Finite Group Databasebeta

← #82422

Subgroups — #82422

Order 384.

620 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 141 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×C4×D4C211yesmaximal, characteristic, complemented
2C2×#930C211yesmaximal, characteristic, complemented
3#5223C212yesmaximal, complemented
4#5213C212yesmaximal, complemented
5#5211C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4986C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4959C212yesmaximal, complemented
10#4949C211yesmaximal, characteristic, complemented
11#4940C212yesmaximal, complemented
12#2635—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2212yes—
17C22×#67C2212yes—
18C22×D12C2211yescharacteristic
19C3×C4×D4—24no—
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C42C2211yescharacteristic
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—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#121—31no—
39C2×C4×D4D311yescharacteristic, complemented
40#400—31no—
41#399—31no—
42#394—31no—
43#394—32no—
44#392—32no—
45#392—31no—
46#388—32no—
47#386—32no—
48#381—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C23×D3C2312yes—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C22×C3×C4—22no—
56C22×C3×C4—42no—
57C22×C3×C4—42no—
58C22×C3×C4D414yescomplemented
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C2×#67—24no—
65C2×#67—24no—
66C2×#67—42no—
67C2×#67—42no—
68C2×#67—42no—
69C2×#67—42no—
70C22×Dic3C2312yes—
71C22×Dic3C2312yes—
72C22×Dic3C2312yes—
73C2×D12—22no—
74C2×D12—42no—
75C3×C4⋊C4—22no—
76C3×C22⋊C4—22no—
77C3×C22⋊C4—22no—
78C3×C42—22no—
79#217—42no—
80#217—42no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—42no—
84(C2×D3)⋊C4—42no—
85C4⋊Dic3—22no—
86Dic3⋊C4—42no—
87C22×D4—32no—
88C22×D4—32no—
89C22×D4—31no—
90C22×D4C2×D311yescharacteristic
91C23×C4C2×D311yescharacteristic
92C23×C4C2×D311yescharacteristic
93#121—64no—
94C4×D4—24no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4C2×D311yescharacteristic
105C2×C22⋊C4C2×D311yescharacteristic
106C2×C42C2×D311yescharacteristic
107#95—32no—
108#95—31no—
109#95—31no—
110#95—32no—
111#95—32no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3C2×D412yes—
121C23×C3#4112yes—
122C23×C3C2411yescharacteristic
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C3×D4—28no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—44no—
140C2×C3×C4—44no—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4#4112yes—
144#67—48no—
145#67—48no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155D12—44no—
156C24—32no—
157C24C22×D311yescharacteristic
158C24C22×D311yescharacteristic
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—64no—
163C2×D4—64no—
164C2×D4—62no—
165C2×D4—122no—
166C2×D4—122no—
167C2×D4—122no—
168C2×D4—122no—
169C2×D4—122no—
170C22×C4—22no—
171C22×C4—32no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4#6714yescomplemented
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D311yescharacteristic
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—62no—
184C4⋊C4—122no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—64no—
188C22⋊C4—64no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C22⋊C4—122no—
192C22⋊C4—122no—
193C42—22no—
194C22×C3—22no—
195C22×C3—22no—
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—24no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3C22×D411yescharacteristic
205C22×C3C2×#4111yescharacteristic
206C22×C3C2×#4111yescharacteristic
207C22×C3H(5,𝔽2)12yes—
208C22×C3H(5,𝔽2)12yes—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C3×C4—22no—
217C3×C4—22no—
218C3×C4—22no—
219C3×C4—41no—
220C3×C4—41no—
221Dic3—42no—
222Dic3—42no—
223Dic3—42no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23#23512yes—
240C23C2×#6712yes—
241C23C23×D311yescharacteristic
242D4—28no—
243D4—124no—
244D4—128no—
245D4—128no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4#23512yes—
266C2×C4C2×#6712yes—
267C2×C4C2×#6712yes—
268C2×C3—42no—
269C2×C3—42no—
270C2×C3#52511yescharacteristic
271C2×C3#53411yescharacteristic
272C2×C3#53511yescharacteristic
273C2×C3#53812yes—
274C2×C3#54012yes—
275D3—82no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—24no—
280C22—22no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22—122no—
290C22—122no—
291C22—122no—
292C22—122no—
293C22C2×#23511yescharacteristic
294C22C2×#23511yescharacteristic
295C22#100412yes—
296C22#100912yes—
297C22C22×#6711yescharacteristic
298C4—22no—
299C4—22no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—122no—
304C4—122no—
305C4—122no—
306C3#263511yescharacteristic, complemented, Sylow
307C2—42no—
308C2—42no—
309C2—242no—
310C2#556012yes—
311C2#556212yes—
312C2#579111yescharacteristic
313C2#580411yescharacteristic
314C2#583011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
hypercenterC2×C4×D4D3
hypocenterC3#2635
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. C2×#930C2semidirect
3. #5223C2semidirect
4. #5213C2semidirect
5. #5211C2semidirect
6. #4991C2semidirect
7. #4986C2semidirect
8. #4960C2semidirect
9. #4959C2semidirect
10. #4949C2semidirect
11. #4940C2semidirect
39. C2×C4×D4D3semidirect
58. C22×C3×C4D4semidirect
176. C22×C4#67semidirect
306. C3#2635semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9302
#52232
#52132
#52112
#49912
#49862
#49602
#49592
#49492
#49402
#26353