Finite Group Databasebeta

← #77209

Subgroups — #77209

Order 384.

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

normal subgroups only

This group has 534 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×#947C211yesmaximal, characteristic, complemented
2C2×#898C211yesmaximal, characteristic, complemented
3C3×#395C211yesmaximal, characteristic, complemented
4#5231C211yesmaximal, characteristic, complemented
5#4994C212yesmaximal, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4988C212yesmaximal, complemented
8#4672C212yesmaximal, complemented
9#4671C211yesmaximal, characteristic
10#4669C212yesmaximal, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2662—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×D12C2212yes—
15C22×D12C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18#947—24no—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2211yescharacteristic
31C2×C4×Dic3C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33#898—24no—
34C3×#95C2211yescharacteristic
35C3×#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×#124—31no—
40#396—32no—
41#395D311yescharacteristic, complemented
42#394—31no—
43#394—32no—
44#393—31no—
45#392—31no—
46#390—32no—
47#388—32no—
48C4×C4⋊C4—31no—
49C23×D3C2311yescharacteristic
50C23×D3C2312yes—
51C2×C3×Q8—22no—
52C2×C3×Q8—42no—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×D12—22no—
63C2×D12—24no—
64C2×D12—42no—
65C2×D12—42no—
66C2×D12—42no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—24no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78C4⋊Dic3—22no—
79Dic3⋊C4—24no—
80Dic3⋊C4—22no—
81C4×Dic3—42no—
82C4×Dic3D414yescomplemented
83C22×Q8C2×D311yescharacteristic
84C22×D4—31no—
85C22×D4—32no—
86#126—64no—
87#124—64no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C4⋊C4C2×D312yes—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C42—31no—
103C2×C42—32no—
104#95—31no—
105#95—32no—
106#95C2×D311yescharacteristic
107#95C2×D312yes—
108C23×C3C2411yescharacteristic
109C22×D3—21no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—22no—
121C3×Q8—44no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4C2×D412yes—
134C2×C3×C4#4114yes—
135C2×Dic3—22no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3C2×D412yes—
144C2×Dic3C2×D412yes—
145D12—44no—
146D12—48no—
147C24—31no—
148C24—32no—
149C2×Q8—22no—
150C2×Q8—42no—
151C2×D4—62no—
152C2×D4—64no—
153C2×D4—122no—
154C2×D4—122no—
155C2×D4—122no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4C22×D312yes—
161C22×C4C22×D311yescharacteristic
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D312yes—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—64no—
170C22⋊C4—64no—
171C22⋊C4—64no—
172C22⋊C4—62no—
173C22⋊C4—62no—
174C22⋊C4—62no—
175C22⋊C4—64no—
176C22⋊C4—122no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—34no—
180C42—122no—
181C22×C3C22×D411yescharacteristic
182C22×C3C2×#4112yes—
183C22×C3H(5,𝔽2)12yes—
184C22×C3H(5,𝔽2)11yescharacteristic
185C22×C32−1+411yescharacteristic
186C2×D3—41no—
187C2×D3—42no—
188C2×D3—42no—
189C2×D3—41no—
190C2×D3—41no—
191C2×D3—42no—
192C2×D3—42no—
193C2×D3—42no—
194C2×D3—42no—
195C2×D3—42no—
196C2×D3—42no—
197C2×D3—42no—
198C3×C4—22no—
199C3×C4—24no—
200C3×C4—41no—
201C3×C4—42no—
202C3×C4—41no—
203Dic3—22no—
204Dic3—22no—
205Dic3—41no—
206Dic3—42no—
207C23—61no—
208C23—62no—
209C23—62no—
210C23—62no—
211C23—62no—
212C23—62no—
213C23—62no—
214C23—62no—
215C23—61no—
216C23—62no—
217C23—62no—
218C23—61no—
219C23C23×D311yescharacteristic
220Q8—44no—
221D4—128no—
222D4—124no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—62no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4D3×D412yes—
245C2×C4#23914yes—
246C2×C3#53211yescharacteristic
247C2×C3#53511yescharacteristic
248C2×C3#53711yescharacteristic
249C2×C3#53812yes—
250C2×C3#54212yes—
251D3—82no—
252D3—81no—
253C22—122no—
254C22—122no—
255C22—122no—
256C22—121no—
257C22—122no—
258C22—122no—
259C22—122no—
260C22—122no—
261C22—122no—
262C22—121no—
263C22—121no—
264C22—122no—
265C22C2×D3×D411yescharacteristic
266C22#100412yes—
267C22C2×#23912yes—
268C22#100711yescharacteristic
269C22#100911yescharacteristic
270C4—24no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—62no—
276C4—62no—
277C4—122no—
278C4—121no—
279C3#266211yescharacteristic, complemented, Sylow
280C2—241no—
281C2—242no—
282C2#559211yescharacteristic
283C2#562911yescharacteristic
284C2#564412yes—
285C2#572511yescharacteristic
286C2#573312yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radical#4671C2
hypercenter#395D3
hypocenterC3#2662
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#947C2semidirect
2. C2×#898C2semidirect
3. C3×#395C2semidirect
4. #5231C2semidirect
5. #4994C2semidirect
6. #4991C2semidirect
7. #4988C2semidirect
8. #4672C2semidirect
10. #4669C2semidirect
11. #4649C2semidirect
41. #395D3semidirect
82. C4×Dic3D4semidirect
279. C3#2662semidirect

Maximal subgroups

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

subgroupindex
C2×#9472
C2×#8982
C3×#3952
#52312
#49942
#49912
#49882
#46722
#46712
#46692
#46492
#26623