Finite Group Databasebeta

← #27185242

Subgroups — #27185242

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#20265909C213yesmaximal, complemented
2#20265906C211yesmaximal, characteristic, complemented
3#20265892C213yesmaximal, complemented
4#20265874C213yesmaximal, complemented
5#20265861C213yesmaximal, complemented
6#20265846C211yesmaximal, characteristic, complemented
7#20265838C211yesmaximal, characteristic, complemented
8#20265796C213yesmaximal, complemented
9#20265780C213yesmaximal, complemented
10#20265713C213yesmaximal, complemented
11#20265707C213yesmaximal, complemented
12#20265659C211yesmaximal, characteristic, complemented
13#20265655C211yesmaximal, characteristic, complemented
14#41×Dic101C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17#15692255C2213yes—
18#15692251C2213yes—
19#15692250C2213yes—
20#15692248C2216yes—
21#15692248C2211yescharacteristic
22#15692246C2213yes—
23#15692246C2213yes—
24#15692245C2213yes—
25#15692244C2213yes—
26#15692241C2216yes—
27#15692241C2213yes—
28#15692239C2216yes—
29#15692239C2213yes—
30#15692237C2216yes—
31#15692236C2213yes—
32#15692232C2213yes—
33#15692231C2213yes—
34#15692229C2213yes—
35#15692229C2213yes—
36#15692227C2213yes—
37#15692226C2216yes—
38#15692226C2213yes—
39Q8×Dic101C2213yes—
40Q8×Dic101C2211yescharacteristic
41Q8×Dic101C2211yescharacteristic
42Q8×Dic101C2213yes—
43Q8×Dic101C2211yescharacteristic
44C4×Dic202C2213yes—
45C4×D404C2213yes—
46D4×Dic101C2213yes—
47D4×Dic101C2213yes—
48#15692217C2213yes—
49C4×C22⋊D101C2213yes—
50C4×C22⋊D101C2213yes—
51#15692215C2216yes—
52#15692214C2213yes—
53#15692214C2213yes—
54#15692214C2213yes—
55#15692213C2216yes—
56#15692212C2213yes—
57C4⋊C4×D101C2213yes—
58C2×C4×Dic101C2213yes—
59C2×#41×C101C2211yescharacteristic
60C22×Q8×C101C2211yescharacteristic
61#124×C101C2213yes—
62C22⋊Q8×C101C2213yes—
63#121×C101C2213yes—
64C4×Q8×C101C2211yescharacteristic
65C4×D4×C101C2213yes—
66#13108075—24no—
67C2×Dic202C2313yes—
68C2×D404C2313yes—
69C2×C22⋊D101C2316yes—
70C2×C22⋊D101C2313yes—
71C2×C22⋊D101C2313yes—
72C22×Dic101C2313yes—
73C22×Dic101C2313yes—
74C2×C4×D101C2313yes—
75C2×C4×D101C2316yes—
76C4⋊Dic101C2313yes—
77C4⋊Dic101C2316yes—
78C4⋊Dic101C2313yes—
79C4⋊Dic101C2313yes—
80C4⋊Dic101C2313yes—
81C4⋊Dic101C2313yes—
82C4⋊Dic101C2311yescharacteristic
83#13108053C2313yes—
84#13108053C2311yescharacteristic
85#13108053C2313yes—
86#13108053C2311yescharacteristic
87#13108053C2313yes—
88#13108053C2313yes—
89C22⋊Dic101C2313yes—
90C22⋊Dic101C2311yescharacteristic
91C22⋊Dic101C2316yes—
92C22⋊Dic101C2311yescharacteristic
93C22⋊Dic101C2313yes—
94#13108051C2313yes—
95#13108051C2313yes—
96#13108051C2313yes—
97#13108051C2316yes—
98#13108051C2313yes—
99C4×Dic101C2311yescharacteristic
100C4×Dic101C2313yes—
101C4×Dic101C2313yes—
102C4×Dic101C2313yes—
103C4×Dic101C2316yes—
104C4×Dic101C2313yes—
