Finite Group Databasebeta

← #27180929

Subgroups — #27180929

Order 12928.

440 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 153 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
1#20265734C211yesmaximal, characteristic, complemented
2#20265477C214yesmaximal, complemented
3#20265467C212yesmaximal, complemented
4#20265457C212yesmaximal, complemented
5#20265217C212yesmaximal, complemented
6#20265201C211yesmaximal, characteristic, complemented
7#20265182C212yesmaximal
8C2×C4×D4×C101C211yesmaximal, characteristic
9C2×#13108058C2212yes—
10#15692255C2211yescharacteristic, complemented
11#15692255C2211yescharacteristic, complemented
12#15692247C2212yescomplemented
13#15692234—21no—
14#15692234—21no—
15C4×Dic202—21no—
16C4×Dic202—21no—
17#15692214C2212yes—
18#15692188C2214yescomplemented
19#15692186C2214yescomplemented
20C4⋊#11717579C2212yescomplemented
21C4⋊#11717579C2212yescomplemented
22#15692176C2214yes—
23#15692176C2214yes—
24C22×D4×C101C2211yescharacteristic
25C23×C4×C101—21no—
26C4×D4×C101—21no—
27C4×D4×C101—21no—
28C4×D4×C101C2212yes—
29C4×D4×C101C2212yes—
30C2×C4⋊C4×C101C2211yescharacteristic
31C2×C22⋊C4×C101—21no—
32C2×C42×C101C2211yescharacteristic
33C2×Dic202—21no—
34C2×Dic202—21no—
35#13108058—24no—
36C2×#11717579C2314yes—
37C4⋊Dic101—21no—
38C4⋊Dic101—21no—
39C4⋊Dic101C2312yes—
40C4⋊Dic101C2312yes—
41#13108053—22no—
42#13108053—22no—
43C22⋊Dic101—22no—
44C4×Dic101—22no—
45C24×C101—21no—
46C2×D4×C101—21no—
47C2×D4×C101—22no—
48C2×D4×C101Q814yescomplemented
49C2×D4×C101C2312yes—
50C22×C4×C101—21no—
51C22×C4×C101—22no—
52C22×C4×C101—22no—
53C22×C4×C101—44no—
54C22×C4×C101D411yescharacteristic
55C22×C4×C101D411yescharacteristic
56C22×C4×C101C2311yescharacteristic
57C4⋊C4×C101—21no—
58C4⋊C4×C101C2312yes—
59C22⋊C4×C101—22no—
60C22⋊C4×C101—22no—
61C42×C101D411yescharacteristic
62C42×C101D411yescharacteristic
63C42×C101C2311yescharacteristic
64C42×C101C2311yescharacteristic
65C23×C101—24no—
66C23×C101—22no—
67C23×C101—42no—
68C23×C101—42no—
69C23×C101C2×D411yescharacteristic
70D4×C101—24no—
71D4×C101—22no—
72D4×C101C2×Q814yes—
73C2×C4×C101—22no—
74C2×C4×C101—22no—
75C2×C4×C101—41no—
76C2×C4×C101—44no—
77C2×C4×C101—42no—
78C2×C4×C101—44no—
79C2×C4×C101C2×D411yescharacteristic
80C2×C4×C101C2×D411yescharacteristic
81C2×C4×C101C2×D411yescharacteristic
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2×Q812yes—
85C2×C4×C101#4112yes—
86C2×C4×C101C2411yescharacteristic
87C2×Dic101—22no—
88C2×Dic101—22no—
89Dic202—42no—
90Dic202—42no—
91#11717579—24no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—24no—
95C22×C101—48no—
96C22×C101—42no—
97C22×C101—42no—
98C22×C101C22⋊Q812yes—
99C22×C101C22×D411yescharacteristic
100C4×C101—21no—
101C4×C101—21no—
102C4×C101—21no—
103C4×C101—42no—
104C4×C101C22⋊Q812yes—
105C4×C101AGL(1,ℤ/8)12yes—
106C4×C101C22×Q811yescharacteristic
107C4×C101C2×#4111yescharacteristic
108Dic101—42no—
109Dic101—42no—
110C2×C101—22no—
111C2×C101—44no—
112C2×C101C2×C22⋊Q811yescharacteristic
113C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C101#57511yescharacteristic
115#3077—1011nomaximal, Sylow
116C101#307711yescharacteristic, complemented, Sylow
