Finite Group Databasebeta

← #44562275

Subgroups — #44562275

Order 32128.

386 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#30632408C211yesmaximal, characteristic, complemented
2#30632336C211yesmaximal, characteristic
3#30632036C211yesmaximal, characteristic, complemented
4#30632029C211yesmaximal, characteristic, complemented
5#30632026C211yesmaximal, characteristic, complemented
6#30631965C211yesmaximal, characteristic, complemented
7#30631926C211yesmaximal, characteristic, complemented
8#30631925C211yesmaximal, characteristic, complemented
9#30631895C211yesmaximal, characteristic, complemented
10#30631846C211yesmaximal, characteristic, complemented
11SD16×Dic251C211yesmaximal, characteristic, complemented
12(C22⋊D251)⋊C8C211yesmaximal, characteristic, complemented
13#30631683C211yesmaximal, characteristic, complemented
14#30631680C211yesmaximal, characteristic, complemented
15#452×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×#16860800C2211yescharacteristic
18C22×C251⋊C8C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548714C2211yescharacteristic
21#22548713C2211yescharacteristic
22#22548710—21no—
23#22548708—21no—
24#22548701—21no—
25#22548698—21no—
26#22548696—21no—
27#22548694C2211yescharacteristic
28#22548691C2211yescharacteristic
29#22548691C2211yescharacteristic
30#22548687—21no—
31Q8×Dic251C2211yescharacteristic
32#22548680—21no—
33D4×Dic251C2211yescharacteristic
34C4×C22⋊D251C2211yescharacteristic
35#22548674C2211yescharacteristic
36#22548673C2211yescharacteristic
37C2×Dic251⋊C4—21no—
38#22548649C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548647C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548639C2211yescharacteristic
46#22548639C2211yescharacteristic
47#22548638C2211yescharacteristic
48#22548636C2211yescharacteristic
49#22548636C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×#41×C251C2211yescharacteristic
53C2×Q16×C251C2211yescharacteristic
54C2×SD16×C251C2211yescharacteristic
55C22⋊Q8×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic
59#16860809—22no—
60#16860809—41no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251D411yescharacteristic
65C22×Dic251—21no—
66C22×Dic251—21no—
67C2×C4×D251C2311yescharacteristic
68#16860800—41no—
69C2×C251⋊C8—41no—
70C2×C251⋊C8C2311yescharacteristic
71C2×C251⋊C8C2311yescharacteristic
72C8×D251—41no—
73C4⋊Dic251—21no—
74C4⋊Dic251C2311yescharacteristic
75C4⋊Dic251C2311yescharacteristic
76C4⋊Dic251C2311yescharacteristic
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84C22⋊Dic251—21no—
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87C22⋊Dic251D411yescharacteristic
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4D411yescharacteristic
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251C2311yescharacteristic
96#41×C251—22no—
97#41×C251—41no—
98C2×Q8×C251C2311yescharacteristic
99C2×Q8×C251C2311yescharacteristic
100C2×D4×C251—21no—
101C2×D4×C251C2311yescharacteristic
102C22×C4×C251—21no—
103C22×C4×C251C2311yescharacteristic
104Q16×C251—41no—
105SD16×C251—22no—
106C2×C8×C251C2311yescharacteristic
107C2×C8×C251C2311yescharacteristic
108C4⋊C4×C251—21no—
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C22×D251C2×D411yescharacteristic
112Dic502—21no—
113Dic502—21no—
114Dic502—42no—
115C22⋊D251—22no—
116C22⋊D251—42no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—42no—
122C2×Dic251—42no—
123C2×Dic251C2×D411yescharacteristic
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C4×D251—21no—
127C4×D251—21no—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C23×C251—21no—
131C23×C251C2×D411yescharacteristic
132Q8×C251—21no—
133Q8×C251—21no—
134Q8×C251—21no—
135Q8×C251—21no—
136D4×C251—21no—
137D4×C251—42no—
138D4×C251#4112yes—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—42no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2411yescharacteristic
147C8×C251—21no—
148C8×C251—21no—
149C22×C251—21no—
150C22×C251—22no—
151C22×C251—21no—
152C22×C251—21no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251C22×D411yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—21no—
159C4×C251—21no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251C2×#4111yescharacteristic
164C4×C251H(5,𝔽2)11yescharacteristic
