Finite Group Databasebeta

← #27182585

Subgroups — #27182585

Order 12928.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265790C211yesmaximal, characteristic
2#20265785C211yesmaximal, characteristic
3#20265487C211yesmaximal, characteristic, complemented
4#20265483C211yesmaximal, characteristic, complemented
5#20265481C211yesmaximal, characteristic, complemented
6#20265436C211yesmaximal, characteristic
7#20265434C211yesmaximal, characteristic
8#20265419C211yesmaximal, characteristic
9#20265417C211yesmaximal, characteristic
10#20265388C211yesmaximal, characteristic, complemented
11#20265387C211yesmaximal, characteristic, complemented
12#20265386C211yesmaximal, characteristic, complemented
13#20265386C211yesmaximal, characteristic, complemented
14#20265202C211yesmaximal, characteristic
15#479×C101C211yesmaximal, characteristic, complemented
16C2×Dic404C2211yescharacteristic
17C2×Dic404C2211yescharacteristic
18C2×#13108064C2211yescharacteristic
19C2×#13108064C2211yescharacteristic
20C2×#13108057C2211yescharacteristic
21C2×C8×D101C2211yescharacteristic
22#15692255C2211yescharacteristic
23#15692253C2211yescharacteristic
24#15692250—21no—
25#15692250—21no—
26#15692250—21no—
27#15692249—21no—
28#15692247—21no—
29#15692246—21no—
30#15692246—21no—
31#15692246C2211yescharacteristic
32#15692245—21no—
33#15692235C2211yescharacteristic
34#15692235C2211yescharacteristic
35#15692235C2211yescharacteristic
36#15692231C2211yescharacteristic
37C4×Dic202C2211yescharacteristic
38#15692220C2211yescharacteristic
39#15692220C2211yescharacteristic
40#15692212C2211yescharacteristic
41#15692188C2211yescharacteristic
42#15692186C2211yescharacteristic
43C4⋊#11717579C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692180C2211yescharacteristic
47#15692180C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692178C2211yescharacteristic
50#15692178C2211yescharacteristic
51#15692178C2211yescharacteristic
52#125×C101C2211yescharacteristic
53C4×Q8×C101C2211yescharacteristic
54#107×C101C2211yescharacteristic
55#106×C101C2211yescharacteristic
56C4⋊C8×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic
59C2×Dic202C2311yescharacteristic
60C2×Dic202C2311yescharacteristic
61C2×Dic202C2311yescharacteristic
62C2×C4×D101C2311yescharacteristic
63Dic404—41no—
64Dic404—41no—
65#13108064—41no—
66#13108064—41no—
67#13108057—41no—
68C2×#11717579C2311yescharacteristic
69C2×#11717579C2311yescharacteristic
70C8×D101—41no—
71C4⋊Dic101—21no—
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76C4⋊Dic101—21no—
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053D411yescharacteristic
86#13108053D411yescharacteristic
87#13108051—21no—
88#13108051—21no—
89#13108051—21no—
90#13108051—21no—
91#13108051D411yescharacteristic
92#13108051D411yescharacteristic
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101—21no—
96C4×Dic101C2311yescharacteristic
97C2×Q8×C101C2311yescharacteristic
98C2×C8×C101C2311yescharacteristic
99C2×C8×C101C2311yescharacteristic
100C4⋊C4×C101—21no—
101C4⋊C4×C101—21no—
102C4⋊C4×C101—21no—
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C4⋊C4×C101C2311yescharacteristic
106C42×C101—21no—
107C42×C101C2311yescharacteristic
108C22×D101C2×D411yescharacteristic
109Q8×C101—21no—
110Q8×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101C2×D411yescharacteristic
116C2×C4×C101C2×D411yescharacteristic
117C2×C4×C101C2411yescharacteristic
118C2×Dic101—21no—
119C2×Dic101—21no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101C2×D411yescharacteristic
123C2×Dic101C2×D411yescharacteristic
124C2×Dic101C2×D411yescharacteristic
125C4×D101—21no—
126C4×D101—21no—
127Dic202—21no—
128Dic202—21no—
129Dic202—21no—
130Dic202—21no—
131Dic202—21no—
132Dic202—21no—
133C8×C101—21no—
134C8×C101—21no—
135#11717579—21no—
136#11717579—21no—
137C22×C101C22×D411yescharacteristic
138C2×D101—21no—
139C2×D101—21no—
140C2×D101—21no—
141C4×C101—21no—
142C4×C101—21no—
143C4×C101—41no—
144C4×C101—41no—
145C4×C101—41no—
146C4×C101—41no—
147C4×C101H(5,𝔽2)11yescharacteristic
