Finite Group Databasebeta

← #84052

Subgroups — #84052

Order 384.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C23×C3×D4C211yesmaximal, characteristic, complemented
2C23×#67C211yesmaximal, characteristic, complemented
3C2×#953C212yesmaximal, complemented
4C2×#937C212yesmaximal, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5223C214yesmaximal, complemented
7#5220C212yesmaximal, complemented
8#5213C212yesmaximal, complemented
9#2403—31nomaximal, Sylow
10C25×C3C2211yescharacteristic
11C25×C3C2211yescharacteristic, complemented
12C24×D3C2211yescharacteristic, complemented
13C22×C3×D4—24no—
14C22×C3×D4—28no—
15C22×C3×D4C2212yes—
16C22×C3×D4C2212yescomplemented
17C23×C3×C4C2211yescharacteristic, complemented
18C22×#67—24no—
19C22×#67—28no—
20C22×#67C2212yes—
21C22×#67C2212yescomplemented
22C22×#67C2214yescomplemented
23C23×Dic3C2211yescharacteristic, complemented
24#953—28no—
25C2×#217C2212yescomplemented
26C2×#217C2214yescomplemented
27C2×#217C2212yescomplemented
28C2×#217C2212yescomplemented
29#937—28no—
30C2×(C2×D3)⋊C4C2214yescomplemented
31C3⋊#95C2214yescomplemented
32C23×D4—31no—
33C23×D4D311yescharacteristic, complemented
34C2×#120—32no—
35C2×#120—32no—
36#392—34no—
37#386—32no—
38#386—32no—
39#379—31no—
40C24×C3—24no—
41C24×C3—24no—
42C24×C3—22no—
43C24×C3—24no—
44C24×C3—24no—
45C24×C3—42no—
46C24×C3—42no—
47C24×C3D414yescomplemented
48C24×C3C2311yescharacteristic
49C24×C3C2312yes—
50C24×C3C2311yescharacteristic
51C24×C3C2311yescharacteristic
52C23×D3—22no—
53C23×D3—24no—
54C23×D3—24no—
55C23×D3—42no—
56C23×D3C2312yes—
57C2×C3×D4—232no—
58C2×C3×D4—24no—
59C2×C3×D4—24no—
60C2×C3×D4—24no—
61C2×C3×D4—24no—
62C2×C3×D4—24no—
63C2×C3×D4D418yescomplemented
64C22×C3×C4—22no—
65C22×C3×C4—24no—
66C22×C3×C4C2312yes—
67C2×#67—232no—
68C2×#67—24no—
69C2×#67—24no—
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—28no—
73C2×#67—28no—
74C2×#67—24no—
75C2×#67—44no—
76C2×#67D418yescomplemented
77C22×Dic3—22no—
78C22×Dic3—24no—
79C22×Dic3C2314yes—
80C22×Dic3C2312yes—
81#217—24no—
82#217—24no—
83#217—28no—
84#217—42no—
85(C2×D3)⋊C4—28no—
86C25—31no—
87C25C2×D311yescharacteristic
88C25C2×D311yescharacteristic, complemented
89C22×D4—24no—
90C22×D4—28no—
91C22×D4—32no—
92C22×D4—32no—
93C22×D4—34no—
94C22×D4—68no—
95C22×D4—64no—
96C22×D4C2×D312yes—
97C22×D4C2×D312yescomplemented
98C23×C4—31no—
99C23×C4C2×D311yescharacteristic, complemented
100#120—68no—
101#120—68no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—34no—
105C2×C22⋊C4—34no—
106C2×C22⋊C4—32no—
107#95—34no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—28no—
113C23×C3—28no—
114C23×C3—28no—
115C23×C3—24no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—24no—
122C23×C3—44no—
123C23×C3—416no—
124C23×C3—416no—
125C23×C3—44no—
126C23×C3—44no—
127C23×C3C2×D414yes—
128C23×C3C2×D412yes—
129C23×C3C2×D414yes—
130C23×C3C2×D412yes—
131C23×C3C2×D412yes—
132C23×C3C2×D414yescomplemented
133C23×C3C2411yescharacteristic
134C22×D3—28no—
135C22×D3—24no—
136C22×D3—22no—
137C22×D3—22no—
138C22×D3—24no—
139C22×D3—416no—
140C22×D3—44no—
141C22×D3—44no—
142C22×D3C2×D414yescomplemented
143C3×D4—216no—
144C3×D4—216no—
145C2×C3×C4—28no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4C2×D414yescomplemented
149#67—216no—
150#67—216no—
151#67—416no—
152C2×Dic3—28no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—24no—
156C2×Dic3—24no—
157C2×Dic3—24no—
158C2×Dic3C2×D414yescomplemented
159C24—24no—
160C24—24no—
161C24—22no—
162C24—24no—
163C24—24no—
164C24—32no—
165C24—42no—
166C24—42no—
167C24—64no—
168C24—62no—
169C24—64no—
170C24—122no—
171C24#6714yescomplemented
172C24C22×D311yescharacteristic
173C24C22×D312yes—
174C24C22×D311yescharacteristic
175C24C22×D311yescharacteristic
176C2×D4—232no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—38no—
183C2×D4—64no—
184C2×D4—64no—
185C2×D4—632no—
186C2×D4—64no—
