Finite Group Databasebeta

← #27187145

Subgroups — #27187145

Order 12928.

380 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×D101⋊C8C211yesmaximal, characteristic
2C2×#15692318C212yesmaximal, complemented
3#20266119C211yesmaximal, characteristic
4#20266050C212yesmaximal, complemented
5C22⋊Q8×D101C211yesmaximal, characteristic
6C2×D101⋊C8—24no—
7C2×D101⋊C8—21no—
8C2×D101⋊C8C2212yes—
9C22×#11717581—21no—
10C2×C4⋊#87938C2212yes—
11C2×C22⋊#87938—21no—
12#15692318—24no—
13(C2×#87938)⋊C4—21no—
14C2×Q8×D101C2211yescharacteristic
15C22×C4×D101C2211yescharacteristic
16#15692235C411yescharacteristic, complemented
17#15692234C411yescharacteristic, complemented
18#15692233—21no—
19#15692232—21no—
20#15692231C411yescharacteristic, complemented
21C4⋊C4×D101—21no—
22C4⋊C4×D101C2211yescharacteristic
23C22⋊C4×D101—21no—
24C22⋊Q8×C101C411yescharacteristic, complemented
25C22×#87938—22no—
26D101⋊C8—24no—
27D101⋊C8—22no—
28D101⋊C8D414yes—
29C2×#11717581—24no—
30C2×#11717581—22no—
31C4⋊#87938—22no—
32C4⋊#87938—22no—
33C22⋊#87938—42no—
34C23×D101D411yescharacteristic
35C2×Dic202—21no—
36C2×Dic202C2×C411yescharacteristic
37Q8×D101—22no—
38Q8×D101—22no—
39C22×Dic101D411yescharacteristic
40C2×C4×D101—22no—
41C2×C4×D101—22no—
42C2×C4×D101—21no—
43C2×C4×D101—21no—
44C2×C4×D101C2×C411yescharacteristic
45C2×C4×D101D411yescharacteristic
46C2×C4×D101D411yescharacteristic
47C2×C4×D101C2311yescharacteristic
48C4⋊Dic101—21no—
49C4⋊Dic101C2×C411yescharacteristic
50#13108053—21no—
51#13108053—21no—
52#13108053—21no—
53C22⋊Dic101—21no—
54#13108051—21no—
55#13108051—21no—
56C2×Q8×C101C2×C411yescharacteristic
57C22×C4×C101C2×C411yescharacteristic
58C4⋊C4×C101—21no—
59C4⋊C4×C101C2×C411yescharacteristic
60C22⋊C4×C101—21no—
61C23×C101C22⋊C411yescharacteristic
62C22×D101—22no—
63C22×D101—22no—
64C22×D101C22⋊C411yescharacteristic
65C22×D101SD1612yes—
66C22×D101Q1612yes—
67C22×D101C2×D411yescharacteristic
68C2×#87938—42no—
69C2×#87938—42no—
70C2×#87938—42no—
71Q8×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101—21no—
74C2×C4×C101—21no—
75C2×C4×C101C22⋊C411yescharacteristic
76C2×C4×C101C22×C411yescharacteristic
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C2×D411yescharacteristic
83C4×D101—22no—
84C4×D101—22no—
85C4×D101—21no—
86C4×D101—21no—
87C4×D101—42no—
88C4×D101—42no—
89C4×D101C2×D412yes—
90C4×D101#4112yes—
91Dic202—22no—
92Dic202—41no—
93Dic202—41no—
94#11717581—24no—
95C22×C101—21no—
96C22×C101—21no—
97C22×C101#10312yescomplemented
98C22×C101C2×C22⋊C411yescharacteristic
99C2×D101—21no—
100C2×D101—21no—
101C2×D101—24no—
102C2×D101—24no—
103C2×D101#10312yescomplemented
104C2×D101#12112yes—
105C2×D101C2×SD1611yescharacteristic
106C2×D101C2×Q1611yescharacteristic
107C2×D101#13512yes—
108#87938—82no—
109C4×C101—21no—
110C4×C101—41no—
111C4×C101—41no—
112C4×C101C4×D412yes—
113Dic101—21no—
114Dic101—41no—
115Dic101—41no—
116Dic101#12011yescharacteristic
117Dic101#12311yescharacteristic
118C2×C101—22no—
119C2×C101#38611yescharacteristic
120C2×C101C2×#10311yescharacteristic
