Finite Group Databasebeta

← #76895

Subgroups — #76895

Order 384.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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×#893C211yesmaximal, characteristic, complemented
3C2×#884C212yesmaximal, complemented
4C2×#882C212yesmaximal, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#4991C211yesmaximal, characteristic, complemented
8#4957C212yesmaximal, complemented
9#4955C212yesmaximal, complemented
10#4671C211yesmaximal, characteristic, complemented
11#4667C211yesmaximal, characteristic, complemented
12#2408—31nomaximal, Sylow
13C24×D3C2212yes—
14C22×C3×D4C2211yescharacteristic
15C22×#67C2212yes—
16C22×#67C2212yescomplemented
17C2×D3×D4—28no—
18C2×D3×D4—24no—
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×D12C2212yescomplemented
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic, complemented
24C2×C3×C4⋊C4C2211yescharacteristic, complemented
25C2×C3×C22⋊C4C2212yescomplemented
26C2×C3×C22⋊C4C2212yescomplemented
27C2×#217C2212yescomplemented
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
31C2×(C2×D3)⋊C4C2212yescomplemented
32C2×(C2×D3)⋊C4C2212yescomplemented
33C2×(C2×D3)⋊C4C2212yescomplemented
34C2×Dic3⋊C4C2212yescomplemented
35C2×C4×Dic3C2211yescharacteristic, complemented
36#893—24no—
37#884—28no—
38#882—28no—
39C3×#95C2211yescharacteristic, complemented
40C3⋊#95C2211yescharacteristic, complemented
41C23×D4—31no—
42C2×#121—32no—
43C2×#120—32no—
44C2×C4×D4—31no—
45#394—31no—
46#394D311yescharacteristic, complemented
47#392—31no—
48#392—31no—
49#390—32no—
50#380—31no—
51#379—32no—
52C24×C3C2312yes—
53C23×D3—24no—
54C23×D3—24no—
55C23×D3—24no—
56C23×D3—24no—
57C23×D3—24no—
58C23×D3—42no—
59C23×D3—42no—
60C23×D3C2312yes—
61C23×D3C2311yescharacteristic
62C23×D3C2311yescharacteristic
63C2×C3×D4—24no—
64C2×C3×D4—22no—
65C22×C3×C4C2312yes—
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C2×#67—24no—
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—24no—
73C2×#67—42no—
74C2×#67D418yescomplemented
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78D3×D4—232no—
79C2×D12—22no—
80C2×D12—24no—
81C2×D12—22no—
82C2×D12—22no—
83C2×D12—24no—
84C2×D12—24no—
85C2×D12—42no—
86C2×D12D414yescomplemented
87C2×C4×D3—24no—
88C2×C4×D3—24no—
89C2×C4×D3—22no—
90C2×C4×D3D414yescomplemented
91C3×C4⋊C4—22no—
92C3×C22⋊C4—24no—
93C3×C22⋊C4—24no—
94#217—42no—
95(C2×D3)⋊C4—24no—
96(C2×D3)⋊C4—24no—
97(C2×D3)⋊C4—22no—
98(C2×D3)⋊C4—22no—
99(C2×D3)⋊C4—24no—
100(C2×D3)⋊C4—42no—
101Dic3⋊C4—24no—
102C4×Dic3—22no—
103C25—32no—
104C22×D4—32no—
105C22×D4—32no—
106C22×D4—31no—
107C22×D4—31no—
108C22×D4—32no—
109C22×D4—64no—
110C22×D4—68no—
111C22×D4C2×D311yescharacteristic
112C23×C4—31no—
113C23×C4—31no—
114#121—68no—
115#120—68no—
116C4×D4—64no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4C2×D311yescharacteristic, complemented
119C2×C22⋊C4—31no—
120C2×C22⋊C4—31no—
121C2×C22⋊C4—32no—
122C2×C22⋊C4—32no—
123C2×C22⋊C4—32no—
124C2×C22⋊C4—32no—
125C2×C22⋊C4—32no—
126C2×C22⋊C4C2×D312yescomplemented
127C2×C22⋊C4C2×D312yescomplemented
128C2×C42—31no—
129#95—31no—
130#95C2×D311yescharacteristic, complemented
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—22no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3C2×D414yes—
138C23×C3C2411yescharacteristic
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—216no—
142C22×D3—21no—
143C22×D3—22no—
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—22no—
147C22×D3—21no—
148C22×D3—22no—
149C22×D3—22no—
150C22×D3—22no—
151C22×D3—22no—
152C22×D3—48no—
153C22×D3—48no—
154C22×D3—48no—
155C22×D3—48no—
156C22×D3—48no—
157C22×D3—48no—
158C22×D3C2×D414yes—
159C22×D3C2×D412yes—
160C22×D3C2×D412yes—
161C22×D3C2×D412yescomplemented
162C22×D3#4112yes—
163C3×D4—28no—
164C2×C3×C4—21no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—22no—
172C2×C3×C4C2×D412yes—
173C2×C3×C4C2×D412yescomplemented
174#67—216no—
175#67—48no—
176C2×Dic3—21no—
177C2×Dic3—22no—
178C2×Dic3—22no—
179C2×Dic3—22no—
180C2×Dic3—22no—
181C2×Dic3C2×D412yes—
182C2×Dic3C2×D414yes—
183D12—28no—
184D12—28no—
185D12—48no—
186C4×D3—28no—
187C4×D3—44no—
188C24—31no—
189C24—32no—
190C24—31no—
191C24—64no—
192C24—64no—
193C24—64no—
194C24—64no—
195C24—64no—
196C24—122no—
197C24—122no—
