Finite Group Databasebeta

← C2×#5032 #82603

Subgroups — C2×#5032 #82603

Order 384.

1018 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 149 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×#1004C211yesmaximal, characteristic, complemented
2C2×C3×#120C211yesmaximal, characteristic, complemented
3C2×#953C212yesmaximal, complemented
4C2×#834C211yesmaximal, characteristic, complemented
5#5032C218yesmaximal, direct product factor
6C2×#806C212yesmaximal, complemented
7C2×B(4,𝔽2)—31nomaximal, Sylow
8C25×C3C2211yescharacteristic, complemented
9C22×C3×D4—21no—
10C22×C3×D4C2211yescharacteristic
11C22×#67—21no—
12C22×#67—22no—
13C22×#67C2212yes—
14#1004—22no—
15#1004—24no—
16#1004C2214yescomplemented
17C2×#237—22no—
18C2×D3×D4—22no—
19C2×#235—21no—
20C3×#120—22no—
21C3×#120C2214yescomplemented
22C2×C3×C22⋊C4—21no—
23C2×C3×C22⋊C4C2211yescharacteristic, complemented
24#953—24no—
25#953C2218yescomplemented
26C2×#217—22no—
27C2×#217C2212yescomplemented
28#834C2214yescomplemented
29#806C2218yescomplemented
30C2×H(5,𝔽2)—31no—
31C2×#120—32no—
32C2×#120D311yescharacteristic, complemented
33B(4,𝔽2)—38no—
34C2×#99—31no—
35C2×#99—32no—
36C24×C3—22no—
37C24×C3—21no—
38C24×C3—22no—
39C24×C3—21no—
40C24×C3—44no—
41C24×C3D411yescharacteristic, complemented
42C24×C3C2311yescharacteristic
43C24×C3C2312yescomplemented
44C23×D3—21no—
45C23×D3D412yescomplemented
46C2×C3×D4—22no—
47C2×C3×D4—22no—
48C2×C3×D4—24no—
49C2×C3×D4—44no—
50C2×C3×D4D412yescomplemented
51C2×C3×D4C2312yes—
52C22×C3×C4—21no—
53C22×C3×C4D411yescharacteristic, complemented
54C2×#67—22no—
55C2×#67—24no—
56C2×#67—22no—
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—22no—
60C2×#67—24no—
61C2×#67—28no—
62C2×#67—48no—
63C2×#67D414yescomplemented
64C2×#67C2314yes—
65C22×Dic3—21no—
66C22×Dic3—22no—
67C22×Dic3D412yescomplemented
68#237—28no—
69#237—28no—
70D3×D4—28no—
71D3×D4—28no—
72#235—24no—
73#235—24no—
74C2×D12—21no—
75C2×C4×D3—22no—
76C2×Dic6—21no—
77C3×C22⋊C4—21no—
78C3×C22⋊C4—24no—
79C3×C22⋊C4C2312yescomplemented
80#217—22no—
81#217—28no—
82#217C2314yescomplemented
83C25C2×D311yescharacteristic, complemented
84H(5,𝔽2)—34no—
85H(5,𝔽2)—64no—
86H(5,𝔽2)—62no—
87C2×#41—62no—
88C2×#41—61no—
89C22×D4—21no—
90C22×D4—32no—
91C22×D4—62no—
92C22×D4—61no—
93C22×D4—62no—
94C22×D4C2×D311yescharacteristic
95#120—22no—
96#120—38no—
97#120—64no—
98#120C2×D314yescomplemented
99C2×C22⋊C4—21no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—62no—
102C2×C22⋊C4C2×D311yescharacteristic, complemented
103#99—34no—
104#99—38no—
105C23×C3—28no—
106C23×C3—24no—
107C23×C3—21no—
108C23×C3—24no—
109C23×C3—22no—
110C23×C3—21no—
111C23×C3—42no—
112C23×C3—44no—
113C23×C3—44no—
114C23×C3—42no—
115C23×C3—48no—
116C23×C3—44no—
117C23×C3—42no—
118C23×C3—42no—
119C23×C3—41no—
120C23×C3—41no—
121C23×C3C2×D412yes—
122C23×C3C2×D412yes—
123C23×C3C2×D411yescharacteristic
124C23×C3C2×D411yescharacteristic
125C23×C3C2×D412yescomplemented
126C23×C3C2411yescharacteristic
127C22×D3—22no—
128C22×D3—24no—
129C22×D3—22no—
130C22×D3—44no—
131C22×D3—82no—
132C22×D3C2×D414yescomplemented
133C3×D4—24no—
134C3×D4—24no—
135C3×D4—48no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D412yescomplemented
141#67—28no—
142#67—24no—
143#67—24no—
144#67—24no—
145#67—24no—
146#67—416no—
147C2×Dic3—24no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—24no—
151C2×Dic3—44no—
152C2×Dic3C2×D414yescomplemented
153D12—24no—
154C4×D3—28no—
155Dic6—24no—
156C24—22no—
157C24—21no—
158C24—22no—
159C24—21no—
160C24—32no—
161C24—44no—
162C24—61no—
163C24#6711yescharacteristic, complemented
164C24C22×D311yescharacteristic
165C24C22×D312yescomplemented
166#41—64no—
167#41—68no—
168#41—68no—
169#41—64no—
170C2×Q8—61no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—24no—
174C2×D4—34no—
175C2×D4—34no—
176C2×D4—44no—
177C2×D4—61no—
178C2×D4—68no—
179C2×D4—68no—
180C2×D4—62no—
