Finite Group Databasebeta

← #27184722

Subgroups — #27184722

Order 12928.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265909C213yesmaximal, complemented
2#20265903C211yesmaximal, characteristic, complemented
3#20265898C213yesmaximal, complemented
4#20265879C213yesmaximal, complemented
5#20265873C213yesmaximal, complemented
6#20265846C211yesmaximal, characteristic, complemented
7#20265838C211yesmaximal, characteristic, complemented
8#20265722C213yesmaximal, complemented
9#20265713C213yesmaximal, complemented
10#20265706C213yesmaximal, complemented
11#41×Dic101C211yesmaximal, characteristic, complemented
12C4×#13108075C211yesmaximal, characteristic, complemented
13C4×#13108073C213yesmaximal, complemented
14C2×Q8×Dic101C211yesmaximal, characteristic, complemented
15#548×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108073C2213yes—
18#15692254C2213yes—
19#15692251C2213yes—
20#15692250C2216yes—
21#15692248C2216yes—
22#15692247C2213yes—
23#15692246C2213yes—
24#15692244C2213yes—
25#15692241C2216yes—
26#15692239C2216yes—
27#15692237C2216yes—
28#15692231C2213yes—
29#15692229C2213yes—
30#15692229C2213yes—
31#15692227C2216yes—
32#15692226C2213yes—
33Q8×Dic101C2211yescharacteristic
34Q8×Dic101C2211yescharacteristic
35Q8×Dic101C2213yes—
36Q8×Dic101C2211yescharacteristic
37Q8×Dic101C2213yes—
38C4×Dic202C2213yes—
39#15692220C2213yes—
40C4×D404C2213yes—
41D4×Dic101C2213yes—
42D4×Dic101C2213yes—
43D4×Dic101C2213yes—
44#15692217C2213yes—
45C4×C22⋊D101C2213yes—
46C4×C22⋊D101C2213yes—
47#15692215C2213yes—
48#15692215C2213yes—
49#15692215C2216yes—
50#15692214C2213yes—
51#15692213C2213yes—
52#15692213C2213yes—
53#15692211C2211yescharacteristic
54#15692211C2213yes—
55C2×C4⋊Dic101C2213yes—
56C2×C4×Dic101C2213yes—
57C2×C4×Dic101C2213yes—
58C42×D101C2213yes—
59C2×#41×C101C2211yescharacteristic
60C22×Q8×C101C2211yescharacteristic
61#124×C101C2213yes—
62C22⋊Q8×C101C2213yes—
63#121×C101C2213yes—
64C4×Q8×C101C2211yescharacteristic
65C4×D4×C101C2213yes—
66#13108075—24no—
67#13108073—212no—
68C2×Dic202C2313yes—
69C2×D404C2313yes—
70C2×C22⋊D101C2313yes—
71C2×C22⋊D101C2313yes—
72C2×C22⋊D101C2316yes—
73C22×Dic101C2313yes—
74C22×Dic101C2313yes—
75C22×Dic101C2313yes—
76C22×Dic101C2311yescharacteristic
77C2×C4×D101C2313yes—
78C2×C4×D101C2313yes—
79C2×C4×D101C2313yes—
80C4⋊Dic101C2313yes—
81C4⋊Dic101C2313yes—
82C4⋊Dic101C2316yes—
83C4⋊Dic101C2313yes—
84C4⋊Dic101C2313yes—
85#13108053C2311yescharacteristic
86#13108053C2313yes—
87#13108053C2311yescharacteristic
88#13108053C2313yes—
89#13108053C2316yes—
90C22⋊Dic101C2311yescharacteristic
91C22⋊Dic101C2313yes—
92C22⋊Dic101C2316yes—
93#13108051C2313yes—
94#13108051C2313yes—
95#13108051C2316yes—
96#13108051C2316yes—
97C4×Dic101C2311yescharacteristic
98C4×Dic101C2313yes—
99C4×Dic101C2313yes—
100C4×Dic101C2311yescharacteristic
101C4×Dic101C2313yes—
102C4×Dic101C2313yes—
103C4×Dic101C2313yes—
104C4×Dic101C2313yes—
105C4×Dic101C2313yes—
106#41×C101—24no—
107C2×Q8×C101—24no—
108C2×Q8×C101C2313yes—