148C4×C1012−1+411yescharacteristic
149Dic101—21no—
150Dic101—21no—
151Dic101—21no—
152Dic101—41no—
153Dic101—41no—
154Dic101—41no—
155Dic101—41no—
156C2×C101#53711yescharacteristic
157C2×C101#57811yescharacteristic
158C2×C101#57811yescharacteristic
159D101—41no—
160#3261—1011nomaximal, Sylow
161C101#326111yescharacteristic, complemented, Sylow
162#541—1011no—
163#541—1011no—
164#479—1011no—
165#479—1011no—
166#479—1011no—
167#479D10111yescharacteristic, complemented
168#470—1011no—
169#470—1011no—
170#467—1011no—
171#467—1011no—
172#462—1011no—
173#462—1011no—
174#462—1011no—
175#462—1011no—
176#424—1011no—
177C2×Q16—1011no—
178C2×Q16—1011no—
179C2×Q16—1011no—
180C2×Q16—1011no—
181C2×M16—1011no—
182C22×C8—1011no—
183C4⋊Q8—1011no—
184C4⋊Q8—1011no—
185#126—2021no—
186#126—2021no—
187#126—2021no—
188#126—2021no—
189#125—1011no—
190#125—2021no—
191#125—2021no—
192#125—2021no—
193#125—2021no—
194#125C2×D10111yescharacteristic
195C22⋊Q8—1011no—
196C22⋊Q8—1011no—
197C22⋊Q8—1011no—
198C22⋊Q8—1011no—
199C4×Q8—1011no—
200C4×Q8—1011no—
201C4×Q8—1011no—
202C4×Q8C2×D10111yescharacteristic
203#117—1011no—
204#107—1011no—
205#107C2×D10111yescharacteristic
206#106—1011no—
207#106C2×D10111yescharacteristic
208C4⋊C8—1011no—
209C4⋊C8—1011no—
210C4⋊C8—1011no—
211C4⋊C8C2×D10111yescharacteristic
212#103—1011no—
213#103—1011no—
214#103—1011no—
215#103—1011no—
216#103—1011no—
217#103—1011no—
218#103C2×D10111yescharacteristic
219#103C2×D10111yescharacteristic
220C2×Q8—1011no—
221C2×Q8—1011no—
222C2×Q8—1011no—
223C2×Q8C22×D10111yescharacteristic
224C22×C4—1011no—
225Q16—4041no—
226Q16—4041no—
227Q16—4041no—
228Q16—4041no—
229M16—4041no—
230C2×C8—1011no—
231C2×C8—1011no—
232C2×C8—4041no—
233C2×C8C22×D10111yescharacteristic
234C2×C8C22×D10111yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—1011no—
239C4⋊C4—1011no—
240C4⋊C4—1011no—
241C4⋊C4—2021no—
242C4⋊C4—2021no—
243C4⋊C4—2021no—
244C4⋊C4—2021no—
245C4⋊C4—2021no—
246C4⋊C4—2021no—
247C4⋊C4—2021no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C4⋊C4—2021no—
251C4⋊C4—2021no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4C22×D10111yescharacteristic
255C4⋊C4C22×D10111yescharacteristic
256C4⋊C4C22×D10111yescharacteristic
257C22⋊C4—1011no—
258C22⋊C4—1011no—
259C22⋊C4—2021no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C42—21no—
264C42—1011no—
265C42—2021no—
266C42—2021no—
267C42—2021no—
268C42C22×D10111yescharacteristic
269C23—1011no—
270Q8—21no—
271Q8—21no—
272Q8—2021no—
273Q8—2021no—
274Q8—2021no—
275Q8—2021no—
276Q8—2021no—
277Q8—2021no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—1011no—
283C2×C4—1011no—
284C2×C4—1011no—
285C2×C4—2021no—
286C2×C4—2021no—
287C2×C4—2021no—
288C2×C4—2021no—
289C2×C4—2021no—
290C2×C4—2021no—
291C2×C4D4×D10111yescharacteristic
292C2×C4D4×D10111yescharacteristic
293C2×C4C23×D10111yescharacteristic
294C8—21no—
295C8—21no—
296C8—2021no—
297C8—2021no—
298C22—2021no—
299C22—2021no—
300C22—2021no—
301C22C2×D4×D10111yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—4041no—
309C4—4041no—
310C4—4041no—
311C4—4041no—
312C4—2021no—
313C4—2021no—
314C4—2021no—
315C4#1569230111yescharacteristic
316C4#1569230311yescharacteristic
317C2—4041no—
318C2#2026575711yescharacteristic
319C2#2026599111yescharacteristic
320C2#2026599311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#479×C101C2
socleC22×C101C22×D4
prime radicalC22×D101C2×D4
hypercenter#479D101
hypocenterC101#3261
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
3. #20265487C2semidirect
4. #20265483C2semidirect
5. #20265481C2semidirect
10. #20265388C2semidirect
11. #20265387C2semidirect
12. #20265386C2semidirect
13. #20265386C2semidirect
15. #479×C101C2semidirect
161. C101#3261semidirect
167. #479D101semidirect

Maximal subgroups

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

subgroupindex
#202657902
#202657852
#202654872
#202654832
#202654812
#202654362
#202654342
#202654192
#202654172
#202653882
#202653872
#202653862
#202652022
#479×C1012
#3261101