Finite Group Databasebeta

← #85506

Subgroups — #85506

Order 384.

365 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 46 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×Q8×A4C211yesmaximal, characteristic, complemented
2C2×C4×S4C211yesmaximal, characteristic
3(C2×S4)⋊C4C212yesmaximal
4C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6#2409—31nomaximal, Sylow
7C22×S4C2211yescharacteristic
8Q8×A4—22no—
9C2×C4×A4C2212yes—
10C2×C4×A4C2211yescharacteristic
11C2×SL(2,ℤ/4)C2211yescharacteristic
12C2×SL(2,ℤ/4)C2212yes—
13C4×S4—22no—
14#946—41nomaximal
15C23×Q8D311yescharacteristic, complemented
16C2×C22⋊Q8—31no—
17C2×C22⋊Q8—31no—
18C2×C4×D4—31no—
19#395—32no—
20#393—32no—
21#391—31no—
22#391—31no—
23#386—32no—
24#385—32no—
25#380—31no—
26C22×A4C2311yescharacteristic
27C2×S4—21no—
28C2×S4—21no—
29C2×S4Q812yescomplemented
30C2×C3×Q8—41no—
31C2×C4×D3—41no—
32C4×A4—22no—
33C4×A4D412yes—
34SL(2,ℤ/4)—21no—
35SL(2,ℤ/4)—22no—
36(C2×D3)⋊C4—42no—
37C4⋊Dic3—41no—
38Dic3⋊C4—42no—
39C22×Q8—22no—
40C22×Q8—31no—
41C22×Q8—31no—
42C22×Q8—32no—
43C22×Q8—62no—
44C22×D4—31no—
45C23×C4—31no—
46C23×C4C2×D312yes—
47C23×C4C2×D311yescharacteristic
48C22⋊Q8—64no—
49C22⋊Q8—61no—
50C22⋊Q8—62no—
51C22⋊Q8—61no—
52C4×D4—62no—
53C4×D4—62no—
54C2×C4⋊C4—31no—
55C2×C4⋊C4—31no—
56C2×C4⋊C4—31no—
57C2×C4⋊C4—32no—
58C2×C4⋊C4—32no—
59C2×C22⋊C4—31no—
60C2×C22⋊C4—32no—
61C2×C22⋊C4—32no—
62C2×C22⋊C4—31no—
63C2×C22⋊C4—32no—
64C2×C42—31no—
65#95—32no—
66#95—32no—
67#95—32no—
68#95—32no—
69#95—32no—
70C22×D3—41no—
71C2×A4C2×D411yescharacteristic
72C2×A4C2×Q811yescharacteristic
73C2×A4#4111yescharacteristic
74S4—22no—
75C3×Q8—82no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×Dic3—41no—
79C2×Dic3—42no—
80C4×D3—82no—
81C24—31no—
82C24C22×D311yescharacteristic
83C2×Q8—34no—
84C2×Q8—32no—
85C2×Q8—31no—
86C2×Q8—61no—
87C2×Q8—62no—
88C2×Q8—62no—
89C2×Q8—62no—
90C2×Q8—64no—
91C2×Q8—64no—
92C2×Q8S411yescharacteristic, complemented
93C2×D4—32no—
94C2×D4—32no—
95C2×D4—61no—
96C2×D4—61no—
97C2×D4—61no—
98C2×D4—61no—
99C22×C4—22no—
100C22×C4—31no—
101C22×C4—31no—
102C22×C4—32no—
103C22×C4—32no—
104C22×C4—31no—
105C22×C4—32no—
106C22×C4—31no—
107C22×C4—32no—
108C22×C4—32no—
109C22×C4—31no—
110C22×C4—62no—
111C22×C4—62no—
112C22×C4—61no—
113C22×C4—62no—
114C22×C4—61no—
115C22×C4#6712yes—
116C4⋊C4—62no—
117C4⋊C4—61no—
118C4⋊C4—62no—
119C4⋊C4—64no—
120C4⋊C4—62no—
121C4⋊C4—62no—
122C4⋊C4—61no—
123C22⋊C4—62no—
124C22⋊C4—64no—
125C22⋊C4—62no—
126C22⋊C4—62no—
127C22⋊C4—61no—
128C22⋊C4—62no—
129C22⋊C4—61no—
130C22⋊C4—62no—
131C42—62no—
132C22×C3—41no—
133C2×D3—42no—
134C2×D3—81no—
135C2×D3—81no—
136A4C22⋊Q811yescharacteristic, complemented
137C3×C4—42no—
138C3×C4—82no—
139Dic3—81no—
140Dic3—82no—
141C23—31no—
142C23—31no—
143C23—31no—
144C23—31no—
145C23—32no—
146C23—61no—
147C23—61no—
148C23—61no—
149C23—61no—
150C23—61no—
151C23—61no—
152C23D3×Q811yescharacteristic
153C23#23911yescharacteristic
154C23C2×#6711yescharacteristic
155Q8—22no—
156Q8—64no—
157Q8—64no—
158Q8—62no—
159D4—62no—
160D4—62no—
161D4—62no—
162D4—62no—
163C2×C4—31no—
164C2×C4—32no—
165C2×C4—32no—
166C2×C4—32no—
167C2×C4—34no—
168C2×C4—32no—
169C2×C4—32no—
170C2×C4—62no—
171C2×C4—62no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—62no—
176C2×C4—61no—
177C2×C4—62no—
178C2×C4—61no—
179C2×C4—62no—
180C2×C4—62no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—62no—
184C2×C4—61no—
185C2×C4—62no—
186C2×C4—62no—
187C2×C4—61no—
188C2×C4—122no—
189C2×C4—122no—
190C2×C4C2×S412yes—
191C2×C4C2×S411yescharacteristic
192C2×C3—41no—
193C2×C3—41no—
194C2×C3—41no—
195D3—82no—
196C22—31no—
197C22—31no—
198C22—31no—
199C22—31no—
200C22—31no—
201C22—31no—
202C22—31no—
203C22—31no—
204C22—31no—
205C22—61no—
206C22—62no—
207C22—62no—
208C22—62no—
209C22—121no—
210C22—121no—
211C22—121no—
212C22—121no—
213C22#94611yescharacteristic, complemented
214C22C22×S411yescharacteristic
215C4—22no—
216C4—32no—
217C4—62no—
218C4—62no—
219C4—122no—
220C4—121no—
221C4—122no—
222C4—121no—
223C4GL(2,ℤ/4)12yes—
224C3—41noSylow
225C2—31no—
226C2—31no—
227C2—31no—
228C2—31no—
229C2—122no—
230C2Q8×S411yescharacteristic
231C2#591911yescharacteristic
232C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×S4
commutator, cosocleC22×A4C23
FittingC23×Q8D3
socleC24C22×D3
prime radicalC22×S4C22
hypercenterC2×Q8S4
hypocenterA4C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×A4C2semidirect
4. C4⋊SL(2,ℤ/4)C2semidirect
5. SL(2,ℤ/4)⋊C4C2semidirect
15. C23×Q8D3semidirect
29. C2×S4Q8semidirect
92. C2×Q8S4semidirect
136. A4C22⋊Q8semidirect
213. C22#946semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×A42
C2×C4×S42
(C2×S4)⋊C42
C4⋊SL(2,ℤ/4)2
SL(2,ℤ/4)⋊C42
#24093
#9464