Finite Group Databasebeta

← #74599

Subgroups — #74599

Order 384.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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×D12C211yesmaximal, characteristic
2C2×#894C212yesmaximal, complemented
3C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#385C211yesmaximal, characteristic, complemented
5#5210C211yesmaximal, characteristic, complemented
6#4991C212yesmaximal, complemented
7#4955C211yesmaximal, characteristic, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4955C211yesmaximal, characteristic, complemented
10#4672C212yesmaximal, complemented
11#4665C212yesmaximal, complemented
12#2405—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C24×D3C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×D12—24no—
17C22×D12—28no—
18C22×D12C2212yes—
19C22×D12C2212yes—
20C22×C4×D3C2211yescharacteristic, complemented
21C22×C4×D3C2211yescharacteristic, complemented
22C2×C3×C4⋊C4C2212yescomplemented
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×#217C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yescomplemented
28C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D3)⋊C4C2212yescomplemented
30C2×(C2×D3)⋊C4C2212yescomplemented
31C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
32C2×(C2×D3)⋊C4C2212yescomplemented
33C2×(C2×D3)⋊C4C2212yescomplemented
34C2×(C2×D3)⋊C4C2212yescomplemented
35C2×Dic3⋊C4C2212yescomplemented
36#894—28no—
37D3×C22⋊C4—24no—
38C3×#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C23×D4—31no—
41C2×#123—32no—
42C22×C22⋊C4—31no—
43#394—32no—
44#394—32no—
45#386—32no—
46#385—31no—
47#385D311yescharacteristic, complemented
48#379—31no—
49#379—31no—
50#379—31no—
51C24×C3C2311yescharacteristic
52C23×D3—22no—
53C23×D3—22no—
54C23×D3—22no—
55C23×D3—24no—
56C23×D3—24no—
57C23×D3—22no—
58C23×D3—22no—
59C23×D3—42no—
60C23×D3—42no—
61C23×D3D414yes—
62C23×D3C2311yescharacteristic
63C23×D3C2312yes—
64C23×D3C2311yescharacteristic
65C22×C3×C4—22no—
66C22×C3×C4—24no—
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2312yes—
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2312yes—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C2×D12—232no—
75C2×D12—24no—
76C2×D12—24no—
77C2×D12—24no—
78C2×D12—24no—
79C2×D12—24no—
80C2×D12D418yes—
81C2×C4×D3—22no—
82C2×C4×D3—22no—
83C2×C4×D3—24no—
84C2×C4×D3—42no—
85C3×C4⋊C4—24no—
86C3×C22⋊C4—22no—
87C3×C22⋊C4—22no—
88#217—22no—
89#217—22no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—24no—
93(C2×D3)⋊C4—24no—
94(C2×D3)⋊C4—24no—
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97(C2×D3)⋊C4—42no—
98Dic3⋊C4—24no—
99C25—31no—
100C25—31no—
101C22×D4—32no—
102C22×D4—32no—
103C22×D4—68no—
104C22×D4—64no—
105C23×C4—31no—
106C23×C4—31no—
107C23×C4C2×D311yescharacteristic
108#123—68no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4C2×D312yescomplemented
111C2×C22⋊C4—31no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—32no—
118C2×C22⋊C4—32no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4—64no—
122C2×C22⋊C4C2×D311yescharacteristic
123C2×C22⋊C4C2×D311yescharacteristic
124#95—32no—
125#95C2×D312yescomplemented
126C23×C3—22no—
127C23×C3—22no—
128C23×C3C2×D412yes—
129C23×C3#4112yes—
130C23×C3#4112yes—
131C23×C3C2411yescharacteristic
132C22×D3—21no—
133C22×D3—21no—
134C22×D3—22no—
135C22×D3—28no—
136C22×D3—28no—
137C22×D3—22no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—21no—
141C22×D3—22no—
142C22×D3—21no—
143C22×D3—22no—
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—22no—
147C22×D3—28no—
148C22×D3—44no—
149C22×D3—48no—
150C22×D3—48no—
151C22×D3—48no—
152C22×D3—48no—
153C22×D3—44no—
154C22×D3—44no—
155C22×D3C2×D412yes—
156C22×D3C2×D414yes—
157C22×D3C2×D414yes—
158C22×D3C2×D412yes—
159C2×C3×C4—28no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4C2×D414yes—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3—21no—
173C2×Dic3—21no—
174C2×Dic3—22no—
175C2×Dic3—22no—
176C2×Dic3—22no—
177D12—216no—
178D12—216no—
179C4×D3—44no—
180C4×D3—44no—
181C24—31no—
182C24—32no—
183C24—31no—
184C24—34no—
185C24—64no—
186C24—64no—
187C24—62no—
