Finite Group Databasebeta

← #27184749

Subgroups — #27184749

Order 12928.

1094 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#20265903C211yesmaximal, characteristic, complemented
2#20265899C213yesmaximal, complemented
3#20265868C213yesmaximal, complemented
4#20265862C213yesmaximal, complemented
5Q8×C22⋊D101C211yesmaximal, characteristic, complemented
6#20265839C211yesmaximal, characteristic, complemented
7#20265838C211yesmaximal, characteristic, complemented
8#20265828C213yesmaximal, complemented
9#20265731C213yesmaximal, complemented
10#20265714C213yesmaximal, complemented
11#20265711C213yesmaximal, complemented
12C4×#13108075C211yesmaximal, characteristic, complemented
13C4×#13108074C213yesmaximal, complemented
14C4×Q8×D101C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C101C211yesmaximal, characteristic, complemented
16C2×#13108075C2211yescharacteristic
17C2×#13108074C2213yes—
18C2×Q8×D101C2211yescharacteristic
19#15692254C2213yes—
20#15692253C2213yes—
21#15692250C2216yes—
22#15692249C2216yes—
23#15692247C2213yes—
24#15692246C2213yes—
25#15692245C2216yes—
26#15692244C2213yes—
27#15692243C2213yes—
28#15692240C2216yes—
29#15692238C2216yes—
30#15692235C2213yes—
31#15692234C2213yes—
32#15692233C2216yes—
33#15692231C2213yes—
34#15692231C2213yes—
35#15692230C2213yes—
36#15692229C2213yes—
37Q8×Dic101C2211yescharacteristic
38Q8×Dic101C2211yescharacteristic
39C4×Dic202C2213yes—
40C4×Dic202C2213yes—
41C4×Dic202C2213yes—
42#15692220C2213yes—
43C4×D404C2213yes—
44C4×D404C2213yes—
45C4×D404C2213yes—
46#15692217C2213yes—
47C4×C22⋊D101C2213yes—
48C4×C22⋊D101C2213yes—
49C4×C22⋊D101C2211yescharacteristic
50C4×C22⋊D101C2213yes—
51#15692214C2213yes—
52#15692213C2213yes—
53#15692212C2216yes—
54#15692212C2213yes—
55#15692212C2213yes—
56C4⋊C4×D101C2213yes—
57C42×D101C2213yes—
58C42×D101C2213yes—
59C22×Q8×C101C2211yescharacteristic
60C4×Q8×C101C2213yes—
61C4×Q8×C101C2211yescharacteristic
62C4×Q8×C101C2213yes—
63C4×Q8×C101C2211yescharacteristic
64C2×C4⋊C4×C101C2213yes—
65C2×C42×C101C2213yes—
66#13108075—24no—
67#13108074—212no—
68C2×Dic202C2313yes—
69C2×Dic202C2313yes—
70Q8×D101—24no—
71C2×D404C2313yes—
72C2×D404C2313yes—
73C2×C22⋊D101C2313yes—
74C2×C22⋊D101C2311yescharacteristic
75C2×C4×D101C2313yes—
76C2×C4×D101C2313yes—
77C2×C4×D101C2313yes—
78C2×C4×D101C2313yes—
79C2×C4×D101C2311yescharacteristic
80C2×C4×D101C2313yes—
81C4⋊Dic101C2313yes—
82C4⋊Dic101C2313yes—
83C4⋊Dic101C2313yes—
84C4⋊Dic101C2313yes—
85#13108053C2316yes—
86#13108053C2311yescharacteristic
87#13108053C2313yes—
88#13108053C2313yes—
89#13108053C2313yes—
90#13108053C2316yes—
91#13108053C2316yes—
92C22⋊Dic101C2311yescharacteristic
93C22⋊Dic101C2313yes—
94#13108051C2311yescharacteristic
95#13108051C2313yes—
96#13108051C2313yes—
97#13108051C2313yes—
98#13108051C2316yes—
99#13108051C2316yes—
100#13108051C2316yes—
101C4×Dic101C2311yescharacteristic
102C4×Dic101C2313yes—
103C4×Dic101C2313yes—
104C4×Dic101C2313yes—
105C4×Dic101C2313yes—
106C4×Dic101C2313yes—
107C2×Q8×C101—24no—
108C2×Q8×C101C2313yes—
