Finite Group Databasebeta

← #77138

Subgroups — #77138

Order 384.

1000 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 149 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×#937C212yesmaximal, complemented
2C2×#882C212yesmaximal, complemented
3C2×#882C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4959C212yesmaximal, complemented
8#4957C212yesmaximal, complemented
9#4672C212yesmaximal, complemented
10#2679—31nomaximal, Sylow
11C24×D3C2212yescomplemented
12C22×C3×D4C2211yescharacteristic
13C22×#67C2212yes—
14C22×#67C2212yes—
15C22×D12C2212yes—
16C22×D12C2211yescharacteristic
17C22×D12C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic, complemented
19C2×C3×C22⋊C4C2212yescomplemented
20C2×C3×C22⋊C4C2212yescomplemented
21C2×#217C2212yescomplemented
22#937—28no—
23C2×(C2×D3)⋊C4C2212yescomplemented
24C2×(C2×D3)⋊C4C2212yescomplemented
25C2×(C2×D3)⋊C4C2212yescomplemented
26C2×(C2×D3)⋊C4C2212yescomplemented
27C2×(C2×D3)⋊C4C2212yescomplemented
28C2×(C2×D3)⋊C4C2212yescomplemented
29C2×Dic3⋊C4C2212yescomplemented
30C2×C4×Dic3C2211yescharacteristic, complemented
31#882—28no—
32#882—28no—
33C3×#95C2211yescharacteristic, complemented
34C3⋊#95C2211yescharacteristic, complemented
35C2×#120—32no—
36C2×#120—32no—
37C2×#120—32no—
38#394—31no—
39#394—32no—
40#394—32no—
41#394D311yescharacteristic, complemented
42#390—31no—
43#390—32no—
44C24×C3C2312yes—
45C23×D3—24no—
46C23×D3—24no—
47C23×D3—24no—
48C23×D3—42no—
49C23×D3—42no—
50C23×D3—42no—
51C23×D3—42no—
52C23×D3C2312yes—
53C23×D3C2312yes—
54C2×C3×D4—24no—
55C2×C3×D4—41no—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—42no—
63C2×#67—42no—
64C2×#67—42no—
65C2×#67D418yescomplemented
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C2×D12—24no—
69C2×D12—24no—
70C2×D12—24no—
71C2×D12—41no—
72C2×D12—41no—
73C2×D12—42no—
74C2×D12—41no—
75C2×D12D414yescomplemented
76C3×C4⋊C4—41no—
77C3×C22⋊C4—24no—
78C3×C22⋊C4—24no—
79#217—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—24no—
86Dic3⋊C4—42no—
87C4×Dic3—41no—
88C25—32no—
89C22×D4—32no—
90C22×D4—32no—
91C22×D4—31no—
92C22×D4—31no—
93C22×D4—32no—
94C22×D4C2×D311yescharacteristic
95#120—68no—
96#120—68no—
97#120—68no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4C2×D311yescharacteristic, complemented
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4C2×D312yescomplemented
108C2×C22⋊C4C2×D312yescomplemented
109C2×C42—31no—
110#95—31no—
111#95C2×D311yescharacteristic, complemented
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3C2×D414yes—
119C23×C3C2411yescharacteristic
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—48no—
133C22×D3—48no—
134C22×D3—48no—
135C22×D3—48no—
136C22×D3—48no—
137C22×D3—48no—
138C22×D3—48no—
139C22×D3C2×D414yes—
140C22×D3C2×D414yes—
141C3×D4—44no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4C2×D412yes—
152#67—48no—
153#67—48no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3C2×D414yes—
160D12—44no—
161D12—44no—
162D12—48no—
163C24—32no—
164C24—32no—
165C24—64no—
166C24—64no—
167C24—64no—
168C24—122no—
169C24—122no—
170C24—122no—
171C24—122no—
172C24C22×D312yes—
173C2×D4—24no—
174C2×D4—38no—
175C2×D4—34no—
176C2×D4—41no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—64no—
180C2×D4—64no—
181C2×D4—64no—
182C2×D4—121no—
183C2×D4—121no—
184C2×D4—121no—
185C2×D4—122no—
186C2×D4—122no—
187C2×D4—122no—
188C2×D4—122no—
189C22×C4—32no—
190C22×C4—32no—
