Finite Group Databasebeta

← #27185244

Subgroups — #27185244

Order 12928.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265912C213yesmaximal, complemented
2#20265893C213yesmaximal, complemented
3#20265880C213yesmaximal, complemented
4#20265865C213yesmaximal, complemented
5#20265849C211yesmaximal, characteristic, complemented
6#20265839C211yesmaximal, characteristic, complemented
7#20265839C211yesmaximal, characteristic, complemented
8#20265801C213yesmaximal, complemented
9#20265783C213yesmaximal, complemented
10#20265714C213yesmaximal, complemented
11#20265712C213yesmaximal, complemented
12#20265659C211yesmaximal, characteristic, complemented
13#20265653C211yesmaximal, characteristic, complemented
14C4×#13108075C211yesmaximal, characteristic, complemented
15#557×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108075C2211yescharacteristic
18C2×Q8×D101C2211yescharacteristic
19#15692252C2213yes—
20#15692250C2216yes—
21#15692250C2213yes—
22#15692250C2211yescharacteristic
23#15692246C2213yes—
24#15692244C2216yes—
25#15692244C2213yes—
26#15692243C2213yes—
27#15692243C2213yes—
28#15692240C2216yes—
29#15692240C2216yes—
30#15692240C2213yes—
31#15692240C2213yes—
32#15692235C2213yes—
33#15692235C2213yes—
34#15692231C2213yes—
35#15692230C2213yes—
36#15692230C2213yes—
37#15692230C2213yes—
38#15692230C2216yes—
39#15692230C2213yes—
40#15692230C2213yes—
41Q8×Dic101C2211yescharacteristic
42C4×Dic202C2213yes—
43#15692220C2213yes—
44C4×D404C2213yes—
45C4×D404C2213yes—
46C4×D404C2213yes—
47C4×D404C2213yes—
48C4×D404C2213yes—
49#15692217C2213yes—
50#15692217C2213yes—
51#15692217C2216yes—
52#15692213C2213yes—
53#15692213C2216yes—
54#15692213C2213yes—
55#15692212C2213yes—
56#15692212C2213yes—
57C4⋊C4×D101C2213yes—
58C42×D101C2213yes—
59C4⋊Q8×C101C2213yes—
60#125×C101C2213yes—
61#125×C101C2213yes—
62C4×Q8×C101C2211yescharacteristic
63C4×Q8×C101C2211yescharacteristic
64C4×Q8×C101C2211yescharacteristic
65C4×Q8×C101C2213yes—
66#13108075—24no—
67#13108075—24no—
68C2×Dic202C2313yes—
69Q8×D101—24no—
70C2×D404C2313yes—
71C2×D404C2313yes—
72C2×D404C2316yes—
73C2×D404C2313yes—
74C2×D404C2313yes—
75C2×D404C2313yes—
76C2×C4×D101C2313yes—
77C2×C4×D101C2313yes—
78C2×C4×D101C2313yes—
79C2×C4×D101C2313yes—
80C2×C4×D101C2313yes—
81C2×C4×D101C2316yes—
82C4⋊Dic101C2313yes—
83C4⋊Dic101C2313yes—
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2313yes—
86#13108053C2313yes—
87#13108053C2311yescharacteristic
88#13108053C2313yes—
89#13108053C2313yes—
90#13108051C2311yescharacteristic
91#13108051C2313yes—
92#13108051C2313yes—
93#13108051C2313yes—
94#13108051C2313yes—
95#13108051C2316yes—
96#13108051C2316yes—
97#13108051C2313yes—
98#13108051C2316yes—
99#13108051C2311yescharacteristic
100#13108051C2311yescharacteristic
101#13108051C2313yes—
102#13108051C2313yes—
103C4×Dic101C2311yescharacteristic
