Finite Group Databasebeta

← #44564444

Subgroups — #44564444

Order 32128.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632420C213yesmaximal, complemented
2Q8×Dic502C211yesmaximal, characteristic, complemented
3#30632392C213yesmaximal, complemented
4#30632356C211yesmaximal, characteristic, complemented
5#30632349C211yesmaximal, characteristic, complemented
6#30632344C211yesmaximal, characteristic, complemented
7#30632342C213yesmaximal, complemented
8#30632338C213yesmaximal, complemented
9#30632308C213yesmaximal, complemented
10#30632302C213yesmaximal, complemented
11#30632234C213yesmaximal, complemented
12#30632215C213yesmaximal, complemented
13#30632169C211yesmaximal, characteristic, complemented
14C4×Q8×D251C211yesmaximal, characteristic, complemented
15#520×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×#16860810C2216yes—
18C2×Q8×D251C2211yescharacteristic
19#22548715C2213yes—
20#22548714C2213yes—
21#22548714C2213yes—
22#22548711C2213yes—
23#22548710C2213yes—
24#22548708C2213yes—
25#22548705C2213yes—
26#22548703C2213yes—
27#22548702C2213yes—
28#22548701C2213yes—
29#22548701C2216yes—
30#22548700C2213yes—
31#22548698C2216yes—
32#22548697C2213yes—
33#22548697C2216yes—
34#22548695C2213yes—
35#22548694C2216yes—
36#22548694C2213yes—
37#22548693C2211yescharacteristic
38#22548693C2216yes—
39#22548690C2213yes—
40#22548689C2213yes—
41#22548688C2213yes—
42#22548686C2213yes—
43#22548686C2213yes—
44Q8×Dic251C2211yescharacteristic
45Q8×Dic251C2211yescharacteristic
46C4×Dic502C2213yes—
47C4×Dic502C2213yes—
48#22548680C2213yes—
49C4×D1004C2213yes—
50C4×D1004C2213yes—
51#22548677C2213yes—
52C4×C22⋊D251C2213yes—
53C4×C22⋊D251C2213yes—
54C4×C22⋊D251C2216yes—
55#22548672C2213yes—
56C4⋊C4×D251C2213yes—
57C4⋊C4×D251C2213yes—
58C42×D251C2213yes—
59C2×#41×C251C2211yescharacteristic
60C4×Q8×C251C2211yescharacteristic
61C4×Q8×C251C2211yescharacteristic
62C4×D4×C251C2213yes—
63C4×D4×C251C2213yes—
64#117×C251C2213yes—
65C2×C4⋊C4×C251C2213yes—
66#16860811—24no—
67#16860810—412no—
68C2×Dic502C2313yes—
69C2×Dic502C2311yescharacteristic
70C2×Dic502C2316yes—
71Q8×D251—24no—
72C2×D1004C2313yes—
73C2×D1004C2313yes—
74C2×C22⋊D251C2313yes—
75C2×C22⋊D251C2313yes—
76C2×C22⋊D251C2316yes—
77C2×C4×D251C2313yes—
78C2×C4×D251C2313yes—
79C2×C4×D251C2313yes—
80C2×C4×D251C2311yescharacteristic
81C2×C4×D251C2316yes—
82C4⋊Dic251C2313yes—
83C4⋊Dic251C2313yes—
84C4⋊Dic251C2313yes—
85C4⋊Dic251C2313yes—
86Dic251⋊C4C2313yes—
87Dic251⋊C4C2316yes—
88Dic251⋊C4C2311yescharacteristic
89Dic251⋊C4C2311yescharacteristic
90Dic251⋊C4C2313yes—
91Dic251⋊C4C2313yes—
92Dic251⋊C4C2313yes—
93C22⋊Dic251C2313yes—
94C22⋊Dic251C2316yes—
95C22⋊Dic251C2313yes—
96(C2×D251)⋊C4C2313yes—
97(C2×D251)⋊C4C2313yes—
98(C2×D251)⋊C4C2311yescharacteristic
99(C2×D251)⋊C4C2313yes—
100(C2×D251)⋊C4C2316yes—
101(C2×D251)⋊C4C2311yescharacteristic
102(C2×D251)⋊C4C2313yes—
103C4×Dic251C2313yes—
104C4×Dic251C2313yes—
105C4×Dic251C2313yes—
106C4×Dic251C2313yes—
107#41×C251—24no—
