Finite Group Databasebeta

← #76375

Subgroups — #76375

Order 384.

420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C3×#390C211yesmaximal, characteristic
2#5221C211yesmaximal, characteristic
3#5221C211yesmaximal, characteristic
4#4984C211yesmaximal, characteristic, complemented
5#4972C211yesmaximal, characteristic, complemented
6#4953C212yesmaximal, complemented
7#4952C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4948C212yesmaximal, complemented
10#4932C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2644—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C3⋊#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2211yescharacteristic
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36#397—32no—
37#397—32no—
38#397—32no—
39#396—32no—
40#390—31no—
41#390—31no—
42#390D311yescharacteristic
43#389—31no—
44#389—31no—
45#389—31no—
46#376—31no—
47C24×C3C2312yes—
48C2×C3×D4—22no—
49C2×C3×D4—22no—
50C2×C3×D4—22no—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C3×C4⋊C4—22no—
62C3×C22⋊C4—42no—
63C3×C22⋊C4—42no—
64C3×C42—22no—
65#217—42no—
66#217—42no—
67#217—42no—
68#217—42no—
69C4⋊Dic3—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3—42no—
72Dic3⋊C4—42no—
73Dic3⋊C4—42no—
74C4×Dic3—22no—
75C4×Dic3—22no—
76C22×D4C2×D311yescharacteristic
77C2×C4⋊C4—31no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4C2×D311yescharacteristic
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4C2×D312yes—
88C2×C22⋊C4C2×D312yes—
89C2×C42—31no—
90C2×C42—31no—
91C2×C42C2×D311yescharacteristic
92#95—31no—
93#95—31no—
94#95—31no—
95#95—31no—
96#95—32no—
97#95—32no—
98#95—32no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—22no—
106C23×C3C2411yescharacteristic
107C3×D4—44no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4#4112yes—
118C2×C3×C4#4112yes—
119C2×C3×C4#4112yes—
120C2×Dic3—21no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C24C22×D312yes—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—32no—
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D311yescharacteristic
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—122no—
156C4⋊C4—122no—
157C4⋊C4—122no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—122no—
161C22⋊C4—122no—
162C22⋊C4—122no—
163C22⋊C4—122no—
164C42—22no—
165C42—62no—
166C42—62no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3—42no—
174C22×C3C2×#4111yescharacteristic
175C22×C3C2×#4111yescharacteristic
176C22×C3C2×#4111yescharacteristic
177C22×C3H(5,𝔽2)11yescharacteristic
178C22×C3H(5,𝔽2)11yescharacteristic
179C22×C3H(5,𝔽2)11yescharacteristic
180C22×C32−1+411yescharacteristic
181C3×C4—42no—
182C3×C4—41no—
183C3×C4—41no—
184C3×C4#12614yes—
185Dic3—41no—
186Dic3—41no—
187Dic3—42no—
188Dic3—42no—
189Dic3—41no—
190Dic3—41no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23C23×D311yescharacteristic
199D4—44no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4#23512yes—
228C2×C4#23712yes—
229C2×C4#23712yes—
230C2×C3—82no—
231C2×C3C2×#12611yescharacteristic
232C2×C3#54011yescharacteristic
233C2×C3#54011yescharacteristic
234C2×C3#54011yescharacteristic
235C2×C3#54211yescharacteristic
236C2×C3#54211yescharacteristic
237C2×C3#54211yescharacteristic
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22C2×#23511yescharacteristic
246C22C2×#23711yescharacteristic
247C22C2×#23711yescharacteristic
248C22#100411yescharacteristic
249C22#100411yescharacteristic
250C22#100911yescharacteristic
251C22#101011yescharacteristic
252C4—42no—
253C4—41no—
254C4—41no—
255C4—121no—
256C4—121no—
257C4—121no—
258C4—121no—
259C4—122no—
260C4—122no—
261C4#87914yes—
262C3#264411yescharacteristic, complemented, Sylow
263C2—82no—
264C2C2×#87911yescharacteristic
265C2#556111yescharacteristic
266C2#556211yescharacteristic
267C2#560311yescharacteristic
268C2#561911yescharacteristic
269C2#568211yescharacteristic
270C2#571911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2644
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
4. #4984C2semidirect
5. #4972C2semidirect
6. #4953C2semidirect
7. #4952C2semidirect
8. #4952C2semidirect
9. #4948C2semidirect
10. #4932C2semidirect
11. #4928C2semidirect
262. C3#2644semidirect

Maximal subgroups

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

subgroupindex
C3×#3902
#52212
#49842
#49722
#49532
#49522
#49482
#49322
#49282
#26443