Finite Group Databasebeta

← #27187002

Subgroups — #27187002

Order 12928.

356 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D101⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
2C2×#15692317C212yesmaximal
3#20266121C211yesmaximal, characteristic, complemented
4#20266050C212yesmaximal, complemented
5C4×D4×D101C211yesmaximal, characteristic
6C2×D101⋊C8C2212yes—
7C2×C4⋊#87938C2212yes—
8#15692322—21no—
9C4⋊#11717581—21no—
10#15692317—24no—
11(C2×#87938)⋊C4—21no—
12(C2×#87938)⋊C4—21no—
13C2×D4×D101C2211yescharacteristic
14C22×C4×D101—21no—
15C4×D404C411yescharacteristic, complemented
16D4×Dic101C411yescharacteristic, complemented
17#15692217C411yescharacteristic, complemented
18C4×C22⋊D101—21no—
19#15692215—21no—
20C4⋊C4×D101C2211yescharacteristic
21C22⋊C4×D101—21no—
22C42×D101C2211yescharacteristic
23C4×D4×C101C411yescharacteristic, complemented
24C22×#87938—22no—
25D101⋊C8—22no—
26D101⋊C8—22no—
27C2×#11717581—22no—
28C4⋊#87938—22no—
29C4⋊#87938—22no—
30C23×D101—21no—
31C2×D404C2×C411yescharacteristic
32C2×C22⋊D101—21no—
33D4×D101—21no—
34D4×D101—21no—
35D4×D101D412yes—
36D4×D101Q812yes—
37C22×Dic101—21no—
38C2×C4×D101—21no—
39C2×C4×D101—21no—
40C2×C4×D101—21no—
41C2×C4×D101—44no—
42C2×C4×D101D411yescharacteristic
43C2×C4×D101D411yescharacteristic
44C2×C4×D101C2311yescharacteristic
45C4⋊Dic101C2×C411yescharacteristic
46#13108053—21no—
47C22⋊Dic101—21no—
48#13108051—21no—
49#13108051—21no—
50C4×Dic101C2×C411yescharacteristic
51C4×Dic101D411yescharacteristic
52C4×Dic101D411yescharacteristic
53C2×D4×C101C2×C411yescharacteristic
54C22×C4×C101—21no—
55C4⋊C4×C101C2×C411yescharacteristic
56C22⋊C4×C101—21no—
57C42×C101C2×C411yescharacteristic
58C23×C101—21no—
59C22×D101—21no—
60C22×D101—22no—
61C22×D101—22no—
62C22×D101—42no—
63C22×D101—42no—
64C22×D101C2×D411yescharacteristic
65C2×#87938—42no—
66C2×#87938—42no—
67C2×#87938—42no—
68D4×C101—21no—
69D4×C101C4⋊C412yes—
70C2×C4×C101—21no—
71C2×C4×C101—42no—
72C2×C4×C101C22⋊C411yescharacteristic
73C2×C4×C101C22⋊C411yescharacteristic
74C2×C4×C101C22×C411yescharacteristic
75C22⋊D101—22no—
76C22⋊D101—22no—
77C2×Dic101—21no—
78C2×Dic101—42no—
79C2×Dic101C22⋊C411yescharacteristic
80C2×Dic101C22⋊C411yescharacteristic
81C2×Dic101C2×D411yescharacteristic
82D404—21no—
83D404C4⋊C412yes—
84C4×D101—22no—
85C4×D101—22no—
86C4×D101—42no—
87C4×D101—42no—
88C4×D101—42no—
89C4×D101C2×D411yescharacteristic
90C4×D101C2×Q811yescharacteristic
91C4×D101#4111yescharacteristic
92C4×D101#4111yescharacteristic
93#11717581—42no—
94C22×C101—22no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101C2×C22⋊C411yescharacteristic
98C2×D101—22no—
99C2×D101—41no—
100C2×D101—41no—
101C2×D101—44no—
102C2×D101—44no—
103C2×D101#12111yescharacteristic
104C2×D101C22⋊Q811yescharacteristic
105C2×D101#13512yes—
106C2×D101AGL(1,ℤ/8)12yes—
107#87938—82no—
108C4×C101—21no—
109C4×C101—21no—
110C4×C101—41no—
111C4×C101C2×C4⋊C411yescharacteristic
112C4×C101#11711yescharacteristic
113Dic101—21no—
114Dic101—21no—
115Dic101—41no—
116Dic101#12111yescharacteristic
117Dic101C22⋊Q811yescharacteristic
118C2×C101—42no—
119C2×C101#38011yescharacteristic
