Finite Group Databasebeta

← #76889

Subgroups — #76889

Order 384.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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×#893C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#4979C211yesmaximal, characteristic, complemented
7#4960C212yesmaximal, complemented
8#4959C212yesmaximal, complemented
9Dic3×C22⋊C4C212yesmaximal, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4666C211yesmaximal, characteristic, complemented
12#2388—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2212yes—
16C23×Dic3C2212yes—
17C2×#237—28no—
18C2×#237—24no—
19C22×D12C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2212yes—
25C2×C3×C22⋊C4C2212yes—
26C2×#217C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34#893—24no—
35#67⋊C4—28no—
36C3×#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×C4×D4—31no—
42C2×C4×D4—32no—
43#395—31no—
44#394—32no—
45#394D311yescharacteristic, complemented
46#393—31no—
47#392—31no—
48#392—32no—
49C4×C22⋊C4—32no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C23×D3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C2×C3×D4—24no—
55C2×C3×D4—22no—
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—24no—
66C22×Dic3—24no—
67C22×Dic3—24no—
68C22×Dic3D418yescomplemented
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2312yes—
71C22×Dic3C2312yes—
72C22×Dic3C2311yescharacteristic
73#237—232no—
74C2×D12—22no—
75C2×D12—22no—
76C2×D12—22no—
77C2×C4×D3—24no—
78C2×C4×D3—24no—
79C2×C4×D3—22no—
80C2×C4×D3D414yescomplemented
81C2×Dic6—22no—
82C2×Dic6—24no—
83C3×C4⋊C4—22no—
84C3×C22⋊C4—24no—
85C3×C22⋊C4—24no—
86#217—24no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—24no—
90(C2×D3)⋊C4—24no—
91Dic3⋊C4—24no—
92C4×Dic3—24no—
93C4×Dic3—22no—
94C2×#41—68no—
95C2×#41—64no—
96C22×Q8—31no—
97C22×D4—32no—
98C22×D4—32no—
99C22×D4—31no—
100C22×D4C2×D311yescharacteristic
101C23×C4—32no—
102C23×C4—31no—
103C23×C4—31no—
104C4×D4—68no—
105C4×D4—64no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4C2×D312yes—
114C2×C22⋊C4C2×D312yes—
115C2×C42—32no—
116C2×C42—31no—
117#95—31no—
118#95—32no—
119#95—32no—
120#95C2×D311yescharacteristic
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3C2×D414yes—
128C23×C3C2411yescharacteristic
129C22×D3—21no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—21no—
134C22×D3—21no—
135C22×D3—21no—
136C22×D3—22no—
137C22×D3C2×D412yes—
138C22×D3#4112yescomplemented
139C3×D4—28no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4C2×D412yes—
149C2×C3×C4#4112yescomplemented
150#67—216no—
151#67—216no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—216no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3C2×D414yes—
160C2×Dic3C2×D414yes—
161C2×Dic3C2×D412yes—
162C2×Dic3#4114yescomplemented
163D12—28no—
164C4×D3—28no—
165C4×D3—44no—
166Dic6—28no—
167C24—31no—
168C24—31no—
169C24C22×D312yes—
170#41—632no—
171C2×Q8—64no—
172C2×Q8—62no—
173C2×D4—24no—
174C2×D4—22no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—64no—
180C2×D4—64no—
181C2×D4—64no—
182C2×D4—64no—
183C2×D4—62no—
184C22×C4—31no—
185C22×C4—38no—
186C22×C4—32no—
187C22×C4—32no—
188C22×C4—34no—
189C22×C4—31no—
190C22×C4—64no—
191C22×C4—64no—
192C22×C4—62no—
193C22×C4—64no—
194C22×C4—64no—
195C22×C4C22×D312yes—
196C22×C4C22×D311yescharacteristic
197C22×C4C22×D311yescharacteristic
198C22×C4C22×D311yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—64no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—64no—
204C22⋊C4—64no—
205C22⋊C4—62no—
206C22⋊C4—64no—
207C22⋊C4—62no—
208C42—62no—
209C42—64no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3C22×D411yescharacteristic
218C22×C3C22×D412yes—
219C22×C3C2×#4111yescharacteristic
220C22×C3C2×#4111yescharacteristic
221C22×C3C2×#4112yes—
222C2×D3—22no—
223C2×D3—22no—
224C2×D3—22no—
225C2×D3—24no—
226C2×D3—22no—
227C2×D3—41no—
228C2×D3—42no—
229C2×D3—41no—
230C2×D3—42no—
231C2×D3—42no—
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—41no—
235C3×C4—42no—
236Dic3—24no—
237Dic3—24no—
238Dic3—41no—
239Dic3#12014yes—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—32no—
247C23—32no—
248C23—62no—
249C23—61no—
250C23—62no—
251C23—61no—
252C23—61no—
253C23—62no—
254C23—62no—
255C23—61no—
256C23D3×D414yes—
257C23C23×D311yescharacteristic
258Q8—68no—
259D4—28no—
260D4—68no—
261D4—616no—
262D4—616no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—34no—
272C2×C4—32no—
273C2×C4—34no—
274C2×C4—34no—
275C2×C4—68no—
276C2×C4—616no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—62no—
283C2×C4—124no—
284C2×C4D3×D412yes—
285C2×C4#23712yescomplemented
286C2×C3—44no—
287C2×C3C2×#12011yescharacteristic
288C2×C3#52512yes—
289C2×C3#52511yescharacteristic
290C2×C3#53211yescharacteristic
291C2×C3#53212yes—
292D3—42no—
293D3—42no—
294C22—24no—
295C22—24no—
296C22—24no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—64no—
302C22—62no—
303C22—62no—
304C22—62no—
305C22—62no—
306C22—122no—
307C22—121no—
308C22—121no—
309C22—122no—
310C22—122no—
311C22C2×D3×D412yes—
312C22C2×D3×D411yescharacteristic
313C22C2×#23711yescharacteristic
314C22D3×#4111yescharacteristic
315C22D3×#4112yes—
316C4—22no—
317C4—22no—
318C4—34no—
319C4—41no—
320C4—42no—
321C4—64no—
322C4—64no—
323C4—121no—
324C3#238811yescharacteristic, complemented, Sylow
325C2—44no—
326C2—122no—
327C2—122no—
328C2D3×#12011yescharacteristic
329C2#559612yes—
330C2#560611yescharacteristic
331C2#565612yes—
332C2#566811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2388
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#893C2semidirect
3. C2×#67⋊C4C2semidirect
4. C3×#394C2semidirect
5. #5223C2semidirect
6. #4979C2semidirect
7. #4960C2semidirect
8. #4959C2semidirect
9. Dic3×C22⋊C4C2semidirect
10. #4673C2semidirect
11. #4666C2semidirect
45. #394D3semidirect
68. C22×Dic3D4semidirect
80. C2×C4×D3D4semidirect
138. C22×D3#41semidirect
149. C2×C3×C4#41semidirect
162. C2×Dic3#41semidirect
285. C2×C4#237semidirect
324. C3#2388semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#8932
C2×#67⋊C42
C3×#3942
#52232
#49792
#49602
#49592
Dic3×C22⋊C42
#46732
#46662
#23883