Finite Group Databasebeta

← #27185211

Subgroups — #27185211

Order 12928.

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#20265910C213yesmaximal, complemented
2Q8×Dic202C211yesmaximal, characteristic, complemented
3#20265882C213yesmaximal, complemented
4#20265846C211yesmaximal, characteristic, complemented
5#20265839C211yesmaximal, characteristic, complemented
6#20265834C211yesmaximal, characteristic, complemented
7#20265832C213yesmaximal, complemented
8#20265828C213yesmaximal, complemented
9#20265798C213yesmaximal, complemented
10#20265792C213yesmaximal, complemented
11#20265724C213yesmaximal, complemented
12#20265705C213yesmaximal, complemented
13#20265659C211yesmaximal, characteristic, complemented
14C4×Q8×D101C211yesmaximal, characteristic, complemented
15#520×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108074C2216yes—
18C2×Q8×D101C2211yescharacteristic
19#15692255C2213yes—
20#15692254C2213yes—
21#15692254C2213yes—
22#15692251C2213yes—
23#15692250C2213yes—
24#15692248C2213yes—
25#15692245C2213yes—
26#15692243C2213yes—
27#15692242C2213yes—
28#15692241C2216yes—
29#15692241C2213yes—
30#15692240C2213yes—
31#15692238C2216yes—
32#15692237C2216yes—
33#15692237C2213yes—
34#15692235C2213yes—
35#15692234C2216yes—
36#15692234C2213yes—
37#15692233C2216yes—
38#15692233C2211yescharacteristic
39#15692230C2213yes—
40#15692229C2213yes—
41#15692228C2213yes—
42#15692226C2213yes—
43#15692226C2213yes—
44Q8×Dic101C2211yescharacteristic
45Q8×Dic101C2211yescharacteristic
46C4×Dic202C2213yes—
47C4×Dic202C2213yes—
48#15692220C2213yes—
49C4×D404C2213yes—
50C4×D404C2213yes—
51#15692217C2213yes—
52C4×C22⋊D101C2213yes—
53C4×C22⋊D101C2213yes—
54C4×C22⋊D101C2216yes—
55#15692212C2213yes—
56C4⋊C4×D101C2213yes—
57C4⋊C4×D101C2213yes—
58C42×D101C2213yes—
59C2×#41×C101C2211yescharacteristic
60C4×Q8×C101C2211yescharacteristic
61C4×Q8×C101C2211yescharacteristic
62C4×D4×C101C2213yes—
63C4×D4×C101C2213yes—
64#117×C101C2213yes—
65C2×C4⋊C4×C101C2213yes—
66#13108075—24no—
67#13108074—412no—
68C2×Dic202C2313yes—
69C2×Dic202C2311yescharacteristic
70C2×Dic202C2316yes—
71Q8×D101—24no—
72C2×D404C2313yes—
73C2×D404C2313yes—
74C2×C22⋊D101C2313yes—
75C2×C22⋊D101C2313yes—
76C2×C22⋊D101C2316yes—
77C2×C4×D101C2313yes—
78C2×C4×D101C2313yes—
79C2×C4×D101C2313yes—
80C2×C4×D101C2311yescharacteristic
81C2×C4×D101C2316yes—
82C4⋊Dic101C2313yes—
83C4⋊Dic101C2313yes—
84C4⋊Dic101C2313yes—
85C4⋊Dic101C2313yes—
86#13108053C2313yes—
87#13108053C2316yes—
88#13108053C2311yescharacteristic
89#13108053C2311yescharacteristic
90#13108053C2313yes—
91#13108053C2313yes—
92#13108053C2313yes—
93C22⋊Dic101C2313yes—
94C22⋊Dic101C2316yes—
95C22⋊Dic101C2313yes—
96#13108051C2313yes—
97#13108051C2313yes—
98#13108051C2311yescharacteristic
99#13108051C2313yes—
100#13108051C2316yes—
101#13108051C2311yescharacteristic
102#13108051C2313yes—
103C4×Dic101C2313yes—
104C4×Dic101C2313yes—
105C4×Dic101C2313yes—
106C4×Dic101C2313yes—
107#41×C101—24no—
