Finite Group Databasebeta

← #27182848

Subgroups — #27182848

Order 12928.

484 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692274C211yesmaximal, characteristic, complemented
2#20265826C212yesmaximal, complemented
3#20265555C211yesmaximal, characteristic, complemented
4#20265418C211yesmaximal, characteristic, complemented
5#20265415C211yesmaximal, characteristic, complemented
6#20265380C212yesmaximal, complemented
7#20265339C212yesmaximal, complemented
8#20265336C212yesmaximal, complemented
9D8×Dic101C211yesmaximal, characteristic, complemented
10C8×C22⋊D101C211yesmaximal, characteristic, complemented
11C2×#135×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2212yes—
13#15692274—24no—
14C2×Dic404C2211yescharacteristic
15C2×#13108064C2212yes—
16C2×#13108062C2212yes—
17C2×C8×D101C2211yescharacteristic
18#15692253C2212yes—
19#15692238—22no—
20#15692236—22no—
21#15692234C2212yes—
22#15692231C2212yes—
23#15692227—22no—
24D4×Dic101C2212yes—
25C4×C22⋊D101C2211yescharacteristic
26C2×#13108053—22no—
27#15692189C2211yescharacteristic
28#15692184C2211yescharacteristic
29#15692180C2212yes—
30#15692179C2212yes—
31#15692176C2212yes—
32C22⋊#11717579C2211yescharacteristic
33#15692167C2211yescharacteristic
34C8×Dic101C2211yescharacteristic
35C2×#41×C101C2212yes—
36#135×C101—24no—
37C2×Q16×C101C2211yescharacteristic
38C2×SD16×C101C2212yes—
39C2×D8×C101C2211yescharacteristic
40C22×C8×C101C2211yescharacteristic
41#13108073—24no—
42#13108073—42no—
43C2×Dic202C2312yes—
44C2×C22⋊D101—22no—
45C2×C22⋊D101D411yescharacteristic
46C22×Dic101—22no—
47C22×Dic101—22no—
48C2×C4×D101C2311yescharacteristic
49Dic404—22no—
50#13108064—42no—
51#13108062—24no—
52C2×#11717579C2311yescharacteristic
53C2×#11717579C2311yescharacteristic
54C8×D101—22no—
55C4⋊Dic101C2312yes—
56#13108053—22no—
57#13108053—22no—
58#13108053—22no—
59#13108053—22no—
60#13108053D411yescharacteristic
61C22⋊Dic101—22no—
62C22⋊Dic101—22no—
63C22⋊Dic101D411yescharacteristic
64#13108051—22no—
65#13108051D411yescharacteristic
66C4×Dic101C2311yescharacteristic
67#41×C101—24no—
68#41×C101—42no—
69C2×Q8×C101C2312yes—
70C2×D4×C101—22no—
71C2×D4×C101C2312yes—
72C22×C4×C101—22no—
73C22×C4×C101C2311yescharacteristic
74Q16×C101—22no—
75SD16×C101—24no—
76D8×C101D414yescomplemented
77C2×C8×C101—22no—
78C2×C8×C101C2311yescharacteristic
79C2×C8×C101C2311yescharacteristic
80C23×C101—22no—
81C23×C101C2×D411yescharacteristic
82C22×D101C2×D411yescharacteristic
83Q8×C101—22no—
84Q8×C101—22no—
85D4×C101—22no—
86D4×C101—44no—
87D4×C101C2×D414yes—
88C2×C4×C101—21no—
89C2×C4×C101—21no—
90C2×C4×C101—22no—
91C2×C4×C101—44no—
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2411yescharacteristic
94C22⋊D101—22no—
95C22⋊D101—44no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101—44no—
99C2×Dic101—44no—
100C2×Dic101C2×D411yescharacteristic
101C2×Dic101C2×D411yescharacteristic
102C2×Dic101C2×D411yescharacteristic
103C4×D101—21no—
104C4×D101—21no—
105Dic202—22no—
106Dic202—22no—
107C8×C101—21no—
108C8×C101C2×D412yes—
109#11717579—21no—
110#11717579—21no—
111C22×C101—24no—
112C22×C101—21no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—42no—
116C22×C101—42no—
117C22×C101C22×D411yescharacteristic
118C2×D101—21no—
119C2×D101—21no—
120C2×D101—21no—
121C4×C101—21no—
122C4×C101—42no—
123C4×C101C22×D411yescharacteristic
124C4×C101H(5,𝔽2)11yescharacteristic
125Dic101—21no—
126Dic101—21no—
127Dic101—42no—
128Dic101—42no—
129Dic101#13512yescomplemented
