Finite Group Databasebeta

← #29833698

Subgroups — #29833698

Order 15616.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21447728C212yesmaximal, complemented
2#21447645C214yesmaximal, complemented
3D4×#11758185C212yesmaximal, complemented
4#21447622C212yesmaximal, complemented
5#21447618C211yesmaximal, characteristic, complemented
6#21447575C211yesmaximal, characteristic, complemented
7C2×C4⋊#11758185C212yesmaximal, complemented
8C8×#41×C61C211yesmaximal, characteristic
9C2×#14519717C2212yes—
10C22×#11758185C2214yes—
11C4⋊#11758185C2214yes—
12C4⋊#11758185C2212yes—
13C4⋊#11758185C2211yescharacteristic
14C4⋊#11758185C2212yes—
15C4⋊#11758185C2211yescharacteristic
16C22⋊#11758185C2212yes—
17C22⋊#11758185C2214yes—
18#16839538C2214yes—
19C4×#11758185C2212yes—
20C4×#41×C61C411yescharacteristic
21C8×Q8×C61C411yescharacteristic
22C8×Q8×C61C2211yescharacteristic
23C8×D4×C61C412yes—
24C8×D4×C61C411yescharacteristic
25C8×D4×C61C2211yescharacteristic
26C8×D4×C61C2212yes—
27C61×#431C411yescharacteristic
28C61×#431C412yes—
29C2×C4×C8×C61C2212yes—
30C2×C4×C8×C61C2211yescharacteristic
31#14519717—42no—
32C2×#11758185—44no—
33C2×#11758185C2312yes—
34C2×#11758185C2312yes—
35C2×#11758185C2314yes—
36C2×#41×C61C811yescharacteristic
37C22×C8×C61C2×C412yes—
38C22×C8×C61C2×C411yescharacteristic
39C22×C8×C61C2312yes—
40C22×C8×C61C2311yescharacteristic
41C4×Q8×C61C811yescharacteristic
42C4×Q8×C61C2×C411yescharacteristic
43C4×D4×C61C812yes—
44C4×D4×C61C811yescharacteristic
45C4×D4×C61C2×C412yes—
46C4×D4×C61C2×C411yescharacteristic
47#117×C61C811yescharacteristic
48#117×C61C812yes—
49C2×C42×C61C2×C412yes—
50C2×C42×C61C2×C411yescharacteristic
51C4⋊C8×C61C2×C412yes—
52C4⋊C8×C61C2×C411yescharacteristic
53C4⋊C8×C61C2×C411yescharacteristic
54C4⋊C8×C61C2×C412yes—
55C22⋊C8×C61C2×C412yes—
56C22⋊C8×C61C2×C412yes—
57C22⋊C8×C61C2×C412yes—
58C4×C8×C61C2×C412yes—
59C4×C8×C61C2×C411yescharacteristic
60C4×C8×C61C2×C412yes—
61C4×C8×C61C2×C411yescharacteristic
62C4×C8×C61C2311yescharacteristic
63C4×C8×C61C2312yes—
64C4×C8×C61C2311yescharacteristic
65#41×C61—24no—
66C2×Q8×C61C2×C811yescharacteristic
67C2×D4×C61C2×C812yes—
68C2×D4×C61C2×C811yescharacteristic
69C22×C4×C61C2×C812yes—
70C22×C4×C61C2×C811yescharacteristic
71C22×C4×C61C2×C811yescharacteristic
72C22×C4×C61C2×C812yes—
73C22×C4×C61C22×C411yescharacteristic
74C22×C4×C61C22×C412yes—
75C2×C8×C61—22no—
76C2×C8×C61—21no—
77C2×C8×C61—22no—
78C2×C8×C61—21no—
79C2×C8×C61—42no—
80C2×C8×C61—41no—
81C2×C8×C61C22×C411yescharacteristic
82C2×C8×C61C22×C412yes—
83C2×C8×C61C22×C411yescharacteristic
84C2×C8×C61C22×C411yescharacteristic
85C2×C8×C61C22×C412yes—
86C2×C8×C61C2411yescharacteristic
87C4⋊C4×C61C2×C812yes—
88C4⋊C4×C61C2×C811yescharacteristic
89C4⋊C4×C61C2×C812yes—
90C4⋊C4×C61C2×C811yescharacteristic
91C22⋊C4×C61C2×C812yes—
92C22⋊C4×C61C2×C812yes—
93C22⋊C4×C61C2×C812yes—
94C42×C61C2×C812yes—
95C42×C61C2×C811yescharacteristic
96C42×C61C2×C812yes—
