Finite Group Databasebeta

← #44558332

Subgroups — #44558332

Order 32128.

408 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30631629C211yesmaximal, characteristic, complemented
2#30631626C211yesmaximal, characteristic, complemented
3#30631623C211yesmaximal, characteristic
4#30631623C211yesmaximal, characteristic
5#30631621C211yesmaximal, characteristic, complemented
6#30631619C211yesmaximal, characteristic, complemented
7#30631586C211yesmaximal, characteristic, complemented
8#30631585C211yesmaximal, characteristic, complemented
9#30631585C211yesmaximal, characteristic, complemented
10#30631584C211yesmaximal, characteristic, complemented
11#30631580C211yesmaximal, characteristic
12#30631562C211yesmaximal, characteristic
13#30631562C211yesmaximal, characteristic
14#30631559C211yesmaximal, characteristic
15#388×C251C211yesmaximal, characteristic
16C23×Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2211yescharacteristic
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×C4⋊Dic251C2211yescharacteristic
23C2×C4⋊Dic251C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×Dic251⋊C4C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×C4×Dic251C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C2×C4⋊C4×C251C2211yescharacteristic
45C2×C22⋊C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C42×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C22×Dic251—41no—
52C22×Dic251—41no—
53C22×Dic251—41no—
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C4⋊Dic251—22no—
63C4⋊Dic251—22no—
64C4⋊Dic251—41no—
65C4⋊Dic251—41no—
66C4⋊Dic251—41no—
67C4⋊Dic251—41no—
68C4⋊Dic251—41no—
69Dic251⋊C4—22no—
70Dic251⋊C4—41no—
71Dic251⋊C4—41no—
72Dic251⋊C4—41no—
73Dic251⋊C4—41no—
74Dic251⋊C4—41no—
75C22⋊Dic251—22no—
76C22⋊Dic251—41no—
77C22⋊Dic251—41no—
78C4×Dic251—41no—
79C24×C251C2311yescharacteristic
80C22×C4×C251C2311yescharacteristic
81C22×C4×C251C2311yescharacteristic
82C22×C4×C251C2311yescharacteristic
83C22×C4×C251C2311yescharacteristic
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C4⋊C4×C251—22no—
87C22⋊C4×C251—22no—
88C22⋊C4×C251—41no—
89C22⋊C4×C251Q814yes—
90C42×C251—41no—
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×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—44no—
115C2×Dic251#4112yes—
116C23×C251—21no—
117C23×C251—21no—
118C23×C251—21no—
119C23×C251—21no—
120C23×C251—21no—
121C23×C251—21no—
122C23×C251C2×Q812yes—
123C23×C251C2411yescharacteristic
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×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251C2×Q812yes—
140C2×C4×C251C2×Q812yes—
141C2×C4×C251#4112yes—
142C22×C251—22no—
143C22×C251—22no—
144C22×C251—22no—
145C22×C251—41no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251—41no—
149C22×C251C22×Q811yescharacteristic
150C22×C251C2×#4111yescharacteristic
151C22×C251C2×#4111yescharacteristic
152C22×C251H(5,𝔽2)11yescharacteristic
153C22×C251H(5,𝔽2)11yescharacteristic
154C22×C251H(5,𝔽2)11yescharacteristic
155C22×C2512−1+411yescharacteristic
156C4×C251—22no—
157C4×C251—22no—
158C4×C251—22no—
159C4×C251—41no—
160C4×C251—41no—
161C4×C251—41no—
162Dic251—22no—
163Dic251—41no—
164Dic251—41no—
165Dic251—41no—
166Dic251—41no—
167Dic251—41no—
168Dic251—41no—
169Dic251—41no—
170C2×C251—42no—
171C2×C251#53911yescharacteristic
172C2×C251#53911yescharacteristic
173C2×C251#54411yescharacteristic
174C2×C251#55411yescharacteristic
175C2×C251#55411yescharacteristic
176C2×C251#55911yescharacteristic
177C2×C251#56011yescharacteristic
178C251#280711yescharacteristic, complemented, Sylow
179#2807—2511nomaximal, Sylow
180#400—2511no—
181#400—2511no—
182#399—2511no—
183#399—2511no—
