Finite Group Databasebeta

← #76754

Subgroups — #76754

Order 384.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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×#237C211yesmaximal, characteristic, complemented
2C2×#937C212yesmaximal, complemented
3C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#392C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5220C212yesmaximal, complemented
7#4958C212yesmaximal, complemented
8#4956C212yesmaximal, complemented
9#4947C211yesmaximal, characteristic, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4664C211yesmaximal, characteristic, complemented
12#2404—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×C3×D4C2211yescharacteristic, complemented
16C22×#67C2212yes—
17C22×#67C2212yes—
18C23×Dic3C2212yescomplemented
19C2×#237—28no—
20C2×#237—24no—
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×Dic6C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2212yescomplemented
26C2×#217C2212yes—
27C2×#217C2211yescharacteristic
28C2×#217C2212yescomplemented
29#937—28no—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34D3×C22⋊C4—24no—
35C3×#95C2211yescharacteristic, complemented
36C3⋊#95C2211yescharacteristic, complemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C22×#41—31no—
41C2×#120—32no—
42C22×C22⋊C4—31no—
43#393—31no—
44#393—31no—
45#392—31no—
46#392D311yescharacteristic, complemented
47#386—32no—
48#386—32no—
49#386—32no—
50#381—31no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C23×D3—22no—
54C23×D3—22no—
55C23×D3—22no—
56C23×D3—24no—
57C23×D3—42no—
58C23×D3C2311yescharacteristic
59C23×D3C2311yescharacteristic
60C2×C3×D4—24no—
61C2×C3×D4—24no—
62C2×C3×D4—24no—
63C2×C3×D4—22no—
64C2×C3×D4—42no—
65C22×C3×C4C2312yes—
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C2×#67—24no—
69C2×#67—24no—
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—42no—
73C2×#67D418yescomplemented
74C22×Dic3—24no—
75C22×Dic3—24no—
76C22×Dic3—24no—
77C22×Dic3C2312yes—
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2312yes—
80C22×Dic3C2311yescharacteristic
81#237—232no—
82C2×C4×D3—24no—
83C2×C4×D3—22no—
84C2×C4×D3—42no—
85C2×C4×D3D414yescomplemented
86C2×Dic6—22no—
87C2×Dic6—24no—
88C3×C22⋊C4—22no—
89C3×C22⋊C4—42no—
90#217—22no—
91#217—24no—
92#217—42no—
93(C2×D3)⋊C4—24no—
94(C2×D3)⋊C4—22no—
95(C2×D3)⋊C4—22no—
96(C2×D3)⋊C4—24no—
97C25—31no—
98C2×#41—64no—
99C2×#41—68no—
100C22×Q8—31no—
101C22×D4—32no—
102C22×D4—32no—
103C22×D4C2×D312yes—
104C22×D4C2×D311yescharacteristic, complemented
105C23×C4—31no—
106C23×C4—32no—
107C23×C4—31no—
108#120—68no—
109C2×C22⋊C4—31no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—64no—
117C2×C22⋊C4C2×D311yescharacteristic
118C2×C22⋊C4C2×D312yescomplemented
119#95—32no—
120#95—31no—
121#95—32no—
122#95—32no—
123#95C2×D311yescharacteristic, complemented
124C23×C3—21no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—21no—
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—22no—
134C23×C3C2×D414yes—
135C23×C3#4112yescomplemented
136C23×C3C2411yescharacteristic
137C22×D3—21no—
138C22×D3—21no—
139C22×D3—22no—
140C22×D3—28no—
141C22×D3—21no—
142C22×D3—22no—
143C22×D3—21no—
144C22×D3—22no—
145C22×D3—48no—
146C22×D3—48no—
147C22×D3—44no—
148C22×D3—44no—
149C22×D3C2×D414yes—
150C22×D3C2×D412yes—
151C22×D3#4112yescomplemented
152C3×D4—28no—
153C3×D4—48no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—21no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4C2×D412yes—
161#67—216no—
162#67—48no—
163C2×Dic3—216no—
164C2×Dic3—22no—
165C2×Dic3—22no—
166C2×Dic3—22no—
167C2×Dic3—21no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3C2×D414yes—
173C2×Dic3C2×D412yes—
174C4×D3—28no—
175C4×D3—44no—
176Dic6—28no—
177C24—31no—
178C24—31no—
179C24—62no—
180C24—62no—