198C24C22×D312yes—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—38no—
202C2×D4—34no—
203C2×D4—62no—
204C2×D4—64no—
205C2×D4—62no—
206C2×D4—64no—
207C2×D4—64no—
208C2×D4—64no—
209C2×D4—64no—
210C2×D4—632no—
211C2×D4—64no—
212C2×D4—64no—
213C2×D4—62no—
214C2×D4—122no—
215C2×D4—122no—
216C22×C4—31no—
217C22×C4—34no—
218C22×C4—32no—
219C22×C4—31no—
220C22×C4—64no—
221C22×C4—62no—
222C22×C4—64no—
223C22×C4C22×D312yes—
224C22×C4C22×D311yescharacteristic
225C22×C4C22×D311yescharacteristic
226C22×C4C22×D311yescharacteristic
227C4⋊C4—22no—
228C4⋊C4—64no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—62no—
232C22⋊C4—64no—
233C22⋊C4—64no—
234C22⋊C4—62no—
235C22⋊C4—64no—
236C22⋊C4—122no—
237C22⋊C4—122no—
238C42—62no—
239C22×C3—24no—
240C22×C3—24no—
241C22×C3—24no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3C22×D411yescharacteristic
247C22×C3C22×D412yes—
248C22×C3C22×D411yescharacteristic
249C22×C3C2×#4111yescharacteristic
250C22×C3H(5,𝔽2)12yes—
251C2×D3—22no—
252C2×D3—22no—
253C2×D3—24no—
254C2×D3—24no—
255C2×D3—24no—
256C2×D3—22no—
257C2×D3—24no—
258C2×D3—24no—
259C2×D3—24no—
260C2×D3—432no—
261C2×D3—41no—
262C2×D3—42no—
263C2×D3—41no—
264C2×D3—42no—
265C2×D3—42no—
266C2×D3—42no—
267C2×D3—42no—
268C2×D3#12114yescomplemented
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—41no—
272C3×C4—42no—
273Dic3—41no—
274Dic3—42no—
275Dic3#12014yescomplemented
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—32no—
283C23—32no—
284C23—32no—
285C23—34no—
286C23—32no—
287C23—616no—
288C23—62no—
289C23—62no—
290C23—61no—
291C23—62no—
292C23—62no—
293C23—62no—
294C23—62no—
295C23—61no—
296C23—62no—
297C23—62no—
298C23—62no—
299C23—62no—
300C23—128no—
301C23—128no—
302C23—128no—
303C23—128no—
304C23—128no—
305C23—128no—
306C23D3×D414yes—
307C23C23×D311yescharacteristic
308D4—28no—
309D4—68no—
310D4—68no—
311D4—616no—
312D4—128no—
313D4—128no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—32no—
323C2×C4—34no—
324C2×C4—68no—
325C2×C4—61no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—124no—
331C2×C4D3×D412yes—
332C2×C4D3×D412yescomplemented
333C2×C3—44no—
334C2×C3C2×#12011yescharacteristic
335C2×C3C2×#12111yescharacteristic
336C2×C3#53511yescharacteristic
337C2×C3D4212yes—
338C2×C3#54612yes—
339D3—24no—
340D3—42no—
341D3—44no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—34no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—64no—
351C22—62no—
352C22—62no—
353C22—64no—
354C22—64no—
355C22—64no—
356C22—62no—
357C22—64no—
358C22—64no—
359C22—1232no—
360C22—122no—
361C22—122no—
362C22—122no—
363C22—121no—
364C22—121no—
365C22—122no—
366C22—122no—
367C22C2×D3×D412yes—
368C22C2×D3×D411yescharacteristic
369C22C2×D3×D411yescharacteristic
370C22D3×#4111yescharacteristic
371C22#100912yes—
372C4—22no—
373C4—22no—
374C4—34no—
375C4—41no—
376C4—42no—
377C4—121no—
378C4—122no—
379C3#240811yescharacteristic, complemented, Sylow
380C2—44no—
381C2—64no—
382C2—124no—
383C2—122no—
384C2D3×#12011yescharacteristic
385C2D3×#12111yescharacteristic
386C2#560912yes—
387C2#565412yes—
388C2#567411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2408
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×#893C2semidirect
3. C2×#884C2semidirect
4. C2×#882C2semidirect
5. C3×#394C2semidirect
6. #5223C2semidirect
7. #4991C2semidirect
8. #4957C2semidirect
9. #4955C2semidirect
10. #4671C2semidirect
11. #4667C2semidirect
16. C22×#67C22semidirect
21. C22×D12C22semidirect
23. C22×C4×D3C22semidirect
24. C2×C3×C4⋊C4C22semidirect
25. C2×C3×C22⋊C4C22semidirect
26. C2×C3×C22⋊C4C22semidirect
27. C2×#217C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×(C2×D3)⋊C4C22semidirect
33. C2×(C2×D3)⋊C4C22semidirect
34. C2×Dic3⋊C4C22semidirect
35. C2×C4×Dic3C22semidirect
39. C3×#95C22semidirect
40. C3⋊#95C22semidirect
46. #394D3semidirect
74. C2×#67D4semidirect
86. C2×D12D4semidirect
90. C2×C4×D3D4semidirect
118. C2×C4⋊C4C2×D3semidirect
126. C2×C22⋊C4C2×D3semidirect
127. C2×C22⋊C4C2×D3semidirect
130. #95C2×D3semidirect
161. C22×D3C2×D4semidirect
173. C2×C3×C4C2×D4semidirect
268. C2×D3#121semidirect
275. Dic3#120semidirect
332. C2×C4D3×D4semidirect
379. C3#2408semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#8932
C2×#8842
C2×#8822
C3×#3942
#52232
#49912
#49572
#49552
#46712
#46672
#24083