Finite Group Databasebeta

← #44558325

Subgroups — #44558325

Order 32128.

392 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30631621C211yesmaximal, characteristic
2#30631621C211yesmaximal, characteristic
3#30631620C211yesmaximal, characteristic
4#30631619C211yesmaximal, characteristic
5#30631608C211yesmaximal, characteristic
6#30631607C211yesmaximal, characteristic
7#30631594C211yesmaximal, characteristic
8#30631594C211yesmaximal, characteristic
9#30631591C211yesmaximal, characteristic
10#30631585C211yesmaximal, characteristic
11#30631584C211yesmaximal, characteristic
12#30631584C211yesmaximal, characteristic
13#30631574C211yesmaximal, characteristic
14#30631571C211yesmaximal, characteristic
15#391×C251C211yesmaximal, characteristic
16C22×Dic502C2211yescharacteristic
17C2×C4⋊Dic251C2211yescharacteristic
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×Dic251⋊C4C2211yescharacteristic
30C2×C4×Dic251C2211yescharacteristic
31C2×C4×Dic251C2211yescharacteristic
32C251⋊#95C2211yescharacteristic
33C251⋊#95C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C22×Q8×C251C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C42×C251C2211yescharacteristic
47#95×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C2×Dic502—22no—
52C2×Dic502—41no—
53C2×Dic502—41no—
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C4⋊Dic251—22no—
63C4⋊Dic251—41no—
64C4⋊Dic251—41no—
65C4⋊Dic251—41no—
66C4⋊Dic251—41no—
67Dic251⋊C4—22no—
68Dic251⋊C4—41no—
69Dic251⋊C4—41no—
70Dic251⋊C4—41no—
71Dic251⋊C4—41no—
72Dic251⋊C4—41no—
73Dic251⋊C4—41no—
74Dic251⋊C4Q814yescomplemented
75C4×Dic251—41no—
76C4×Dic251—41no—
77C2×Q8×C251—41no—
78C2×Q8×C251—41no—
79C2×Q8×C251Q814yescomplemented
80C22×C4×C251C2311yescharacteristic
81C22×C4×C251C2311yescharacteristic
82C22×C4×C251C2311yescharacteristic
83C22×C4×C251C2311yescharacteristic
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C4⋊C4×C251—41no—
88C42×C251—41no—
89Dic502—44no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251—21no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—21no—
96C2×Dic251—21no—
97C2×Dic251—21no—
98C2×Dic251—21no—
99C2×Dic251—21no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251C2×Q812yes—
112C2×Dic251C2×Q812yes—
113C2×Dic251#4112yes—
114C23×C251C2411yescharacteristic
115Q8×C251—44no—
116C2×C4×C251—21no—
117C2×C4×C251—21no—
118C2×C4×C251—21no—
119C2×C4×C251—21no—
120C2×C4×C251—21no—
121C2×C4×C251—21no—
122C2×C4×C251—21no—
123C2×C4×C251—21no—
124C2×C4×C251—21no—
125C2×C4×C251—21no—
126C2×C4×C251—21no—
127C2×C4×C251—21no—
128C2×C4×C251—21no—
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251C2×Q812yes—
134C2×C4×C251C2×Q812yes—
135C2×C4×C251C2×Q812yes—
136C2×C4×C251C2×Q812yes—
137C22×C251C22×Q811yescharacteristic
138C22×C251C22×Q811yescharacteristic
139C22×C251C2×#4111yescharacteristic
140C22×C251H(5,𝔽2)11yescharacteristic
141C22×C251H(5,𝔽2)11yescharacteristic
142C22×C251H(5,𝔽2)11yescharacteristic
143C22×C2512−1+411yescharacteristic
144C4×C251—22no—
145C4×C251—22no—
146C4×C251—22no—
147C4×C251—22no—
148C4×C251—41no—
149C4×C251—41no—
150C4×C251—41no—
151Dic251—22no—
152Dic251—22no—
153Dic251—22no—
154Dic251—41no—
155Dic251—41no—
156Dic251—41no—
157Dic251—41no—
158Dic251—41no—
159C2×C251#53811yescharacteristic
160C2×C251#54311yescharacteristic
161C2×C251#54411yescharacteristic
162C2×C251#55411yescharacteristic
163C2×C251#55711yescharacteristic
164C2×C251Q8211yescharacteristic
165C2×C251#56211yescharacteristic
166C251#280611yescharacteristic, complemented, Sylow
167#2806—2511nomaximal, Sylow