105#41×C101—24no—
106C2×Q8×C101—24no—
107C2×Q8×C101C2313yes—
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101C2313yes—
110C2×D4×C101C2313yes—
111C22×C4×C101C2313yes—
112C22×C4×C101C2313yes—
113C4⋊C4×C101C2313yes—
114C4⋊C4×C101C2313yes—
115C22⋊C4×C101C2311yescharacteristic
116C22⋊C4×C101C2313yes—
117C22⋊C4×C101C2316yes—
118C42×C101C2313yes—
119C23×C101C2413yes—
120C23×C101C2411yescharacteristic
121C22×D101C2413yes—
122Q8×C101—26no—
123Q8×C101#4114yes—
124D4×C101—26no—
125D4×C101—43no—
126C2×C4×C101—26no—
127C2×C4×C101—26no—
128C2×C4×C101C2411yescharacteristic
129C2×C4×C101C2413yes—
130C2×C4×C101C2413yes—
131C2×C4×C101C2411yescharacteristic
132C2×C4×C101C2413yes—
133C22⋊D101—43no—
134C22⋊D101—46no—
135C22⋊D101—43no—
136C2×Dic101—43no—
137C2×Dic101—43no—
138C2×Dic101C2411yescharacteristic
139C2×Dic101C2411yescharacteristic
140C2×Dic101C2413yes—
141C2×Dic101C2413yes—
142C2×Dic101C2411yescharacteristic
143C2×Dic101C2413yes—
144C2×Dic101C2411yescharacteristic
145D404—26no—
146C4×D101—26no—
147C4×D101—46no—
148Dic202—43no—
149C22×C101—21no—
150C22×C101—23no—
151C22×C101—23no—
152C22×C101—23no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101C2511yescharacteristic
156C2×D101—23no—
157C2×D101—23no—
158C2×D101—23no—
159C4×C101—21no—
160C4×C101—23no—
161C4×C101—21no—
162C4×C101—23no—
163C4×C101C2×#4116yes—
164Dic101—21no—
165Dic101—23no—
166Dic101—23no—
167Dic101—23no—
168Dic101—21no—
169Dic101—21no—
170Dic101—21no—
171C2×C101—41no—
172C2×C101—43no—
173C2×C101C22×#4111yescharacteristic
174C2×C101#58511yescharacteristic
175C2×C101#58511yescharacteristic
176D101—43no—
177#3524—1011nomaximal, Sylow
178C101#352411yescharacteristic, complemented, Sylow
179#559—1013no—
180#557—1011no—
181#555—1013no—
182#553—1013no—
183#552—1013no—
184#550—1011no—
185#548—1011no—
186#548D10111yescharacteristic, complemented
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×#41C2×D10111yescharacteristic
196C22×Q8C2×D10111yescharacteristic
197C4⋊Q8—1013no—
198#127—1013no—
199#126—1013no—
200#126—1011no—
201#126—1016no—
202#125—1013no—
203#125—1013no—
204#125—1013no—
205#124—1013no—
206#124—1013no—
207#124—1016no—
208#124C2×D10113yes—
209#123—1013no—
210#123—1016no—
211#123—1013no—
212#123—1016no—
213C22⋊Q8—1013no—
214C22⋊Q8—1013no—
215C22⋊Q8C2×D10113yes—
216#121—1013no—
217#121—1016no—
218#121—1013no—
219#121—1013no—
220#121—1013no—
221#121C2×D10113yes—
222C4×Q8—1011no—
223C4×Q8—1011no—
224C4×Q8—1013no—
225C4×Q8—1011no—
226C4×Q8—1013no—
227C4×Q8—1013no—
228C4×Q8C2×D10111yescharacteristic
229C4×D4—1013no—
230C4×D4—1013no—
231C4×D4—1013no—
232C4×D4—1013no—
233C4×D4—1013no—
234C4×D4—1016no—
235C4×D4—1013no—
236C4×D4C2×D10113yes—
237#117—1013no—
238#117—1013no—
239#117—1013no—
240#117—1016no—
241#117—1013no—
242C2×C4⋊C4—1013no—
243C2×C42—1013no—
244#41—24no—
245#41—2024no—
