Finite Group Databasebeta

← #27187117

Subgroups — #27187117

Order 12928.

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

normal subgroups only

This group has 404 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×C4⋊#87938C211yesmaximal, characteristic
2C2×#15692318C212yesmaximal, complemented
3D101⋊(C22⋊C8)C211yesmaximal, characteristic
4#20266050C212yesmaximal, complemented
5C22⋊Q8×D101C211yesmaximal, characteristic
6C23×#87938—21no—
7C2×D101⋊C8C2212yes—
8C2×C4⋊#87938—24no—
9C2×C4⋊#87938—21no—
10C2×C4⋊#87938C2212yes—
11#15692318—24no—
12#15692312—21no—
13C22⋊#11717581—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—24no—
26C22×#87938—22no—
27C22×#87938—42no—
28C22×#87938—42no—
29D101⋊C8—22no—
30D101⋊C8—22no—
31C2×#11717581—22no—
32C4⋊#87938—24no—
33C4⋊#87938—22no—
34C4⋊#87938D414yes—
35C23×D101D411yescharacteristic
36C2×Dic202—21no—
37C2×Dic202C2×C411yescharacteristic, complemented
38Q8×D101—22no—
39Q8×D101—22no—
40C22×Dic101D411yescharacteristic
41C2×C4×D101—22no—
42C2×C4×D101—22no—
43C2×C4×D101—21no—
44C2×C4×D101—21no—
45C2×C4×D101C2×C411yescharacteristic
46C2×C4×D101D411yescharacteristic
47C2×C4×D101D411yescharacteristic
48C2×C4×D101C2311yescharacteristic
49C4⋊Dic101—21no—
50C4⋊Dic101C2×C411yescharacteristic, complemented
51#13108053—21no—
52#13108053—21no—
53#13108053—21no—
54C22⋊Dic101—21no—
55#13108051—21no—
56#13108051—21no—
57C2×Q8×C101C2×C411yescharacteristic, complemented
58C22×C4×C101C2×C411yescharacteristic
59C4⋊C4×C101—21no—
60C4⋊C4×C101C2×C411yescharacteristic, complemented
61C22⋊C4×C101—21no—
62C23×C101C22⋊C411yescharacteristic
63C22×D101—21no—
64C22×D101—21no—
65C22×D101—21no—
66C22×D101—21no—
67C22×D101C22⋊C411yescharacteristic
68C22×D101SD1612yes—
69C22×D101Q1612yes—
70C22×D101C2×D411yescharacteristic
71C2×#87938—24no—
72C2×#87938—48no—
73C2×#87938—42no—
74C2×#87938—42no—
75Q8×C101—22no—
76C2×C4×C101—22no—
77C2×C4×C101—21no—
78C2×C4×C101—21no—
79C2×C4×C101C22⋊C411yescharacteristic
80C2×C4×C101C22×C411yescharacteristic
81C2×Dic101—22no—
82C2×Dic101—21no—
83C2×Dic101—21no—
84C2×Dic101C22⋊C411yescharacteristic
85C2×Dic101C2×D411yescharacteristic
86C4×D101—21no—
87C4×D101—21no—
88C4×D101—22no—
89C4×D101—22no—
90C4×D101—41no—
91C4×D101—41no—
92C4×D101—41no—
93C4×D101—41no—
94C4×D101C2×D412yes—
95C4×D101#4112yes—
96Dic202—22no—
97Dic202—42no—
98#11717581—42no—
99C22×C101—21no—
100C22×C101—21no—
101C22×C101#10312yescomplemented
102C22×C101C2×C22⋊C411yescharacteristic
103C2×D101—21no—
104C2×D101—21no—
105C2×D101—22no—
106C2×D101—22no—
107C2×D101—22no—
108C2×D101—22no—
109C2×D101#10312yescomplemented
110C2×D101#12011yescharacteristic
111C2×D101#12311yescharacteristic
112C2×D101C2×SD1611yescharacteristic
113C2×D101C2×Q1611yescharacteristic
114C2×D101AGL(1,ℤ/8)11yescharacteristic
115C2×D101#13711yescharacteristic
116#87938—44no—
117C4×C101—21no—
118C4×C101—41no—
119C4×C101—41no—
120C4×C101C4×D412yes—
121Dic101—21no—
122Dic101—41no—
123Dic101—41no—
124Dic101#12112yes—
125C2×C101—22no—
126C2×C101#38611yescharacteristic
127C2×C101C2×#10311yescharacteristic
128C2×C101#41711yescharacteristic
129D101—22no—
130D101#45011yescharacteristic
131D101#45111yescharacteristic