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101C2313yes—
110C22×C4×C101C2313yes—
111C22×C4×C101C2316yes—
112C4⋊C4×C101C2313yes—
113C4⋊C4×C101C2313yes—
114C4⋊C4×C101C2313yes—
115C4⋊C4×C101C2311yescharacteristic
116C22⋊C4×C101C2313yes—
117C22⋊C4×C101C2313yes—
118C42×C101C2313yes—
119C42×C101C2313yes—
120C23×C101C2413yes—
121C22×D101C2413yes—
122C22×D101C2411yescharacteristic
123Q8×C101#4114yescomplemented
124D4×C101—26no—
125C2×C4×C101—26no—
126C2×C4×C101—46no—
127C2×C4×C101C2411yescharacteristic
128C2×C4×C101C2411yescharacteristic
129C2×C4×C101C2413yes—
130C2×C4×C101C2413yes—
131C2×C4×C101C2411yescharacteristic
132C2×C4×C101C2413yes—
133C22⋊D101—26no—
134C22⋊D101—26no—
135C22⋊D101—46no—
136C2×Dic101C2411yescharacteristic
137C2×Dic101C2413yes—
138C2×Dic101C2413yes—
139C2×Dic101C2413yes—
140C2×Dic101C2411yescharacteristic
141C2×Dic101C2411yescharacteristic
142D404—26no—
143D404—26no—
144C4×D101—26no—
145C4×D101—26no—
146C4×D101—26no—
147C4×D101—22no—
148C4×D101—46no—
149Dic202—26no—
150Dic202—22no—
151Dic202—46no—
152C22×C101—23no—
153C22×C101—23no—
154C22×C101—23no—
155C22×C101C2511yescharacteristic
156C2×D101—21no—
157C2×D101—21no—
158C2×D101—23no—
159C2×D101—23no—
160C2×D101—23no—
161C2×D101H(5,𝔽2)12yes—
162C4×C101—21no—
163C4×C101—23no—
164C4×C101—23no—
165C4×C101—21no—
166C4×C101—21no—
167C4×C101C2×#4116yes—
168Dic101—23no—
169Dic101—21no—
170Dic101—23no—
171Dic101—23no—
172Dic101—21no—
173Dic1012−1+412yes—
174C2×C101—43no—
175C2×C101C22×#4111yescharacteristic
176C2×C101C2×H(5,𝔽2)11yescharacteristic
177C2×C101C2×2−1+411yescharacteristic
178D101—22no—
179D101—43no—
180#3522—1011nomaximal, Sylow
181C101#352211yescharacteristic, complemented, Sylow
182#559—1013no—
183Q82—1011no—
184#554—1013no—
185#550—1011no—
186#548—1011no—
187#548—1011no—
188#547—1013no—
189#547—1013no—
190#542—1013no—
191#542—1013no—
192#532—1013no—
193#529—1013no—
194#520—1011no—
195#520D10111yescharacteristic, complemented
196C2×C4×Q8—1011no—
197C2×#41—1011no—
198C2×#41—1016no—
199C2×#41C2×D10111yescharacteristic
200C22×Q8—1011no—
201C4⋊Q8—1013no—
202C4⋊Q8—1013no—
203C4⋊Q8—1013no—
204#127—1013no—
205#126—1013no—
206#126—1013no—
207#125—1013no—
208#124—1013no—
209#124—1013no—
210#124—1016no—
211#124—1013no—
212#123—1016no—
213#123—1013no—
214#123—1013no—
215#123—1016no—
216C22⋊Q8—1016no—
217C22⋊Q8—1013no—
218C22⋊Q8—1011no—
219C22⋊Q8—1016no—
220C22⋊Q8—1013no—
221#121—1013no—
222#121—1013no—
223#121—1013no—
224#121—1013no—
225#121—1013no—
226C4×Q8—1011no—
227C4×Q8—1011no—
228C4×Q8—1013no—
229C4×Q8—1013no—
230C4×Q8—1013no—
231C4×Q8C2×D10111yescharacteristic
232C4×Q8C2×D10111yescharacteristic
233C4×D4—1013no—
234C4×D4—1013no—
235C4×D4—1016no—
236C4×D4—1013no—
237C4×D4—1013no—
238C4×D4—1013no—
239C4×D4C2×D10113yes—
240C4×D4C2×D10113yes—
241#117—1013no—
242#117C2×D10113yes—
243C2×C4⋊C4—1013no—
244C2×C4⋊C4—1013no—
245C2×C4⋊C4C2×D10113yes—
246C2×C42—1013no—
247#41—24no—
248#41—40412no—
249#41—2024no—
