Finite Group Databasebeta

← C2×#7134 #124127

Subgroups — C2×#7134 #124127

Order 512.

542 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 138 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×#1994C211yesmaximal, characteristic
2C2×#1276C212yesmaximal
3#7134C214yesmaximal, direct product factor
4C2×#533C411yescharacteristic
5C2×#431—22no—
6C2×#423C411yescharacteristic
7C2×C43C2211yescharacteristic
8#1994—22no—
9#1994C412yes—
10#1994C2212yes—
11C2×#339—22no—
12#1276—22no—
13#1276C2214yes—
14C22×M16—21no—
15#533—21no—
16#533—21no—
17#533—41no—
18#533—41no—
19#533C812yescomplemented
20#533C2×C412yes—
21C2×C4⋊Q8—21no—
22C2×#125C812yescomplemented
23C2×C22⋊Q8—41no—
24C2×C4×Q8—41no—
25C2×#117—21no—
26C22×C42—41no—
27C22×C42C2×C411yescharacteristic
28C22×C42D412yes—
29#431—24no—
30#431—24no—
31#431—42no—
32#431—42no—
33#423—21no—
34#423—41no—
35#423C2×C412yes—
36C2×C4⋊C8—24no—
37C2×C4⋊C8—41no—
38C2×C22⋊C8—42no—
39C2×C4×C8—42no—
40C43—41no—
41C43C2×C411yescharacteristic
42C43D412yes—
43C43C2311yescharacteristic
44#339—24no—
45#339—24no—
46C22×Q8—41no—
47C23×C4—21no—
48C23×C4—22no—
49C23×C4C22⋊C411yescharacteristic
50C2×M16—22no—
51C2×M16—41no—
52C2×M16—42no—
53C2×M16—42no—
54C22×C8—44no—
55C22×C8—41no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—41no—
59C4⋊Q8—41no—
60#125—22no—
61#125—42no—
62#125C2×C814yescomplemented
63C22⋊Q8—42no—
64C22⋊Q8—42no—
65C22⋊Q8—42no—
66C22⋊Q8—42no—
67C4×Q8—42no—
68C4×Q8—42no—
69C4×Q8—42no—
70C4×Q8—42no—
71#117—22no—
72#117—22no—
73#117—41no—
74#117—41no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C22⋊C4—41no—
79C2×C42—21no—
80C2×C42—22no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—41no—
86C2×C42—42no—
87C2×C42—41no—
88C2×C42—48no—
89C2×C42—41no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C22⋊C412yes—
92C2×C42C2×C811yescharacteristic
93C2×C42C2×C811yescharacteristic
94C2×C42M1611yescharacteristic
95C2×C42M1611yescharacteristic
96C2×C42SD1612yes—
97C2×C42SD1612yes—
98C2×C42Q1612yes—
99C2×C42Q1612yes—
100C2×C42C22×C411yescharacteristic
101C2×C42C2×D412yes—
102C4⋊C8—28no—
103C4⋊C8—42no—
104C4⋊C8—44no—
105C4⋊C8—42no—
106C22⋊C8—44no—
107C22⋊C8—44no—
108C4×C8—44no—
109C4×C8—44no—
110C24#9911yescharacteristic
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—84no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—48no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4C22⋊C811yescharacteristic
130C22×C4C22⋊C811yescharacteristic
131C22×C4#9911yescharacteristic
132C22×C4#10011yescharacteristic
133C22×C4#10011yescharacteristic
134C22×C4#10312yes—
135C22×C4#10312yes—
136C22×C4#10412yes—
137C22×C4#10412yes—
138C22×C4C2×C22⋊C411yescharacteristic
139M16—44no—
140M16—82no—
141C2×C8—48no—
142C2×C8—42no—
143C2×C8—88no—
144C2×C8—81no—
145C2×C8—81no—
146C4⋊C4—24no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—22no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42—21no—
163C42—21no—
164C42—22no—
165C42—22no—
166C42—44no—
167C42—48no—
168C42—44no—
169C42—42no—
170C42—42no—
171C42C22⋊C811yescharacteristic
172C42C22⋊C811yescharacteristic
173C42#10312yes—
174C42#10312yes—
175C42C22×C811yescharacteristic
176C42C2×M1611yescharacteristic
177C42C2×SD1612yes—
178C42C2×Q1612yes—
179C23—42no—
180C23#32311yescharacteristic
181C23#32311yescharacteristic
182C23#32812yes—
183C23#32812yes—
184C23C2×#9911yescharacteristic
185Q8—84no—
186Q8—84no—
187C2×C4—28no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—84no—
209C2×C4#32311yescharacteristic
210C2×C4#32311yescharacteristic
211C2×C4Q8⋊C812yes—
212C2×C4Q8⋊C812yes—
213C2×C4Q8⋊C812yes—
214C2×C4Q8⋊C812yes—
215C2×C4#32812yes—
216C2×C4#32812yes—
217C2×C4#32911yescharacteristic
218C2×C4#32911yescharacteristic
219C2×C4#32912yes—
220C2×C4#33312yes—
221C2×C4#33311yescharacteristic
222C2×C4#33311yescharacteristic
223C2×C4C2×C22⋊C811yescharacteristic
224C2×C4C2×#10011yescharacteristic
225C2×C4C2×#10312yes—
226C2×C4C2×#10412yes—
227C8—88no—
228C8—82no—
229C22—22no—
230C22—24no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22#127211yescharacteristic
235C22#127212yes—
236C22#127211yescharacteristic
237C22C2×#32311yescharacteristic
238C22C2×#32812yes—
239C4—24no—
240C4—24no—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—41no—
245C4—42no—
246C4—42no—
247C4—84no—
248C4#127211yescharacteristic
249C4#127212yes—
250C4#127211yescharacteristic
251C4C2×Q8⋊C812yes—
252C4C2×#32911yescharacteristic
253C4C2×#33311yescharacteristic
254C2—21no—
255C2—22no—
256C2—21no—
257C2—22no—
258C2#713412yesdirect product factor
259C2C2×#127211yescharacteristic
subgroupG/H
centerC2×C4#333
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC22#1272
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7134C2direct (N × H)
19. #533C8semidirect
22. C2×#125C8semidirect
62. #125C2×C8semidirect
258. C2#7134direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#19942
C2×#12762
#71342