187C2×D4—64no—
188C2×D4—64no—
189C2×D4—68no—
190C2×D4—68no—
191C2×D4—124no—
192C2×D4#6718yescomplemented
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—32no—
196C22×C4—34no—
197C22×C4—64no—
198C22×C4—62no—
199C22×C4C22×D312yes—
200C22⋊C4—68no—
201C22⋊C4—64no—
202C22⋊C4—64no—
203C22⋊C4—68no—
204C22⋊C4—122no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—28no—
210C22×C3—28no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—24no—
214C22×C3—22no—
215C22×C3—28no—
216C22×C3—416no—
217C22×C3—416no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—44no—
221C22×C3—44no—
222C22×C3#12014yescomplemented
223C22×C3#12014yescomplemented
224C22×C3C22×D412yes—
225C22×C3C22×D411yescharacteristic
226C22×C3C22×D412yes—
227C22×C3H(5,𝔽2)12yes—
228C2×D3—24no—
229C2×D3—28no—
230C2×D3—416no—
231C2×D3—42no—
232C2×D3—42no—
233C2×D3—44no—
234C3×C4—24no—
235Dic3—24no—
236Dic3—44no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—28no—
242C23—28no—
243C23—28no—
244C23—24no—
245C23—22no—
246C23—22no—
247C23—24no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—34no—
252C23—44no—
253C23—416no—
254C23—416no—
255C23—44no—
256C23—44no—
257C23—68no—
258C23—62no—
259C23—64no—
260C23—64no—
261C23—62no—
262C23—1216no—
263C23—124no—
264C23—124no—
265C23D3×D414yes—
266C23C2×#6712yes—
267C23C2×#6712yes—
268C23C2×#6714yes—
269C23C2×#6712yes—
270C23C2×#6714yescomplemented
271C23C23×D311yescharacteristic
272D4—216no—
273D4—216no—
274D4—616no—
275D4—616no—
276D4—1216no—
277C2×C4—28no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—34no—
281C2×C4—68no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—64no—
285C2×C4—64no—
286C2×C4—64no—
287C2×C4C2×#6714yescomplemented
288C2×C3—24no—
289C2×C3—24no—
290C2×C3—42no—
291C2×C3—44no—
292C2×C3C2×#12011yescharacteristic
293C2×C3C2×#12011yescharacteristic
294C2×C3#53411yescharacteristic
295C2×C3D4214yes—
296D3—44no—
297C22—22no—
298C22—28no—
299C22—28no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—28no—
305C22—22no—
306C22—22no—
307C22—24no—
308C22—416no—
309C22—416no—
310C22—42no—
311C22—44no—
312C22—44no—
313C22—42no—
314C22—64no—
315C22—68no—
316C22—1216no—
317C22—124no—
318C22—122no—
319C22—122no—
320C22#93714yescomplemented
321C22#95314yescomplemented
322C22C2×D3×D412yes—
323C22#100412yes—
324C22C22×#6711yescharacteristic
325C22C22×#6712yes—
326C4—24no—
327C4—64no—
328C4—124no—
329C3#240311yescharacteristic, complemented, Sylow
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—124no—
335C2C2×#93711yescharacteristic
336C2D4×#6714yes—
337C2#580411yescharacteristic
338C2C2×#95311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC23×C3×D4C2
hypercenterC23×D4D3
hypocenterC3#2403
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×D4C2semidirect
2. C23×#67C2semidirect
3. C2×#953C2semidirect
4. C2×#937C2semidirect
5. #5247C2semidirect
6. #5223C2semidirect
7. #5220C2semidirect
8. #5213C2semidirect
11. C25×C3C22semidirect
12. C24×D3C22semidirect
16. C22×C3×D4C22semidirect
17. C23×C3×C4C22semidirect
21. C22×#67C22semidirect
22. C22×#67C22semidirect
23. C23×Dic3C22semidirect
25. C2×#217C22semidirect
26. C2×#217C22semidirect
27. C2×#217C22semidirect
28. C2×#217C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C3⋊#95C22semidirect
33. C23×D4D3semidirect
47. C24×C3D4semidirect
63. C2×C3×D4D4semidirect
76. C2×#67D4semidirect
88. C25C2×D3semidirect
97. C22×D4C2×D3semidirect
99. C23×C4C2×D3semidirect
132. C23×C3C2×D4semidirect
142. C22×D3C2×D4semidirect
148. C2×C3×C4C2×D4semidirect
158. C2×Dic3C2×D4semidirect
171. C24#67semidirect
192. C2×D4#67semidirect
222. C22×C3#120semidirect
223. C22×C3#120semidirect
270. C23C2×#67semidirect
287. C2×C4C2×#67semidirect
320. C22#937semidirect
321. C22#953semidirect
329. C3#2403semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D42
C23×#672
C2×#9532
C2×#9372
#52472
#52232
#52202
#52132
#24033