Finite Group Databasebeta

← C2×D4⋊#622 #11774174

Subgroups — C2×D4⋊#622 #11774174

Order 1152.

770 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 114 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×D4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C4⋊#622C211yesmaximal, characteristic, complemented
3C2×C4×#622C211yesmaximal, characteristic, complemented
4D4⋊#622C214yesmaximal, direct product factor
5C22×C32×D4C411yescharacteristic
6C23×C3⋊Dic3—21no—
7C22×#622—21no—
8C22×#622—21no—
9C2×#3757—21no—
10D4×C3⋊Dic3—21no—
11D4×C3⋊Dic3—21no—
12D4×C3⋊Dic3C412yes—
13D4×C3⋊Dic3C2212yes—
14C2×C4⋊(C3⋊Dic3)C411yescharacteristic
15C2×C4×C3⋊Dic3C2211yescharacteristic
16C4⋊#622—21no—
17C4⋊#622C2212yescomplemented
18C4×#622—21no—
19C4×#622C2212yescomplemented
20C2×D4×Dic3—62no—
21C24×C32—21no—
22C2×C32×D4—21no—
23C2×C32×D4—22no—
24C2×C32×D4C814yescomplemented
25C2×C32×D4C2×C412yes—
26C22×C32×C4C2×C411yescharacteristic
27C22×C3⋊Dic3—21no—
28C22×C3⋊Dic3—22no—
29C22×C3⋊Dic3—22no—
30C22×C3⋊Dic3—42no—
31C22×C3⋊Dic3—42no—
32C22×C3⋊Dic3D411yescharacteristic
33C22×C3⋊Dic3D411yescharacteristic
34C2×#622—24no—
35C2×#622—22no—
36C2×#622—22no—
37C2×#622—42no—
38#3757—22no—
39#3757—22no—
40C4⋊(C3⋊Dic3)—21no—
41C4⋊(C3⋊Dic3)C2×C412yes—
42C4×C3⋊Dic3C2×C411yescharacteristic
43C4×C3⋊Dic3D411yescharacteristic
44C4×C3⋊Dic3D411yescharacteristic
45C4×C3⋊Dic3C2311yescharacteristic
46C2×D4⋊C8—91nomaximal, Sylow
47C22×C3×D4—22no—
48C23×Dic3—64no—
49C2×#217—64no—
50D4×Dic3—64no—
51D4×Dic3—64no—
52D4×Dic3—64no—
53D4×Dic3—64no—
54C2×C4⋊Dic3—62no—
55C2×C4×Dic3—62no—
56C23×C32—24no—
57C23×C32—22no—
58C23×C32—42no—
59C23×C32—42no—
60C23×C32C22⋊C411yescharacteristic
61C32×D4—24no—
62C32×D4—22no—
63C32×D4C2×C814yescomplemented
64C2×C32×C4C22⋊C411yescharacteristic
65C2×C32×C4C2×C812yes—
66C2×C32×C4M1612yes—
67C2×C32×C4C22×C411yescharacteristic
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3—42no—
71C2×C3⋊Dic3—44no—
72C2×C3⋊Dic3—42no—
73C2×C3⋊Dic3—42no—
74C2×C3⋊Dic3C22⋊C411yescharacteristic
75C2×C3⋊Dic3C22⋊C411yescharacteristic
76C2×C3⋊Dic3D812yes—
77C2×C3⋊Dic3SD1612yes—
78C2×C3⋊Dic3C2×D411yescharacteristic
79C2×C3⋊Dic3C2×D411yescharacteristic
80#622—24no—
81#622—42no—
82C2×C4×D4—91no—
83C2×C4⋊C8—91no—
84C2×C4×C8—91no—
85D4⋊C8—94no—
86C24×C3—24no—
87C2×C3×D4—24no—
88C2×C3×D4—28no—
89C2×C3×D4—28no—
90C2×C3×D4—24no—
91C22×C3×C4—22no—
92C22×Dic3—62no—
93C22×Dic3—62no—
94C22×Dic3—64no—
95C22×Dic3—68no—
96C22×Dic3—68no—
97C22×Dic3—128no—
98C22×Dic3—128no—
99#217—68no—
100#217—68no—
101C4⋊Dic3—64no—
102C4⋊Dic3—64no—
103C4×Dic3—62no—
104C4×Dic3—62no—
105C4×Dic3—62no—
106C4×Dic3—62no—
107C22×C32—24no—
108C22×C32—48no—
109C22×C32—42no—
110C22×C32—42no—
111C22×C32C22⋊C812yes—
112C22×C32#10212yes—
113C22×C32#10412yes—
114C22×C32C2×C22⋊C411yescharacteristic
115C32×C4C22⋊C812yes—
116C32×C4C22×C811yescharacteristic
117C32×C4C2×M1611yescharacteristic
118C3⋊Dic3—21no—
119C3⋊Dic3—21no—
120C3⋊Dic3—42no—
121C3⋊Dic3#10212yes—
122C3⋊Dic3C2×D811yescharacteristic
123C3⋊Dic3C2×SD1611yescharacteristic
124C22×D4#15711yescharacteristic
125C23×C4—181no—
126C22×C8—181no—
127C22×C8—181no—
128C4×D4—92no—
129C4×D4—92no—
130C4×D4—181no—