109C2×Q8×C101C2311yescharacteristic
110C22×C4×C101C2313yes—
111C22×C4×C101C2311yescharacteristic
112C22×C4×C101C2313yes—
113C4⋊C4×C101C2313yes—
114C4⋊C4×C101C2313yes—
115C4⋊C4×C101C2316yes—
116C42×C101C2313yes—
117C42×C101C2313yes—
118C42×C101C2313yes—
119C42×C101C2313yes—
120C23×C101C2411yescharacteristic
121C22×D101C2413yes—
122C22×D101C2411yescharacteristic
123Q8×C101—26no—
124Q8×C101#4114yescomplemented
125C2×C4×C101—26no—
126C2×C4×C101—26no—
127C2×C4×C101—22no—
128C2×C4×C101C2411yescharacteristic
129C2×C4×C101C2413yes—
130C2×C4×C101C2413yes—
131C2×C4×C101C2413yes—
132C2×C4×C101C2413yes—
133C2×C4×C101C2411yescharacteristic
134C22⋊D101—26no—
135C22⋊D101#4114yescomplemented
136C2×Dic101C2411yescharacteristic
137C2×Dic101C2411yescharacteristic
138C2×Dic101C2413yes—
139C2×Dic101C2413yes—
140C2×Dic101C2413yes—
141C2×Dic101C2411yescharacteristic
142D404—26no—
143D404—26no—
144C4×D101—26no—
145C4×D101—26no—
146C4×D101—26no—
147C4×D101—22no—
148C4×D101—26no—
149C4×D101—26no—
150Dic202—26no—
151Dic202—26no—
152C22×C101—21no—
153C22×C101—21no—
154C22×C101C2×#4112yes—
155C22×C101C2511yescharacteristic
156C2×D101—21no—
157C2×D101—21no—
158C2×D101—23no—
159C2×D101—23no—
160C2×D101—23no—
161C2×D101C2×#4112yes—
162C4×C101—23no—
163C4×C101—23no—
164C4×C101—23no—
165C4×C101—21no—
166C4×C101C2×#4116yes—
167C4×C1012−1+412yes—
168Dic101—23no—
169Dic101—23no—
170Dic101—21no—
171Dic101—21no—
172Dic101—23no—
173Dic101C2×#4112yes—
174C2×C101—22no—
175C2×C101C22×#4111yescharacteristic
176C2×C101C22×#4111yescharacteristic
177C2×C101C2×2−1+411yescharacteristic
178D101—22no—
179D101—43no—
180#3482—1011nomaximal, Sylow
181C101#348211yescharacteristic, complemented, Sylow
182#557—1011no—
183#556—1013no—
184#552—1013no—
185#552—1013no—
186D4×Q8—1011no—
187#548—1011no—
188#548—1011no—
189#547—1013no—
190#533—1013no—
191#529—1013no—
192#529—1013no—
193C4×#41—1011no—
194C4×#41—1013no—
195C2×C4×Q8—1011no—
196C2×C4×Q8D10111yescharacteristic, complemented
197C2×#41—1011no—
198C2×#41—1013no—
199C22×Q8—1011no—
200C22×Q8C2×D10111yescharacteristic
201C4⋊Q8—1013no—
202C4⋊Q8—1013no—
203#126—1016no—
204#126—1016no—
205#125—1013no—
206#125—1016no—
207#125—1013no—
208#124—1013no—
209#124—1013no—
210#123—1016no—
211#123—1016no—
212C22⋊Q8—1013no—
213C22⋊Q8—1013no—
214C22⋊Q8—1013no—
215C22⋊Q8—1016no—
216C22⋊Q8—1013no—
217#121—1013no—
218#121—1013no—
219C4×Q8—1011no—
220C4×Q8—1011no—
221C4×Q8—1013no—
222C4×Q8—1013no—
223C4×Q8—1013no—
224C4×Q8—1013no—
225C4×Q8C2×D10111yescharacteristic
226C4×Q8C2×D10113yes—
227C4×Q8C2×D10111yescharacteristic
228C4×Q8C2×D10113yes—
229C4×D4—1011no—
230C4×D4—1013no—
231C4×D4—1013no—
232C4×D4—1013no—
233C4×D4—1013no—
234C4×D4—1013no—
235C4×D4—1013no—
236C4×D4—1013no—
237#117—1013no—
238#117—1013no—
239#117—1013no—
240#117—1013no—
241#117—1016no—
242C2×C4⋊C4—1013no—
243C2×C4⋊C4C2×D10113yes—
244C2×C42—1013no—
245C2×C42—1013no—
246C2×C42C2×D10113yes—
247#41—20212no—
248#41—2024no—
249C2×Q8—24no—
250C2×Q8—1013no—
251C2×Q8—1013no—
252C2×Q8—2024no—