109C2×Q8×C101C2311yescharacteristic
110C2×D4×C101C2313yes—
111C2×D4×C101C2313yes—
112C22×C4×C101C2313yes—
113C22×C4×C101C2313yes—
114C4⋊C4×C101C2313yes—
115C4⋊C4×C101C2313yes—
116C22⋊C4×C101C2311yescharacteristic
117C22⋊C4×C101C2313yes—
118C22⋊C4×C101C2316yes—
119C42×C101C2313yes—
120C23×C101C2413yes—
121C23×C101C2411yescharacteristic
122C22×D101C2413yes—
123Q8×C101—26no—
124Q8×C101#4114yescomplemented
125D4×C101—26no—
126D4×C101—26no—
127C2×C4×C101—26no—
128C2×C4×C101—26no—
129C2×C4×C101C2411yescharacteristic
130C2×C4×C101C2413yes—
131C2×C4×C101C2413yes—
132C2×C4×C101C2411yescharacteristic
133C2×C4×C101C2413yes—
134C22⋊D101—26no—
135C22⋊D101—26no—
136C22⋊D101—46no—
137C2×Dic101—26no—
138C2×Dic101—43no—
139C2×Dic101—43no—
140C2×Dic101#4114yescomplemented
141C2×Dic101C2411yescharacteristic
142C2×Dic101C2413yes—
143C2×Dic101C2411yescharacteristic
144C2×Dic101C2413yes—
145C2×Dic101C2413yes—
146C2×Dic101C2411yescharacteristic
147C2×Dic101C2411yescharacteristic
148D404—26no—
149C4×D101—26no—
150C4×D101—26no—
151C4×D101—43no—
152Dic202—26no—
153C22×C101—21no—
154C22×C101—21no—
155C22×C101—23no—
156C22×C101—23no—
157C22×C101—23no—
158C22×C101C2×#4112yes—
159C22×C101C2511yescharacteristic
160C2×D101—23no—
161C2×D101—23no—
162C2×D101—23no—
163C4×C101—23no—
164C4×C101—21no—
165C4×C101—23no—
166C4×C101—21no—
167C4×C101C2×#4116yes—
168Dic101—23no—
169Dic101—21no—
170Dic101—23no—
171Dic101—21no—
172Dic101—23no—
173Dic101C2×#4112yes—
174Dic101C2×#4112yes—
175C2×C101—22no—
176C2×C101—43no—
177C2×C101C22×#4111yescharacteristic
178C2×C101C22×#4111yescharacteristic
179C2×C101#58511yescharacteristic
180D101—43no—
181#3480—1011nomaximal, Sylow
182C101#348011yescharacteristic, complemented, Sylow
183#559—1013no—
184#557—1011no—
185#556—1013no—
186#553—1013no—
187#553—1013no—
188#550—1011no—
189#548—1011no—
190#548D10111yescharacteristic, complemented
191#532—1013no—
192#529—1013no—
193#529—1013no—
194C4×#41—1011no—
195C4×#41—1011no—
196C4×#41—1013no—
197C2×C4×Q8—1011no—
198C2×#41—1011no—
199C2×#41—1013no—
200C2×#41C2×D10111yescharacteristic
201C22×Q8C2×D10111yescharacteristic
202C4⋊Q8—1013no—
203#127—1013no—
204#126—1016no—
205#126—1016no—
206#125—1013no—
207#125—1013no—
208#124—1013no—
209#124—1016no—
210#124C2×D10113yes—
211#123—1016no—
212#123—1016no—
213C22⋊Q8—1013no—
214C22⋊Q8C2×D10113yes—
215#121—1013no—
216#121—1016no—
217#121—1013no—
218#121—1013no—
219#121C2×D10113yes—
220C4×Q8—1011no—
221C4×Q8—1013no—
222C4×Q8—1011no—
223C4×Q8—1013no—
224C4×Q8—1011no—
225C4×Q8—1013no—
226C4×Q8—1013no—
227C4×Q8C2×D10111yescharacteristic
228C4×D4—1013no—
229C4×D4—1013no—
230C4×D4—1013no—
231C4×D4—1013no—
232C4×D4—1013no—
233C4×D4—1013no—
234C4×D4—1013no—
235C4×D4—1013no—
236C4×D4—1016no—
237C4×D4—1013no—
238C4×D4C2×D10113yes—
239#117—1013no—
240#117—1013no—
241#117—1013no—
242#117—1011no—
243#117—1013no—
244C2×C4⋊C4—1013no—
245C2×C42—1013no—
246C2×C42—1013no—
247C2×C42—1013no—
248#41—24no—
249#41—20212no—
250#41—2024no—