117#533—1011no—
118C2×C4×D4D10111yescharacteristic
119#478—1014no—
120#476—1012no—
121#474—1012no—
122#428—1012no—
123#423—1011no—
124#418—1012no—
125C22×D4C2×D10111yescharacteristic
126C23×C4—21no—
127C2×M16—1012no—
128C4⋊Q8—1011no—
129C4⋊Q8—1011no—
130#125—1012no—
131C22⋊Q8—2021no—
132C22⋊Q8—2021no—
133C4×Q8—2021no—
134C4×Q8—2021no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4C2×D10112yes—
138C4×D4C2×D10112yes—
139#117—1012no—
140C2×C4⋊C4C2×D10111yescharacteristic
141C2×C22⋊C4—21no—
142C2×C42C2×D10111yescharacteristic
143#107—1014no—
144#106—1014no—
145C4⋊C8—1012no—
146C4⋊C8—1012no—
147#102—1014no—
148#102—1014no—
149C24—21no—
150C2×Q8—2021no—
151C2×Q8—2021no—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4Dic20214yescomplemented
155C2×D4C22×D10112yes—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4C22⋊D10111yescharacteristic
161C22×C4C22⋊D10111yescharacteristic
162C22×C4C22×D10111yescharacteristic
163M16—2024no—
164C2×C8—1014no—
165C4⋊C4—21no—
166C4⋊C4—1012no—
167C4⋊C4—1012no—
168C4⋊C4—2022no—
169C4⋊C4—2022no—
170C4⋊C4—2021no—
171C4⋊C4—2021no—
172C4⋊C4C22×D10112yes—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—2022no—
176C42—2022no—
177C42C22⋊D10111yescharacteristic
178C42C22⋊D10111yescharacteristic
179C42C22×D10111yescharacteristic
180C42C22×D10111yescharacteristic
181C23—22no—
182C23—24no—
183C23—42no—
184C23—42no—
185C23C2×C22⋊D10111yescharacteristic
186Q8—4042no—
187Q8—4042no—
188D4—22no—
189D4—24no—
190D4C2×Dic20214yes—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—2022no—
198C2×C4—2022no—
199C2×C4C2×C22⋊D10111yescharacteristic
200C2×C4C2×C22⋊D10111yescharacteristic
201C2×C4C2×C22⋊D10111yescharacteristic
202C2×C4C2×C22⋊D10111yescharacteristic
203C2×C4C2×C22⋊D10111yescharacteristic
204C2×C4C2×Dic20212yes—
205C2×C4#1310807412yes—
206C2×C4C23×D10111yescharacteristic
207C8—2024no—
208C22—21no—
209C22—21no—
210C22—24no—
211C22—48no—
212C22—42no—
213C22—42no—
214C22#1569223412yes—
215C22C22×C22⋊D10111yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—4042no—
221C4—4042no—
222C4#1569223412yes—
223C4#1569228112yes—
224C4C22×Dic20211yescharacteristic
225C4C2×#1310807411yescharacteristic
226C2—22no—
227C2—44no—
228C2C2×#1569223411yescharacteristic
229C2C2×#1569228111yescharacteristic
230C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×C4×D4×C101C2
socleC22×C101C22×D4
prime radicalC22×D4×C101C22
hypercenterC2×C4×D4D101
hypocenterC101#3077
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20265734C2semidirect
2. #20265477C2semidirect
3. #20265467C2semidirect
4. #20265457C2semidirect
5. #20265217C2semidirect
6. #20265201C2semidirect
10. #15692255C22semidirect
11. #15692255C22semidirect
12. #15692247C22semidirect
18. #15692188C22semidirect
19. #15692186C22semidirect
20. C4⋊#11717579C22semidirect
21. C4⋊#11717579C22semidirect
48. C2×D4×C101Q8semidirect
116. C101#3077semidirect
154. C2×D4Dic202semidirect

Maximal subgroups

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

subgroupindex
#202657342
#202654772
#202654672
#202654572
#202652172
#202652012
#202651822
C2×C4×D4×C1012
#3077101