253C2×Q8C22×D10113yes—
254C2×Q8C22×D10111yescharacteristic
255C2×D4—1011no—
256C2×D4—1013no—
257C2×D4—1013no—
258C2×D4—1013no—
259C22×C4—1013no—
260C22×C4—1013no—
261C22×C4—1013no—
262C22×C4—1013no—
263C22×C4—1011no—
264C22×C4—1013no—
265C22×C4C22×D10113yes—
266C22×C4C22×D10113yes—
267C22×C4C22×D10111yescharacteristic
268C4⋊C4—1016no—
269C4⋊C4—1011no—
270C4⋊C4—1013no—
271C4⋊C4—1013no—
272C4⋊C4—1013no—
273C4⋊C4—1016no—
274C4⋊C4—1016no—
275C4⋊C4—1013no—
276C4⋊C4—1013no—
277C4⋊C4—1013no—
278C4⋊C4—1013no—
279C4⋊C4C22×D10113yes—
280C4⋊C4C22×D10113yes—
281C4⋊C4C22×D10116yes—
282C22⋊C4—1011no—
283C22⋊C4—1013no—
284C22⋊C4—1011no—
285C22⋊C4—1013no—
286C22⋊C4—1016no—
287C22⋊C4—1013no—
288C22⋊C4—1013no—
289C22⋊C4—1016no—
290C22⋊C4—1016no—
291C42—1013no—
292C42—1013no—
293C42—1013no—
294C42—1013no—
295C42—1011no—
296C42—1013no—
297C42C22×D10113yes—
298C42C22×D10113yes—
299C42C22×D10113yes—
300C42C22×D10113yes—
301C23—1011no—
302C23—1013no—
303C23C23×D10111yescharacteristic
304Q8—26no—
305Q8—2026no—
306Q8—2026no—
307Q8#1310807414yescomplemented
308D4—1014no—
309D4—2026no—
310D4—2026no—
311D4—2026no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—1011no—
316C2×C4—1011no—
317C2×C4—1011no—
318C2×C4—1013no—
319C2×C4—1013no—
320C2×C4—1013no—
321C2×C4—2026no—
322C2×C4—2026no—
323C2×C4—2026no—
324C2×C4—2026no—
325C2×C4—2026no—
326C2×C4—2022no—
327C2×C4C23×D10113yes—
328C2×C4C23×D10111yescharacteristic
329C2×C4C23×D10113yes—
330C2×C4C23×D10113yes—
331C2×C4C23×D10113yes—
332C2×C4C23×D10111yescharacteristic
333C22—21no—
334C22—21no—
335C22—1012no—
336C22—2023no—
337C22—2023no—
338C22—2023no—
339C22—2021no—
340C22—2021no—
341C22#41×D10112yes—
342C22C24×D10111yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—1012no—
348C4—2023no—
349C4—2021no—
350C4—2023no—
351C4—2021no—
352C4—2023no—
353C4C2×#1310807416yes—
354C4#1569230312yes—
355C2—22no—
356C2—4043no—
357C2—2022no—
358C2C2×#41×D10111yescharacteristic
359C2C22×#1310807411yescharacteristic
360C2C2×#1569230311yescharacteristic
subgroupG/H
centerC2×C4C23×D101
commutator, socle, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC2×C4×Q8×C101C2
hypercenterC2×C4×Q8D101
hypocenterC101#3482
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20265903C2semidirect
2. #20265899C2semidirect
3. #20265868C2semidirect
4. #20265862C2semidirect
5. Q8×C22⋊D101C2semidirect
6. #20265839C2semidirect
7. #20265838C2semidirect
8. #20265828C2semidirect
9. #20265731C2semidirect
10. #20265714C2semidirect
11. #20265711C2semidirect
12. C4×#13108075C2semidirect
13. C4×#13108074C2semidirect
14. C4×Q8×D101C2semidirect
15. C2×C4×Q8×C101C2semidirect
124. Q8×C101#41semidirect
135. C22⋊D101#41semidirect
181. C101#3482semidirect
196. C2×C4×Q8D101semidirect
307. Q8#13108074semidirect

Maximal subgroups

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

subgroupindex
#202659032
#202658992
#202658682
#202658622
Q8×C22⋊D1012
#202658392
#202658382
#202658282
#202657312
#202657142
#202657112
C4×#131080752
C4×#131080742
C4×Q8×D1012
C2×C4×Q8×C1012
#3482101