Finite Group Databasebeta

← #83175

Subgroups — #83175

Order 384.

1348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×D3×D4C211yesmaximal, characteristic, complemented
2C2×C3×#127C211yesmaximal, characteristic, complemented
3C2×#940C211yesmaximal, characteristic, complemented
4C2×#938C211yesmaximal, characteristic, complemented
5C2×#937C214yesmaximal, complemented
6#5223C214yesmaximal, complemented
7#5221C211yesmaximal, characteristic, complemented
8#4939C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#2433—31nomaximal, Sylow
11C24×D3C2212yes—
12C22×C3×D4C2211yescharacteristic
13C22×C3×D4C2211yescharacteristic, complemented
14C22×C3×D4C2214yescomplemented
15C22×#67C2212yes—
16C22×#67C2212yes—
17C22×#67C2214yescomplemented
18C2×D3×D4—24no—
19C2×D3×D4—24no—
20C2×D3×D4—24no—
21C22×D12C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C3×#127—24no—
24C2×C3×C42C2211yescharacteristic, complemented
25C2×#217C2214yescomplemented
26C2×#217C2212yescomplemented
27#940—24no—
28#938—24no—
29#937—216no—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2214yescomplemented
32C2×C4⋊Dic3C2211yescharacteristic, complemented
33C2×C4×Dic3C2211yescharacteristic, complemented
34C3⋊#95C2212yescomplemented
35C23×D4—31no—
36C2×#127—31no—
37C2×#127D311yescharacteristic, complemented
38C2×#121—31no—
39C2×#120—34no—
40#392—34no—
41#390—31no—
42#390—31no—
43C4×C22⋊C4—31no—
44C24×C3C2312yes—
45C24×C3C2312yes—
46C23×D3—24no—
47C23×D3—24no—
48C23×D3—24no—
49C23×D3—28no—
50C23×D3—44no—
51C23×D3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×D4—28no—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—22no—
59C2×C3×D4—28no—
60C2×C3×D4—44no—
61C2×C3×D4D414yescomplemented
62C22×C3×C4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C2×#67—24no—
65C2×#67—24no—
66C2×#67—28no—
67C2×#67—24no—
68C2×#67—28no—
69C2×#67—24no—
70C2×#67—28no—
71C2×#67—44no—
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C22×Dic3C2312yes—
75D3×D4—232no—
76C2×D12—22no—
77C2×D12—22no—
78C2×D12D414yescomplemented
79C2×C4×D3—22no—
80C2×C4×D3—22no—
81C2×C4×D3D414yescomplemented
82C3×C42—22no—
83#217—24no—
84#217—28no—
85(C2×D3)⋊C4—28no—
86(C2×D3)⋊C4D418yescomplemented
87C4⋊Dic3—22no—
88C4×Dic3—22no—
89C25—32no—
90C22×D4—32no—
91C22×D4—32no—
92C22×D4—34no—
93C22×D4—31no—
94C22×D4—64no—
95C22×D4—64no—
96C22×D4—64no—
97C22×D4C2×D311yescharacteristic
98C22×D4C2×D311yescharacteristic, complemented
99C22×D4C2×D314yescomplemented
100C23×C4—31no—
101#127—24no—
102#127—64no—
103#121—64no—
104#120—616no—
105C2×C4⋊C4—31no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—34no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—34no—
110C2×C42—31no—
111C2×C42C2×D311yescharacteristic, complemented
112#95—32no—
113C23×C3—24no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—24no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—24no—
120C23×C3—22no—
121C23×C3—24no—
122C23×C3C2×D414yes—
123C23×C3C2411yescharacteristic
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—22no—
127C22×D3—216no—
128C22×D3—24no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—24no—
132C22×D3—416no—
133C22×D3—416no—
134C22×D3—48no—
135C22×D3—48no—
136C22×D3C2×D414yes—
137C22×D3C2×D414yes—
138C22×D3C2×D412yes—
139C3×D4—28no—
140C3×D4—416no—
141C3×D4—44no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4C2×D412yes—
145C2×C3×C4C2×D412yes—
146C2×C3×C4C2×D412yes—
147C2×C3×C4C2×D414yescomplemented
148#67—216no—
149#67—416no—
150#67—48no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—24no—
157C2×Dic3C2×D412yes—
158C2×Dic3C2×D414yescomplemented
159D12—28no—
160C4×D3—28no—
161C24—31no—
162C24—32no—
163C24—64no—
164C24—64no—
165C24—68no—
166C24—64no—
167C24—124no—
168C24C22×D312yes—
169C24C22×D312yes—
170C2×D4—28no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—28no—
176C2×D4—22no—
177C2×D4—34no—