168#398—2511no—
169#398—2511no—
170#398—2511no—
171#398—2511no—
172#395—2511no—
173#395—2511no—
174#391—2511no—
175#391—2511no—
176#391—2511no—
177#391D25111yescharacteristic
178#389—2511no—
179#389—2511no—
180#389—2511no—
181#387—2511no—
182#387—2511no—
183C22×Q8—2511no—
184C22×Q8C2×D25111yescharacteristic
185C2×C4⋊C4—2511no—
186C2×C4⋊C4—2511no—
187C2×C4⋊C4—2511no—
188C2×C4⋊C4—2511no—
189C2×C4⋊C4—2511no—
190C2×C4⋊C4—2511no—
191C2×C4⋊C4—2511no—
192C2×C4⋊C4—2511no—
193C2×C4⋊C4—2511no—
194C2×C4⋊C4—2511no—
195C2×C4⋊C4—2511no—
196C2×C4⋊C4—2511no—
197C2×C4⋊C4—2511no—
198C2×C4⋊C4C2×D25111yescharacteristic
199C2×C42—2511no—
200C2×C42—2511no—
201C2×C42C2×D25111yescharacteristic
202#95—2511no—
203#95—2511no—
204#95—2511no—
205#95—2511no—
206#95—2511no—
207#95—2511no—
208#95—2511no—
209#95—2511no—
210#95—2511no—
211#95—2511no—
212#95—2511no—
213#95—2511no—
214#95C2×D25111yescharacteristic
215#95C2×D25111yescharacteristic
216#95C2×D25111yescharacteristic
217#95C2×D25111yescharacteristic
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—10041no—
221C2×Q8—10041no—
222C2×Q8—5022no—
223C2×Q8Dic50214yescomplemented
224C22×C4—2511no—
225C22×C4—2511no—
226C22×C4—2511no—
227C22×C4—2511no—
228C22×C4—2511no—
229C22×C4—2511no—
230C22×C4—2511no—
231C22×C4—2511no—
232C22×C4C22×D25111yescharacteristic
233C22×C4C22×D25111yescharacteristic
234C22×C4C22×D25111yescharacteristic
235C22×C4C22×D25111yescharacteristic
236C22×C4C22×D25111yescharacteristic
237C22×C4C22×D25111yescharacteristic
238C22×C4C22×D25111yescharacteristic
239C4⋊C4—41no—
240C4⋊C4—10041no—
241C4⋊C4—10041no—
242C4⋊C4—10041no—
243C4⋊C4—10041no—
244C4⋊C4—10041no—
245C4⋊C4—10041no—
246C4⋊C4—10041no—
247C4⋊C4—10041no—
248C4⋊C4—10041no—
249C4⋊C4—10041no—
250C4⋊C4—5022no—
251C4⋊C4—5022no—
252C4⋊C4—2514no—
253C42—41no—
254C42—10041no—
255C42—10041no—
256C23C23×D25111yescharacteristic
257Q8—44no—
258Q8—10044no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—5021no—
277C2×C4—5021no—
278C2×C4—5021no—
279C2×C4—5021no—
280C2×C4—5021no—
281C2×C4—5021no—
282C2×C4—5021no—
283C2×C4—5021no—
284C2×C4—5021no—
285C2×C4—5021no—
286C2×C4—5021no—
287C2×C4—5021no—
288C2×C4—5021no—
289C2×C4—5021no—
290C2×C4—5021no—
291C2×C4—5021no—
292C2×C4—5021no—
293C2×C4—5021no—
294C2×C4—5021no—
295C2×C4—5021no—
296C2×C4—5021no—
297C2×C4—2512no—
298C2×C4—2512no—
299C2×C4—2512no—
300C2×C4Q8×D25112yes—
301C2×C4C2×Dic50212yes—
302C2×C4C2×Dic50212yes—
303C2×C4C2×Dic50212yes—
304C22C2×Q8×D25111yescharacteristic
305C22C22×Dic50211yescharacteristic
306C22#41×D25111yescharacteristic
307C22#2254876011yescharacteristic
308C22#2254876011yescharacteristic
309C22#2254876111yescharacteristic
310C22#2254876311yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—10041no—
319C4—10041no—
320C4—10041no—
321C4—10041no—
322C4—10041no—
323C4—5022no—
324C4—5022no—
325C4—5022no—
326C2#3063227111yescharacteristic
327C2#3063231311yescharacteristic
328C2#3063231411yescharacteristic
329C2#3063239111yescharacteristic
330C2#3063241311yescharacteristic
331C2Q8×Dic50211yescharacteristic
332C2#3063243311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#391×C251C2
hypercenter#391D251
hypocenterC251#2806
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
74. Dic251⋊C4Q8semidirect
79. C2×Q8×C251Q8semidirect
166. C251#2806semidirect
223. C2×Q8Dic502semidirect

Maximal subgroups

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

subgroupindex
#306316212
#306316202
#306316192
#306316082
#306316072
#306315942
#306315912
#306315852
#306315842
#306315742
#306315712
#391×C2512
#2806251