Finite Group Databasebeta

← C2×#7443 #150314

Subgroups — C2×#7443 #150314

Order 512.

465 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2060C211yesmaximal, characteristic
2C2×#1325C211yesmaximal, characteristic
3C2×#1296C211yesmaximal, characteristic
4#7443C214yesmaximal, direct product factor
5C2×#544C411yescharacteristic
6C2×#428—21no—
7C22×C4⋊C8—21no—
8C2×#384C2211yescharacteristic
9#2060—21no—
10#2060—21no—
11#2060C412yes—
12#2060C2212yes—
13C2×#343C411yescharacteristic
14C2×#339—21no—
15C2×#324—21no—
16#1325—21no—
17#1325C2212yes—
18#1296—21no—
19#1296C2212yes—
20C22×M16—21no—
21C23×C8—41no—
22#544—21no—
23#544—41no—
24#544—42no—
25#544C2×C412yes—
26C2×C4⋊Q8—41no—
27C2×#125—41no—
28C2×C22⋊Q8—41no—
29C2×#117—21no—
30C22×C4⋊C4—41no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34#428—22no—
35#428—22no—
36#428—42no—
37C2×#107—41no—
38C2×#106—41no—
39C2×C4⋊C8—24no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—41no—
42C2×C4⋊C8—42no—
43C2×C22⋊C8—41no—
44#384—41no—
45#384C2×C411yescharacteristic
46#384D411yescharacteristic
47#384D411yescharacteristic
48#384C2311yescharacteristic
49#343—21no—
50#343C2×C412yes—
51#339—22no—
52#339—22no—
53#324—22no—
54#324—41no—
55C22×Q8—41no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C22⋊C411yescharacteristic
60C2×M16—24no—
61C2×M16—22no—
62C2×M16—41no—
63C2×M16—42no—
64C22×C8—41no—
65C22×C8—44no—
66C22×C8—42no—
67C22×C8—82no—
68C22×C8—82no—
69C22×C8—81no—
70C4⋊Q8—42no—
71C4⋊Q8—42no—
72C4⋊Q8—44no—
73#125—42no—
74#125—42no—
75#125—44no—
76C22⋊Q8—42no—
77C22⋊Q8—42no—
78C22⋊Q8—42no—
79C22⋊Q8—42no—
80#117—22no—
81#117—22no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—82no—
99C2×C4⋊C4—82no—
100C2×C4⋊C4C22⋊C411yescharacteristic
101C2×C4⋊C4C22⋊C411yescharacteristic
102C2×C4⋊C4C22×C411yescharacteristic
103C2×C4⋊C4C2×D411yescharacteristic
104C2×C22⋊C4—41no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—41no—
108C2×C42C22⋊C411yescharacteristic
109C2×C42SD1612yes—
110C2×C42Q1612yes—
111C2×C42C2×D411yescharacteristic
112#107—42no—
113#107—42no—
114#106—42no—
115#106—42no—
116C4⋊C8—24no—
117C4⋊C8—44no—
118C4⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—81no—
121C24#9911yescharacteristic
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—162no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4#9911yescharacteristic
149C22×C4#10312yes—
150C22×C4#10412yes—
151C22×C4C2×C22⋊C411yescharacteristic
152M16—24no—
153M16—44no—
154M16—42no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—82no—
159C2×C8—82no—
160C2×C8—88no—
161C2×C8—82no—
162C2×C8—162no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—42no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C42—22no—
186C42—22no—
187C42—21no—
188C42—21no—
189C42—42no—
190C42—42no—
191C42#10312yes—
192C42C2×SD1611yescharacteristic
193C42C2×Q1611yescharacteristic
194C23—22no—
195C23—22no—
196C23#32812yes—
197C23AΣL(1,𝔽16)11yescharacteristic
198C23AΣL(1,𝔽16)11yescharacteristic
199C23#35612yes—
200C23C2×#9911yescharacteristic
201Q8—162no—
202Q8—162no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—161no—
239C2×C4—161no—
240C2×C4#32812yes—
241C2×C4C2×#10311yescharacteristic
242C2×C4C2×#10411yescharacteristic
243C8—44no—
244C8—84no—
245C8—162no—
246C22—21no—
247C22—21no—
248C22—24no—
249C22—24no—
250C22—21no—
251C22—21no—
252C22#135412yes—
253C22#135412yes—
254C22#140212yes—
255C22#140512yes—
256C22C2×#32811yescharacteristic
257C22C2×AΣL(1,𝔽16)11yescharacteristic
258C22C2×#35611yescharacteristic
259C4—21no—
260C4—21no—
261C4—24no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—22no—
266C4—21no—
267C4—82no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—82no—
273C2—21no—
274C2—22no—
275C2—21no—
276C2#744314yesdirect product factor
277C2C2×#135411yescharacteristic
278C2C2×#140211yescharacteristic
279C2C2×#140511yescharacteristic
subgroupG/H
center, socleC23AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7443C2direct (N × H)
276. C2#7443direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#20602
C2×#13252
C2×#12962
#74432