Finite Group Databasebeta

← #44562252

Subgroups — #44562252

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#30632377C211yesmaximal, characteristic, complemented
2#30632041C211yesmaximal, characteristic, complemented
3#30632025C211yesmaximal, characteristic, complemented
4#30631998C211yesmaximal, characteristic, complemented
5#30631994C211yesmaximal, characteristic, complemented
6#30631926C211yesmaximal, characteristic, complemented
7#30631914C211yesmaximal, characteristic, complemented
8#30631906C211yesmaximal, characteristic, complemented
9#30631849C211yesmaximal, characteristic, complemented
10#30631843C211yesmaximal, characteristic, complemented
11C4×#16860800C211yesmaximal, characteristic, complemented
12D4×C251⋊C8C211yesmaximal, characteristic, complemented
13#30631683C211yesmaximal, characteristic, complemented
14C2×#22548639C211yesmaximal, characteristic, complemented
15#548×C251C211yesmaximal, characteristic
16C2×#16860800C2211yescharacteristic
17C2×#16860800C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C22×C251⋊C8C2211yescharacteristic
20C22×C251⋊C8C2211yescharacteristic
21#22548708—21no—
22#22548707C2211yescharacteristic
23#22548705—21no—
24#22548696—21no—
25#22548694C2211yescharacteristic
26#22548694C2211yescharacteristic
27C4×Dic502C2211yescharacteristic
28#22548674C2211yescharacteristic
29#22548671—21no—
30C2×C4⋊Dic251C2211yescharacteristic
31#22548648C2211yescharacteristic
32#22548648C2211yescharacteristic
33#22548646C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35#22548639C2211yescharacteristic
36#22548639C2211yescharacteristic
37#22548639C2211yescharacteristic
38#22548639C2211yescharacteristic
39#22548639C2211yescharacteristic
40#22548639C2211yescharacteristic
41#22548638C2211yescharacteristic
42#22548636C2211yescharacteristic
43#22548636C2211yescharacteristic
44#22548636C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48C2×#41×C251C2211yescharacteristic
49C22×Q8×C251C2211yescharacteristic
50#124×C251—21no—
51#124×C251C2211yescharacteristic
52C22⋊Q8×C251—21no—
53C22⋊Q8×C251C2211yescharacteristic
54#121×C251—21no—
55#121×C251C2211yescharacteristic
56C4×Q8×C251C2211yescharacteristic
57C4×D4×C251—21no—
58C4×D4×C251C2211yescharacteristic
59C2×Dic502C2311yescharacteristic
60C22×Dic251—21no—
61#16860800—22no—
62#16860800—41no—
63#16860798—41no—
64C2×C251⋊C8—41no—
65C2×C251⋊C8—41no—
66C2×C251⋊C8C2311yescharacteristic
67C2×C251⋊C8C2311yescharacteristic
68C2×C251⋊C8C2311yescharacteristic
69C2×C251⋊C8C2311yescharacteristic
70C4⋊Dic251—21no—
71C4⋊Dic251C2311yescharacteristic
72C4⋊Dic251C2311yescharacteristic
73C4⋊Dic251C2311yescharacteristic
74Dic251⋊C4—21no—
75Dic251⋊C4—21no—
76Dic251⋊C4—21no—
77Dic251⋊C4—21no—
78Dic251⋊C4—21no—
79C22⋊Dic251—21no—
80C22⋊Dic251—21no—
81C22⋊Dic251—21no—
82C4×Dic251—21no—
83C4×Dic251—21no—
84#41×C251—22no—
85#41×C251—41no—
86C2×Q8×C251—21no—
87C2×Q8×C251—22no—
88C2×Q8×C251—41no—
89C2×Q8×C251C2311yescharacteristic
90C2×Q8×C251C2311yescharacteristic
91C2×D4×C251—21no—
92C2×D4×C251—21no—
93C2×D4×C251D411yescharacteristic
94C2×D4×C251C2311yescharacteristic
95C22×C4×C251—21no—
96C22×C4×C251—21no—
97C22×C4×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C4⋊C4×C251—21no—
100C4⋊C4×C251—21no—
101C4⋊C4×C251D411yescharacteristic
102C4⋊C4×C251C2311yescharacteristic
103C22⋊C4×C251—21no—
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C22⋊C4×C251D411yescharacteristic
108C22⋊C4×C251D411yescharacteristic
109C42×C251—21no—
110C42×C251C2311yescharacteristic
111Dic502—21no—
112Dic502—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—42no—
118C251⋊C8—21no—
119C251⋊C8—21no—
120C251⋊C8—21no—
121C251⋊C8—21no—
122C23×C251—21no—
123C23×C251C2×D411yescharacteristic
124C23×C251C2×D411yescharacteristic
125Q8×C251—21no—
126Q8×C251—21no—
127Q8×C251—21no—
128Q8×C251—42no—
129Q8×C251#4112yes—
130D4×C251—22no—
131D4×C251—21no—
132D4×C251—21no—
133D4×C251—42no—
134D4×C251—42no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—42no—
143C2×C4×C251—42no—
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2×D411yescharacteristic
148C2×C4×C251C2411yescharacteristic
