Finite Group Databasebeta

← C2×#7132 #124120

Subgroups — C2×#7132 #124120

Order 512.

978 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 186 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×#1445C211yesmaximal, characteristic
2C2×#1280C212yesmaximal
3#7132C214yesmaximal, direct product factor
4C23×C4⋊C4C411yescharacteristic
5C22×C22⋊C8—22no—
6C22×#95—22no—
7C22×#95C411yescharacteristic
8C22×#95C2211yescharacteristic
9#1445—22no—
10#1445C412yes—
11#1445C2212yes—
12#1280—22no—
13#1280C2214yes—
14C24×C4—21no—
15C24×C4—21no—
16C24×C4C2×C411yescharacteristic
17C24×C4D412yes—
18C23×C8—42no—
19C22×C4⋊C4—21no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4—41no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4—42no—
24C22×C4⋊C4C814yescomplemented
25C22×C4⋊C4C2×C412yes—
26C2×C22⋊C8—28no—
27C2×C22⋊C8—24no—
28C2×C22⋊C8—42no—
29C2×C22⋊C8—44no—
30C2×#95—21no—
31C2×#95—24no—
32C2×#95—24no—
33C2×#95—41no—
34C2×#95—41no—
35C2×#95—42no—
36C2×#95—48no—
37C2×#95C2×C412yes—
38C2×#95C2×C411yescharacteristic
39C2×#95D412yes—
40C2×#95C2311yescharacteristic
41C25C22⋊C411yescharacteristic
42C23×C4—21no—
43C23×C4—21no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—24no—
48C23×C4—24no—
49C23×C4—24no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—44no—
53C23×C4—44no—
54C23×C4—44no—
55C23×C4—44no—
56C23×C4—42no—
57C23×C4—44no—
58C23×C4—44no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—42no—
62C23×C4—42no—
63C23×C4—44no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C2×C812yes—
67C23×C4M1612yes—
68C23×C4SD1614yes—
69C23×C4Q1614yes—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D412yes—
72C22×C8—48no—
73C22×C8—44no—
74C22×C8—84no—
75C22×C8—84no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4C2×C814yescomplemented
87C22⋊C8—28no—
88C22⋊C8—48no—
89C22⋊C8—44no—
90#95—41no—
91#95—42no—
92#95—41no—
93#95—42no—
94#95—42no—
95#95—48no—
96#95—48no—
97C24—24no—
98C24—22no—
99C24—22no—
100C24C22⋊C812yes—
101C24#9911yescharacteristic
102C24#9911yescharacteristic
103C24#10012yes—
104C24#10314yes—
105C24#10414yes—
106C24C2×C22⋊C411yescharacteristic
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—48no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—48no—
138C22×C4—48no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—48no—
142C22×C4—44no—
143C22×C4—416no—
144C22×C4—42no—
145C22×C4—416no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—82no—
154C22×C4—88no—
155C22×C4—82no—
156C22×C4—84no—
157C22×C4—88no—
158C22×C4—88no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—84no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4C22⋊C812yes—
165C22×C4#10314yes—
166C22×C4C22×C811yescharacteristic
167C22×C4C2×M1611yescharacteristic
168C22×C4C2×SD1612yes—
169C22×C4C2×Q1612yes—
170C2×C8—48no—
171C2×C8—84no—
172C2×C8—816no—
173C2×C8—84no—
174C4⋊C4—44no—
175C4⋊C4—48no—
176C4⋊C4—44no—
177C23—24no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—22no—
182C23—22no—
183C23—28no—
184C23—28no—
185C23—28no—
186C23—28no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23#32312yes—
192C23#32312yes—
193C23Q8⋊C818yes—
194C23#32814yes—
195C23#32814yes—
196C23#32914yes—
197C23#33314yes—
198C23#35214yes—
199C23#35312yes—
200C23#35512yes—
201C23C2×C22⋊C811yescharacteristic
202C23C2×#9911yescharacteristic
203C23C2×#10011yescharacteristic
204C23C2×#10312yes—
205C23C2×#10412yes—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—48no—
223C2×C4—48no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—48no—
234C2×C4—48no—
235C2×C4—44no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4—816no—
243C2×C4—84no—
244C2×C4—88no—
245C2×C4—88no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C8—88no—
250C22—21no—
251C22—22no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—24no—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—28no—
263C22—216no—
264C22—28no—
265C22#127214yes—
266C22#127214yes—
267C22#131714yes—
268C22#131814yes—
269C22#135218yes—
270C22#135318yes—
271C22C2×#32311yescharacteristic
272C22C2×Q8⋊C812yes—
273C22C2×#32812yes—
274C22C2×#32911yescharacteristic
275C22C2×#33311yescharacteristic
276C22C2×#35212yes—
277C22C2×#35311yescharacteristic
278C22C2×#35511yescharacteristic
279C4—42no—
280C4—41no—
281C4—41no—
282C4—42no—
283C4—44no—
284C4—42no—
285C4—84no—
286C4—84no—
287C2—21no—
288C2—21no—
289C2—22no—
290C2—24no—
291C2#713218yesdirect product factor
292C2C2×#127211yescharacteristic
293C2C2×#131711yescharacteristic
294C2C2×#131811yescharacteristic
295C2C2×#135212yes—
296C2C2×#135312yes—
subgroupG/H
center, socleC24#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7132C2direct (N × H)
24. C22×C4⋊C4C8semidirect
86. C2×C4⋊C4C2×C8semidirect
291. C2#7132direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#14452
C2×#12802
#71322