Finite Group Databasebeta

← #27180916

Subgroups — #27180916

Order 12928.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265730C211yesmaximal, characteristic, complemented
2#41×Dic101C211yesmaximal, characteristic
3#20265486C211yesmaximal, characteristic, complemented
4#20265482C211yesmaximal, characteristic, complemented
5#20265479C211yesmaximal, characteristic, complemented
6#20265475C211yesmaximal, characteristic, complemented
7#20265470C211yesmaximal, characteristic, complemented
8#20265465C211yesmaximal, characteristic, complemented
9#20265460C211yesmaximal, characteristic, complemented
10#20265455C211yesmaximal, characteristic, complemented
11#20265215C211yesmaximal, characteristic, complemented
12#20265213C211yesmaximal, characteristic, complemented
13C2×#15692184C211yesmaximal, characteristic, complemented
14#20265173C211yesmaximal, characteristic
15#416×C101C211yesmaximal, characteristic
16C22×#11717579C2211yescharacteristic
17#15692254C2211yescharacteristic
18#15692254C2211yescharacteristic
19#15692246C2211yescharacteristic
20#15692246C2211yescharacteristic
21#15692232—21no—
22#15692232—21no—
23Q8×Dic101C2211yescharacteristic
24Q8×Dic101C2211yescharacteristic
25C4×Dic202—21no—
26C4×Dic202—21no—
27D4×Dic101—21no—
28D4×Dic101—21no—
29D4×Dic101C2211yescharacteristic
30D4×Dic101C2211yescharacteristic
31#15692214—21no—
32#15692214C2211yescharacteristic
33#15692214C2211yescharacteristic
34C2×C4×Dic101—21no—
35C2×C4×Dic101C2211yescharacteristic
36#15692189C2211yescharacteristic
37#15692189C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692188C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692187C2211yescharacteristic
42#15692186C2211yescharacteristic
43#15692186C2211yescharacteristic
44#15692184C2211yescharacteristic
45#15692184C2211yescharacteristic
46#15692184C2211yescharacteristic
47#15692184C2211yescharacteristic
48#15692179C2211yescharacteristic
49#15692179C2211yescharacteristic
50#15692176C2211yescharacteristic
51#15692176C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C22×C8×C101C2211yescharacteristic
54#117×C101C2211yescharacteristic
55#103×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58#102×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202—21no—
61C22×Dic101—21no—
62C22×Dic101—21no—
63C22×Dic101D411yescharacteristic
64C22×Dic101D411yescharacteristic
65C2×#11717579—22no—
66C2×#11717579C2311yescharacteristic
67C2×#11717579C2311yescharacteristic
68C4⋊Dic101—21no—
69C4⋊Dic101—21no—
70C4⋊Dic101—21no—
71C4⋊Dic101—21no—
72C4⋊Dic101C2311yescharacteristic
73C4⋊Dic101C2311yescharacteristic
74C4⋊Dic101C2311yescharacteristic
75C4⋊Dic101C2311yescharacteristic
76#13108053—21no—
77#13108053—21no—
78#13108053—21no—
79#13108053—21no—
80C22⋊Dic101—21no—
81C22⋊Dic101—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C4×Dic101—21no—
85C4×Dic101—21no—
86C4×Dic101—21no—
87C4×Dic101—21no—
88C4×Dic101D411yescharacteristic
89C4×Dic101D411yescharacteristic
90C4×Dic101C2311yescharacteristic
91C4×Dic101C2311yescharacteristic
92#41×C101—22no—
93#41×C101Q814yescomplemented
94C2×Q8×C101C2311yescharacteristic
95C2×D4×C101—21no—
96C2×D4×C101C2311yescharacteristic
97C22×C4×C101—21no—
98C22×C4×C101C2311yescharacteristic
99C2×C8×C101—22no—
100C2×C8×C101C2311yescharacteristic
101C2×C8×C101C2311yescharacteristic
102C4⋊C4×C101C2311yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C22⋊C4×C101—21no—
105C42×C101—21no—
106C23×C101—21no—
107C23×C101C2×D411yescharacteristic
108Q8×C101—21no—
109Q8×C101C2×Q812yes—
110D4×C101—24no—
111D4×C101—21no—
112D4×C101C2×Q812yes—
113C2×C4×C101—21no—
114C2×C4×C101—24no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101C2×D411yescharacteristic
118C2×C4×C101C2×Q812yes—
119C2×C4×C101#4112yes—
120C2×C4×C101C2411yescharacteristic
121C2×Dic101—22no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—41no—
127C2×Dic101—42no—
128C2×Dic101—42no—
129C2×Dic101C2×D411yescharacteristic
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133Dic202—42no—
134Dic202—42no—
135C8×C101—21no—
136C8×C101—21no—
137#11717579—21no—
138#11717579—21no—
139C22×C101—21no—
140C22×C101—21no—
141C22×C101—22no—
142C22×C101—41no—
143C22×C101—41no—
144C22×C101C22⋊Q812yes—
145C22×C101C22×D411yescharacteristic
146C4×C101—22no—
147C4×C101—41no—
148C4×C101—41no—
149C4×C101C22⋊Q812yes—
150C4×C101C22×Q811yescharacteristic
