Finite Group Databasebeta

← #76759

Subgroups — #76759

Order 384.

724 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#938C212yesmaximal, complemented
2C2×#938C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4C3×#392C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#5220C212yesmaximal, complemented
7#4956C212yesmaximal, complemented
8#4953C211yesmaximal, characteristic, complemented
9#4951C212yesmaximal, complemented
10D3×#95C211yesmaximal, characteristic, complemented
11#4661C211yesmaximal, characteristic, complemented
12#2543—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C22×#67C2212yes—
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C22×C4×D3C2212yes—
19C22×C4×D3C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26#938—28no—
27#938—24no—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×C4⋊Dic3C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33#67⋊C4—24no—
34C3×#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×#121—31no—
40C2×#121—32no—
41C2×C4×D4—31no—
42#398—31no—
43#397—31no—
44#394—31no—
45#392D311yescharacteristic, complemented
46#388—32no—
47#386—32no—
48#386—32no—
49C2×#95—31no—
50C24×C3C2312yes—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C2×C3×D4—24no—
54C2×C3×D4—22no—
55C2×C3×D4—24no—
56C2×C3×D4—42no—
57C2×C3×D4—42no—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C2×#67—22no—
62C2×#67—22no—
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—24no—
66C2×#67D414yescomplemented
67C22×Dic3—22no—
68C22×Dic3—42no—
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2312yes—
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2311yescharacteristic
74C2×C4×D3—24no—
75C2×C4×D3—24no—
76C2×C4×D3—42no—
77C2×C4×D3—42no—
78C2×C4×D3D414yescomplemented
79C3×C22⋊C4—22no—
80C3×C22⋊C4—24no—
81#217—24no—
82#217—24no—
83#217—24no—
84#217—42no—
85(C2×D3)⋊C4—22no—
86C4⋊Dic3—22no—
87C4⋊Dic3—24no—
88Dic3⋊C4—22no—
89C4×Dic3—22no—
90C22×D4—31no—
91C22×D4—32no—
92C22×D4C2×D312yes—
93C22×D4C2×D311yescharacteristic
94C23×C4—31no—
95C23×C4—31no—
96C23×C4—32no—
97#121—64no—
98#121—68no—
99C4×D4—64no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—31no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊C4C2×D312yes—
110C2×C42—31no—
111#95—32no—
112#95—31no—
113#95—32no—
114#95—32no—
115#95C2×D311yescharacteristic
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3C2×D412yes—
128C23×C3C2411yescharacteristic
129C22×D3—22no—
130C22×D3—22no—
131C22×D3C2×D412yes—
132C22×D3C2×D412yes—
133C22×D3#4112yes—
134C3×D4—48no—
135C3×D4—44no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4C2×D412yes—
142C2×C3×C4#4114yes—
143#67—28no—
144#67—48no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—44no—
157C2×Dic3C2×D412yes—
158C2×Dic3C2×D412yes—
159C4×D3—44no—
160C4×D3—48no—
161C24—31no—
162C24C22×D312yes—
163C24C22×D311yescharacteristic
164C2×D4—24no—
165C2×D4—22no—
166C2×D4—24no—
167C2×D4—34no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—64no—
173C2×D4—64no—
174C2×D4—64no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—34no—
178C22×C4—32no—
179C22×C4—31no—
180C22×C4—32no—
181C22×C4—64no—
182C22×C4—62no—
183C22×C4—64no—
184C22×C4—122no—
185C22×C4—122no—
186C22×C4—122no—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D312yes—
190C4⋊C4—64no—
191C4⋊C4—62no—
192C4⋊C4—62no—
193C22⋊C4—22no—
194C22⋊C4—24no—
195C22⋊C4—62no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—122no—
200C42—62no—
201C22×C3—22no—
202C22×C3—24no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3C22×D411yescharacteristic
214C22×C3C22×D411yescharacteristic
215C22×C3C2×#4112yes—
216C22×C3C2×#4111yescharacteristic
217C22×C3H(5,𝔽2)12yes—
218C2×D3—22no—
219C2×D3—22no—
220C2×D3—24no—
221C2×D3—24no—
222C2×D3#12114yes—
223C3×C4—22no—
224C3×C4—24no—
225C3×C4—41no—
226Dic3—22no—
227Dic3—22no—
228Dic3—42no—
229Dic3—41no—
230Dic3—42no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—32no—
243C23—32no—
244C23—32no—
245C23—62no—
246C23—62no—
247C23D3×D412yes—
248C23C23×D311yescharacteristic
249D4—48no—
250D4—44no—
251D4—68no—
252D4—128no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—32no—
259C2×C4—32no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—124no—
272C2×C4—128no—
273C2×C4—124no—
274C2×C4D3×D412yes—
275C2×C4#23714yes—
276C2×C3—42no—
277C2×C3—82no—
278C2×C3C2×#12111yescharacteristic
279C2×C3#53211yescharacteristic
280C2×C3#53411yescharacteristic
281C2×C3#54612yes—
282C2×C3#55312yes—
283D3—24no—
284C22—24no—
285C22—22no—
286C22—34no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—62no—
298C22—62no—
299C22—64no—
300C22—64no—
301C22C2×D3×D411yescharacteristic
302C22C2×D3×D411yescharacteristic
303C22C2×#23712yes—
304C22#100412yes—
305C22D3×#4111yescharacteristic
306C4—22no—
307C4—24no—
308C4—41no—
309C4—62no—
310C4—62no—
311C4—121no—
312C4—122no—
313C4—122no—
314C3#254311yescharacteristic, complemented, Sylow
315C2—42no—
316C2—64no—
317C2—82no—
318C2#558911yescharacteristic
319C2#559112yes—
320C2D3×#12111yescharacteristic
321C2#561412yes—
322C2#571611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2543
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#938C2semidirect
2. C2×#938C2semidirect
3. C2×#67⋊C4C2semidirect
4. C3×#392C2semidirect
5. #5222C2semidirect
6. #5220C2semidirect
7. #4956C2semidirect
8. #4953C2semidirect
9. #4951C2semidirect
10. D3×#95C2semidirect
11. #4661C2semidirect
45. #392D3semidirect
66. C2×#67D4semidirect
78. C2×C4×D3D4semidirect
314. C3#2543semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×#67⋊C42
C3×#3922
#52222
#52202
#49562
#49532
#49512
D3×#952
#46612
#25433