108C2×Q8×C251C2311yescharacteristic
109C2×D4×C251C2313yes—
110C22×C4×C251C2313yes—
111C22×C4×C251C2316yes—
112C4⋊C4×C251C2313yes—
113C4⋊C4×C251C2313yes—
114C4⋊C4×C251C2313yes—
115C4⋊C4×C251C2311yescharacteristic
116C22⋊C4×C251C2313yes—
117C22⋊C4×C251C2313yes—
118C42×C251C2313yes—
119C42×C251C2313yes—
120C22×D251C2413yes—
121C22×D251C2411yescharacteristic
122Dic502—26no—
123Dic502—22no—
124Dic502—46no—
125D1004—26no—
126D1004—26no—
127C22⋊D251—26no—
128C22⋊D251—26no—
129C22⋊D251—46no—
130C2×Dic251C2411yescharacteristic
131C2×Dic251C2413yes—
132C2×Dic251C2413yes—
133C2×Dic251C2413yes—
134C2×Dic251C2411yescharacteristic
135C2×Dic251C2411yescharacteristic
136C4×D251—26no—
137C4×D251—26no—
138C4×D251—26no—
139C4×D251—22no—
140C4×D251—46no—
141C23×C251C2413yes—
142Q8×C251#4114yescomplemented
143D4×C251—26no—
144C2×C4×C251—26no—
145C2×C4×C251—46no—
146C2×C4×C251C2411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C2×C4×C251C2413yes—
149C2×C4×C251C2413yes—
150C2×C4×C251C2411yescharacteristic
151C2×C4×C251C2413yes—
152C22×C251—23no—
153C22×C251—23no—
154C22×C251—23no—
155C22×C251C2511yescharacteristic
156C2×D251—21no—
157C2×D251—21no—
158C2×D251—23no—
159C2×D251—23no—
160C2×D251—23no—
161C2×D251H(5,𝔽2)12yes—
162C4×C251—21no—
163C4×C251—23no—
164C4×C251—23no—
165C4×C251—21no—
166C4×C251—21no—
167C4×C251C2×#4116yes—
168Dic251—21no—
169Dic251—23no—
170Dic251—23no—
171Dic251—23no—
172Dic251—21no—
173Dic2512−1+412yes—
174C2×C251—43no—
175C2×C251C22×#4111yescharacteristic
176C2×C251C2×H(5,𝔽2)11yescharacteristic
177C2×C251C2×2−1+411yescharacteristic
178D251—22no—
179D251—43no—
180C251#352211yescharacteristic, complemented, Sylow
181#3522—2511nomaximal, Sylow
182#559—2513no—
183Q82—2511no—
184#554—2513no—
185#550—2511no—
186#548—2511no—
187#548—2511no—
188#547—2513no—
189#547—2513no—
190#542—2513no—
191#542—2513no—
192#532—2513no—
193#529—2513no—
194#520—2511no—
195#520D25111yescharacteristic, complemented
196C2×C4×Q8—2511no—
197C2×#41—2511no—
198C2×#41—2516no—
199C2×#41C2×D25111yescharacteristic
200C22×Q8—2511no—
201C4⋊Q8—2513no—
202C4⋊Q8—2513no—
203C4⋊Q8—2513no—
204#127—2513no—
205#126—2513no—
206#126—2513no—
207#125—2513no—
208#124—2513no—
209#124—2513no—
210#124—2516no—
211#124—2513no—
212#123—2516no—
213#123—2513no—
214#123—2513no—
215#123—2516no—
216C22⋊Q8—2516no—
217C22⋊Q8—2513no—
218C22⋊Q8—2511no—
219C22⋊Q8—2516no—
220C22⋊Q8—2513no—
221#121—2513no—
222#121—2513no—
223#121—2513no—
224#121—2513no—
225#121—2513no—
226C4×Q8—2511no—
227C4×Q8—2511no—
228C4×Q8—2513no—
229C4×Q8—2513no—
230C4×Q8—2513no—
231C4×Q8C2×D25111yescharacteristic
232C4×Q8C2×D25111yescharacteristic
233C4×D4—2513no—
234C4×D4—2513no—
235C4×D4—2516no—
236C4×D4—2513no—
237C4×D4—2513no—
238C4×D4—2513no—
239C4×D4C2×D25113yes—
240C4×D4C2×D25113yes—
241#117—2513no—
242#117C2×D25113yes—
243C2×C4⋊C4—2513no—
244C2×C4⋊C4—2513no—
245C2×C4⋊C4C2×D25113yes—
246C2×C42—2513no—
247#41—24no—
248#41—100412no—
249#41—5024no—
250C2×Q8—5024no—
251C2×Q8—2516no—
