Finite Group Databasebeta

← #76905

Subgroups — #76905

Order 384.

1088 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 161 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
2C2×#896C211yesmaximal, characteristic, complemented
3C2×#885C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4957C212yesmaximal, complemented
8#4955C212yesmaximal, complemented
9#4955C212yesmaximal, complemented
10#4672C211yesmaximal, characteristic, complemented
11#4666C211yesmaximal, characteristic, complemented
12#2435—31nomaximal, Sylow
13C24×D3C2212yes—
14C22×C3×D4C2211yescharacteristic
15C22×#67C2212yes—
16C2×D3×D4—28no—
17C2×D3×D4—24no—
18C22×D12C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic, complemented
21C2×C3×C4⋊C4C2211yescharacteristic, complemented
22C2×C3×C22⋊C4C2212yescomplemented
23C2×C3×C22⋊C4C2212yescomplemented
24C2×#217C2212yescomplemented
25C2×#217C2212yescomplemented
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yescomplemented
30C2×(C2×D3)⋊C4C2212yescomplemented
31C2×(C2×D3)⋊C4C2212yescomplemented
32C2×(C2×D3)⋊C4C2212yescomplemented
33C2×Dic3⋊C4C2211yescharacteristic, complemented
34C2×Dic3⋊C4C2211yescharacteristic, complemented
35C2×C4×Dic3C2212yescomplemented
36#896—24no—
37#885—28no—
38C3×#95C2211yescharacteristic, complemented
39C3⋊#95C2211yescharacteristic, complemented
40C23×D4—31no—
41C2×#124—32no—
42C2×C22⋊Q8—31no—
43#394—31no—
44#394—31no—
45#394—31no—
46#394D311yescharacteristic, complemented
47#390—32no—
48#379—32no—
49#379—32no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C23×D3—24no—
53C23×D3—24no—
54C23×D3—24no—
55C23×D3—24no—
56C23×D3—24no—
57C23×D3—42no—
58C23×D3—42no—
59C23×D3C2312yes—
60C23×D3C2311yescharacteristic
61C2×C3×D4—24no—
62C2×C3×D4—22no—
63C22×C3×C4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C2×#67—24no—
68C2×#67—24no—
69C2×#67D418yes—
70C22×Dic3C2312yes—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2311yescharacteristic
74D3×D4—232no—
75C2×D12—24no—
76C2×D12—22no—
77C2×C4×D3—24no—
78C2×C4×D3D414yes—
79C2×Dic6—24no—
80C2×Dic6—22no—
81C3×C4⋊C4—22no—
82C3×C22⋊C4—24no—
83C3×C22⋊C4—42no—
84#217—24no—
85#217—42no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—22no—
90(C2×D3)⋊C4—24no—
91(C2×D3)⋊C4—42no—
92(C2×D3)⋊C4—42no—
93Dic3⋊C4—22no—
94Dic3⋊C4—22no—
95C4×Dic3—24no—
96C25—32no—
97C22×Q8—31no—
98C22×D4—32no—
99C22×D4—31no—
100C22×D4—64no—
101C22×D4—68no—
102C22×D4C2×D311yescharacteristic
103C23×C4—31no—
104#124—68no—
105C22⋊Q8—64no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4C2×D311yescharacteristic, complemented
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—32no—
118C2×C22⋊C4C2×D312yescomplemented
119C2×C22⋊C4C2×D312yescomplemented
120C2×C42—32no—
121#95—31no—
122#95C2×D311yescharacteristic, complemented
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3C2×D414yes—
130C23×C3C2411yescharacteristic
131C22×D3—22no—
132C22×D3—21no—
133C22×D3—21no—
134C22×D3—216no—
135C22×D3—22no—
136C22×D3—22no—
137C22×D3—22no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—48no—
142C22×D3—48no—
143C22×D3—48no—
144C22×D3—48no—
145C22×D3—48no—
146C22×D3—48no—
147C22×D3C2×D414yes—
148C22×D3C2×D412yes—
149C22×D3#4114yes—
150C3×D4—28no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—21no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4C2×D412yes—
161#67—216no—
162C2×Dic3—22no—
163C2×Dic3—21no—
164C2×Dic3—21no—
165C2×Dic3—22no—
166C2×Dic3—22no—
167C2×Dic3—21no—
168C2×Dic3—22no—
169C2×Dic3—21no—
170C2×Dic3C2×D414yes—
171C2×Dic3C2×D412yes—
172D12—28no—
173C4×D3—28no—
174Dic6—44no—
175C24—31no—
176C24—32no—
177C24—64no—
178C24—64no—
179C24—64no—
180C24—64no—
181C24—64no—
182C24—122no—
183C24—122no—
184C24C22×D312yes—
185C2×Q8—62no—