191C22×C4C22×D311yescharacteristic
192C22×C4C22×D311yescharacteristic
193C22×C4C22×D312yes—
194C22×C4C22×D311yescharacteristic
195C4⋊C4—41no—
196C4⋊C4—122no—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—64no—
200C22⋊C4—64no—
201C22⋊C4—64no—
202C22⋊C4—64no—
203C22⋊C4—64no—
204C22⋊C4—64no—
205C22⋊C4—64no—
206C42—121no—
207C22×C3—24no—
208C22×C3—24no—
209C22×C3—24no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3C22×D411yescharacteristic
215C22×C3C22×D412yes—
216C22×C3H(5,𝔽2)11yescharacteristic
217C22×C3H(5,𝔽2)11yescharacteristic
218C22×C3H(5,𝔽2)12yes—
219C2×D3—24no—
220C2×D3—24no—
221C2×D3—24no—
222C2×D3—24no—
223C2×D3—24no—
224C2×D3—24no—
225C2×D3—432no—
226C2×D3—42no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—42no—
231C2×D3—42no—
232C2×D3—42no—
233C2×D3—42no—
234C3×C4—22no—
235C3×C4—41no—
236C3×C4—42no—
237C3×C4—41no—
238Dic3—24no—
239Dic3—42no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—34no—
247C23—34no—
248C23—62no—
249C23—62no—
250C23—62no—
251C23—62no—
252C23—62no—
253C23—62no—
254C23—62no—
255C23—62no—
256C23—62no—
257C23—62no—
258C23—62no—
259C23—62no—
260C23—128no—
261C23—128no—
262C23—128no—
263C23—128no—
264C23—128no—
265C23—128no—
266C23—128no—
267C23D3×D414yes—
268C23C23×D311yescharacteristic
269D4—44no—
270D4—124no—
271D4—124no—
272D4—128no—
273D4—128no—
274D4—128no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—34no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4D3×D412yes—
291C2×C3—44no—
292C2×C3#53412yes—
293C2×C3#53411yescharacteristic
294C2×C3D4211yescharacteristic
295C2×C3D4212yes—
296C2×C3H(3,𝔽4)11yescharacteristic
297D3—44no—
298D3—44no—
299C22—24no—
300C22—24no—
301C22—24no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—64no—
307C22—64no—
308C22—64no—
309C22—64no—
310C22—64no—
311C22—64no—
312C22—1232no—
313C22—122no—
314C22—122no—
315C22—122no—
316C22—122no—
317C22—122no—
318C22—122no—
319C22—122no—
320C22—122no—
321C22C2×D3×D412yes—
322C22C2×D3×D411yescharacteristic
323C22#100411yescharacteristic
324C22#100911yescharacteristic
325C22#100912yes—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—42no—
330C4—64no—
331C4—122no—
332C3#267911yescharacteristic, complemented, Sylow
333C2—44no—
334C2—124no—
335C2—124no—
336C2#559011yescharacteristic
337C2#560511yescharacteristic
338C2#560912yes—
339C2#565312yes—
340C2#568311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2679
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#937C2semidirect
2. C2×#882C2semidirect
3. C2×#882C2semidirect
4. C3×#394C2semidirect
5. #4991C2semidirect
6. #4988C2semidirect
7. #4959C2semidirect
8. #4957C2semidirect
9. #4672C2semidirect
11. C24×D3C22semidirect
18. C2×C3×C4⋊C4C22semidirect
19. C2×C3×C22⋊C4C22semidirect
20. C2×C3×C22⋊C4C22semidirect
21. C2×#217C22semidirect
23. C2×(C2×D3)⋊C4C22semidirect
24. C2×(C2×D3)⋊C4C22semidirect
25. C2×(C2×D3)⋊C4C22semidirect
26. C2×(C2×D3)⋊C4C22semidirect
27. C2×(C2×D3)⋊C4C22semidirect
28. C2×(C2×D3)⋊C4C22semidirect
29. C2×Dic3⋊C4C22semidirect
30. C2×C4×Dic3C22semidirect
33. C3×#95C22semidirect
34. C3⋊#95C22semidirect
41. #394D3semidirect
65. C2×#67D4semidirect
75. C2×D12D4semidirect
99. C2×C4⋊C4C2×D3semidirect
107. C2×C22⋊C4C2×D3semidirect
108. C2×C22⋊C4C2×D3semidirect
111. #95C2×D3semidirect
332. C3#2679semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8822
C3×#3942
#49912
#49882
#49592
#49572
#46722
#26793