131C4×D4—181no—
132C2×C4⋊C4—91no—
133C2×C22⋊C4—181no—
134C2×C42—91no—
135C4⋊C8—92no—
136C4⋊C8—181no—
137C4×C8—92no—
138C4×C8—181no—
139C23×C3—28no—
140C23×C3—216no—
141C23×C3—22no—
142C23×C3—48no—
143C23×C3—48no—
144C3×D4—28no—
145C3×D4—216no—
146C3×D4—28no—
147C2×C3×C4—22no—
148C2×C3×C4—24no—
149C2×C3×C4—24no—
150C2×C3×C4—22no—
151C2×Dic3—62no—
152C2×Dic3—64no—
153C2×Dic3—64no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—68no—
158C2×Dic3—124no—
159C2×Dic3—1216no—
160C2×Dic3—128no—
161C2×Dic3—128no—
162C2×Dic3—128no—
163C2×C32—44no—
164C2×C32D4⋊C814yescomplemented
165C2×C32C2×C22⋊C811yescharacteristic
166C2×C32C2×#10211yescharacteristic
167C2×C32C2×#10411yescharacteristic
168C24—21no—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4#62214yescomplemented
172C2×D4C2×#15712yes—
173C22×C4—91no—
174C22×C4—91no—
175C22×C4—182no—
176C22×C4—182no—
177C22×C4—181no—
178C22×C4—362no—
179C22×C4—362no—
180C22×C4C2×#15711yescharacteristic
181C2×C8—182no—
182C2×C8—184no—
183C2×C8—182no—
184C2×C8—362no—
185C4⋊C4—92no—
186C4⋊C4—181no—
187C22⋊C4—182no—
188C22⋊C4—182no—
189C42—91no—
190C42—91no—
191C42—91no—
192C42—91no—
193C22×C3—216no—
194C22×C3—22no—
195C22×C3—24no—
196C22×C3—24no—
197C22×C3—24no—
198C22×C3—48no—
199C22×C3—432no—
200C22×C3—48no—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—24no—
204Dic3—64no—
205Dic3—62no—
206Dic3—62no—
207Dic3—122no—
208Dic3—122no—
209Dic3—128no—
210C32C2×D4⋊C811yescharacteristic, complemented, Sylow
211C23—22no—
212C23—24no—
213C23—42no—
214C23—42no—
215C23C22⋊#15711yescharacteristic
216D4—22no—
217D4—24no—
218D4C2×#62214yescomplemented
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—92no—
222C2×C4—92no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—182no—
226C2×C4—182no—
227C2×C4—362no—
228C2×C4—362no—
229C2×C4—364no—
230C2×C4—362no—
231C2×C4C2×#62212yes—
232C2×C4#379212yes—
233C2×C4C22⋊#15711yescharacteristic
234C2×C4C22×#15711yescharacteristic
235C8—184no—
236C8—362no—
237C2×C3—28no—
238C2×C3—22no—
239C2×C3—22no—
240C2×C3—22no—
241C2×C3—416no—
242C22—24no—
243C22—48no—
244C22—42no—
245C22—42no—
246C22#6378212yes—
247C22#6378312yes—
248C22C22⋊#62212yes—
249C22C2×C22⋊#15711yescharacteristic
250C4—91no—
251C4—91no—
252C4—92no—
253C4—181no—
254C4—181no—
255C4—362no—
256C4C22⋊#62212yes—
257C4C22×#62211yescharacteristic
258C4C2×#379211yescharacteristic
259C3—22no—
260C2—44no—
261C2D4⋊#62214yesdirect product factor
262C2C2×#6378211yescharacteristic
263C2C2×#6378311yescharacteristic
264C2C2×C22⋊#62211yescharacteristic
subgroupG/H
centerC23C22⋊#157
commutatorC32×C4C22×C8
FrattiniC2×C4C22×#157
Fitting, prime radicalC22×C32×D4C4
socleC23×C32C22⋊C4
cosocleC4×C3⋊Dic3C23
hypercenterC22×D4#157
hypocenterC32C2×D4⋊C8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#622C2semidirect
3. C2×C4×#622C2semidirect
4. D4⋊#622C2direct (N × H)
17. C4⋊#622C22semidirect
19. C4×#622C22semidirect
24. C2×C32×D4C8semidirect
63. C32×D4C2×C8semidirect
164. C2×C32D4⋊C8semidirect
171. C2×D4#622semidirect
210. C32C2×D4⋊C8semidirect
218. D4C2×#622semidirect
261. C2D4⋊#622direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊Dic32
C2×C4⋊#6222
C2×C4×#6222
D4⋊#6222
C2×D4⋊C89