Finite Group Databasebeta

← #27185219

Subgroups — #27185219

Order 12928.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265903C211yesmaximal, characteristic, complemented
2#20265902C213yesmaximal, complemented
3#20265898C213yesmaximal, complemented
4#20265869C213yesmaximal, complemented
5#20265867C213yesmaximal, complemented
6Q8×C22⋊D101C211yesmaximal, characteristic, complemented
7Q8×C22⋊D101C211yesmaximal, characteristic, complemented
8#20265788C213yesmaximal, complemented
9#20265787C213yesmaximal, complemented
10#20265729C213yesmaximal, complemented
11#20265707C213yesmaximal
12#20265655C211yesmaximal, characteristic, complemented
13#20265653C211yesmaximal, characteristic, complemented
14C2×Q8×Dic101C211yesmaximal, characteristic, complemented
15D4×Q8×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17#15692254C2213yes—
18#15692253C2213yes—
19#15692253C2213yes—
20#15692250C2213yes—
21#15692250C2216yes—
22#15692248C2213yes—
23#15692248C2216yes—
24#15692247C2213yes—
25#15692246C2216yes—
26#15692246C2213yes—
27#15692245C2216yes—
28#15692245C2216yes—
29#15692245C2213yes—
30#15692241C2213yes—
31#15692239C2213yes—
32#15692236C2213yes—
33#15692234C2213yes—
34#15692234C2213yes—
35#15692232C2213yes—
36#15692231C2213yes—
37#15692231C2213yes—
38#15692227C2211yescharacteristic
39Q8×Dic101C2213yes—
40Q8×Dic101C2213yes—
41Q8×Dic101C2211yescharacteristic
42Q8×Dic101C2211yescharacteristic
43Q8×Dic101C2213yes—
44Q8×Dic101C2213yes—
45Q8×Dic101C2211yescharacteristic
46C4×Dic202C2213yes—
47C4×Dic202C2213yes—
48#15692220C2213yes—
49#15692220C2216yes—
50C4×C22⋊D101C2213yes—
51C4×C22⋊D101C2213yes—
52#15692214C2213yes—
53#15692214C2213yes—
54#15692213C2213yes—
55#15692212C2213yes—
56#15692211C2216yes—
57C2×C4⋊Dic101C2213yes—
58C2×C4×Dic101C2213yes—
59C22×Q8×C101C2211yescharacteristic
60C22×Q8×C101C2211yescharacteristic
61C4⋊Q8×C101C2213yes—
62C22⋊Q8×C101C2213yes—
63C22⋊Q8×C101C2213yes—
64C4×Q8×C101C2211yescharacteristic
65C4×D4×C101C2213yes—
66C2×Dic202C2313yes—
67C2×Dic202C2313yes—
68Q8×D101—24no—
69C2×C22⋊D101C2311yescharacteristic
70C2×C22⋊D101C2311yescharacteristic
71C22×Dic101C2311yescharacteristic
72C22×Dic101C2313yes—
73C2×C4×D101C2313yes—
74C4⋊Dic101C2313yes—
75C4⋊Dic101C2313yes—
76C4⋊Dic101C2316yes—
77C4⋊Dic101C2313yes—
78C4⋊Dic101C2313yes—
79C4⋊Dic101C2313yes—
80C4⋊Dic101C2313yes—
81#13108053C2316yes—
82#13108053C2316yes—
83#13108053C2313yes—
84#13108053C2313yes—
85#13108053C2313yes—
86#13108053C2313yes—
87#13108053C2316yes—
88#13108053C2311yescharacteristic
89#13108053C2313yes—
90C22⋊Dic101C2313yes—
91C22⋊Dic101C2311yescharacteristic
92C22⋊Dic101C2313yes—
93C22⋊Dic101C2313yes—
94#13108051C2311yescharacteristic
95#13108051C2313yes—
96#13108051C2313yes—
97#13108051C2313yes—
98C4×Dic101C2313yes—
99C4×Dic101C2313yes—
100C4×Dic101C2313yes—
101C4×Dic101C2313yes—
102C4×Dic101C2316yes—
103C4×Dic101C2316yes—
