Finite Group Databasebeta

← #82698

Subgroups — #82698

Order 384.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×C3×#121C211yesmaximal, characteristic
2C2×#936C211yesmaximal, characteristic, complemented
3C2×#924C211yesmaximal, characteristic, complemented
4C22×C4×Dic3C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5220C214yesmaximal, complemented
7#4977C211yesmaximal, characteristic, complemented
8#4973C211yesmaximal, characteristic, complemented
9#4945C214yesmaximal, complemented
10#2850—31nomaximal, Sylow
11C22×C3×D4C2212yes—
12C22×C3×D4C2211yescharacteristic
13C23×C3×C4C2211yescharacteristic
14C23×Dic3C2212yes—
15C22×Dic6C2211yescharacteristic
16C3×#121—24no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2214yes—
20C2×#217C2214yes—
21C2×#217C2212yes—
22#936—24no—
23C2×C4⋊Dic3C2211yescharacteristic
24#924—24no—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3—24no—
27C2×C4×Dic3C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2211yescharacteristic
30C3⋊#95C2214yes—
31C3⋊#95C2214yes—
32C2×#124—31no—
33C2×C22⋊Q8—31no—
34C2×#121D311yescharacteristic
35C22×C42—31no—
36#391—31no—
37#390—31no—
38#388—34no—
39#386—34no—
40#382—31no—
41C24×C3C2312yes—
42C24×C3C2311yescharacteristic
43C2×C3×D4—24no—
44C2×C3×D4—28no—
45C2×C3×D4—22no—
46C2×C3×D4—22no—
47C2×C3×D4—22no—
48C22×C3×C4—22no—
49C22×C3×C4—22no—
50C22×C3×C4D414yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3—24no—
55C22×Dic3—24no—
56C22×Dic3—24no—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C2×Dic6—22no—
62C2×Dic6—22no—
63C2×Dic6—22no—
64C3×C4⋊C4—22no—
65C3×C22⋊C4—24no—
66#217—24no—
67#217—28no—
68#217—28no—
69C4⋊Dic3—22no—
70Dic3⋊C4—24no—
71C4×Dic3—22no—
72C4×Dic3—24no—
73C4×Dic3D414yescomplemented
74C22×Q8—31no—
75C22×D4C2×D311yescharacteristic
76C22×D4C2×D312yes—
77C23×C4—32no—
78C23×C4C2×D311yescharacteristic
79#124—64no—
80C22⋊Q8—64no—
81#121—24no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4C2×D311yescharacteristic
85C2×C22⋊C4—32no—
86C2×C22⋊C4—34no—
87C2×C22⋊C4—34no—
88C2×C22⋊C4C2×D312yes—
89C2×C42—31no—
90C2×C42—32no—
91C2×C42—31no—
92C2×C42—64no—
93#95—34no—
94#95—34no—
95C23×C3—21no—
96C23×C3—22no—
97C23×C3—21no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—24no—
101C23×C3—22no—
102C23×C3—24no—
103C23×C3C2×D412yes—
104C23×C3#4114yes—
105C23×C3C2411yescharacteristic
106C3×D4—48no—
107C3×D4—44no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—28no—
111C2×C3×C4—22no—
112C2×C3×C4—24no—
113C2×C3×C4C2×D412yes—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2×D412yes—
116C2×C3×C4#4112yes—
117C2×Dic3—216no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—24no—
126C2×Dic3—22no—
127C2×Dic3C2×D414yes—
128C2×Dic3#4114yes—
129Dic6—44no—
130C24C22×D312yes—
131C24C22×D311yescharacteristic
132C2×Q8—62no—
133C2×Q8—62no—
134C2×Q8—62no—
135C2×D4—28no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—32no—
146C22×C4—64no—
147C22×C4—64no—
148C22×C4—64no—
149C22×C4#6714yes—
150C22×C4C22×D312yes—
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—64no—
155C4⋊C4—62no—
156C22⋊C4—24no—
157C22⋊C4—68no—
158C22⋊C4—68no—
159C22⋊C4—64no—
160C42—34no—
161C42—64no—
162C42—62no—
163C22×C3—22no—
164C22×C3—24no—
165C22×C3—24no—
166C22×C3—22no—
167C22×C3—42no—
168C22×C3—44no—
169C22×C3—42no—
170C22×C3—44no—
171C22×C3—42no—
172C22×C3#12414yescomplemented
173C22×C3C22×D411yescharacteristic
174C22×C3C22×D411yescharacteristic
175C22×C3C2×#4111yescharacteristic
176C22×C3C2×#4112yes—
177C22×C3C2×#4112yes—
178C3×C4—22no—
179C3×C4—42no—
180C3×C4#12114yes—
181Dic3—24no—
182Dic3—24no—
183Dic3—42no—
184Dic3—42no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—24no—
190C23—22no—
191C23—24no—
192C23—22no—
193C23#23714yes—
194C23C2×#6712yes—
195C23C23×D311yescharacteristic
196Q8—124no—
197D4—48no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—34no—
205C2×C4—34no—
206C2×C4—616no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—64no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4D3×D412yes—
217C2×C4#23712yes—
218C2×C4C2×#6712yes—
219C2×C4C2×#6712yes—
220C2×C3—24no—
221C2×C3—82no—
222C2×C3C2×#12111yescharacteristic
223C2×C3#52512yes—
224C2×C3C2×#12411yescharacteristic
225C2×C3#52912yes—
226C2×C3#53211yescharacteristic
227C22—22no—
228C22—22no—
229C22—24no—
230C22—24no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—44no—
235C22—44no—
236C22#93614yescomplemented
237C22C2×D3×D411yescharacteristic
238C22C2×#23712yes—
239C22C2×#23711yescharacteristic
240C22D3×#4112yes—
241C22C22×#6711yescharacteristic
242C4—22no—
243C4—42no—
244C4—64no—
245C4—64no—
246C4—122no—
247C4—122no—
248C4#93914yes—
249C3#285011yescharacteristic, complemented, Sylow
250C2—24no—
251C2—82no—
252C2#560012yes—
253C2#560612yes—
254C2C2×#93611yescharacteristic
255C2C2×#93911yescharacteristic
256C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2850
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#936C2semidirect
3. C2×#924C2semidirect
4. C22×C4×Dic3C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. #4977C2semidirect
8. #4973C2semidirect
9. #4945C2semidirect
73. C4×Dic3D4semidirect
172. C22×C3#124semidirect
236. C22#936semidirect
249. C3#2850semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9362
C2×#9242
C22×C4×Dic32
#52212
#52202
#49772
#49732
#49452
#28503