Finite Group Databasebeta

← #76665

Subgroups — #76665

Order 384.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 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×D4×Dic3C212yesmaximal, complemented
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4C3×#392C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5222C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4947C211yesmaximal, characteristic, complemented
9Dic3×C22⋊C4C212yesmaximal, complemented
10#4666C211yesmaximal, characteristic, complemented
11#4656C211yesmaximal, characteristic, complemented
12#2388—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×#67C2212yes—
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2212yes—
19C2×#237—28no—
20C2×#237—24no—
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×C22⋊C4C2212yes—
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×#217C2212yes—
26C2×#217C2212yes—
27C2×#217C2212yes—
28D4×Dic3—28no—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×C4⋊Dic3C2212yes—
32C2×Dic3⋊C4C2211yescharacteristic
33C2×C4×Dic3C2212yes—
34C2×C4×Dic3C2211yescharacteristic
35#67⋊C4—24no—
36C3×#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C22×#41—31no—
41C2×C4×D4—32no—
42C2×C4×D4—31no—
43#395—31no—
44#394—31no—
45#394—32no—
46#393—31no—
47#392—32no—
48#392D311yescharacteristic, complemented
49C4×C22⋊C4—32no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C2×C3×D4—24no—
55C2×C3×D4—24no—
56C2×C3×D4—24no—
57C2×C3×D4—22no—
58C2×C3×D4—24no—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C2×#67—24no—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—24no—
66C2×#67—24no—
67C2×#67—22no—
68C22×Dic3—24no—
69C22×Dic3—24no—
70C22×Dic3—22no—
71C22×Dic3—24no—
72C22×Dic3D418yescomplemented
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2311yescharacteristic
78#237—232no—
79C2×C4×D3—24no—
80C2×C4×D3D414yescomplemented
81C2×Dic6—22no—
82C2×Dic6—24no—
83C3×C22⋊C4—22no—
84C3×C22⋊C4—24no—
85#217—24no—
86#217—24no—
87#217—24no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—24no—
90C4⋊Dic3—24no—
91Dic3⋊C4—22no—
92C4×Dic3—22no—
93C4×Dic3—24no—
94C2×#41—68no—
95C2×#41—64no—
96C22×Q8—31no—
97C22×D4—31no—
98C22×D4—32no—
99C22×D4C2×D312yes—
100C22×D4C2×D311yescharacteristic
101C23×C4—32no—
102C23×C4—31no—
103C23×C4—31no—
104C4×D4—64no—
105C4×D4—68no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4C2×D311yescharacteristic
114C2×C22⋊C4C2×D312yes—
115C2×C42—31no—
116C2×C42—32no—
117#95—32no—
118#95—32no—
119#95—31no—
120#95C2×D311yescharacteristic
121C23×C3—22no—
122C23×C3—21no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—21no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—22no—
131C23×C3C2×D414yes—
132C23×C3#4112yescomplemented
133C23×C3C2411yescharacteristic
134C22×D3—22no—
135C22×D3—22no—
136C22×D3—21no—
137C22×D3—21no—
138C22×D3C2×D412yes—
139C3×D4—28no—
140C3×D4—216no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—21no—
145C2×C3×C4—22no—
146C2×C3×C4C2×D412yes—
147C2×C3×C4#4114yescomplemented
148#67—216no—
149#67—28no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—216no—
155C2×Dic3—22no—
156C2×Dic3—21no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—44no—
161C2×Dic3C2×D414yes—
162C2×Dic3C2×D414yes—
163C2×Dic3C2×D412yes—
164C2×Dic3#4112yescomplemented
165C4×D3—28no—
166Dic6—28no—
167C24—31no—
168C24C22×D312yes—
169C24C22×D311yescharacteristic
170#41—632no—
171C2×Q8—64no—
172C2×Q8—62no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—24no—
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—64no—
182C2×D4—64no—
183C2×D4—62no—
184C22×C4—31no—
185C22×C4—38no—
186C22×C4—31no—
187C22×C4—32no—
188C22×C4—34no—
189C22×C4—32no—
190C22×C4—31no—
191C22×C4—64no—
192C22×C4—64no—
193C22×C4—64no—
194C22×C4—62no—
195C22×C4—64no—
196C22×C4C22×D311yescharacteristic
197C22×C4C22×D311yescharacteristic
198C22×C4C22×D312yes—
199C4⋊C4—64no—
200C4⋊C4—62no—
201C22⋊C4—22no—
202C22⋊C4—24no—
203C22⋊C4—62no—
204C22⋊C4—64no—
205C22⋊C4—64no—
206C22⋊C4—64no—
207C22⋊C4—64no—
208C42—64no—
209C42—62no—
210C22×C3—24no—
211C22×C3—22no—
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—24no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—41no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3—42no—
222C22×C3C22×D411yescharacteristic
223C22×C3C22×D412yes—
224C22×C3C2×#4111yescharacteristic
225C22×C3C2×#4112yes—
226C22×C3C2×#4111yescharacteristic
227C2×D3—22no—
228C2×D3—22no—
229C2×D3—22no—
230C2×D3—42no—
231C2×D3—42no—
232C3×C4—22no—
233C3×C4—24no—
234C3×C4—41no—
235Dic3—24no—
236Dic3—22no—
237Dic3—41no—
238Dic3—42no—
239Dic3#12014yes—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—32no—
251C23—62no—
252C23—62no—
253C23—61no—
254C23—61no—
255C23D3×D414yes—
256C23#23712yescomplemented
257C23C23×D311yescharacteristic
258Q8—68no—
259D4—28no—
260D4—216no—
261D4—616no—
262D4—68no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—32no—
269C2×C4—34no—
270C2×C4—34no—
271C2×C4—32no—
272C2×C4—68no—
273C2×C4—616no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—124no—
284C2×C4D3×D412yes—
285C2×C4#23714yescomplemented
286C2×C3—44no—
287C2×C3—42no—
288C2×C3C2×#12011yescharacteristic
289C2×C3#52512yes—
290C2×C3#52511yescharacteristic
291C2×C3#53211yescharacteristic
292C2×C3#53212yes—
293D3—42no—
294C22—24no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—41no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—62no—
307C22—62no—
308C22—62no—
309C22—122no—
310C22—122no—
311C22C2×D3×D411yescharacteristic
312C22C2×D3×D412yes—
313C22C2×#23712yes—
314C22C2×#23711yescharacteristic
315C22D3×#4111yescharacteristic
316C4—24no—
317C4—22no—
318C4—34no—
319C4—41no—
320C4—64no—
321C4—62no—
322C4—121no—
323C4—122no—
324C3#238811yescharacteristic, complemented, Sylow
325C2—44no—
326C2—42no—
327C2—122no—
328C2#558612yes—
329C2D3×#12011yescharacteristic
330C2#559612yes—
331C2#560611yescharacteristic
332C2#571611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2388
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×D4×Dic3C2semidirect
3. C2×#67⋊C4C2semidirect
4. C3×#392C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #4961C2semidirect
8. #4947C2semidirect
9. Dic3×C22⋊C4C2semidirect
10. #4666C2semidirect
11. #4656C2semidirect
48. #392D3semidirect
72. C22×Dic3D4semidirect
80. C2×C4×D3D4semidirect
132. C23×C3#41semidirect
147. C2×C3×C4#41semidirect
164. C2×Dic3#41semidirect
256. C23#237semidirect
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×D4×Dic32
C2×#67⋊C42
C3×#3922
#52232
#52222
#49612
#49472
Dic3×C22⋊C42
#46662
#46562
#23883