104C4×Dic101C2313yes—
105C4×Dic101C2313yes—
106C4×Dic101C2316yes—
107C2×Q8×C101C2313yes—
108C2×Q8×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C4⋊C4×C101C2313yes—
111C4⋊C4×C101C2313yes—
112C4⋊C4×C101C2313yes—
113C4⋊C4×C101C2313yes—
114C4⋊C4×C101C2313yes—
115C4⋊C4×C101C2316yes—
116C42×C101C2313yes—
117C42×C101C2313yes—
118C42×C101C2313yes—
119C22×D101C2411yescharacteristic
120C22×D101C2413yes—
121C22×D101C2413yes—
122Q8×C101—43no—
123Q8×C101#4114yes—
124C2×C4×C101C2411yescharacteristic
125C2×C4×C101C2411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C2×C4×C101C2413yes—
128C2×C4×C101C2413yes—
129C2×C4×C101C2413yes—
130C2×C4×C101C2413yes—
131C2×Dic101C2411yescharacteristic
132C2×Dic101C2413yes—
133C2×Dic101C2411yescharacteristic
134C2×Dic101C2413yes—
135C2×Dic101C2411yescharacteristic
136D404—26no—
137D404—26no—
138D404—43no—
139D404—43no—
140D404—43no—
141D404—46no—
142C4×D101—26no—
143C4×D101—26no—
144C4×D101—26no—
145C4×D101—43no—
146C4×D101—46no—
147C4×D101—43no—
148Dic202—26no—
149C22×C101C2511yescharacteristic
150C2×D101—21no—
151C2×D101—23no—
152C2×D101—23no—
153C2×D101—23no—
154C2×D101—23no—
155C2×D101—23no—
156C2×D101—21no—
157C2×D101—23no—
158C2×D101—21no—
159C4×C101—21no—
160C4×C101—23no—
161C4×C101—23no—
162C4×C101—21no—
163C4×C101—23no—
164C4×C101—21no—
165C4×C101C2×#4116yes—
166Dic101—23no—
167Dic101—21no—
168Dic101—21no—
169Dic101—21no—
170Dic101—23no—
171C2×C101C22×#4111yescharacteristic
172C2×C101#58511yescharacteristic
173C2×C101#58511yescharacteristic
174D101—41no—
175D101—43no—
176D101—43no—
177#3524—1011nomaximal, Sylow
178C101#352411yescharacteristic, complemented, Sylow
179#559—1013no—
180#557D10111yescharacteristic, complemented
181#555—1013no—
182#553—1013no—
183#552—1013no—
184#550—1011no—
185#548—1011no—
186#548—1011no—
187#542—1013no—
188#540—1013no—
189#529—1013no—
190#529—1013no—
191#520—1011no—
192#519—1011no—
193C4×#41—1011no—
194C2×#41—1011no—
195C2×#41—1011no—
196C22×Q8—1011no—
197C4⋊Q8C2×D10113yes—
198#127—1013no—
199#126—1011no—
200#126—1013no—
201#126—1016no—
202#125—1013no—
203#125C2×D10113yes—
204#125C2×D10113yes—
205#124—1013no—
206#124—1016no—
207#124—1013no—
208#124—1013no—
209#123—1016no—
210#123—1013no—
211#123—1013no—
212#123—1016no—
213C22⋊Q8—1013no—
214C22⋊Q8—1013no—
215C22⋊Q8—1013no—
216#121—1013no—
217#121—1013no—
218#121—1013no—
219#121—1013no—
220#121—1016no—
221#121—1013no—
222C4×Q8—1011no—
223C4×Q8—1013no—
224C4×Q8—1013no—
225C4×Q8C2×D10111yescharacteristic
226C4×Q8C2×D10111yescharacteristic
227C4×Q8C2×D10111yescharacteristic
228C4×Q8C2×D10113yes—
229C4×D4—1013no—
230C4×D4—1013no—
231C4×D4—1016no—
232C4×D4—1013no—
233C4×D4—1013no—
234C4×D4—1013no—
235C4×D4—1013no—
236C4×D4—1013no—
237#117—1013no—
238#117—1013no—
239#117—1016no—
240#117—1013no—
241#117—1013no—
242C2×C4⋊C4—1013no—
243C2×C42—1013no—
244#41—2024no—
