Finite Group Databasebeta

← C2×#5912 #87474

Subgroups — C2×#5912 #87474

Order 384.

895 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×C4×A4C211yesmaximal, characteristic, complemented
2C2×GL(2,ℤ/4)C212yesmaximal, complemented
3#5912C218yesmaximal, direct product factor
4C2×C4⋊S4C211yesmaximal, characteristic, complemented
5C2×C4×S4C212yesmaximal, complemented
6C2×A4⋊Q8C211yesmaximal, characteristic, complemented
7C2×#525—31nomaximal, Sylow
8C23×A4C2211yescharacteristic
9C22×S4C2212yes—
10C2×#235—41nomaximal
11C2×C4×A4C2211yescharacteristic
12C2×C4×A4C2211yescharacteristic
13C2×C4×A4C2214yescomplemented
14GL(2,ℤ/4)C2218yescomplemented
15C2×SL(2,ℤ/4)C2212yes—
16C4⋊S4C2214yescomplemented
17C4×S4C2218yescomplemented
18A4⋊Q8C2214yescomplemented
19C22×#41—31no—
20C24×C4D311yescharacteristic, complemented
21#525—38no—
22#525—38no—
23C2×#123—32no—
24C2×C22⋊Q8—31no—
25C2×C22⋊Q8—31no—
26C2×#121—31no—
27C2×#121—31no—
28C2×#120—32no—
29C2×C4×D4—32no—
30C2×C4×D4—32no—
31C2×#117—31no—
32C22×A4—22no—
33C22×A4C2312yes—
34C22×A4C2311yescharacteristic
35C2×S4—24no—
36C2×S4C2314yes—
37C22×C3×C4—41no—
38C2×#67—42no—
39#235—48no—
40C2×D12—41no—
41C2×C4×D3—42no—
42C2×Dic6—41no—
43C4×A4C2312yes—
44C4×A4C2312yes—
45SL(2,ℤ/4)C2314yes—
46C25C2×D311yescharacteristic
47C2×#41—38no—
48C2×#41—61no—
49C2×#41—61no—
50C2×#41—61no—
51C2×#41—64no—
52C2×#41—61no—
53C22×Q8—31no—
54C22×D4—32no—
55C22×D4—32no—
56C22×D4—31no—
57C22×D4—32no—
58C23×C4—31no—
59C23×C4—31no—
60C23×C4—31no—
61C23×C4—31no—
62C23×C4—34no—
63C23×C4—32no—
64C23×C4C2×D311yescharacteristic
65C23×C4C2×D311yescharacteristic
66C23×C4C2×D314yescomplemented
67#123—316no—
68C22⋊Q8—34no—
69C22⋊Q8—38no—
70C22⋊Q8—34no—
71#121—34no—
72#121—38no—
73#121—34no—
74#120—38no—
75#120—38no—
76C4×D4—38no—
77C4×D4—316no—
78C4×D4—38no—
79#117—38no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C42—32no—
90C23×C3—41no—
91C22×D3—42no—
92C2×A4—22no—
93C2×A4#4112yescomplemented
94C2×A4C2411yescharacteristic
95S4—24no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—44no—
99#67—48no—
100C2×Dic3—42no—
101D12—44no—
102C4×D3—48no—
103Dic6—44no—
104C24—22no—
105C24—31no—
106C24—31no—
107C24—32no—
108C24—32no—
109C24—62no—
110C24C22×D312yes—
111C24C22×D311yescharacteristic
112#41—68no—
113#41—68no—
114#41—68no—
115#41—68no—
116C2×Q8—34no—
117C2×Q8—61no—
118C2×Q8—62no—
119C2×Q8—61no—
120C2×D4—34no—
121C2×D4—38no—
122C2×D4—38no—
123C2×D4—34no—
124C2×D4—34no—
125C2×D4—61no—
126C2×D4—62no—
127C2×D4—64no—
128C2×D4—68no—
129C2×D4—64no—
130C2×D4—62no—
131C2×D4—64no—
132C2×D4—62no—
133C2×D4—61no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—34no—
140C22×C4—34no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—34no—
145C22×C4—32no—
146C22×C4—34no—
147C22×C4—34no—
148C22×C4—32no—
149C22×C4—32no—
150C22×C4—32no—
151C22×C4—38no—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4—61no—
155C22×C4—62no—