188C24—62no—
189C24—62no—
190C24—62no—
191C24—62no—
192C24—122no—
193C24—122no—
194C24C22×D311yescharacteristic
195C2×D4—38no—
196C2×D4—64no—
197C2×D4—64no—
198C2×D4—64no—
199C2×D4—64no—
200C2×D4—632no—
201C2×D4—64no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—31no—
205C22×C4—32no—
206C22×C4—31no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4—64no—
210C22×C4—122no—
211C22×C4C22×D311yescharacteristic
212C22×C4C22×D312yes—
213C22×C4C22×D311yescharacteristic
214C22×C4C22×D312yes—
215C4⋊C4—24no—
216C4⋊C4—64no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—62no—
220C22⋊C4—62no—
221C22⋊C4—64no—
222C22⋊C4—64no—
223C22⋊C4—64no—
224C22⋊C4—62no—
225C22⋊C4—62no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C22⋊C4—122no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3#12314yescomplemented
234C22×C3C22×D411yescharacteristic
235C22×C3C22×D412yes—
236C22×C3C2×#4111yescharacteristic
237C22×C3C2×#4111yescharacteristic
238C22×C3H(5,𝔽2)12yes—
239C2×D3—22no—
240C2×D3—22no—
241C2×D3—24no—
242C2×D3—24no—
243C2×D3—22no—
244C2×D3—24no—
245C2×D3—22no—
246C2×D3—22no—
247C2×D3—24no—
248C2×D3—24no—
249C2×D3—416no—
250C2×D3—416no—
251C2×D3—42no—
252C2×D3—42no—
253C2×D3—42no—
254C2×D3—42no—
255C2×D3—42no—
256C2×D3—42no—
257C2×D3#12014yes—
258C3×C4—24no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—42no—
262Dic3—41no—
263Dic3—41no—
264Dic3—42no—
265C23—22no—
266C23—22no—
267C23—34no—
268C23—32no—
269C23—34no—
270C23—32no—
271C23—68no—
272C23—68no—
273C23—61no—
274C23—61no—
275C23—62no—
276C23—68no—
277C23—61no—
278C23—62no—
279C23—62no—
280C23—62no—
281C23—62no—
282C23—61no—
283C23—62no—
284C23—62no—
285C23—62no—
286C23—62no—
287C23—124no—
288C23—128no—
289C23—128no—
290C23—128no—
291C23—128no—
292C23—124no—
293C23—124no—
294C23#23512yes—
295C23D3×D412yes—
296C23#23912yes—
297C23C23×D311yescharacteristic
298D4—616no—
299D4—616no—
300C2×C4—28no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—62no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4—124no—
318C2×C4—124no—
319C2×C4D3×D414yes—
320C2×C3—24no—
321C2×C3C2×#12011yescharacteristic
322C2×C3C2×#12311yescharacteristic
323C2×C3#53411yescharacteristic
324C2×C3#54612yes—
325C2×C3#54612yes—
326D3—24no—
327D3—42no—
328D3—44no—
329C22—22no—
330C22—24no—
331C22—22no—
332C22—24no—
333C22—34no—
334C22—62no—
335C22—62no—
336C22—64no—
337C22—64no—
338C22—62no—
339C22—62no—
340C22—64no—
341C22—64no—
342C22—62no—
343C22—64no—
344C22—1216no—
345C22—1216no—
346C22—122no—
347C22—122no—
348C22—122no—
349C22—122no—
350C22—122no—
351C22—122no—
352C22#89414yescomplemented
353C22C2×#23511yescharacteristic
354C22C2×D3×D411yescharacteristic
355C22C2×D3×D412yes—
356C22C2×#23911yescharacteristic
357C22#100912yes—
358C4—24no—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—121no—
363C4—121no—
364C4—122no—
365C3#240511yescharacteristic, complemented, Sylow
366C2—24no—
367C2—64no—
368C2—122no—
369C2—124no—
370C2C2×#89411yescharacteristic
371C2#555012yes—
372C2D3×#12011yescharacteristic
373C2#563012yes—
374C2#565311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC23×D12C2
hypercenter#385D3
hypocenterC3#2405
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#894C2semidirect
3. C2×D3×C22⋊C4C2semidirect
4. C3×#385C2semidirect
5. #5210C2semidirect
6. #4991C2semidirect
7. #4955C2semidirect
8. #4955C2semidirect
9. #4955C2semidirect
10. #4672C2semidirect
11. #4665C2semidirect
20. C22×C4×D3C22semidirect
21. C22×C4×D3C22semidirect
22. C2×C3×C4⋊C4C22semidirect
27. C2×(C2×D3)⋊C4C22semidirect
28. C2×(C2×D3)⋊C4C22semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×(C2×D3)⋊C4C22semidirect
33. C2×(C2×D3)⋊C4C22semidirect
34. C2×(C2×D3)⋊C4C22semidirect
35. C2×Dic3⋊C4C22semidirect
38. C3×#95C22semidirect
39. C3⋊#95C22semidirect
47. #385D3semidirect
110. C2×C4⋊C4C2×D3semidirect
125. #95C2×D3semidirect
233. C22×C3#123semidirect
352. C22#894semidirect
365. C3#2405semidirect

Maximal subgroups

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

subgroupindex
C23×D122
C2×#8942
C2×D3×C22⋊C42
C3×#3852
#52102
#49912
#49552
#46722
#46652
#24053