×C8#1175822212yes—
304C2×C8C23×D6111yescharacteristic
305C4⋊C4—612no—
306C4⋊C4—612no—
307C4⋊C4—611no—
308C4⋊C4—611no—
309C4⋊C4C8×D6111yescharacteristic
310C4⋊C4C8×D6111yescharacteristic
311C22⋊C4—612no—
312C22⋊C4—612no—
313C22⋊C4—612no—
314C22⋊C4C8×D6112yes—
315C42—612no—
316C42—612no—
317C42—612no—
318C42C2×C4×D6112yes—
319C16—21no—
320C16—21no—
321C16—22no—
322C16—1221no—
323C16—1221no—
324C16—1222no—
325C23—611no—
326C23—611no—
327C23—1222no—
328C23—1222no—
329C23—2441no—
330C23C2×C8×D6111yescharacteristic
331C2×C4—611no—
332C2×C4—612no—
333C2×C4—612no—
334C2×C4—611no—
335C2×C4—1224no—
336C2×C4—1222no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—2442no—
344C2×C4#1451973412yes—
345C2×C4C2×C8×D6112yes—
346C2×C4C2×C8×D6112yes—
347C2×C4C2×C8×D6111yescharacteristic
348C2×C4C2×#1175818912yes—
349C2×C4C22×C4×D6111yescharacteristic
350C8—1221no—
351C8—1221no—
352C8—1222no—
353C8#1451973412yes—
354C8C2×#1175818914yes—
355C8C2×#1175822211yescharacteristic
356C8C2×#1175822211yescharacteristic
357C22—21no—
358C22—21no—
359C22—612no—
360C22—1221no—
361C22—1221no—
362C22—1221no—
363C22—1221no—
364C22—1221no—
365C22—2444no—
366C22#1683981312yes—
367C22C22×C8×D6111yescharacteristic
368C4—22no—
369C4—22no—
370C4—1221no—
371C4—1221no—
372C4—1222no—
373C4—1222no—
374C4#1683981312yes—
375C4C2×#1451973411yescharacteristic
376C4C22×#1175818911yescharacteristic
377C2—22no—
378C2—1222no—
379C2—2441no—
380C2C2×#1683981311yescharacteristic
381C2#504×D6111yescharacteristic
382C2#2145892011yescharacteristic
subgroupG/H
center, FrattiniC2×C8C23×D61
commutator, socleC22×C61C23×C8
FittingC61×#2108C2
prime radicalC23×D61C2×C8
cosocleC2×C8×C61C24
hypercenter#2108D61
hypocenterC61#33422
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C8 ◁ #2108

Complements

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

NH ≅ G/Ntype
2. #21447627C2semidirect
3. #21447624C2semidirect
4. #21447620C2semidirect
5. #21447616C2semidirect
6. #21447579C2semidirect
7. #21447577C2semidirect
10. #21447416C2semidirect
11. C61×#2108C2semidirect
24. #16839580C22semidirect
25. #16839580C22semidirect
26. #16839580C22semidirect
27. #16839580C22semidirect
34. #16839538C22semidirect
35. C4×#11758185C22semidirect
39. C61×#346C22semidirect
40. C4×C16×C61C22semidirect
196. #2108D61semidirect
229. #346C2×D61semidirect
231. C4×C16C2×D61semidirect
232. C61#33422semidirect

Maximal subgroups

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

subgroupindex
#405×D612
#214476272
#214476242
#214476202
#214476162
#214475792
#214475772
#214474292
C2×#168395412
#214474162
C61×#21082
#3342261