165Dic251—21no—
166Dic251—21no—
167Dic251—41no—
168Dic251—41no—
169Dic251—41no—
170Dic251—41no—
171Dic251#13512yes—
172C2×C251—41no—
173C2×C251—42no—
174C2×C251#54611yescharacteristic
175C2×C251C2×#13511yescharacteristic
176C2×C251#57811yescharacteristic
177D251—41no—
178C251#330211yescharacteristic, complemented, Sylow
179#3302—2511nomaximal, Sylow
180#556—2511no—
181#547—2511no—
182#487—2511no—
183#485—2511no—
184#485—2511no—
185#474—2511no—
186#467—2511no—
187#467—2511no—
188#462—2511no—
189#452—2511no—
190#452D25111yescharacteristic, complemented
191C4×SD16—2511no—
192C8×D4—2511no—
193#416—2511no—
194#416—2511no—
195C2×#41—2511no—
196C2×#41C2×D25111yescharacteristic
197C2×Q16—2511no—
198C2×Q16C2×D25111yescharacteristic
199C2×SD16C2×D25111yescharacteristic
200C22×C8—2511no—
201C22×C8—2511no—
202C4⋊Q8—2511no—
203C4⋊Q8—2511no—
204#126—5021no—
205#126—5021no—
206#124—5021no—
207#123—5021no—
208#123—5021no—
209C22⋊Q8—2511no—
210C22⋊Q8—2511no—
211C22⋊Q8—2511no—
212C22⋊Q8C2×D25111yescharacteristic
213#121—5021no—
214C4×Q8—5021no—
215C4×Q8—2511no—
216C4×D4—2511no—
217C4×D4—2511no—
218#117—2511no—
219#117—2511no—
220C2×C4⋊C4—5021no—
221#107—2511no—
222#107—2511no—
223#106—2511no—
224C4⋊C8—2511no—
225#103—2511no—
226#103—2511no—
227#103—2511no—
228#103—2511no—
229#103—2511no—
230#103—2511no—
231#103C2×D25111yescharacteristic
232#102—2511no—
233#102—2511no—
234#102C2×D25111yescharacteristic
235C22⋊C8—2511no—
236C22⋊C8C2×D25111yescharacteristic
237C4×C8—2511no—
238#41—22no—
239#41—41no—
240#41—10041no—
241#41—5022no—
242C2×Q8—5021no—
243C2×Q8—2511no—
244C2×Q8C22×D25111yescharacteristic
245C2×Q8C22×D25111yescharacteristic
246C2×D4—21no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4C22×D25111yescharacteristic
250C22×C4—21no—
251C22×C4—5021no—
252C22×C4—5021no—
253C22×C4—2511no—
254C22×C4C22×D25111yescharacteristic
255Q16—41no—
256Q16—10041no—
257SD16—22no—
258C2×C8—10041no—
259C2×C8—10041no—
260C2×C8—2511no—
261C2×C8—2511no—
262C2×C8C22×D25111yescharacteristic
263C2×C8C22×D25111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—5021no—
266C4⋊C4—5021no—
267C4⋊C4—5021no—
268C4⋊C4—5021no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—2511no—
273C4⋊C4—2511no—
274C4⋊C4—2511no—
275C4⋊C4—2511no—
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4—2511no—
285C22⋊C4—2511no—
286C42—5021no—
287C42—5021no—
288C42—5021no—
289C42—2511no—
290C23—21no—
291C23—2511no—
292C23D4×D25111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—10042no—
298Q8—5021no—
299Q8—5021no—
300D4—21no—
301D4—42no—
302D4—10042no—
303D4—5022no—
304D4#1686080912yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—10042no—
312C2×C4—10042no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—2511no—
320C2×C4—2511no—
321C2×C4—2511no—
322C2×C4D4×D25111yescharacteristic
323C2×C4C23×D25111yescharacteristic
324C8—21no—
325C8—21no—
326C8—5021no—
327C8—5021no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—5021no—
335C22—5021no—
336C22—5021no—
337C22C2×D4×D25111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—10041no—
343C4—10041no—
344C4—10041no—
345C4—10041no—
346C4—5021no—
347C4—5021no—
348C4—2512no—
349C4C2×#1686080911yescharacteristic
350C4#2254876011yescharacteristic
351C2—41no—
352C2—42no—
353C2—10041no—
354C2#3063232411yescharacteristic
355C2#135×D25111yescharacteristic
356C2#3063250311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#452×C251C2
socleC22×C251C22×D4
prime radical#30632336C2
hypercenter#452D251
hypocenterC251#3302
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #30632408C2semidirect
3. #30632036C2semidirect
4. #30632029C2semidirect
5. #30632026C2semidirect
6. #30631965C2semidirect
7. #30631926C2semidirect
8. #30631925C2semidirect
9. #30631895C2semidirect
10. #30631846C2semidirect
11. SD16×Dic251C2semidirect
12. (C22⋊D251)⋊C8C2semidirect
13. #30631683C2semidirect
14. #30631680C2semidirect
15. #452×C251C2semidirect
178. C251#3302semidirect
190. #452D251semidirect

Maximal subgroups

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

subgroupindex
#306324082
#306323362
#306320362
#306320292
#306320262
#306319652
#306319262
#306319252
#306318952
#306318462
SD16×Dic2512
(C22⋊D251)⋊C82
#306316832
#306316802
#452×C2512
#3302251