120C2×C101#41611yescharacteristic
121C2×C101#41811yescharacteristic
122D101—42no—
123D101#46312yes—
124D101#47812yes—
125#1804—1011nomaximal, Sylow
126C101#180411yescharacteristic, complemented, Sylow
127C2×C4×D4—1011no—
128C2×C4⋊C8—1011no—
129C2×#102—1012no—
130#382—1011no—
131#340—1012no—
132C22×D4—1011no—
133C23×C4—2021no—
134C22×C8—1012no—
135C4×D4—1011no—
136C4×D4—1011no—
137C4×D4—1011no—
138C4×D4—2021no—
139C4×D4—2021no—
140C4×D4#8793811yescharacteristic, complemented
141C2×C4⋊C4—1011no—
142C2×C4⋊C4—1012no—
143C2×C22⋊C4—2021no—
144C2×C42—1011no—
145C4⋊C8—2021no—
146C4⋊C8—2021no—
147#102—2024no—
148#95—2021no—
149#95—2021no—
150C24—2021no—
151C2×D4—1012no—
152C2×D4—1012no—
153C2×D4—1011no—
154C2×D4—2021no—
155C2×D4—2021no—
156C2×D4—2021no—
157C2×D4C2×#8793811yescharacteristic
158C22×C4—21no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—4044no—
163C22×C4—2021no—
164C22×C4—2021no—
165C22×C4—2021no—
166C22×C4—2022no—
167C22×C4—2021no—
168C2×C8—2022no—
169C2×C8—2022no—
170C2×C8—2022no—
171C4⋊C4—1011no—
172C4⋊C4—2021no—
173C4⋊C4—2022no—
174C4⋊C4—2022no—
175C4⋊C4C2×#8793811yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—2021no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C42—1011no—
181C42—1011no—
182C42—1011no—
183C42C2×#8793811yescharacteristic
184C23—21no—
185C23—1011no—
186C23—4042no—
187C23—4042no—
188C23—2022no—
189C23—2022no—
190C23—2021no—
191D4—21no—
192D4—1012no—
193D4—2022no—
194D4—2022no—
195D4—2021no—
196D4C4⋊#8793812yes—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—1011no—
200C2×C4—1011no—
201C2×C4—1011no—
202C2×C4—1011no—
203C2×C4—1011no—
204C2×C4—1011no—
205C2×C4—1011no—
206C2×C4—4042no—
207C2×C4—4042no—
208C2×C4—4042no—
209C2×C4—4042no—
210C2×C4—4042no—
211C2×C4—4042no—
212C2×C4—4042no—
213C2×C4—2022no—
214C2×C4—2022no—
215C2×C4—2021no—
216C2×C4C22⋊#8793811yescharacteristic
217C2×C4C22⋊#8793811yescharacteristic
218C2×C4C22×#8793811yescharacteristic
219C8—4042no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—1011no—
224C22—1011no—
225C22—1012no—
226C22—1012no—
227C22—4044no—
228C22—4044no—
229C22—4041no—
230C22—4041no—
231C22—2022no—
232C22C2×C22⋊#8793811yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—8082no—
237C4—1011no—
238C4—1011no—
239C4—4041no—
240C4—2021no—
241C4—2021no—
242C4C2×C4⋊#8793811yescharacteristic
243C4#1569233411yescharacteristic
244C2—42no—
245C2—1012no—
246C2—1012no—
247C2—4042no—
248C2#2026611911yescharacteristic
249C2#2026616711yescharacteristic
250C2#2026616911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC4×D4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C22
cosocleC2×C4×D101C23
hypercenterC4×D4#87938
hypocenterC101#1804
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. D101⋊(C4⋊C8)C2semidirect
3. #20266121C2semidirect
4. #20266050C2semidirect
15. C4×D404C4semidirect
16. D4×Dic101C4semidirect
17. #15692217C4semidirect
23. C4×D4×C101C4semidirect
126. C101#1804semidirect
140. C4×D4#87938semidirect

Maximal subgroups

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

subgroupindex
D101⋊(C4⋊C8)2
C2×#156923172
#202661212
#202660502
C4×D4×D1012
#1804101