Finite Group Databasebeta

← #76893

Subgroups — #76893

Order 384.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic3C211yesmaximal, characteristic
2C2×#887C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic
4#5222C211yesmaximal, characteristic
5#4986C211yesmaximal, characteristic, complemented
6#4952C212yesmaximal, complemented
7#4947C212yesmaximal, complemented
8Dic3×C22⋊C4C212yesmaximal, complemented
9Dic3×C22⋊C4C212yesmaximal, complemented
10#4659C211yesmaximal, characteristic, complemented
11#4648C211yesmaximal, characteristic, complemented
12#2398—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×Dic3C2212yes—
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22D4×Dic3—24no—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2211yescharacteristic
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30#887—24no—
31C3×#95C2211yescharacteristic
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×Q8—31no—
39C2×C4×D4—31no—
40#400—31no—
41#400—31no—
42#397—32no—
43#394—31no—
44#394D311yescharacteristic
45#393—32no—
46C4×C22⋊C4—32no—
47C4×C22⋊C4—32no—
48#376—31no—
49C24×C3C2312yes—
50C2×C3×D4—24no—
51C2×C3×D4—22no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3—24no—
57C22×Dic3—24no—
58C22×Dic3—24no—
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C2×Dic6—22no—
66C2×Dic6—22no—
67C2×Dic6—22no—
68C3×C4⋊C4—22no—
69C3×C22⋊C4—24no—
70C3×C22⋊C4—24no—
71#217—24no—
72#217—24no—
73#217—24no—
74C4⋊Dic3—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77C4×Dic3—22no—
78C4×Dic3—22no—
79C4×Dic3—24no—
80C4×Dic3D414yes—
81C22×Q8—31no—
82C22×D4C2×D311yescharacteristic
83C23×C4—32no—
84C4×Q8—64no—
85C4×D4—64no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4C2×D311yescharacteristic
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4C2×D312yes—
94C2×C22⋊C4C2×D312yes—
95C2×C42—31no—
96C2×C42—31no—
97C2×C42—31no—
98C2×C42—32no—
99#95—32no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95—32no—
105#95C2×D311yescharacteristic
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3#4114yes—
113C23×C3C2411yescharacteristic
114C3×D4—28no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4C2×D412yes—
124C2×C3×C4#4112yes—
125C2×Dic3—21no—
126C2×Dic3—21no—
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—48no—
138C2×Dic3C2×D412yes—
139C2×Dic3C2×D412yes—
140C2×Dic3#4112yes—
141C2×Dic3#4114yes—
142Dic6—28no—
143C24C22×D312yes—
144C2×Q8—62no—
145C2×Q8—62no—
146C2×Q8—62no—
147C2×D4—24no—
148C2×D4—22no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—32no—
153C22×C4—32no—
154C22×C4—31no—
155C22×C4—64no—
156C22×C4—64no—
157C22×C4—64no—
158C22×C4C22×D312yes—
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D311yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C22⋊C4—24no—
167C22⋊C4—24no—
168C22⋊C4—64no—
169C22⋊C4—64no—
170C22⋊C4—64no—
171C42—34no—
172C42—62no—
173C42—62no—
174C42—64no—
175C22×C3—24no—
176C22×C3—24no—
177C22×C3—24no—
178C22×C3—42no—
179C22×C3—42no—
180C22×C3—42no—
181C22×C3—42no—
182C22×C3C22×D411yescharacteristic
183C22×C3C2×#4112yes—
184C22×C3C2×#4111yescharacteristic
185C22×C3C2×#4111yescharacteristic
186C22×C3C2×#4112yes—
187C3×C4—22no—
188C3×C4—22no—
189C3×C4—41no—
190C3×C4—42no—
191Dic3—22no—
192Dic3—22no—
193Dic3—41no—
194Dic3—42no—
195Dic3—42no—
196Dic3#12414yes—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23#23714yes—
204C23C23×D311yescharacteristic
205Q8—68no—
206D4—28no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—32no—
216C2×C4—34no—
217C2×C4—32no—
218C2×C4—32no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—128no—
232C2×C4D3×D412yes—
233C2×C4#23712yes—
234C2×C3—44no—
235C2×C3#52511yescharacteristic
236C2×C3#52511yescharacteristic
237C2×C3C2×#12411yescharacteristic
238C2×C3#52912yes—
239C2×C3#52912yes—
240C22—24no—
241C22—24no—
242C22—24no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22C2×D3×D411yescharacteristic
248C22C2×#23712yes—
249C22C2×#23711yescharacteristic
250C22D3×#4111yescharacteristic
251C22D3×#4112yes—
252C4—22no—
253C4—22no—
254C4—34no—
255C4—41no—
256C4—42no—
257C4—62no—
258C4—62no—
259C4—122no—
260C4—122no—
261C4—121no—
262C3#239811yescharacteristic, complemented, Sylow
263C2—44no—
264C2#558611yescharacteristic
265C2#560012yes—
266C2#560611yescharacteristic
267C2#564912yes—
268C2D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×C3×D4C22
hypercenter#394D3
hypocenterC3#2398
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#887C2semidirect
5. #4986C2semidirect
6. #4952C2semidirect
7. #4947C2semidirect
8. Dic3×C22⋊C4C2semidirect
9. Dic3×C22⋊C4C2semidirect
10. #4659C2semidirect
11. #4648C2semidirect
262. C3#2398semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C2×#8872
C3×#3942
#52222
#49862
#49522
#49472
Dic3×C22⋊C42
#46592
#46482
#23983