251C2×Q8—24no—
252C2×Q8—1013no—
253C2×Q8C22×D10113yes—
254C2×Q8C22×D10111yescharacteristic
255C2×D4—1016no—
256C2×D4—1013no—
257C2×D4—1013no—
258C2×D4—1013no—
259C2×D4C22×D10113yes—
260C2×D4C22×D10113yes—
261C22×C4—1011no—
262C22×C4—1013no—
263C22×C4—1013no—
264C22×C4—1013no—
265C22×C4—1013no—
266C22×C4—1013no—
267C22×C4—1013no—
268C22×C4C22×D10113yes—
269C22×C4C22×D10113yes—
270C4⋊C4—1011no—
271C4⋊C4—1011no—
272C4⋊C4—1013no—
273C4⋊C4—1013no—
274C4⋊C4—1016no—
275C4⋊C4—1013no—
276C4⋊C4—1013no—
277C4⋊C4—1016no—
278C4⋊C4—1013no—
279C4⋊C4—1013no—
280C4⋊C4C22×D10113yes—
281C4⋊C4C22×D10113yes—
282C22⋊C4—1011no—
283C22⋊C4—1013no—
284C22⋊C4—1016no—
285C22⋊C4—1013no—
286C22⋊C4—1016no—
287C22⋊C4—1013no—
288C22⋊C4—1016no—
289C22⋊C4C22×D10111yescharacteristic
290C22⋊C4C22×D10113yes—
291C22⋊C4C22×D10116yes—
292C42—1011no—
293C42—1011no—
294C42—1013no—
295C42—1013no—
296C42—1013no—
297C42—1013no—
298C42—1013no—
299C42—1013no—
300C42—1013no—
301C42C22×D10113yes—
302C23—1013no—
303C23C23×D10113yes—
304C23C23×D10111yescharacteristic
305Q8—26no—
306Q8—2026no—
307Q8#1310807314yescomplemented
308D4—26no—
309D4—26no—
310D4—4046no—
311D4—2026no—
312D4—2026no—
313D4—2026no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—1014no—
317C2×C4—1011no—
318C2×C4—1011no—
319C2×C4—1011no—
320C2×C4—1011no—
321C2×C4—1013no—
322C2×C4—1013no—
323C2×C4—1013no—
324C2×C4—4043no—
325C2×C4—4043no—
326C2×C4—4043no—
327C2×C4—2026no—
328C2×C4—2026no—
329C2×C4—2026no—
330C2×C4C23×D10111yescharacteristic
331C2×C4C23×D10113yes—
332C2×C4C23×D10111yescharacteristic
333C2×C4C23×D10113yes—
334C2×C4C23×D10113yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—2023no—
341C22—2023no—
342C22—2023no—
343C22#41×D10112yes—
344C22C24×D10111yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—1012no—
350C4—1012no—
351C4—2023no—
352C4—2021no—
353C4—2023no—
354C4—2021no—
355C4—2023no—
356C4C2×#1310807316yes—
357C2—22no—
358C2—43no—
359C2—4043no—
360C2C2×#41×D10111yescharacteristic
361C2C22×#1310807311yescharacteristic
362C2#2026601411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#548×C101C2
hypercenter#548D101
hypocenterC101#3480
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20265909C2semidirect
2. #20265903C2semidirect
3. #20265898C2semidirect
4. #20265879C2semidirect
5. #20265873C2semidirect
6. #20265846C2semidirect
7. #20265838C2semidirect
8. #20265722C2semidirect
9. #20265713C2semidirect
10. #20265706C2semidirect
11. #41×Dic101C2semidirect
12. C4×#13108075C2semidirect
13. C4×#13108073C2semidirect
14. C2×Q8×Dic101C2semidirect
15. #548×C101C2semidirect
124. Q8×C101#41semidirect
140. C2×Dic101#41semidirect
182. C101#3480semidirect
190. #548D101semidirect
307. Q8#13108073semidirect

Maximal subgroups

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

subgroupindex
#202659092
#202659032
#202658982
#202658792
#202658732
#202658462
#202658382
#202657222
#202657132
#202657062
#41×Dic1012
C4×#131080752
C4×#131080732
C2×Q8×Dic1012
#548×C1012
#3480101