181C24—64no—
182C24—62no—
183C24—122no—
184C24C22×D312yes—
185C24C22×D311yescharacteristic
186#41—632no—
187C2×Q8—64no—
188C2×Q8—62no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—22no—
193C2×D4—38no—
194C2×D4—42no—
195C2×D4—64no—
196C2×D4—64no—
197C2×D4—64no—
198C2×D4—64no—
199C2×D4—122no—
200C22×C4—31no—
201C22×C4—32no—
202C22×C4—34no—
203C22×C4—32no—
204C22×C4—31no—
205C22×C4—64no—
206C22×C4—62no—
207C22×C4—64no—
208C22×C4—64no—
209C22×C4—64no—
210C22×C4—122no—
211C22×C4C22×D312yes—
212C22×C4C22×D311yescharacteristic
213C22×C4C22×D311yescharacteristic
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—62no—
217C22⋊C4—64no—
218C22⋊C4—64no—
219C22⋊C4—62no—
220C22⋊C4—64no—
221C22⋊C4—62no—
222C22⋊C4—122no—
223C22×C3—24no—
224C22×C3—22no—
225C22×C3—24no—
226C22×C3—24no—
227C22×C3—24no—
228C22×C3—41no—
229C22×C3—42no—
230C22×C3—41no—
231C22×C3—42no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3C22×D411yescharacteristic
236C22×C3C22×D412yes—
237C22×C3C2×#4111yescharacteristic
238C22×C3C2×#4111yescharacteristic
239C22×C3H(5,𝔽2)12yes—
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—24no—
243C2×D3—24no—
244C2×D3—22no—
245C2×D3—24no—
246C2×D3—416no—
247C2×D3—41no—
248C2×D3—41no—
249C2×D3—42no—
250C2×D3#12014yes—
251C3×C4—22no—
252C3×C4—41no—
253C3×C4—42no—
254Dic3—24no—
255Dic3—22no—
256Dic3—41no—
257Dic3—42no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—34no—
269C23—32no—
270C23—32no—
271C23—68no—
272C23—61no—
273C23—62no—
274C23—61no—
275C23—61no—
276C23—62no—
277C23—61no—
278C23—62no—
279C23—128no—
280C23—124no—
281C23—128no—
282C23—124no—
283C23D3×D414yes—
284C23#23712yescomplemented
285C23C23×D311yescharacteristic
286Q8—68no—
287D4—28no—
288D4—48no—
289D4—616no—
290D4—128no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—32no—
298C2×C4—34no—
299C2×C4—68no—
300C2×C4—616no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—61no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—124no—
310C2×C4D3×D412yes—
311C2×C3—42no—
312C2×C3—44no—
313C2×C3C2×#12011yescharacteristic
314C2×C3#52511yescharacteristic
315C2×C3#53511yescharacteristic
316C2×C3#54612yes—
317C2×C3#54612yes—
318D3—24no—
319D3—42no—
320C22—24no—
321C22—24no—
322C22—24no—
323C22—22no—
324C22—24no—
325C22—34no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—62no—
334C22—64no—
335C22—62no—
336C22—64no—
337C22—62no—
338C22—64no—
339C22—1216no—
340C22—121no—
341C22—121no—
342C22—122no—
343C22C2×D3×D411yescharacteristic
344C22C2×D3×D412yes—
345C22C2×#23711yescharacteristic
346C22#100412yes—
347C22D3×#4111yescharacteristic
348C4—22no—
349C4—41no—
350C4—42no—
351C4—62no—
352C4—64no—
353C4—121no—
354C4—122no—
355C3#240411yescharacteristic, complemented, Sylow
356C2—42no—
357C2—44no—
358C2—64no—
359C2—122no—
360C2D3×#12011yescharacteristic
361C2#559112yes—
362C2#560611yescharacteristic
363C2#561012yes—
364C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2404
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#937C2semidirect
3. C2×D3×C22⋊C4C2semidirect
4. C3×#392C2semidirect
5. #5223C2semidirect
6. #5220C2semidirect
7. #4958C2semidirect
8. #4956C2semidirect
9. #4947C2semidirect
10. #4673C2semidirect
11. #4664C2semidirect
15. C22×C3×D4C22semidirect
18. C23×Dic3C22semidirect
23. C22×Dic6C22semidirect
25. C2×C3×C22⋊C4C22semidirect
28. C2×#217C22semidirect
35. C3×#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
46. #392D3semidirect
73. C2×#67D4semidirect
85. C2×C4×D3D4semidirect
104. C22×D4C2×D3semidirect
118. C2×C22⋊C4C2×D3semidirect
123. #95C2×D3semidirect
135. C23×C3#41semidirect
151. C22×D3#41semidirect
284. C23#237semidirect
355. C3#2404semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#9372
C2×D3×C22⋊C42
C3×#3922
#52232
#52202
#49582
#49562
#49472
#46732
#46642
#24043