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251#13512yes—
158C22×C251C22×D411yescharacteristic
159C4×C251—21no—
160C4×C251—21no—
161C4×C251—22no—
162C4×C251—21no—
163C4×C251—21no—
164C4×C251—41no—
165C4×C251—41no—
166C4×C251C2×#4111yescharacteristic
167C4×C251H(5,𝔽2)11yescharacteristic
168Dic251—41no—
169Dic251—41no—
170Dic251—41no—
171Dic251—41no—
172C2×C251—22no—
173C2×C251—41no—
174C2×C251—81no—
175C2×C251#54611yescharacteristic
176C2×C251C2×#13511yescharacteristic
177C2×C251#57811yescharacteristic
178C251#330011yescharacteristic, complemented, Sylow
179#3300—2511nomaximal, Sylow
180#552—2511no—
181#548D25111yescharacteristic
182#488—2511no—
183#485—2511no—
184#480—2511no—
185#479—2511no—
186#467—2511no—
187#465—2511no—
188#464—2511no—
189#452—2511no—
190#451—2511no—
191C4×Q16—2511no—
192C8×D4—2511no—
193#416—2511no—
194C2×#103—2511no—
195C2×#41C2×D25111yescharacteristic
196C22×Q8C2×D25111yescharacteristic
197C2×Q16—2511no—
198C2×Q16—2511no—
199C2×SD16—2511no—
200C22×C8—2511no—
201C22×C8—2511no—
202#126—5021no—
203#125—5021no—
204#125—2511no—
205#124—21no—
206#124C2×D25111yescharacteristic
207#123—5021no—
208C22⋊Q8—21no—
209C22⋊Q8—2511no—
210C22⋊Q8—2511no—
211C22⋊Q8C2×D25111yescharacteristic
212#121—21no—
213#121C2×D25111yescharacteristic
214C4×Q8—2511no—
215C4×Q8C2×D25111yescharacteristic
216C4×D4—21no—
217C4×D4C2×D25111yescharacteristic
218#117—5021no—
219#117—2511no—
220C2×C4⋊C4—2511no—
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#103—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235C22⋊C8—2511no—
236C22⋊C8—2511no—
237C4×C8—2511no—
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—2511no—
244C2×Q8C22×D25111yescharacteristic
245C2×Q8C22×D25111yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊D25111yescharacteristic
249C2×D4C22×D25111yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—5021no—
253C22×C4C22×D25111yescharacteristic
254C22×C4C22×D25111yescharacteristic
255Q16—10041no—
256Q16—5022no—
257SD16—10041no—
258C2×C8—10041no—
259C2×C8—10041no—
260C2×C8—2511no—
261C2×C8—2511no—
262C2×C8—2511no—
263C2×C8—2511no—
264C4⋊C4—21no—
265C4⋊C4—21no—
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⋊C4C22⋊D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4—5021no—
284C22⋊C4C22⋊D25111yescharacteristic
285C22⋊C4C22⋊D25111yescharacteristic
286C42—21no—
287C42—5021no—
288C42—5021no—
289C42C22×D25111yescharacteristic
290C23—21no—
291C23C2×C22⋊D25111yescharacteristic
292C23C2×C22⋊D25111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—5021no—
298Q8—5021no—
299Q8#1686080912yes—
300D4—22no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—10042no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4C2×C22⋊D25111yescharacteristic
320C2×C4C2×C22⋊D25111yescharacteristic
321C2×C4C2×C22⋊D25111yescharacteristic
322C2×C4C2×C22⋊D25111yescharacteristic
323C2×C4C23×D25111yescharacteristic
324C8—5021no—
325C8—5021no—
326C8—5021no—
327C8—5021no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#2254873512yes—
337C22C22×C22⋊D25111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—10041no—
346C4—10041no—
347C4—10041no—
348C4—10041no—
349C4C2×#1686080911yescharacteristic
350C4#2254876011yescharacteristic
351C2—22no—
352C2—41no—
353C2—81no—
354C2#3063233111yescharacteristic
355C2C2×#2254873511yescharacteristic
356C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting, prime radical#548×C251C2
socleC22×C251C22×D4
hypercenter#548D251
hypocenterC251#3300
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #30632377C2semidirect
2. #30632041C2semidirect
3. #30632025C2semidirect
4. #30631998C2semidirect
5. #30631994C2semidirect
6. #30631926C2semidirect
7. #30631914C2semidirect
8. #30631906C2semidirect
9. #30631849C2semidirect
10. #30631843C2semidirect
11. C4×#16860800C2semidirect
12. D4×C251⋊C8C2semidirect
13. #30631683C2semidirect
14. C2×#22548639C2semidirect
178. C251#3300semidirect

Maximal subgroups

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

subgroupindex
#306323772
#306320412
#306320252
#306319982
#306319942
#306319262
#306319142
#306319062
#306318492
#306318432
C4×#168608002
D4×C251⋊C82
#306316832
C2×#225486392
#548×C2512
#3300251