252C2×Q8—2511no—
253C2×Q8—2513no—
254C2×Q8C22×D25111yescharacteristic
255C2×D4—2516no—
256C2×D4—2513no—
257C2×D4—2513no—
258C2×D4—2513no—
259C2×D4—2513no—
260C2×D4C22×D25113yes—
261C22×C4—2516no—
262C22×C4—2511no—
263C22×C4—2513no—
264C22×C4—2513no—
265C22×C4—2513no—
266C22×C4C22×D25113yes—
267C22×C4C22×D25116yes—
268C4⋊C4—2513no—
269C4⋊C4—2516no—
270C4⋊C4—2513no—
271C4⋊C4—2513no—
272C4⋊C4—2511no—
273C4⋊C4—2511no—
274C4⋊C4—2513no—
275C4⋊C4—2513no—
276C4⋊C4—2513no—
277C4⋊C4—2513no—
278C4⋊C4—2513no—
279C4⋊C4C22×D25113yes—
280C4⋊C4C22×D25113yes—
281C4⋊C4C22×D25113yes—
282C4⋊C4C22×D25111yescharacteristic
283C22⋊C4—2513no—
284C22⋊C4—2516no—
285C22⋊C4—2513no—
286C22⋊C4—2513no—
287C22⋊C4—2513no—
288C22⋊C4—2511no—
289C22⋊C4—2511no—
290C22⋊C4—2513no—
291C22⋊C4—2516no—
292C22⋊C4—2513no—
293C22⋊C4C22×D25113yes—
294C22⋊C4C22×D25113yes—
295C42—2513no—
296C42—2513no—
297C42—2513no—
298C42—2513no—
299C42C22×D25113yes—
300C42C22×D25113yes—
301C23—2513no—
302C23—2511no—
303C23C23×D25113yes—
304Q8—10046no—
305Q8—5022no—
306Q8—5026no—
307Q8#1686081014yescomplemented
308D4—26no—
309D4—10046no—
310D4—5026no—
311D4—5026no—
312D4—5026no—
313D4—5026no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—10046no—
317C2×C4—5026no—
318C2×C4—5022no—
319C2×C4—5026no—
320C2×C4—5026no—
321C2×C4—2511no—
322C2×C4—2513no—
323C2×C4—2513no—
324C2×C4—2513no—
325C2×C4—2511no—
326C2×C4—2511no—
327C2×C4C23×D25111yescharacteristic
328C2×C4C23×D25113yes—
329C2×C4C23×D25111yescharacteristic
330C2×C4C23×D25113yes—
331C2×C4C23×D25111yescharacteristic
332C2×C4C23×D25113yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—5023no—
337C22—5023no—
338C22—5023no—
339C22—5021no—
340C22—5021no—
341C22—2512no—
342C22C24×D25111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—5021no—
349C4—5023no—
350C4—5023no—
351C4—5023no—
352C4—5021no—
353C4—2512no—
354C4C2×#1686081016yes—
355C2—43no—
356C2—10043no—
357C2—5022no—
358C2C22×#1686081011yescharacteristic
359C2H(5,𝔽2)×D25111yescharacteristic
360C22−1+4×D25111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D251
commutator, socle, cosocleC22×C251C25
Fitting#520×C251C2
hypercenter#520D251
hypocenterC251#3522
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #30632420C2semidirect
2. Q8×Dic502C2semidirect
3. #30632392C2semidirect
4. #30632356C2semidirect
5. #30632349C2semidirect
6. #30632344C2semidirect
7. #30632342C2semidirect
8. #30632338C2semidirect
9. #30632308C2semidirect
10. #30632302C2semidirect
11. #30632234C2semidirect
12. #30632215C2semidirect
13. #30632169C2semidirect
14. C4×Q8×D251C2semidirect
15. #520×C251C2semidirect
142. Q8×C251#41semidirect
180. C251#3522semidirect
195. #520D251semidirect
307. Q8#16860810semidirect

Maximal subgroups

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

subgroupindex
#306324202
Q8×Dic5022
#306323922
#306323562
#306323492
#306323442
#306323422
#306323382
#306323082
#306323022
#306322342
#306322152
#306321692
C4×Q8×D2512
#520×C2512
#3522251