97C42×C61C2×C811yescharacteristic
98C42×C61C22×C411yescharacteristic
99C42×C61C22×C412yes—
100C42×C61C22×C411yescharacteristic
101#11758185—22no—
102#11758185—24no—
103#11758185—22no—
104C23×C61C22×C811yescharacteristic
105C23×C61C22×C812yes—
106Q8×C61—22no—
107D4×C61—24no—
108D4×C61M3214yescomplemented
109C2×C4×C61—24no—
110C2×C4×C61—22no—
111C2×C4×C61—21no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—24no—
116C2×C4×C61—22no—
117C2×C4×C61C22×C811yescharacteristic
118C2×C4×C61C22×C812yes—
119C2×C4×C61C22×C811yescharacteristic
120C2×C4×C61C22×C812yes—
121C2×C4×C61C22×C811yescharacteristic
122C2×C4×C61C22×C812yes—
123C2×C4×C61C22×C811yescharacteristic
124C2×C4×C61C22×C811yescharacteristic
125C2×C4×C61C23×C411yescharacteristic
126C8×C61—22no—
127C8×C61—21no—
128C8×C61—21no—
129C8×C61—21no—
130C8×C61—22no—
131C8×C61H(5,𝔽2)11yescharacteristic
132C8×C612−1+411yescharacteristic
133#33504—611nomaximal, Sylow
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61C2×M3214yes—
140C22×C61C23×C811yescharacteristic
141C4×C61—21no—
142C4×C61—22no—
143C4×C61—22no—
144C4×C61—21no—
145C4×C61—22no—
146C4×C61—21no—
147C4×C61C2×M3212yes—
148C4×C61#50412yes—
149C4×C61#52011yescharacteristic
150C4×C61#56811yescharacteristic
151C8×#41D6111yescharacteristic
152#2183—612no—
153#2168—614no—
154D4×C16—612no—
155#2163—612no—
156#2162—611no—
157#2150—611no—
158C2×C4⋊C16—612no—
159C2×C61—24no—
160C2×C61—41no—
161C2×C61#298811yescharacteristic
162C2×C61C22×M3211yescharacteristic
163C2×C61C2×#50411yescharacteristic
164C4×#41Dic6111yescharacteristic
165C2×M32—612no—
166C22×C16—614no—
167C8×Q8Dic6111yescharacteristic
168C8×Q8C2×D6111yescharacteristic
169C8×D4Dic6112yes—
170C8×D4Dic6111yescharacteristic
171C8×D4C2×D6111yescharacteristic
172C8×D4C2×D6112yes—
173#431Dic6111yescharacteristic
174#431Dic6112yes—
175C2×C4×C8C2×D6112yes—
176C2×C4×C8C2×D6111yescharacteristic
177C4⋊C16—611no—
178C4⋊C16—612no—
179C4⋊C16—614no—
180C4⋊C16—612no—
181C4⋊C16—611no—
182C22⋊C16—612no—
183C22⋊C16—614no—
184#346—614no—
185C4×C16—612no—
186C61#3350411yescharacteristic, complemented, Sylow
187C2×#41#9241811yescharacteristic
188C22×C8C2×Dic6112yes—
189C22×C8C2×Dic6111yescharacteristic
190C22×C8C22×D6112yes—
191C22×C8C22×D6111yescharacteristic
192C4×Q8#9241811yescharacteristic
193C4×Q8C2×Dic6111yescharacteristic
194C4×D4#9241812yes—
195C4×D4#9241811yescharacteristic
196C4×D4C2×Dic6112yes—
197C4×D4C2×Dic6111yescharacteristic
198#117#9241811yescharacteristic
199#117#9241812yes—
200C2×C42C2×Dic6112yes—
201C2×C42C2×Dic6111yescharacteristic
202M32—2442no—
203C2×C16—612no—
204C2×C16—612no—
205C2×C16—614no—
206C2×C16—2444no—
207C4⋊C8C2×Dic6112yes—
208C4⋊C8C2×Dic6111yescharacteristic
209C4⋊C8C2×Dic6111yescharacteristic
210C4⋊C8C2×Dic6112yes—
211C22⋊C8C2×Dic6112yes—
212C22⋊C8C2×Dic6112yes—
213C22⋊C8C2×Dic6112yes—
214C4×C8C2×Dic6112yes—
215C4×C8C2×Dic6111yescharacteristic
216C4×C8C2×Dic6112yes—
217C4×C8C2×Dic6111yescharacteristic
