Finite Group Databasebeta

← #44553147

Subgroups — #44553147

Order 32128.

476 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548708C211yesmaximal, characteristic
2#30631584C211yesmaximal, characteristic, complemented
3#30631584C211yesmaximal, characteristic, complemented
4#30631581C211yesmaximal, characteristic
5#30631581C211yesmaximal, characteristic
6#30631572C211yesmaximal, characteristic, complemented
7#30631571C211yesmaximal, characteristic, complemented
8#30631568C211yesmaximal, characteristic, complemented
9#30631562C211yesmaximal, characteristic
10#30631559C211yesmaximal, characteristic
11#30631552C211yesmaximal, characteristic, complemented
12C4×C4⋊Dic251C211yesmaximal, characteristic, complemented
13#30631538C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic251C211yesmaximal, characteristic
15#388×C251C211yesmaximal, characteristic
16C23×Dic251C2211yescharacteristic
17#22548708C418yescomplemented
18C2×C4⋊Dic251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C4×Dic251C2211yescharacteristic
31C2×C4×Dic251C2211yescharacteristic
32C2×C4×Dic251C2211yescharacteristic
33C2×C4×Dic251C2211yescharacteristic
34C2×C4×Dic251C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C2×C4⋊C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C42×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C22×Dic251—22no—
53C22×Dic251—41no—
54C22×Dic251—41no—
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C4⋊Dic251—22no—
64C4⋊Dic251—41no—
65C4⋊Dic251C2×C414yescomplemented
66Dic251⋊C4—22no—
67Dic251⋊C4—22no—
68Dic251⋊C4—41no—
69Dic251⋊C4—41no—
70Dic251⋊C4C2×C414yescomplemented
71Dic251⋊C4C2×C414yescomplemented
72C22⋊Dic251—41no—
73C22⋊Dic251C2×C414yes—
74C22⋊Dic251C2×C414yes—
75C4×Dic251—41no—
76C4×Dic251—41no—
77C4×Dic251—41no—
78C4×Dic251—41no—
79C4×Dic251C2×C414yescomplemented
80C24×C251C2311yescharacteristic
81C22×C4×C251C2311yescharacteristic
82C22×C4×C251C2311yescharacteristic
83C22×C4×C251C2311yescharacteristic
84C22×C4×C251C2311yescharacteristic
85C22×C4×C251C2311yescharacteristic
86C22×C4×C251C2311yescharacteristic
87C4⋊C4×C251—41no—
88C22⋊C4×C251—22no—
89C22⋊C4×C251—41no—
90C22⋊C4×C251C2×C414yes—
91C42×C251—22no—
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—44no—
112C2×Dic251C22×C412yes—
113C2×Dic251C22×C412yes—
114C2×Dic251C22×C412yes—
115C2×Dic251C22×C412yes—
116C2×Dic251#4112yes—
117C23×C251—21no—
118C23×C251—21no—
119C23×C251—21no—
120C23×C251—21no—
121C23×C251—21no—
122C23×C251—21no—
123C23×C251C22×C412yes—
124C23×C251C2411yescharacteristic
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×C251—21no—
140C2×C4×C251C22×C412yes—
141C2×C4×C251C22×C412yes—
142C2×C4×C251#4112yes—
143C22×C251—22no—
144C22×C251—22no—
145C22×C251—22no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251C23×C411yescharacteristic
151C22×C251C2×#4111yescharacteristic
152C22×C251C2×#4111yescharacteristic
153C22×C251H(5,𝔽2)11yescharacteristic
154C22×C251H(5,𝔽2)11yescharacteristic
155C22×C2512−1+411yescharacteristic
156C22×C2512−1+411yescharacteristic
157C4×C251—22no—
158C4×C251—22no—
159C4×C251—22no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251—41no—
163Dic251—22no—
164Dic251—22no—
165Dic251—22no—
166Dic251—22no—
167Dic251—22no—
168Dic251—41no—
169Dic251—41no—
170Dic251—41no—
171C2×C251—42no—
172C2×C251C4×#4111yescharacteristic
173C2×C251#52011yescharacteristic
174C2×C251#52011yescharacteristic
175C2×C251#53811yescharacteristic
176C2×C251#53911yescharacteristic
177C2×C251#54111yescharacteristic
178C2×C251#54211yescharacteristic
179C251#233411yescharacteristic, complemented, Sylow
180#2334—2511nomaximal, Sylow
