Finite Group Databasebeta

← #76760

Subgroups — #76760

Order 384.

1134 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 153 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×#937C212yesmaximal, complemented
3C2×#883C211yesmaximal, characteristic, complemented
4C3×#392C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5222C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4955C212yesmaximal, complemented
10#4672C211yesmaximal, characteristic, complemented
11#4664C211yesmaximal, characteristic, complemented
12#2613—31nomaximal, Sylow
13C24×D3C2212yes—
14C22×C3×D4C2211yescharacteristic
15C22×C3×D4C2212yescomplemented
16C22×#67C2212yes—
17C22×#67C2211yescharacteristic
18C22×#67C2212yescomplemented
19C2×D3×D4—28no—
20C2×D3×D4—24no—
21C22×D12C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2212yescomplemented
25C2×#217C2212yescomplemented
26C2×#217C2211yescharacteristic, complemented
27C2×#217C2212yescomplemented
28#937—28no—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2212yescomplemented
32C2×(C2×D3)⋊C4C2212yescomplemented
33C2×(C2×D3)⋊C4C2212yescomplemented
34C2×C4⋊Dic3C2211yescharacteristic, complemented
35C2×Dic3⋊C4C2211yescharacteristic, complemented
36#883—24no—
37C3×#95C2211yescharacteristic, complemented
38C3⋊#95C2211yescharacteristic, complemented
39C3⋊#95C2212yescomplemented
40C23×D4—31no—
41C2×#123—31no—
42C2×#120—32no—
43#394—31no—
44#394—32no—
45#394—31no—
46#392—32no—
47#392—31no—
48#392D311yescharacteristic, complemented
49#381—31no—
50#379—32no—
51C24×C3C2311yescharacteristic
52C24×C3C2312yes—
53C23×D3—24no—
54C23×D3—24no—
55C23×D3—42no—
56C23×D3—42no—
57C23×D3—42no—
58C23×D3—42no—
59C23×D3D418yescomplemented
60C23×D3C2312yes—
61C23×D3C2311yescharacteristic
62C2×C3×D4—24no—
63C2×C3×D4—24no—
64C2×C3×D4—22no—
65C2×C3×D4—42no—
66C2×C3×D4—42no—
67C22×C3×C4C2312yes—
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—22no—
73C2×#67—24no—
74C2×#67—24no—
75C2×#67—24no—
76C2×#67—42no—
77C2×#67—42no—
78C22×Dic3C2312yes—
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82D3×D4—232no—
83C2×D12—24no—
84C2×D12—22no—
85C2×C4×D3—24no—
86C2×C4×D3D414yescomplemented
87C3×C22⋊C4—22no—
88C3×C22⋊C4—42no—
89#217—24no—
90#217—22no—
91#217—42no—
92(C2×D3)⋊C4—22no—
93(C2×D3)⋊C4—24no—
94(C2×D3)⋊C4—24no—
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97C4⋊Dic3—22no—
98Dic3⋊C4—22no—
99C25—32no—
100C22×D4—31no—
101C22×D4—32no—
102C22×D4—32no—
103C22×D4—31no—
104C22×D4—64no—
105C22×D4—68no—
106C22×D4C2×D311yescharacteristic
107C22×D4C2×D312yescomplemented
108C23×C4—31no—
109#123—64no—
110#120—68no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—31no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—31no—
118C2×C22⋊C4—32no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4C2×D311yescharacteristic, complemented
122C2×C22⋊C4C2×D312yescomplemented
123#95—32no—
124#95—31no—
125#95C2×D311yescharacteristic, complemented
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—22no—
134C23×C3—21no—
135C23×C3—22no—
136C23×C3—21no—
137C23×C3C2×D414yes—
138C23×C3C2411yescharacteristic
139C22×D3—22no—
140C22×D3—21no—
141C22×D3—21no—
142C22×D3—22no—
143C22×D3—22no—
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—22no—
147C22×D3—22no—
148C22×D3—216no—
149C22×D3—48no—
150C22×D3—48no—
151C22×D3—48no—
152C22×D3—48no—
153C22×D3—48no—
154C22×D3—48no—
155C22×D3C2×D414yes—
156C22×D3C2×D414yes—
157C22×D3C2×D412yes—
158C3×D4—28no—
159C3×D4—48no—
160C2×C3×C4—22no—
161C2×C3×C4—21no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4C2×D412yes—
167#67—216no—
168#67—48no—
169#67—44no—
170C2×Dic3—22no—
171C2×Dic3—21no—
172C2×Dic3—21no—
173C2×Dic3—22no—
174C2×Dic3—22no—
175C2×Dic3—21no—
176C2×Dic3—22no—
177C2×Dic3—21no—
178C2×Dic3—22no—
179C2×Dic3C2×D412yes—
180D12—28no—
181C4×D3—28no—
182C24—31no—
183C24—32no—
184C24—38no—
185C24—64no—
186C24—64no—
187C24—122no—
188C24—122no—
189C24—122no—
190C24—122no—
191C24C22×D311yescharacteristic
