Finite Group Databasebeta

← C2×#7829 #165753

Subgroups — C2×#7829 #165753

Order 512.

1017 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1017 conjugacy classes of subgroups — too many to draw. Only its 218 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×#2734C211yesmaximal, characteristic
2C2×#2100C212yesmaximal
3C2×#2095C211yesmaximal, characteristic
4#7829C218yesmaximal, direct product factor
5C2×#1272C211yesmaximal, characteristic
6C2×#1271C212yesmaximal
7C22×C4⋊Q8—21no—
8C22×#107—21no—
9C22×#103—21no—
10#2734—22no—
11#2734—24no—
12#2734C2214yes—
13C2×#398—21no—
14C2×#395—22no—
15C2×#391C2211yescharacteristic
16C2×#389C2212yes—
17C2×#387—22no—
18C2×#384C2211yescharacteristic
19#2100—24no—
20#2100C2218yes—
21#2095—22no—
22#2095C2214yes—
23C2×#340—22no—
24C2×#340—22no—
25C2×#336C2211yescharacteristic
26C2×#336C2212yes—
27#1272C2214yes—
28#1271C2218yes—
29C23×Q8—21no—
30C23×Q8D411yescharacteristic
31C23×C8—21no—
32C23×C8—22no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—44no—
36C2×C4⋊Q8—44no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2311yescharacteristic
47C2×#107—24no—
48C2×#107—44no—
49C2×#103—24no—
50C2×#103—44no—
51#398—24no—
52#398—24no—
53#395—28no—
54#395—28no—
55#391—22no—
56#391D412yes—
57#391C2312yes—
58#389—22no—
59#389—22no—
60#389D414yes—
61#389C2314yes—
62#387—28no—
63#387—28no—
64#384—21no—
65#384—22no—
66#384C2312yes—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71#340—28no—
72#340—28no—
73#336—21no—
74#336—22no—
75#336C2312yes—
76#336C2314yes—
77C22×Q8—24no—
78C22×Q8—21no—
79C22×Q8—22no—
80C22×Q8—44no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8—82no—
84C22×Q8—82no—
85C22×Q8SD1614yes—
86C22×Q8C2×D412yes—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—24no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—44no—
100C22×C8—44no—
101C22×C8—44no—
102C4⋊Q8—48no—
103C4⋊Q8—416no—
104C4⋊Q8—48no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—48no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4Q1618yes—
127C2×C4⋊C4C2×D414yes—
128C2×C4⋊C4C2×D414yes—
129C2×C4⋊C4C2×D412yes—
130C2×C42—24no—
131C2×C42—42no—
132C2×C42—44no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D412yes—
135C2×C42C2411yescharacteristic
136#107—48no—
137#103—48no—
138#95—24no—
139#95—28no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144C24#12011yescharacteristic
145C2×Q8—22no—
146C2×Q8—24no—
147C2×Q8—48no—
148C2×Q8—42no—
149C2×Q8—48no—
150C2×Q8—42no—
151C2×Q8—84no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—84no—
155C2×Q8—84no—
156C2×Q8—816no—
157C2×Q8C2×SD1614yes—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—28no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—82no—
179C22×C4—84no—
180C22×C4#12011yescharacteristic
181C22×C4#12012yes—
182C22×C4C2×SD1612yes—
183C22×C4C2×Q1614yes—
184C22×C4AGL(1,ℤ/8)12yes—
185C22×C4#13714yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D412yes—
188C2×C8—24no—
189C2×C8—48no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—88no—
196C4⋊C4—24no—
197C4⋊C4—28no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—416no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—48no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—81no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4—84no—
212C4⋊C4C2×Q1618yes—
213C42—42no—
214C42—41no—
215C42—41no—
216C42—48no—
217C23#45012yes—
218C23#45114yes—
219C23#45312yes—
220C23#45414yes—
221C23#45812yes—
222C23C2×#12011yescharacteristic
223Q8—44no—
224Q8—82no—
225Q8—84no—
226Q8—816no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—28no—
230C2×C4—28no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—28no—
241C2×C4—48no—
242C2×C4—416no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—88no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4—164no—
257C2×C4#45012yes—
258C2×C4#45114yes—
259C2×C4C22×SD1611yescharacteristic
260C2×C4C22×Q1612yes—
261C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
262C2×C4C2×#13712yes—
263C8—44no—
264C8—84no—
265C22#161314yes—
266C22#161718yes—
267C22#163314yes—
268C22#163718yes—
269C22#165714yes—
270C22C2×#45011yescharacteristic
271C22C2×#45112yes—
272C22C2×#45311yescharacteristic
273C22C2×#45412yes—
274C22C2×#45811yescharacteristic
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—22no—
280C4—24no—
281C4—48no—
282C4—44no—
283C4—82no—
284C4—84no—
285C4—162no—
286C2#782918yesdirect product factor
287C2C2×#161311yescharacteristic
288C2C2×#161712yes—
289C2C2×#163712yes—
290C2C2×#163311yescharacteristic
291C2C2×#165711yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7829C2direct (N × H)
286. C2#7829direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#21002
C2×#20952
#78292
C2×#12722
C2×#12712