218C4×C8C22×D6111yescharacteristic
219C4×C8C22×D6112yes—
220C4×C8C22×D6111yescharacteristic
221#41—24no—
222C2×Q8C2×#9241811yescharacteristic
223C2×D4C2×#9241812yes—
224C2×D4C2×#9241811yescharacteristic
225C22×C4C2×#9241812yes—
226C22×C4C2×#9241811yescharacteristic
227C22×C4C2×#9241811yescharacteristic
228C22×C4C2×#9241812yes—
229C22×C4C22×Dic6111yescharacteristic
230C22×C4C22×Dic6112yes—
231C2×C8—22no—
232C2×C8—21no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—42no—
236C2×C8—41no—
237C2×C8C22×Dic6111yescharacteristic
238C2×C8C22×Dic6112yes—
239C2×C8C22×Dic6111yescharacteristic
240C2×C8C22×Dic6111yescharacteristic
241C2×C8C22×Dic6112yes—
242C2×C8C23×D6111yescharacteristic
243C4⋊C4C2×#9241812yes—
244C4⋊C4C2×#9241811yescharacteristic
245C4⋊C4C2×#9241812yes—
246C4⋊C4C2×#9241811yescharacteristic
247C22⋊C4C2×#9241812yes—
248C22⋊C4C2×#9241812yes—
249C22⋊C4C2×#9241812yes—
250C42C2×#9241812yes—
251C42C2×#9241811yescharacteristic
252C42C2×#9241812yes—
253C42C2×#9241811yescharacteristic
254C42C22×Dic6111yescharacteristic
255C42C22×Dic6112yes—
256C42C22×Dic6111yescharacteristic
257C16—1222no—
258C16—1224no—
259C16—1222no—
260C23C22×#9241811yescharacteristic
261C23C22×#9241812yes—
262Q8—22no—
263D4—24no—
264D4#1451971714yescomplemented
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4C22×#9241811yescharacteristic
274C2×C4C22×#9241812yes—
275C2×C4C22×#9241811yescharacteristic
276C2×C4C22×#9241812yes—
277C2×C4C22×#9241811yescharacteristic
278C2×C4C22×#9241812yes—
279C2×C4C22×#9241811yescharacteristic
280C2×C4C22×#9241811yescharacteristic
281C2×C4C23×Dic6111yescharacteristic
282C8—22no—
283C8—21no—
284C8—21no—
285C8—21no—
286C8—22no—
287C8#1451982311yescharacteristic
288C8#1451982411yescharacteristic
289C22—22no—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22C2×#1451971714yes—
295C22C23×#9241811yescharacteristic
296C4—22no—
297C4—21no—
298C4—21no—
299C4—22no—
300C4—22no—
301C4—21no—
302C4C2×#1451971712yes—
303C4#1684017912yes—
304C4#1684024511yescharacteristic
305C4#1684052911yescharacteristic
306C2—24no—
307C2—41no—
308C2#2145493011yescharacteristic
309C2C22×#1451971711yescharacteristic
310C2C2×#1684017911yescharacteristic
subgroupG/H
center, FrattiniC2×C8C23×D61
commutator, socleC22×C61C23×C8
FittingC8×#41×C61C2
prime radicalC2×#41×C61C8
cosocleC2×C8×C61C24
hypercenterC8×#41D61
hypocenterC61#33504
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C8 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. #21447728C2semidirect
2. #21447645C2semidirect
3. D4×#11758185C2semidirect
4. #21447622C2semidirect
5. #21447618C2semidirect
6. #21447575C2semidirect
7. C2×C4⋊#11758185C2semidirect
108. D4×C61M32semidirect
186. C61#33504semidirect
264. D4#14519717semidirect

Maximal subgroups

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

subgroupindex
#214477282
#214476452
D4×#117581852
#214476222
#214476182
#214475752
C2×C4⋊#117581852
C8×#41×C612
#3350461