246C2×Q8—24no—
247C2×Q8—1013no—
248C2×Q8C22×D10113yes—
249C2×Q8C22×D10111yescharacteristic
250C2×D4—1013no—
251C2×D4—1016no—
252C2×D4—1013no—
253C2×D4—1013no—
254C2×D4C22×D10113yes—
255C2×D4C22×D10113yes—
256C22×C4—1013no—
257C22×C4—1013no—
258C22×C4—1016no—
259C22×C4—1013no—
260C22×C4C22×D10113yes—
261C22×C4C22×D10113yes—
262C4⋊C4—1013no—
263C4⋊C4—1011no—
264C4⋊C4—1011no—
265C4⋊C4—1013no—
266C4⋊C4—1013no—
267C4⋊C4—1013no—
268C4⋊C4—1013no—
269C4⋊C4—1013no—
270C4⋊C4—1013no—
271C4⋊C4—1013no—
272C4⋊C4—1013no—
273C4⋊C4—1016no—
274C4⋊C4—1011no—
275C4⋊C4C22×D10113yes—
276C4⋊C4C22×D10113yes—
277C22⋊C4—1011no—
278C22⋊C4—1013no—
279C22⋊C4—1016no—
280C22⋊C4—1011no—
281C22⋊C4—1013no—
282C22⋊C4—1013no—
283C22⋊C4—1013no—
284C22⋊C4—1013no—
285C22⋊C4—1016no—
286C22⋊C4—1013no—
287C22⋊C4C22×D10116yes—
288C22⋊C4C22×D10111yescharacteristic
289C22⋊C4C22×D10113yes—
290C42—1011no—
291C42—1013no—
292C42—1013no—
293C42—1013no—
294C42—1016no—
295C42—1013no—
296C42C22×D10113yes—
297C23—1013no—
298C23C23×D10111yescharacteristic
299C23C23×D10113yes—
300Q8—26no—
301Q8—4043no—
302Q8#1310807314yes—
303D4—26no—
304D4—43no—
305D4—4043no—
306D4—4043no—
307D4—4046no—
308D4—2026no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—1011no—
312C2×C4—1013no—
313C2×C4—1011no—
314C2×C4—1013no—
315C2×C4—1011no—
316C2×C4—1013no—
317C2×C4—1011no—
318C2×C4—4046no—
319C2×C4—4043no—
320C2×C4—4043no—
321C2×C4—2026no—
322C2×C4C23×D10111yescharacteristic
323C2×C4C23×D10113yes—
324C2×C4C23×D10111yescharacteristic
325C2×C4C23×D10113yes—
326C2×C4C23×D10113yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—23no—
332C22—21no—
333C22—2023no—
334C22—2023no—
335C22—2023no—
336C22C24×D10111yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—2021no—
342C4—2023no—
343C4—2021no—
344C4—2021no—
345C4—2021no—
346C4—2023no—
347C4—2023no—
348C4C2×#1310807316yes—
349C2—41no—
350C2—43no—
351C2—4043no—
352C2C22×#1310807311yescharacteristic
353C2#2026601311yescharacteristic
354C2#2026601411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#548×C101C2
hypercenter#548D101
hypocenterC101#3524
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20265909C2semidirect
2. #20265906C2semidirect
3. #20265892C2semidirect
4. #20265874C2semidirect
5. #20265861C2semidirect
6. #20265846C2semidirect
7. #20265838C2semidirect
8. #20265796C2semidirect
9. #20265780C2semidirect
10. #20265713C2semidirect
11. #20265707C2semidirect
12. #20265659C2semidirect
13. #20265655C2semidirect
14. #41×Dic101C2semidirect
15. #548×C101C2semidirect
178. C101#3524semidirect
186. #548D101semidirect

Maximal subgroups

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

subgroupindex
#202659092
#202659062
#202658922
#202658742
#202658612
#202658462
#202658382
#202657962
#202657802
#202657132
#202657072
#202656592
#202656552
#41×Dic1012
#548×C1012
#3524101