Finite Group Databasebeta

← #82889

Subgroups — #82889

Order 384.

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

normal subgroups only

This group has 478 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
1C2×C3×#123C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3#5222C212yesmaximal
4#5212C211yesmaximal, characteristic, complemented
5#4982C212yesmaximal, complemented
6#4981C211yesmaximal, characteristic, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4953C211yesmaximal, characteristic, complemented
9#4952C212yesmaximal, complemented
10#4948C212yesmaximal, complemented
11#4947C211yesmaximal, characteristic, complemented
12#2814—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C3×#123—24no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2212yes—
28#924—24no—
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#123D311yescharacteristic
39C2×C22⋊Q8—31no—
40#398—31no—
41#398—32no—
42#397—32no—
43#397—31no—
44#396—32no—
45#395—31no—
46#394—32no—
47#393—31no—
48#380—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C2×C3×D4—22no—
52C2×C3×D4—42no—
53C22×C3×C4—42no—
54C22×C3×C4D414yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×Dic3C2312yes—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C2×Dic6—22no—
65C2×Dic6—42no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68C3×C22⋊C4—22no—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—22no—
71#217—24no—
72#217—42no—
73#217—42no—
74#217—42no—
75C4⋊Dic3—22no—
76C4⋊Dic3—42no—
77Dic3⋊C4—24no—
78Dic3⋊C4—42no—
79Dic3⋊C4—42no—
80C22×Q8—31no—
81C22×D4C2×D311yescharacteristic
82C23×C4C2×D311yescharacteristic
83#123—24no—
84C22⋊Q8—64no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D311yescharacteristic
97C2×C22⋊C4C2×D311yescharacteristic
98C2×C22⋊C4C2×D311yescharacteristic
99#95—31no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95—32no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—21no—
109C23×C3—21no—
110C23×C3—21no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—21no—
114C23×C3C2×D412yes—
115C23×C3C2411yescharacteristic
116C3×D4—44no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—44no—
127C2×C3×C4C2×D412yes—
128C2×C3×C4C2×D412yes—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141Dic6—44no—
142C24C22×D311yescharacteristic
143C24C22×D311yescharacteristic
144C2×Q8—62no—
145C2×Q8—122no—
146C2×D4—22no—
147C2×D4—42no—
148C22×C4—32no—
149C22×C4—32no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—42no—
153C22×C4#6714yes—
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D311yescharacteristic
156C22×C4C22×D311yescharacteristic
157C22×C4C22×D311yescharacteristic
158C22×C4C22×D311yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—64no—
162C4⋊C4—62no—
163C4⋊C4—122no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—64no—
170C22⋊C4—122no—
171C22⋊C4—122no—
172C22⋊C4—122no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3—22no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—42no—
182C22×C3—42no—
183C22×C3C22×D411yescharacteristic
184C22×C3H(5,𝔽2)12yes—
185C22×C3H(5,𝔽2)11yescharacteristic
186C22×C32−1+412yes—
187C22×C32−1+411yescharacteristic
188C3×C4—22no—
189C3×C4—22no—
190C3×C4—41no—
191C3×C4—41no—
192C3×C4—41no—
193Dic3—42no—
194Dic3—42no—
195Dic3—42no—
196Dic3—42no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23—21no—
201C23—22no—
202C23—21no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23C2×#6712yes—
207C23C23×D311yescharacteristic
208Q8—124no—
209D4—44no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4C2×#6712yes—
233C2×C4C2×#6712yes—
234C2×C3—42no—
235C2×C3—81no—
236C2×C3#53411yescharacteristic
237C2×C3#53611yescharacteristic
238C2×C3#53711yescharacteristic
239C2×C3#56212yes—
240C2×C3#56312yes—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22#100412yes—
252C22#100911yescharacteristic
253C22#101011yescharacteristic
254C22#101012yes—
255C22C22×#6711yescharacteristic
256C4—22no—
257C4—22no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—122no—
262C4—122no—
263C4—122no—
264C4—122no—
265C3#281411yescharacteristic, complemented, Sylow
266C2—42no—
267C2—81no—
268C2#565112yes—
269C2#566712yes—
270C2#580411yescharacteristic
271C2#582511yescharacteristic
272C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
prime radicalC22×C3×D4C22
hypercenterC2×#123D3
hypocenterC3#2814
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
4. #5212C2semidirect
5. #4982C2semidirect
6. #4981C2semidirect
7. #4979C2semidirect
8. #4953C2semidirect
9. #4952C2semidirect
10. #4948C2semidirect
11. #4947C2semidirect
265. C3#2814semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9242
#52222
#52122
#49822
#49812
#49792
#49532
#49522
#49482
#49472
#28143