151C4×C101C2×#4111yescharacteristic
152Dic101—21no—
153Dic101—21no—
154Dic101—21no—
155Dic101—41no—
156Dic101—41no—
157Dic101—41no—
158Dic101—41no—
159Dic101#13512yes—
160C2×C101—22no—
161C2×C101—42no—
162C2×C101C2×C22⋊Q811yescharacteristic
163C2×C101C2×#13511yescharacteristic
164C2×C101#57511yescharacteristic
165#3076—1011nomaximal, Sylow
166C101#307611yescharacteristic, complemented, Sylow
167#533—1011no—
168C4×#41—1011no—
169#479—1011no—
170#479—1011no—
171#478—1011no—
172#478—1011no—
173#477—1011no—
174#476—1011no—
175#475—1011no—
176#474—1011no—
177#427—1011no—
178#427—1011no—
179C2×C4⋊C8—1011no—
180#416—1011no—
181#416D10111yescharacteristic
182C2×#41C2×D10111yescharacteristic
183C22×C8—1011no—
184C22×C8C2×D10111yescharacteristic
185C4⋊Q8—1011no—
186C4⋊Q8—1011no—
187#125—1011no—
188#125—1011no—
189C22⋊Q8—2021no—
190C22⋊Q8—2021no—
191C4×Q8—1011no—
192C4×Q8—1011no—
193C4×Q8—2021no—
194C4×Q8—2021no—
195C4×D4—1011no—
196C4×D4—1011no—
197C4×D4—2021no—
198C4×D4—2021no—
199#117—1011no—
200#117—1011no—
201#117—2021no—
202#117C2×D10111yescharacteristic
203C2×C42—1011no—
204C2×C42—2021no—
205#107—1011no—
206#107—1011no—
207#107—1011no—
208#107—1011no—
209#106—1011no—
210#106—1011no—
211#106—1011no—
212#106—1011no—
213C4⋊C8—1011no—
214C4⋊C8—1011no—
215C4⋊C8—1011no—
216C4⋊C8—1011no—
217#103—1011no—
218#103—1011no—
219#103C2×D10111yescharacteristic
220#103C2×D10111yescharacteristic
221#102—1011no—
222#102—1011no—
223#102C2×D10111yescharacteristic
224#102C2×D10111yescharacteristic
225#41—22no—
226#41Dic20214yescomplemented
227C2×Q8—2021no—
228C2×Q8—2021no—
229C2×Q8C22×D10111yescharacteristic
230C2×D4—21no—
231C2×D4C22×D10111yescharacteristic
232C22×C4—21no—
233C22×C4—1011no—
234C22×C4—1011no—
235C22×C4—2021no—
236C22×C4—2021no—
237C22×C4C22×D10111yescharacteristic
238C2×C8—22no—
239C2×C8—1011no—
240C2×C8—1011no—
241C2×C8—2022no—
242C2×C8C22×D10111yescharacteristic
243C2×C8C22×D10111yescharacteristic
244C4⋊C4—1011no—
245C4⋊C4—1011no—
246C4⋊C4—1011no—
247C4⋊C4—1011no—
248C4⋊C4—2021no—
249C4⋊C4—2021no—
250C4⋊C4—2021no—
251C4⋊C4—2021no—
252C4⋊C4—2021no—
253C4⋊C4—2021no—
254C4⋊C4—2021no—
255C4⋊C4—2021no—
256C4⋊C4C22×D10111yescharacteristic
257C4⋊C4C22×D10111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—2021no—
260C22⋊C4—2021no—
261C22⋊C4—2021no—
262C22⋊C4—2021no—
263C42—21no—
264C42—1011no—
265C42—1011no—
266C42—1011no—
267C42—1011no—
268C42—2021no—
269C42—2021no—
270C42—2021no—
271C42—2021no—
272C23—21no—
273C23D4×D10111yescharacteristic
274Q8—21no—
275Q8—4042no—
276Q8—4042no—
277Q8C2×Dic20212yes—
278D4—24no—
279D4—21no—
280D4C2×Dic20212yes—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—1011no—
286C2×C4—1011no—
287C2×C4—1011no—
288C2×C4—1011no—
289C2×C4—4042no—
290C2×C4—4042no—
291C2×C4—4041no—
292C2×C4—2022no—
293C2×C4—2021no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4D4×D10111yescharacteristic
298C2×C4C2×Dic20212yes—
299C2×C4#1310807312yes—
300C2×C4C23×D10111yescharacteristic
301C8—21no—
302C8—21no—
303C8—2021no—
304C8—2021no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—41no—
309C22—41no—
310C22#1569223212yes—
311C22C2×D4×D10111yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—1012no—
316C4—4041no—
317C4—4041no—
318C4—4041no—
319C4—4041no—
320C4—2021no—
321C4—2021no—
322C4—2021no—
323C4#1569223212yes—
324C4C22×Dic20211yescharacteristic
325C4C2×#1310807311yescharacteristic
326C2—22no—
327C2—42no—
328C2C2×#1569223211yescharacteristic
329C2#135×D10111yescharacteristic
330C2#2026597411yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#416×C101C2
socleC22×C101C22×D4
prime radicalC2×#41×C101C22
hypercenter#416D101
hypocenterC101#3076
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20265730C2semidirect
3. #20265486C2semidirect
4. #20265482C2semidirect
5. #20265479C2semidirect
6. #20265475C2semidirect
7. #20265470C2semidirect
8. #20265465C2semidirect
9. #20265460C2semidirect
10. #20265455C2semidirect
11. #20265215C2semidirect
12. #20265213C2semidirect
13. C2×#15692184C2semidirect
93. #41×C101Q8semidirect
166. C101#3076semidirect
226. #41Dic202semidirect

Maximal subgroups

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

subgroupindex
#202657302
#41×Dic1012
#202654862
#202654822
#202654792
#202654752
#202654702
#202654652
#202654602
#202654552
#202652152
#202652132
C2×#156921842
#202651732
#416×C1012
#3076101