192C24C22×D312yes—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—22no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—64no—
199C2×D4—64no—
200C2×D4—632no—
201C2×D4—64no—
202C2×D4—64no—
203C2×D4—64no—
204C2×D4—64no—
205C2×D4—62no—
206C2×D4—62no—
207C2×D4—122no—
208C2×D4—122no—
209C22×C4—31no—
210C22×C4—34no—
211C22×C4—32no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—64no—
215C22×C4C22×D312yes—
216C22×C4C22×D311yescharacteristic
217C22×C4C22×D311yescharacteristic
218C4⋊C4—62no—
219C4⋊C4—62no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—64no—
223C22⋊C4—64no—
224C22⋊C4—62no—
225C22⋊C4—62no—
226C22⋊C4—64no—
227C22⋊C4—122no—
228C22⋊C4—122no—
229C22⋊C4—122no—
230C22×C3—24no—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3—42no—
237C22×C3—41no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3C22×D412yes—
243C22×C3C22×D411yescharacteristic
244C22×C3H(5,𝔽2)12yes—
245C22×C3H(5,𝔽2)11yescharacteristic
246C22×C3H(5,𝔽2)11yescharacteristic
247C2×D3—24no—
248C2×D3—24no—
249C2×D3—22no—
250C2×D3—22no—
251C2×D3—24no—
252C2×D3—24no—
253C2×D3—24no—
254C2×D3—432no—
255C2×D3—42no—
256C2×D3—42no—
257C2×D3—42no—
258C2×D3—42no—
259C2×D3#12014yes—
260C3×C4—22no—
261C3×C4—42no—
262C3×C4—41no—
263Dic3—22no—
264Dic3—41no—
265Dic3—42no—
266Dic3—41no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—22no—
275C23—21no—
276C23—22no—
277C23—21no—
278C23—32no—
279C23—34no—
280C23—34no—
281C23—616no—
282C23—62no—
283C23—61no—
284C23—61no—
285C23—62no—
286C23—62no—
287C23—62no—
288C23—62no—
289C23—62no—
290C23—62no—
291C23—128no—
292C23—128no—
293C23—128no—
294C23—128no—
295C23—128no—
296C23—128no—
297C23D3×D414yes—
298C23C23×D311yescharacteristic
299D4—28no—
300D4—48no—
301D4—68no—
302D4—616no—
303D4—124no—
304D4—128no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—32no—
312C2×C4—68no—
313C2×C4—62no—
314C2×C4—61no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—62no—
320C2×C4—61no—
321C2×C4—62no—
322C2×C4D3×D412yes—
323C2×C3—44no—
324C2×C3—81no—
325C2×C3C2×#12011yescharacteristic
326C2×C3#53412yes—
327C2×C3#53412yes—
328C2×C3#53511yescharacteristic
329C2×C3#53511yescharacteristic
330D3—24no—
331D3—44no—
332C22—24no—
333C22—24no—
334C22—24no—
335C22—34no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—62no—
346C22—64no—
347C22—64no—
348C22—62no—
349C22—64no—
350C22—64no—
351C22—64no—
352C22—1232no—
353C22—122no—
354C22—122no—
355C22—122no—
356C22—122no—
357C22C2×D3×D411yescharacteristic
358C22C2×D3×D412yes—
359C22#100412yes—
360C22#100411yescharacteristic
361C22#100911yescharacteristic
362C4—22no—
363C4—41no—
364C4—42no—
365C4—62no—
366C4—121no—
367C4—122no—
368C4—121no—
369C3#261311yescharacteristic, complemented, Sylow
370C2—44no—
371C2—64no—
372C2—81no—
373C2—124no—
374C2D3×#12011yescharacteristic
375C2#558912yes—
376C2#560512yes—
377C2#560711yescharacteristic
378C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2613
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×#937C2semidirect
3. C2×#883C2semidirect
4. C3×#392C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #4961C2semidirect
8. #4960C2semidirect
9. #4955C2semidirect
10. #4672C2semidirect
11. #4664C2semidirect
15. C22×C3×D4C22semidirect
18. C22×#67C22semidirect
23. C2×C3×C22⋊C4C22semidirect
24. C2×C3×C22⋊C4C22semidirect
25. C2×#217C22semidirect
26. C2×#217C22semidirect
27. C2×#217C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×(C2×D3)⋊C4C22semidirect
33. C2×(C2×D3)⋊C4C22semidirect
34. C2×C4⋊Dic3C22semidirect
35. C2×Dic3⋊C4C22semidirect
37. C3×#95C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
48. #392D3semidirect
59. C23×D3D4semidirect
86. C2×C4×D3D4semidirect
107. C22×D4C2×D3semidirect
121. C2×C22⋊C4C2×D3semidirect
122. C2×C22⋊C4C2×D3semidirect
125. #95C2×D3semidirect
369. C3#2613semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#9372
C2×#8832
C3×#3922
#52232
#52222
#49612
#49602
#49552
#46722
#46642
#26133