181C2×D4—64no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—64no—
187C2×D4—68no—
188C2×D4—128no—
189C2×D4#6712yescomplemented
190C2×D4C22×D312yes—
191C22×C4—21no—
192C22×C4—32no—
193C22×C4—62no—
194C22×C4—61no—
195C22×C4—62no—
196C22×C4#6711yescharacteristic, complemented
197C22⋊C4—21no—
198C22⋊C4—24no—
199C22⋊C4—34no—
200C22⋊C4—68no—
201C22⋊C4—62no—
202C22⋊C4C22×D312yescomplemented
203C22×C3—24no—
204C22×C3—22no—
205C22×C3—28no—
206C22×C3—24no—
207C22×C3—22no—
208C22×C3—42no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—44no—
213C22×C3—42no—
214C22×C3—48no—
215C22×C3—44no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—44no—
219C22×C3—81no—
220C22×C3#12012yes—
221C22×C3#12011yescharacteristic
222C22×C3#12011yescharacteristic
223C22×C3C22×D412yes—
224C22×C3C22×D411yescharacteristic
225C2×D3—24no—
226C2×D3—44no—
227C2×D3—44no—
228C2×D3—82no—
229C2×D3—84no—
230C3×C4—22no—
231C3×C4—42no—
232Dic3—24no—
233Dic3—44no—
234C23—24no—
235C23—28no—
236C23—21no—
237C23—24no—
238C23—22no—
239C23—21no—
240C23—34no—
241C23—42no—
242C23—44no—
243C23—42no—
244C23—44no—
245C23—44no—
246C23—48no—
247C23—42no—
248C23—41no—
249C23—42no—
250C23—41no—
251C23—64no—
252C23—62no—
253C23—62no—
254C23—124no—
255C23—242no—
256C23D3×D412yes—
257C23D3×D412yes—
258C23C2×#6711yescharacteristic
259C23C2×#6711yescharacteristic
260C23C2×#6712yescomplemented
261C23C23×D311yescharacteristic
262Q8—64no—
263D4—24no—
264D4—24no—
265D4—48no—
266D4—64no—
267D4—68no—
268D4—64no—
269D4—64no—
270D4—64no—
271D4—64no—
272D4—1216no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—34no—
277C2×C4—42no—
278C2×C4—68no—
279C2×C4—64no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—64no—
283C2×C4—124no—
284C2×C4C2×#6712yescomplemented
285C2×C3—22no—
286C2×C3—21no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—42no—
290C2×C3—44no—
291C2×C3B(4,𝔽2)12yescomplemented
292C2×C3C2×#12011yescharacteristic
293D3—44no—
294C22—22no—
295C22—24no—
296C22—24no—
297C22—28no—
298C22—22no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—44no—
305C22—44no—
306C22—48no—
307C22—42no—
308C22—44no—
309C22—42no—
310C22—64no—
311C22—81no—
312C22—124no—
313C22—124no—
314C22—242no—
315C22—244no—
316C22#93711yescharacteristic
317C22#93712yes—
318C22#93711yescharacteristic
319C22C2×D3×D412yes—
320C22C22×#6711yescharacteristic
321C4—22no—
322C4—42no—
323C4—64no—
324C4—124no—
325C3C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
326C2—21no—
327C2—22no—
328C2—21no—
329C2—22no—
330C2—44no—
331C2—42no—
332C2—124no—
333C2#503212yesdirect product factor
334C2C2×#93711yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC2×C3×#120C2
socleC22×C3#120
hypercenterC2×#120D3
hypocenterC3C2×B(4,𝔽2)
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C2×C3×#120C2semidirect
3. C2×#953C2semidirect
4. C2×#834C2semidirect
5. #5032C2direct (N × H)
6. C2×#806C2semidirect
8. C25×C3C22semidirect
16. #1004C22semidirect
21. C3×#120C22semidirect
23. C2×C3×C22⋊C4C22semidirect
25. #953C22semidirect
27. C2×#217C22semidirect
28. #834C22semidirect
29. #806C22semidirect
32. C2×#120D3semidirect
41. C24×C3D4semidirect
43. C24×C3C23semidirect
45. C23×D3D4semidirect
50. C2×C3×D4D4semidirect
53. C22×C3×C4D4semidirect
63. C2×#67D4semidirect
67. C22×Dic3D4semidirect
79. C3×C22⋊C4C23semidirect
82. #217C23semidirect
83. C25C2×D3semidirect
98. #120C2×D3semidirect
102. C2×C22⋊C4C2×D3semidirect
125. C23×C3C2×D4semidirect
132. C22×D3C2×D4semidirect
140. C2×C3×C4C2×D4semidirect
152. C2×Dic3C2×D4semidirect
163. C24#67semidirect
165. C24C22×D3semidirect
189. C2×D4#67semidirect
196. C22×C4#67semidirect
202. C22⋊C4C22×D3semidirect
260. C23C2×#67semidirect
284. C2×C4C2×#67semidirect
291. C2×C3B(4,𝔽2)semidirect
325. C3C2×B(4,𝔽2)semidirect
333. C2#5032direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#10042
C2×C3×#1202
C2×#9532
C2×#8342
#50322
C2×#8062
C2×B(4,𝔽2)3