104C4×Dic101C2313yes—
105C2×Q8×C101—24no—
106C2×Q8×C101—24no—
107C2×Q8×C101C2313yes—
108C2×Q8×C101C2313yes—
109C2×Q8×C101C2311yescharacteristic
110C2×D4×C101C2311yescharacteristic
111C22×C4×C101C2313yes—
112C22×C4×C101C2313yes—
113C4⋊C4×C101C2313yes—
114C4⋊C4×C101C2313yes—
115C4⋊C4×C101C2316yes—
116C22⋊C4×C101C2313yes—
117C22⋊C4×C101C2313yes—
118C42×C101C2313yes—
119C23×C101C2411yescharacteristic
120C23×C101C2411yescharacteristic
121C22×D101C2411yescharacteristic
122Q8×C101—26no—
123Q8×C101—26no—
124Q8×C101#4114yes—
125D4×C101—22no—
126C2×C4×C101—26no—
127C2×C4×C101—26no—
128C2×C4×C101C2411yescharacteristic
129C2×C4×C101C2413yes—
130C2×C4×C101C2413yes—
131C2×C4×C101C2413yes—
132C2×C4×C101C2413yes—
133C22⋊D101—22no—
134C22⋊D101—22no—
135C2×Dic101—26no—
136C2×Dic101—22no—
137C2×Dic101C2411yescharacteristic
138C2×Dic101C2413yes—
139C2×Dic101C2413yes—
140C2×Dic101C2411yescharacteristic
141C2×Dic101C2413yes—
142C2×Dic101C2413yes—
143C2×Dic101C2411yescharacteristic
144C4×D101—26no—
145Dic202—26no—
146Dic202—26no—
147C22×C101—21no—
148C22×C101—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C1012−1+412yes—
153C22×C101C2511yescharacteristic
154C2×D101—21no—
155C2×D101—21no—
156C2×D101—21no—
157C4×C101—23no—
158C4×C101—23no—
159C4×C101—23no—
160C4×C101—21no—
161C4×C101C2×#4116yes—
162Dic101—23no—
163Dic101—23no—
164Dic101—21no—
165Dic101—23no—
166Dic101—23no—
167Dic101—21no—
168Dic1012−1+412yes—
169C2×C101—22no—
170C2×C101—41no—
171C2×C101C22×#4111yescharacteristic
172C2×C101C2×2−1+411yescharacteristic
173C2×C101C2×2−1+411yescharacteristic
174D101—41no—
175#3523—1011nomaximal, Sylow
176C101#352311yescharacteristic, complemented, Sylow
177#557—1013no—
178#557—1011no—
179#556—1013no—
180#552—1013no—
181#552—1013no—
182D4×Q8—1011no—
183D4×Q8—1011no—
184D4×Q8D10111yescharacteristic, complemented
185#541—1013no—
186#541—1013no—
187#533—1013no—
188#529—1013no—
189#519—1011no—
190#519—1011no—
191C2×C4×Q8—1011no—
192C22×Q8—1011no—
193C22×Q8C2×D10111yescharacteristic
194C22×Q8C2×D10111yescharacteristic
195C4⋊Q8—1013no—
196C4⋊Q8—1013no—
197C4⋊Q8—1013no—
198C4⋊Q8C2×D10113yes—
199#126—1013no—
200#126—1016no—
201#126—1013no—
202#126—1016no—
203#125—1013no—
204#125—1016no—
205#125—1016no—
206#125—1016no—
207#125—1013no—
208#125—1013no—
209#124—1013no—
210#123—1013no—
211#123—1013no—
212C22⋊Q8—1013no—
213C22⋊Q8—1013no—
214C22⋊Q8—1013no—
215C22⋊Q8—1013no—
216C22⋊Q8—1013no—
217C22⋊Q8C2×D10113yes—
218C22⋊Q8C2×D10113yes—
219#121—1011no—
220C4×Q8—1011no—
221C4×Q8—1013no—
222C4×Q8—1011no—
223C4×Q8—1013no—
224C4×Q8—1013no—
225C4×Q8—1011no—
226C4×Q8—1013no—
227C4×Q8—1013no—
228C4×Q8—1016no—
229C4×Q8—1013no—
230C4×Q8—1013no—
231C4×Q8C2×D10111yescharacteristic
232C4×D4—1013no—
233C4×D4—1013no—
234C4×D4C2×D10113yes—
235#117—1013no—
236#117—1013no—
237#117—1013no—
238#117—1016no—
239#117—1013no—
240C2×C4⋊C4—1013no—
