Finite Group Databasebeta

← #82867

Subgroups — #82867

Order 384.

598 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 137 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, complemented
2C2×#930C211yesmaximal, characteristic, complemented
3#5223C212yesmaximal, complemented
4C4×#217C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4990C212yesmaximal, complemented
7#4981C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4959C212yesmaximal, complemented
10#4957C212yesmaximal, complemented
11#4953C211yesmaximal, characteristic, complemented
12#2661—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2212yes—
16C22×#67C2212yes—
17C22×D12C2211yescharacteristic
18C3×#123—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2212yes—
26#930—24no—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×C4⋊Dic3C2211yescharacteristic
32C2×Dic3⋊C4C2212yes—
33C2×C4×Dic3C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#123D311yescharacteristic, complemented
39C2×#121—31no—
40#398—31no—
41#397—31no—
42#394—31no—
43#394—32no—
44#392—32no—
45#392—31no—
46#390—32no—
47#388—32no—
48C4×C22⋊C4—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C23×D3C2312yes—
52C2×C3×D4—22no—
53C2×C3×D4—42no—
54C22×C3×C4—42no—
55C22×C3×C4D414yescomplemented
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—42no—
64C2×#67—42no—
65C2×#67—42no—
66C2×#67—42no—
67C22×Dic3C2312yes—
68C22×Dic3C2312yes—
69C22×Dic3C2312yes—
70C2×D12—22no—
71C2×D12—42no—
72C3×C4⋊C4—22no—
73C3×C4⋊C4—22no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—22no—
76C3×C22⋊C4—22no—
77#217—24no—
78#217—42no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—42no—
82(C2×D3)⋊C4—42no—
83C4⋊Dic3—22no—
84Dic3⋊C4—42no—
85C4×Dic3—42no—
86C22×D4—32no—
87C22×D4—32no—
88C22×D4—31no—
89C22×D4C2×D311yescharacteristic
90C23×C4C2×D311yescharacteristic
91#123—24no—
92#121—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C4⋊C4C2×D311yescharacteristic
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D311yescharacteristic
105C2×C22⋊C4C2×D311yescharacteristic
106C2×C42—32no—
107#95—31no—
108#95—31no—
109#95—32no—
110#95—32no—
111C23×C3—22no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—22no—
115C23×C3—21no—
116C23×C3—22no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—22no—
120C23×C3C2×D412yes—
121C23×C3C2411yescharacteristic
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C3×D4—44no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—44no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2×D412yes—
142#67—48no—
143#67—48no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3#4114yes—
153D12—44no—
154C24—32no—
155C24C22×D311yescharacteristic
156C24C22×D311yescharacteristic
157C2×D4—22no—
158C2×D4—42no—
159C2×D4—64no—
160C2×D4—64no—
161C2×D4—62no—
162C2×D4—122no—
163C2×D4—122no—
164C2×D4—122no—
165C2×D4—122no—
166C2×D4—122no—
167C22×C4—32no—
168C22×C4—32no—
169C22×C4—32no—
170C22×C4—42no—
171C22×C4#6714yescomplemented
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—64no—
185C22⋊C4—64no—
186C22⋊C4—64no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C42—122no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3C22×D411yescharacteristic
202C22×C3C2×#4112yes—
203C22×C3H(5,𝔽2)12yes—
204C22×C3H(5,𝔽2)11yescharacteristic
205C22×C3H(5,𝔽2)11yescharacteristic
206C2×D3—42no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C3×C4—22no—
214C3×C4—22no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—41no—
218Dic3—24no—
219Dic3—42no—
220Dic3—42no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23C2×#6712yes—
238C23C23×D311yescharacteristic
239D4—44no—
240D4—124no—
241D4—128no—
242D4—128no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—34no—
253C2×C4—44no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4C2×#6712yes—
263C2×C4C2×#6712yes—
264C2×C3—42no—
265C2×C3—81no—
266C2×C3#53211yescharacteristic
267C2×C3#53411yescharacteristic
268C2×C3#53511yescharacteristic
269C2×C3#53812yes—
270C2×C3#54012yes—
271D3—82no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22#100412yes—
290C22D3×#4112yes—
291C22#100911yescharacteristic
292C22#100911yescharacteristic
293C22C22×#6711yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—64no—
300C4—122no—
301C4—122no—
302C3#266111yescharacteristic, complemented, Sylow
303C2—42no—
304C2—81no—
305C2—242no—
306C2#565712yes—
307C2#566312yes—
308C2#580411yescharacteristic
309C2#583011yescharacteristic
310C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
hypercenterC2×#123D3
hypocenterC3#2661
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
2. C2×#930C2semidirect
3. #5223C2semidirect
4. C4×#217C2semidirect
5. #4991C2semidirect
6. #4990C2semidirect
7. #4981C2semidirect
8. #4960C2semidirect
9. #4959C2semidirect
10. #4957C2semidirect
11. #4953C2semidirect
38. C2×#123D3semidirect
55. C22×C3×C4D4semidirect
171. C22×C4#67semidirect
302. C3#2661semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9302
#52232
C4×#2172
#49912
#49902
#49812
#49602
#49592
#49572
#49532
#26613