184#398—2511no—
185#398—2511no—
186#389—2511no—
187#389—2511no—
188#389—2511no—
189#389—2511no—
190#388—2511no—
191#388D25111yescharacteristic
192#385—2511no—
193#385—2511no—
194#385—2511no—
195C23×C4—2511no—
196C2×C4⋊C4—2511no—
197C2×C4⋊C4—2511no—
198C2×C4⋊C4—2511no—
199C2×C4⋊C4—2511no—
200C2×C4⋊C4—2511no—
201C2×C4⋊C4—2511no—
202C2×C4⋊C4—2511no—
203C2×C4⋊C4—2511no—
204C2×C4⋊C4—2511no—
205C2×C4⋊C4—2511no—
206C2×C4⋊C4—2511no—
207C2×C4⋊C4—2511no—
208C2×C4⋊C4—2511no—
209C2×C4⋊C4C2×D25111yescharacteristic
210C2×C22⋊C4—2511no—
211C2×C22⋊C4—2511no—
212C2×C22⋊C4—2511no—
213C2×C22⋊C4C2×D25111yescharacteristic
214C2×C22⋊C4C2×D25111yescharacteristic
215C2×C22⋊C4C2×D25111yescharacteristic
216C2×C42—2511no—
217C2×C42C2×D25111yescharacteristic
218#95—2511no—
219#95—2511no—
220#95—2511no—
221#95—2511no—
222#95—2511no—
223#95—2511no—
224#95—2511no—
225#95—2511no—
226#95—2511no—
227#95—2511no—
228#95C2×D25111yescharacteristic
229#95C2×D25111yescharacteristic
230C24C22×D25111yescharacteristic
231C22×C4—10041no—
232C22×C4—10041no—
233C22×C4—10041no—
234C22×C4—2511no—
235C22×C4—2511no—
236C22×C4—2511no—
237C22×C4—2511no—
238C22×C4—2511no—
239C22×C4—2511no—
240C22×C4—2511no—
241C22×C4—2511no—
242C22×C4C22×D25111yescharacteristic
243C22×C4C22×D25111yescharacteristic
244C22×C4C22×D25111yescharacteristic
245C22×C4C22×D25111yescharacteristic
246C22×C4C22×D25111yescharacteristic
247C22×C4C22×D25111yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—10041no—
250C4⋊C4—10041no—
251C4⋊C4—10041no—
252C4⋊C4—10041no—
253C4⋊C4—10041no—
254C4⋊C4—10041no—
255C4⋊C4—10041no—
256C4⋊C4—10041no—
257C4⋊C4—10041no—
258C4⋊C4—10041no—
259C4⋊C4—5022no—
260C4⋊C4—5022no—
261C4⋊C4—5022no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—10041no—
265C22⋊C4—10041no—
266C22⋊C4—5022no—
267C22⋊C4Dic50214yes—
268C42—41no—
269C42—10041no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23C2×Dic50212yes—
277C23C23×D25111yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—10044no—
294C2×C4—5021no—
295C2×C4—5021no—
296C2×C4—5021no—
297C2×C4—5021no—
298C2×C4—5021no—
299C2×C4—5021no—
300C2×C4—5021no—
301C2×C4—5021no—
302C2×C4—5021no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5021no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—5021no—
311C2×C4—5021no—
312C2×C4—5021no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—2512no—
318C2×C4C2×Dic50212yes—
319C2×C4C2×Dic50212yes—
320C2×C4#1686080912yes—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22C22×Dic50211yescharacteristic
329C22#41×D25111yescharacteristic
330C22C2×#1686080911yescharacteristic
331C22#2254876011yescharacteristic
332C22#2254876111yescharacteristic
333C22#2254876111yescharacteristic
334C22#2254876211yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—10041no—
342C4—10041no—
343C4—10041no—
344C4—10041no—
345C4—10041no—
346C4—10041no—
347C4—10041no—
348C4—5022no—
349C2—42no—
350C2#3063227811yescharacteristic
351C2#3063228211yescharacteristic
352C2#3063231511yescharacteristic
353C2#3063239411yescharacteristic
354C2#3063239611yescharacteristic
355C2#3063241911yescharacteristic
356C2#3063242711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#388×C251C2
prime radicalC24×C251C23
hypercenter#388D251
hypocenterC251#2807
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631626C2semidirect
5. #30631621C2semidirect
6. #30631619C2semidirect
7. #30631586C2semidirect
8. #30631585C2semidirect
9. #30631585C2semidirect
10. #30631584C2semidirect
178. C251#2807semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316262
#306316232
#306316212
#306316192
#306315862
#306315852
#306315842
#306315802
#306315622
#306315592
#388×C2512
#2807251