245#41—2024no—
246C2×Q8—1013no—
247C2×Q8—2024no—
248C2×Q8C22×D10113yes—
249C2×Q8C22×D10111yescharacteristic
250C2×D4—1013no—
251C2×D4—1013no—
252C2×D4—1016no—
253C2×D4—1013no—
254C2×D4—1013no—
255C2×D4—1013no—
256C22×C4—1013no—
257C22×C4—1013no—
258C22×C4—1016no—
259C22×C4—1013no—
260C22×C4—1013no—
261C22×C4—1013no—
262C4⋊C4—1013no—
263C4⋊C4—1011no—
264C4⋊C4—1013no—
265C4⋊C4—1013no—
266C4⋊C4—1013no—
267C4⋊C4—1013no—
268C4⋊C4—1011no—
269C4⋊C4—1013no—
270C4⋊C4C22×D10113yes—
271C4⋊C4C22×D10113yes—
272C4⋊C4C22×D10113yes—
273C4⋊C4C22×D10116yes—
274C4⋊C4C22×D10113yes—
275C4⋊C4C22×D10111yescharacteristic
276C4⋊C4C22×D10113yes—
277C22⋊C4—1011no—
278C22⋊C4—1013no—
279C22⋊C4—1013no—
280C22⋊C4—1013no—
281C22⋊C4—1016no—
282C22⋊C4—1013no—
283C22⋊C4—1016no—
284C22⋊C4—1011no—
285C22⋊C4—1013no—
286C22⋊C4—1011no—
287C22⋊C4—1013no—
288C22⋊C4—1016no—
289C22⋊C4—1013no—
290C42—1011no—
291C42—1013no—
292C42—1013no—
293C42—1016no—
294C42C22×D10113yes—
295C42C22×D10113yes—
296C42C22×D10113yes—
297C23—1011no—
298C23—1013no—
299C23—1013no—
300Q8—43no—
301Q8—2026no—
302Q8#1310807514yes—
303D4—4043no—
304D4—4046no—
305D4—4043no—
306D4—4043no—
307D4—2026no—
308D4—2026no—
309C2×C4—1011no—
310C2×C4—1011no—
311C2×C4—1013no—
312C2×C4—1013no—
313C2×C4—1011no—
314C2×C4—4043no—
315C2×C4—4043no—
316C2×C4—4046no—
317C2×C4—2026no—
318C2×C4—2026no—
319C2×C4—2026no—
320C2×C4C23×D10113yes—
321C2×C4C23×D10111yescharacteristic
322C2×C4C23×D10113yes—
323C2×C4C23×D10111yescharacteristic
324C2×C4C23×D10111yescharacteristic
325C2×C4C23×D10113yes—
326C2×C4C23×D10113yes—
327C22—2021no—
328C22—2023no—
329C22—2023no—
330C22—2023no—
331C22—2021no—
332C22—2023no—
333C22—2023no—
334C22—2023no—
335C22—2021no—
336C22C24×D10111yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—2021no—
344C4—2021no—
345C4—2023no—
346C4—2021no—
347C4—2023no—
348C4C2×#1310807516yes—
349C2—4041no—
350C2—4043no—
351C2—4043no—
352C2C22×#1310807511yescharacteristic
353C2#2026601311yescharacteristic
354C2#2026601411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#557×C101C2
hypercenter#557D101
hypocenterC101#3524
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #20265912C2semidirect
2. #20265893C2semidirect
3. #20265880C2semidirect
4. #20265865C2semidirect
5. #20265849C2semidirect
6. #20265839C2semidirect
7. #20265839C2semidirect
8. #20265801C2semidirect
9. #20265783C2semidirect
10. #20265714C2semidirect
11. #20265712C2semidirect
12. #20265659C2semidirect
13. #20265653C2semidirect
14. C4×#13108075C2semidirect
15. #557×C101C2semidirect
178. C101#3524semidirect
180. #557D101semidirect

Maximal subgroups

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

subgroupindex
#202659122
#202658932
#202658802
#202658652
#202658492
#202658392
#202658012
#202657832
#202657142
#202657122
#202656592
#202656532
C4×#131080752
#557×C1012
#3524101