181C2×#126—2511no—
182#389—2511no—
183#389—2511no—
184#388—2511no—
185#388—2511no—
186#388D25111yescharacteristic
187#387—2511no—
188#387—2511no—
189#387—2511no—
190#385—2511no—
191#385—2511no—
192#382—2511no—
193C4×C4⋊C4—2511no—
194C4×C4⋊C4—2511no—
195C4×C22⋊C4—2511no—
196C23×C4—2511no—
197#126—2518no—
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⋊C4C2×D25111yescharacteristic
208C2×C22⋊C4—2511no—
209C2×C22⋊C4—2511no—
210C2×C22⋊C4—2511no—
211C2×C22⋊C4C2×D25111yescharacteristic
212C2×C22⋊C4C2×D25111yescharacteristic
213C2×C22⋊C4C2×D25111yescharacteristic
214C2×C42—2511no—
215C2×C42—2511no—
216C2×C42—2511no—
217C2×C42—2511no—
218C2×C42—2511no—
219C2×C42C2×D25111yescharacteristic
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#95—2511no—
229#95—2511no—
230#95C2×D25111yescharacteristic
231#95C2×D25111yescharacteristic
232C24C22×D25111yescharacteristic
233C22×C4—10041no—
234C22×C4—10041no—
235C22×C4—5022no—
236C22×C4—2511no—
237C22×C4—2511no—
238C22×C4—2511no—
239C22×C4—2511no—
240C22×C4—2511no—
241C22×C4—2511no—
242C22×C4—2511no—
243C22×C4—2511no—
244C22×C4C22×D25111yescharacteristic
245C22×C4C22×D25111yescharacteristic
246C22×C4C22×D25111yescharacteristic
247C22×C4C22×D25111yescharacteristic
248C22×C4C22×D25111yescharacteristic
249C22×C4C22×D25111yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—10041no—
252C4⋊C4—10041no—
253C4⋊C4—10041no—
254C4⋊C4—5022no—
255C4⋊C4—5022no—
256C4⋊C4—5022no—
257C4⋊C4—2514no—
258C4⋊C4—2514no—
259C4⋊C4—2514no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—10041no—
263C22⋊C4—2514no—
264C22⋊C4—2514no—
265C22⋊C4C4×D25114yes—
266C42—22no—
267C42—10041no—
268C42—10041no—
269C42—10041no—
270C42—10041no—
271C42—2514no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D25112yes—
279C23C23×D25111yescharacteristic
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—21no—
294C2×C4—21no—
295C2×C4—10044no—
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—2512no—
316C2×C4—2512no—
317C2×C4—2512no—
318C2×C4—2512no—
319C2×C4—2512no—
320C2×C4C2×C4×D25112yes—
321C2×C4C2×C4×D25112yes—
322C2×C4#1686080912yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D25111yescharacteristic
331C22#41×D25111yescharacteristic
332C22C2×#1686080911yescharacteristic
333C22#2254876011yescharacteristic
334C22#2254876111yescharacteristic
335C22#2254876211yescharacteristic
336C22#2254876211yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—10041no—
344C4—10041no—
345C4—10041no—
346C4—5022no—
347C4—5022no—
348C4—5022no—
349C4—5022no—
350C4—5022no—
351C2—42no—
352C2C4×#1686080911yescharacteristic
353C2#3063216611yescharacteristic
354C2#3063216711yescharacteristic
355C2#3063227111yescharacteristic
356C2#3063228111yescharacteristic
357C2#3063229411yescharacteristic
358C2#3063230411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#388×C251C2
socle, cosocleC23×C251C24
prime radicalC24×C251C23
hypercenter#388D251
hypocenterC251#2334
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #30631584C2semidirect
3. #30631584C2semidirect
6. #30631572C2semidirect
7. #30631571C2semidirect
8. #30631568C2semidirect
11. #30631552C2semidirect
12. C4×C4⋊Dic251C2semidirect
13. #30631538C2semidirect
17. #22548708C4semidirect
65. C4⋊Dic251C2×C4semidirect
70. Dic251⋊C4C2×C4semidirect
71. Dic251⋊C4C2×C4semidirect
79. C4×Dic251C2×C4semidirect
179. C251#2334semidirect

Maximal subgroups

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

subgroupindex
C2×#225487082
#306315842
#306315812
#306315722
#306315712
#306315682
#306315622
#306315592
#306315522
C4×C4⋊Dic2512
#306315382
C22⋊C4×Dic2512
#388×C2512
#2334251