186C2×Q8—64no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—38no—
190C2×D4—64no—
191C2×D4—64no—
192C2×D4—64no—
193C2×D4—632no—
194C2×D4—62no—
195C22×C4—31no—
196C22×C4—34no—
197C22×C4—32no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—64no—
201C22×C4C22×D312yes—
202C22×C4C22×D311yescharacteristic
203C22×C4C22×D311yescharacteristic
204C22×C4C22×D311yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—62no—
207C4⋊C4—62no—
208C22⋊C4—24no—
209C22⋊C4—42no—
210C22⋊C4—62no—
211C22⋊C4—62no—
212C22⋊C4—64no—
213C22⋊C4—64no—
214C22⋊C4—64no—
215C22⋊C4—64no—
216C22⋊C4—122no—
217C22⋊C4—122no—
218C22⋊C4—122no—
219C42—64no—
220C22×C3—24no—
221C22×C3—24no—
222C22×C3—24no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3C22×D411yescharacteristic
228C22×C3C22×D412yes—
229C22×C3C2×#4112yes—
230C22×C3H(5,𝔽2)11yescharacteristic
231C22×C3H(5,𝔽2)11yescharacteristic
232C2×D3—22no—
233C2×D3—22no—
234C2×D3—24no—
235C2×D3—24no—
236C2×D3—24no—
237C2×D3—24no—
238C2×D3—24no—
239C2×D3—432no—
240C2×D3—42no—
241C2×D3—42no—
242C2×D3—42no—
243C2×D3—42no—
244C2×D3#12414yescomplemented
245C3×C4—22no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—42no—
249Dic3—41no—
250Dic3—41no—
251Dic3—42no—
252Dic3#12014yes—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—32no—
260C23—34no—
261C23—34no—
262C23—616no—
263C23—62no—
264C23—61no—
265C23—61no—
266C23—62no—
267C23—62no—
268C23—62no—
269C23—62no—
270C23—62no—
271C23—62no—
272C23—128no—
273C23—128no—
274C23—128no—
275C23—128no—
276C23—128no—
277C23—128no—
278C23D3×D414yes—
279C23C23×D311yescharacteristic
280Q8—124no—
281D4—28no—
282D4—68no—
283D4—616no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—34no—
294C2×C4—32no—
295C2×C4—68no—
296C2×C4—62no—
297C2×C4—62no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—61no—
304C2×C4D3×D412yes—
305C2×C3—44no—
306C2×C3C2×#12011yescharacteristic
307C2×C3C2×#12411yescharacteristic
308C2×C3#53511yescharacteristic
309C2×C3#54612yes—
310C2×C3#54612yes—
311D3—24no—
312D3—44no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—34no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—64no—
322C22—62no—
323C22—62no—
324C22—64no—
325C22—64no—
326C22—64no—
327C22—64no—
328C22—1232no—
329C22—122no—
330C22—122no—
331C22—122no—
332C22—122no—
333C22C2×D3×D412yes—
334C22C2×D3×D411yescharacteristic
335C22#100411yescharacteristic
336C22D3×#4112yes—
337C22#100911yescharacteristic
338C4—22no—
339C4—34no—
340C4—41no—
341C4—41no—
342C4—42no—
343C4—122no—
344C4—121no—
345C4—121no—
346C3#243511yescharacteristic, complemented, Sylow
347C2—44no—
348C2—64no—
349C2—124no—
350C2D3×#12011yescharacteristic
351C2#560711yescharacteristic
352C2#561012yes—
353C2#565412yes—
354C2D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC22×D3×D4C2
hypercenter#394D3
hypocenterC3#2435
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#896C2semidirect
3. C2×#885C2semidirect
4. C3×#394C2semidirect
5. #5222C2semidirect
6. #4991C2semidirect
7. #4957C2semidirect
8. #4955C2semidirect
9. #4955C2semidirect
10. #4672C2semidirect
11. #4666C2semidirect
20. C22×Dic6C22semidirect
21. C2×C3×C4⋊C4C22semidirect
22. C2×C3×C22⋊C4C22semidirect
23. C2×C3×C22⋊C4C22semidirect
24. C2×#217C22semidirect
25. C2×#217C22semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×(C2×D3)⋊C4C22semidirect
33. C2×Dic3⋊C4C22semidirect
34. C2×Dic3⋊C4C22semidirect
35. C2×C4×Dic3C22semidirect
38. C3×#95C22semidirect
39. C3⋊#95C22semidirect
46. #394D3semidirect
108. C2×C4⋊C4C2×D3semidirect
118. C2×C22⋊C4C2×D3semidirect
119. C2×C22⋊C4C2×D3semidirect
122. #95C2×D3semidirect
244. C2×D3#124semidirect
346. C3#2435semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#8962
C2×#8852
C3×#3942
#52222
#49912
#49572
#49552
#46722
#46662
#24353