121C2×C101#41611yescharacteristic
122D101—22no—
123D101#46512yes—
124D101#46712yes—
125#1892—1011nomaximal, Sylow
126C101#189211yescharacteristic, complemented, Sylow
127C23×C8—1011no—
128C2×C22⋊Q8—1011no—
129C2×#103—1012no—
130#380—1011no—
131#340—1012no—
132C22×Q8—1011no—
133C23×C4—1011no—
134C22×C8—1012no—
135C22×C8—2024no—
136C22×C8—2021no—
137C22×C8—2021no—
138C22⋊Q8—1011no—
139C22⋊Q8—1011no—
140C22⋊Q8—1011no—
141C22⋊Q8—2021no—
142C22⋊Q8—2021no—
143C22⋊Q8#8793811yescharacteristic, complemented
144C2×C4⋊C4—1011no—
145C2×C4⋊C4—1012no—
146C2×C4⋊C4—2021no—
147C2×C22⋊C4—2021no—
148C2×C22⋊C4—2021no—
149#103—2024no—
150#95—2021no—
151C24—1011no—
152C2×Q8—1011no—
153C2×Q8—2021no—
154C2×Q8—2022no—
155C2×Q8—2022no—
156C2×Q8C2×#8793811yescharacteristic
157C22×C4—1011no—
158C22×C4—1011no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—2022no—
163C22×C4—2022no—
164C22×C4—2021no—
165C22×C4—2021no—
166C22×C4—2022no—
167C22×C4C2×#8793811yescharacteristic
168C2×C8—1014no—
169C2×C8—2022no—
170C2×C8—2022no—
171C2×C8—2024no—
172C2×C8—2024no—
173C4⋊C4—21no—
174C4⋊C4—1011no—
175C4⋊C4—2021no—
176C4⋊C4—2021no—
177C4⋊C4—2021no—
178C4⋊C4—2022no—
179C4⋊C4—2022no—
180C4⋊C4—2021no—
181C4⋊C4C2×#8793811yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—4042no—
184C22⋊C4—2021no—
185C22⋊C4—2021no—
186C22⋊C4—2021no—
187C23—1011no—
188C23—1012no—
189C23—1012no—
190C23—1011no—
191C23—2022no—
192C23—2022no—
193C23C22⋊#8793811yescharacteristic
194Q8—22no—
195Q8—4041no—
196Q8—4041no—
197Q8—2022no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—1011no—
202C2×C4—1012no—
203C2×C4—1012no—
204C2×C4—1011no—
205C2×C4—4042no—
206C2×C4—4042no—
207C2×C4—4042no—
208C2×C4—4042no—
209C2×C4—4042no—
210C2×C4—2021no—
211C2×C4—2021no—
212C2×C4—2022no—
213C2×C4—2021no—
214C2×C4—2021no—
215C2×C4—2021no—
216C2×C4—2021no—
217C2×C4—2022no—
218C2×C4C22⋊#8793811yescharacteristic
219C2×C4C22×#8793811yescharacteristic
220C8—2024no—
221C22—21no—
222C22—21no—
223C22—1012no—
224C22—1011no—
225C22—1011no—
226C22—1012no—
227C22—1012no—
228C22—2024no—
229C22—2024no—
230C22—2021no—
231C22—2021no—
232C22#1569231812yescomplemented
233C22C2×C22⋊#8793811yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—8082no—
238C4—1011no—
239C4—1011no—
240C4—4041no—
241C4—4041no—
242C4—2021no—
243C4D4×#8793812yes—
244C2—22no—
245C2—1012no—
246C2—1012no—
247C2—2022no—
248C2#2026612511yescharacteristic
249C2C2×#1569231811yescharacteristic
250C2#2026616711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenterC22⋊Q8#87938
hypocenterC101#1892
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15692318C2semidirect
4. #20266050C2semidirect
16. #15692235C4semidirect
17. #15692234C4semidirect
20. #15692231C4semidirect
24. C22⋊Q8×C101C4semidirect
97. C22×C101#103semidirect
103. C2×D101#103semidirect
126. C101#1892semidirect
143. C22⋊Q8#87938semidirect
232. C22#15692318semidirect

Maximal subgroups

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

subgroupindex
C22×D101⋊C82
C2×#156923182
#202661192
#202660502
C22⋊Q8×D1012
#1892101