132D101#48311yescharacteristic
133D101#48411yescharacteristic
134#1872—1011nomaximal, Sylow
135C101#187211yescharacteristic, complemented, Sylow
136C2×C22⋊Q8—1011no—
137C22×C4⋊C4—1011no—
138C2×#103—1012no—
139C2×C22⋊C8—1011no—
140#340—1012no—
141C22×Q8—1011no—
142C23×C4—1011no—
143C23×C4—2021no—
144C22×C8—1012no—
145C22⋊Q8—1011no—
146C22⋊Q8—1011no—
147C22⋊Q8—1011no—
148C22⋊Q8—2021no—
149C22⋊Q8—2021no—
150C22⋊Q8#8793811yescharacteristic, complemented
151C2×C4⋊C4—1012no—
152C2×C4⋊C4—1011no—
153C2×C4⋊C4—2021no—
154C2×C4⋊C4—2024no—
155C2×C4⋊C4—2021no—
156C2×C22⋊C4—2021no—
157#103—2024no—
158C22⋊C8—2021no—
159C22⋊C8—2021no—
160C24—1011no—
161C2×Q8—1011no—
162C2×Q8—2021no—
163C2×Q8—2022no—
164C2×Q8—2022no—
165C2×Q8C2×#8793811yescharacteristic, complemented
166C22×C4—1011no—
167C22×C4—1011no—
168C22×C4—1011no—
169C22×C4—1011no—
170C22×C4—1011no—
171C22×C4—4042no—
172C22×C4—4042no—
173C22×C4—2022no—
174C22×C4—2022no—
175C22×C4—2021no—
176C22×C4—2021no—
177C22×C4—2022no—
178C22×C4—2024no—
179C22×C4C2×#8793811yescharacteristic
180C2×C8—2022no—
181C2×C8—2022no—
182C2×C8—2022no—
183C4⋊C4—21no—
184C4⋊C4—1014no—
185C4⋊C4—1011no—
186C4⋊C4—2021no—
187C4⋊C4—2021no—
188C4⋊C4—2021no—
189C4⋊C4—2022no—
190C4⋊C4—2024no—
191C4⋊C4—2021no—
192C4⋊C4C2×#8793811yescharacteristic, complemented
193C22⋊C4—21no—
194C22⋊C4—2021no—
195C22⋊C4—2021no—
196C22⋊C4—2021no—
197C23—1011no—
198C23—1012no—
199C23—1012no—
200C23—1011no—
201C23—2021no—
202C23—2021no—
203C23—2021no—
204C23—2021no—
205C23C22⋊#8793811yescharacteristic
206Q8—22no—
207Q8—4042no—
208Q8—2022no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—1011no—
213C2×C4—1012no—
214C2×C4—1012no—
215C2×C4—1011no—
216C2×C4—4041no—
217C2×C4—4041no—
218C2×C4—4041no—
219C2×C4—4041no—
220C2×C4—4042no—
221C2×C4—4042no—
222C2×C4—4048no—
223C2×C4—2022no—
224C2×C4—2021no—
225C2×C4—2021no—
226C2×C4—2022no—
227C2×C4—2024no—
228C2×C4—2021no—
229C2×C4—2021no—
230C2×C4—2022no—
231C2×C4C22⋊#8793811yescharacteristic
232C2×C4C22×#8793811yescharacteristic
233C8—4042no—
234C22—21no—
235C22—21no—
236C22—1011no—
237C22—1011no—
238C22—1011no—
239C22—1011no—
240C22—1011no—
241C22—1011no—
242C22—1012no—
243C22—2022no—
244C22—2022no—
245C22—2022no—
246C22—2022no—
247C22—2021no—
248C22—2021no—
249C22#1569231812yescomplemented
250C22C2×C22⋊#8793811yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—1012no—
255C4—4044no—
256C4—4041no—
257C4—4041no—
258C4—2021no—
259C4D4×#8793812yes—
260C2—22no—
261C2—1011no—
262C2—1011no—
263C2—1011no—
264C2—1011no—
265C2—2022no—
266C2#2026612511yescharacteristic
267C2C2×#1569231811yescharacteristic
268C2#2026616811yescharacteristic
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#1872
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
37. C2×Dic202C2×C4semidirect
50. C4⋊Dic101C2×C4semidirect
57. C2×Q8×C101C2×C4semidirect
60. C4⋊C4×C101C2×C4semidirect
101. C22×C101#103semidirect
109. C2×D101#103semidirect
135. C101#1872semidirect
150. C22⋊Q8#87938semidirect
165. C2×Q8C2×#87938semidirect
192. C4⋊C4C2×#87938semidirect
249. C22#15692318semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#879382
C2×#156923182
D101⋊(C22⋊C8)2
#202660502
C22⋊Q8×D1012
#1872101