130C2×C101—41no—
131C2×C101—44no—
132C2×C101D4211yescharacteristic
133C2×C101C2×#13511yescharacteristic
134C2×C101#57811yescharacteristic
135D101—41no—
136#3283—1011nomaximal, Sylow
137C101#328311yescharacteristic, complemented, Sylow
138C2×#135—1011no—
139C2×#135D10111yescharacteristic, complemented
140#547—1012no—
141#494—1011no—
142#467—1011no—
143#467—1011no—
144#461—1012no—
145#452—1012no—
146#452—1012no—
147C4×D8—1011no—
148C8×D4—1011no—
149C2×#41—1012no—
150C2×#41C2×D10112yes—
151#135—24no—
152#135—2024no—
153C2×Q16—1012no—
154C2×Q16—1011no—
155C2×Q16C2×D10111yescharacteristic
156C2×SD16—1012no—
157C2×SD16C2×D10112yes—
158C2×D8C2×D10111yescharacteristic
159C22×C8—1011no—
160C22×C8C2×D10111yescharacteristic
161C4⋊Q8—1012no—
162#123—2022no—
163#123—2022no—
164C22⋊Q8—1012no—
165C22⋊Q8—1012no—
166#121—2022no—
167C4×D4—1011no—
168C4×D4—1012no—
169C2×C4⋊C4—2022no—
170#107—1011no—
171C4⋊C8—1011no—
172#103—1012no—
173#103—1012no—
174#102—1012no—
175C22⋊C8—1011no—
176C22⋊C8—1011no—
177C4×C8—1011no—
178#41—24no—
179#41—42no—
180#41—4042no—
181#41—2024no—
182C2×Q8—1012no—
183C2×Q8C22×D10112yes—
184C2×D4—22no—
185C2×D4—1011no—
186C2×D4—2022no—
187C2×D4C22×D10112yes—
188C22×C4—22no—
189C22×C4—1011no—
190C22×C4—2022no—
191C22×C4—2022no—
192C22×C4C22×D10111yescharacteristic
193Q16—22no—
194Q16—4042no—
195Q16—2022no—
196SD16—24no—
197SD16—2024no—
198D8C22⋊D10114yescomplemented
199C2×C8—22no—
200C2×C8—1011no—
201C2×C8—1011no—
202C2×C8—2022no—
203C2×C8C22×D10111yescharacteristic
204C2×C8C22×D10111yescharacteristic
205C4⋊C4—1011no—
206C4⋊C4—1012no—
207C4⋊C4—2022no—
208C4⋊C4—2022no—
209C4⋊C4—2022no—
210C4⋊C4—2022no—
211C22⋊C4—1011no—
212C22⋊C4—1011no—
213C22⋊C4—2022no—
214C22⋊C4—2022no—
215C22⋊C4—2022no—
216C42—1011no—
217C23—22no—
218C23—1011no—
219C23D4×D10111yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—2022no—
223Q8—2022no—
224D4—22no—
225D4—44no—
226D4—4044no—
227D4—2022no—
228D4C2×C22⋊D10114yes—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—1011no—
234C2×C4—1011no—
235C2×C4—1011no—
236C2×C4—4044no—
237C2×C4—4044no—
238C2×C4—2022no—
239C2×C4—2022no—
240C2×C4—2021no—
241C2×C4—2021no—
242C2×C4D4×D10111yescharacteristic
243C2×C4C23×D10111yescharacteristic
244C8—21no—
245C8—2021no—
246C8—2021no—
247C8C2×C22⋊D10112yes—
248C22—21no—
249C22—21no—
250C22—24no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22—2021no—
255C22—2021no—
256C22—2021no—
257C22C2×D4×D10111yescharacteristic
258C4—21no—
259C4—42no—
260C4—1012no—
261C4—4042no—
262C4—4042no—
263C4—2021no—
264C4—2021no—
265C4C22×C22⋊D10111yescharacteristic
266C4#1569230011yescharacteristic
267C2—41no—
268C2—44no—
269C2—4041no—
270C2D4×C22⋊D10111yescharacteristic
271C2#135×D10111yescharacteristic
272C2#2026599111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC22×C101C22×D4
hypercenterC2×#135D101
hypocenterC101#3283
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15692274C2semidirect
2. #20265826C2semidirect
3. #20265555C2semidirect
4. #20265418C2semidirect
5. #20265415C2semidirect
6. #20265380C2semidirect
7. #20265339C2semidirect
8. #20265336C2semidirect
9. D8×Dic101C2semidirect
10. C8×C22⋊D101C2semidirect
11. C2×#135×C101C2semidirect
76. D8×C101D4semidirect
129. Dic101#135semidirect
137. C101#3283semidirect
139. C2×#135D101semidirect
198. D8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922742
#202658262
#202655552
#202654182
#202654152
#202653802
#202653392
#202653362
D8×Dic1012
C8×C22⋊D1012
C2×#135×C1012
#3283101