250C2×Q8—1016no—
251C2×Q8—1011no—
252C2×Q8—1013no—
253C2×Q8—2024no—
254C2×Q8C22×D10111yescharacteristic
255C2×D4—1016no—
256C2×D4—1013no—
257C2×D4—1013no—
258C2×D4—1013no—
259C2×D4—1013no—
260C2×D4C22×D10113yes—
261C22×C4—1016no—
262C22×C4—1011no—
263C22×C4—1013no—
264C22×C4—1013no—
265C22×C4—1013no—
266C22×C4C22×D10113yes—
267C22×C4C22×D10116yes—
268C4⋊C4—1013no—
269C4⋊C4—1016no—
270C4⋊C4—1011no—
271C4⋊C4—1011no—
272C4⋊C4—1013no—
273C4⋊C4—1013no—
274C4⋊C4—1013no—
275C4⋊C4—1013no—
276C4⋊C4—1013no—
277C4⋊C4—1013no—
278C4⋊C4—1013no—
279C4⋊C4C22×D10113yes—
280C4⋊C4C22×D10113yes—
281C4⋊C4C22×D10113yes—
282C4⋊C4C22×D10111yescharacteristic
283C22⋊C4—1013no—
284C22⋊C4—1016no—
285C22⋊C4—1013no—
286C22⋊C4—1013no—
287C22⋊C4—1013no—
288C22⋊C4—1011no—
289C22⋊C4—1011no—
290C22⋊C4—1013no—
291C22⋊C4—1016no—
292C22⋊C4—1013no—
293C22⋊C4C22×D10113yes—
294C22⋊C4C22×D10113yes—
295C42—1013no—
296C42—1013no—
297C42—1013no—
298C42—1013no—
299C42C22×D10113yes—
300C42C22×D10113yes—
301C23—1011no—
302C23—1013no—
303C23C23×D10113yes—
304Q8—4046no—
305Q8—2022no—
306Q8—2026no—
307Q8#1310807414yescomplemented
308D4—26no—
309D4—4046no—
310D4—2026no—
311D4—2026no—
312D4—2026no—
313D4—2026no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—1011no—
317C2×C4—1011no—
318C2×C4—1013no—
319C2×C4—1013no—
320C2×C4—1013no—
321C2×C4—1011no—
322C2×C4—4046no—
323C2×C4—2022no—
324C2×C4—2026no—
325C2×C4—2026no—
326C2×C4—2026no—
327C2×C4C23×D10111yescharacteristic
328C2×C4C23×D10113yes—
329C2×C4C23×D10111yescharacteristic
330C2×C4C23×D10113yes—
331C2×C4C23×D10111yescharacteristic
332C2×C4C23×D10113yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—1012no—
337C22—2023no—
338C22—2023no—
339C22—2023no—
340C22—2021no—
341C22—2021no—
342C22C24×D10111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—1012no—
349C4—2021no—
350C4—2023no—
351C4—2023no—
352C4—2023no—
353C4—2021no—
354C4C2×#1310807416yes—
355C2—43no—
356C2—4043no—
357C2—2022no—
358C2C22×#1310807411yescharacteristic
359C2H(5,𝔽2)×D10111yescharacteristic
360C22−1+4×D10111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#520×C101C2
hypercenter#520D101
hypocenterC101#3522
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20265910C2semidirect
2. Q8×Dic202C2semidirect
3. #20265882C2semidirect
4. #20265846C2semidirect
5. #20265839C2semidirect
6. #20265834C2semidirect
7. #20265832C2semidirect
8. #20265828C2semidirect
9. #20265798C2semidirect
10. #20265792C2semidirect
11. #20265724C2semidirect
12. #20265705C2semidirect
13. #20265659C2semidirect
14. C4×Q8×D101C2semidirect
15. #520×C101C2semidirect
123. Q8×C101#41semidirect
181. C101#3522semidirect
195. #520D101semidirect
307. Q8#13108074semidirect

Maximal subgroups

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

subgroupindex
#202659102
Q8×Dic2022
#202658822
#202658462
#202658392
#202658342
#202658322
#202658282
#202657982
#202657922
#202657242
#202657052
#202656592
C4×Q8×D1012
#520×C1012
#3522101