156C22×C4—62no—
157C22×C4—64no—
158C22×C4—62no—
159C22×C4S411yescharacteristic, complemented
160C22×C4C22×D312yes—
161C22×C4C22×D312yes—
162C4⋊C4—34no—
163C4⋊C4—34no—
164C4⋊C4—38no—
165C4⋊C4—38no—
166C22⋊C4—38no—
167C22⋊C4—38no—
168C22⋊C4—34no—
169C22⋊C4—38no—
170C22⋊C4—38no—
171C22⋊C4—34no—
172C42—38no—
173C22×C3—41no—
174C22×C3—42no—
175C22×C3—82no—
176C2×D3—44no—
177C2×D3—84no—
178A4C2×#4111yescharacteristic, complemented
179C3×C4—42no—
180C3×C4—42no—
181Dic3—44no—
182C23—22no—
183C23—31no—
184C23—31no—
185C23—32no—
186C23—32no—
187C23—34no—
188C23—32no—
189C23—32no—
190C23—31no—
191C23—34no—
192C23—62no—
193C23—62no—
194C23—62no—
195C23—61no—
196C23—64no—
197C23—62no—
198C23—64no—
199C23—62no—
200C23—64no—
201C23—64no—
202C23—62no—
203C23#23512yescomplemented
204C23C2×S411yescharacteristic
205C23C23×D311yescharacteristic
206Q8—64no—
207Q8—64no—
208D4—64no—
209D4—68no—
210D4—64no—
211D4—616no—
212D4—68no—
213D4—64no—
214D4—68no—
215D4—64no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—34no—
219C2×C4—32no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—34no—
224C2×C4—34no—
225C2×C4—32no—
226C2×C4—32no—
227C2×C4—34no—
228C2×C4—34no—
229C2×C4—34no—
230C2×C4—61no—
231C2×C4—64no—
232C2×C4—61no—
233C2×C4—68no—
234C2×C4—62no—
235C2×C4—64no—
236C2×C4—64no—
237C2×C4—62no—
238C2×C4—68no—
239C2×C4C2×S411yescharacteristic
240C2×C4C2×S411yescharacteristic
241C2×C4C2×S414yescomplemented
242C2×C3—42no—
243C2×C3—41no—
244C2×C3—82no—
245D3—84no—
246C22—22no—
247C22—32no—
248C22—31no—
249C22—32no—
250C22—32no—
251C22—31no—
252C22—31no—
253C22—34no—
254C22—32no—
255C22—62no—
256C22—62no—
257C22—64no—
258C22—61no—
259C22—62no—
260C22—64no—
261C22—64no—
262C22—64no—
263C22—62no—
264C22—64no—
265C22—124no—
266C22—124no—
267C22C2×#23511yescharacteristic, complemented
268C22C22×S412yes—
269C22C22×S411yescharacteristic
270C4—32no—
271C4—32no—
272C4—64no—
273C4—64no—
274C4—64no—
275C4C22×S412yes—
276C4C22×S412yes—
277C3—41noSylow
278C2—22no—
279C2—31no—
280C2—31no—
281C2—32no—
282C2—62no—
283C2—124no—
284C2#591212yesdirect product factor
285C2C23×S411yescharacteristic
subgroupG/H
centerC2×C4C2×S4
commutator, cosocleC2×A4C24
FrattiniC2C23×S4
FittingC24×C4D3
socleC24C22×D3
hypercenterC22×C4S4
hypocenterA4C2×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×A4C2semidirect
2. C2×GL(2,ℤ/4)C2semidirect
3. #5912C2direct (N × H)
4. C2×C4⋊S4C2semidirect
5. C2×C4×S4C2semidirect
6. C2×A4⋊Q8C2semidirect
13. C2×C4×A4C22semidirect
14. GL(2,ℤ/4)C22semidirect
16. C4⋊S4C22semidirect
17. C4×S4C22semidirect
18. A4⋊Q8C22semidirect
20. C24×C4D3semidirect
66. C23×C4C2×D3semidirect
93. C2×A4#41semidirect
159. C22×C4S4semidirect
178. A4C2×#41semidirect
203. C23#235semidirect
241. C2×C4C2×S4semidirect
267. C22C2×#235semidirect
284. C2#5912direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×A42
C2×GL(2,ℤ/4)2
#59122
C2×C4⋊S42
C2×C4×S42
C2×A4⋊Q82
C2×#5253
C2×#2354