241C2×C42—1013no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—1013no—
245C2×Q8—1013no—
246C2×Q8—2024no—
247C2×Q8C22×D10113yes—
248C2×Q8C22×D10113yes—
249C2×Q8C22×D10111yescharacteristic
250C2×D4—1011no—
251C2×D4—1011no—
252C2×D4C22×D10111yescharacteristic
253C22×C4—1011no—
254C22×C4—1013no—
255C22×C4—1013no—
256C22×C4C22×D10113yes—
257C22×C4C22×D10113yes—
258C4⋊C4—1013no—
259C4⋊C4—1013no—
260C4⋊C4—1016no—
261C4⋊C4—1016no—
262C4⋊C4—1013no—
263C4⋊C4—1013no—
264C4⋊C4—1011no—
265C4⋊C4—1016no—
266C4⋊C4—1013no—
267C4⋊C4—1013no—
268C4⋊C4—1013no—
269C4⋊C4—1016no—
270C4⋊C4—1013no—
271C4⋊C4—1013no—
272C4⋊C4—1013no—
273C4⋊C4—1013no—
274C4⋊C4C22×D10113yes—
275C4⋊C4C22×D10113yes—
276C4⋊C4C22×D10116yes—
277C22⋊C4—1011no—
278C22⋊C4—1013no—
279C22⋊C4—1013no—
280C22⋊C4—1013no—
281C22⋊C4—1011no—
282C22⋊C4—1013no—
283C22⋊C4—1013no—
284C22⋊C4—1013no—
285C22⋊C4C22×D10113yes—
286C22⋊C4C22×D10113yes—
287C42—1013no—
288C42—1013no—
289C42—1013no—
290C42—1013no—
291C42—1016no—
292C42—1016no—
293C42—1013no—
294C42C22×D10113yes—
295C23—1011no—
296C23C23×D10111yescharacteristic
297C23C23×D10111yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—2026no—
301Q8—2026no—
302Q8#1310807314yes—
303D4—22no—
304D4—2022no—
305D4—2022no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—1011no—
309C2×C4—1011no—
310C2×C4—1013no—
311C2×C4—1013no—
312C2×C4—1011no—
313C2×C4—1013no—
314C2×C4—1013no—
315C2×C4—2022no—
316C2×C4—2026no—
317C2×C4—2026no—
318C2×C4C23×D10113yes—
319C2×C4C23×D10113yes—
320C2×C4C23×D10113yes—
321C2×C4C23×D10111yescharacteristic
322C2×C4C23×D10113yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—2021no—
329C22—2021no—
330C22—2021no—
331C22#1569230312yes—
332C22C24×D10111yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—1012no—
338C4—2023no—
339C4—2021no—
340C4—2023no—
341C4—2023no—
342C4—2023no—
343C4—2021no—
344C4C2×#1310807316yes—
345C2—22no—
346C2—41no—
347C2—4041no—
348C2C22×#1310807311yescharacteristic
349C22−1+4×D10111yescharacteristic
350C2C2×#1569230311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
FittingD4×Q8×C101C2
prime radical#15692227C22
hypercenterD4×Q8D101
hypocenterC101#3523
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20265903C2semidirect
2. #20265902C2semidirect
3. #20265898C2semidirect
4. #20265869C2semidirect
5. #20265867C2semidirect
6. Q8×C22⋊D101C2semidirect
7. Q8×C22⋊D101C2semidirect
8. #20265788C2semidirect
9. #20265787C2semidirect
10. #20265729C2semidirect
12. #20265655C2semidirect
13. #20265653C2semidirect
14. C2×Q8×Dic101C2semidirect
15. D4×Q8×C101C2semidirect
176. C101#3523semidirect
184. D4×Q8D101semidirect

Maximal subgroups

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

subgroupindex
#202659032
#202659022
#202658982
#202658692
#202658672
Q8×C22⋊D1012
#202657882
#202657872
#202657292
#202657072
#202656552
#202656532
C2×Q8×Dic1012
D4×Q8×C1012
#3523101