178C2×D4—44no—
179C2×D4—64no—
180C2×D4—68no—
181C2×D4—632no—
182C2×D4—64no—
183C2×D4—64no—
184C2×D4—68no—
185C2×D4—64no—
186C2×D4—68no—
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—124no—
190C2×D4#6714yescomplemented
191C22×C4—31no—
192C22×C4—32no—
193C22×C4—34no—
194C22×C4—32no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4C22×D312yes—
198C22×C4C22×D311yescharacteristic
199C4⋊C4—62no—
200C22⋊C4—38no—
201C22⋊C4—68no—
202C22⋊C4—64no—
203C22⋊C4—68no—
204C42—22no—
205C42—62no—
206C22×C3—24no—
207C22×C3—28no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—44no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—44no—
215C22×C3—44no—
216C22×C3C22×D411yescharacteristic
217C22×C3C22×D411yescharacteristic
218C22×C3C22×D412yes—
219C22×C3C22×D411yescharacteristic
220C22×C3H(5,𝔽2)12yes—
221C2×D3—22no—
222C2×D3—22no—
223C2×D3—24no—
224C2×D3—24no—
225C2×D3—24no—
226C2×D3—28no—
227C2×D3—432no—
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—44no—
231C2×D3#12714yescomplemented
232C3×C4—24no—
233C3×C4#12014yescomplemented
234Dic3—22no—
235Dic3—24no—
236Dic3—42no—
237C23—24no—
238C23—22no—
239C23—22no—
240C23—24no—
241C23—22no—
242C23—24no—
243C23—22no—
244C23—24no—
245C23—22no—
246C23—34no—
247C23—32no—
248C23—34no—
249C23—616no—
250C23—61no—
251C23—62no—
252C23—61no—
253C23—64no—
254C23—62no—
255C23—64no—
256C23—62no—
257C23—1216no—
258C23—128no—
259C23—1216no—
260C23—128no—
261C23C2×#6714yes—
262C23C23×D311yescharacteristic
263D4—28no—
264D4—416no—
265D4—44no—
266D4—616no—
267D4—68no—
268D4—1216no—
269D4—128no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—34no—
273C2×C4—32no—
274C2×C4—62no—
275C2×C4—61no—
276C2×C4—62no—
277C2×C4—61no—
278C2×C4—68no—
279C2×C4—64no—
280C2×C4—62no—
281C2×C4D3×D412yes—
282C2×C4D3×D412yes—
283C2×C4D3×D414yescomplemented
284C2×C4C2×#6712yes—
285C2×C3—44no—
286C2×C3—82no—
287C2×C3C2×#12011yescharacteristic
288C2×C3C2×#12711yescharacteristic
289C2×C3#53511yescharacteristic
290C2×C3D4212yes—
291C2×C3D4212yes—
292D3—24no—
293D3—44no—
294C22—24no—
295C22—28no—
296C22—34no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—42no—
304C22—44no—
305C22—62no—
306C22—64no—
307C22—62no—
308C22—64no—
309C22—68no—
310C22—64no—
311C22—1232no—
312C22—122no—
313C22—122no—
314C22—124no—
315C22C2×D3×D411yescharacteristic
316C22C2×D3×D412yes—
317C22C2×D3×D411yescharacteristic
318C22#100412yes—
319C22C22×#6711yescharacteristic
320C4—24no—
321C4—64no—
322C4—62no—
323C4—122no—
324C4#93714yescomplemented
325C3#243311yescharacteristic, complemented, Sylow
326C2—44no—
327C2—64no—
328C2—82no—
329C2—124no—
330C2D3×#12711yescharacteristic
331C2#571511yescharacteristic
332C2#571712yes—
333C2C2×#93711yescharacteristic
334C2D4×#6712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#127C2
hypercenterC2×#127D3
hypocenterC3#2433
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×C3×#127C2semidirect
3. C2×#940C2semidirect
4. C2×#938C2semidirect
5. C2×#937C2semidirect
6. #5223C2semidirect
7. #5221C2semidirect
8. #4939C2semidirect
9. C4×(C2×D3)⋊C4C2semidirect
13. C22×C3×D4C22semidirect
14. C22×C3×D4C22semidirect
17. C22×#67C22semidirect
24. C2×C3×C42C22semidirect
25. C2×#217C22semidirect
26. C2×#217C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×C4⋊Dic3C22semidirect
33. C2×C4×Dic3C22semidirect
34. C3⋊#95C22semidirect
37. C2×#127D3semidirect
61. C2×C3×D4D4semidirect
78. C2×D12D4semidirect
81. C2×C4×D3D4semidirect
86. (C2×D3)⋊C4D4semidirect
98. C22×D4C2×D3semidirect
99. C22×D4C2×D3semidirect
111. C2×C42C2×D3semidirect
147. C2×C3×C4C2×D4semidirect
158. C2×Dic3C2×D4semidirect
190. C2×D4#67semidirect
231. C2×D3#127semidirect
233. C3×C4#120semidirect
283. C2×C4D3×D4semidirect
324. C4#937semidirect
325. C3#2433semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×C3×#1272
C2×#9402
C2×#9382
C2×#9372
#52232
#52212
#49392
C4×(C2×D3)⋊C42
#24333