Finite Group Databasebeta

← #77682

Subgroups — #77682

Order 384.

724 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 147 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×#895C211yesmaximal, characteristic, complemented
2C2×#884C212yesmaximal, complemented
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4C3×#397C211yesmaximal, characteristic, complemented
5#4983C211yesmaximal, characteristic, complemented
6#4960C211yesmaximal, characteristic, complemented
7#4958C212yesmaximal, complemented
8#4672C211yesmaximal, characteristic, complemented
9#4669C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11D3×#95C211yesmaximal, characteristic, complemented
12#2543—31nomaximal, Sylow
13C22×#67C2212yes—
14C22×#67C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×D12C2212yes—
17C22×D12C2211yescharacteristic
18C22×C4×D3C2212yes—
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31#895—24no—
32#884—28no—
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#397D311yescharacteristic, complemented
44#394—31no—
45#392—31no—
46#388—32no—
47#386—32no—
48#386—32no—
49C2×#95—31no—
50C24×C3C2311yescharacteristic
51C23×D3C2312yes—
52C23×D3C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—42no—
62C2×#67D414yescomplemented
63C22×Dic3—22no—
64C22×Dic3—42no—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C2×D12—24no—
70C2×D12—24no—
71C2×D12—24no—
72C2×D12—22no—
73C2×D12—42no—
74C2×C4×D3—24no—
75C2×C4×D3—24no—
76C2×C4×D3—42no—
77C2×C4×D3—42no—
78C2×C4×D3D414yescomplemented
79C3×C4⋊C4—22no—
80C3×C22⋊C4—24no—
81C3×C22⋊C4—22no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—42no—
87Dic3⋊C4—22no—
88Dic3⋊C4—24no—
89C4×Dic3—22no—
90C22×D4—31no—
91C22×D4—32no—
92C22×D4—32no—
93C22×D4—31no—
94C23×C4—31no—
95C23×C4—31no—
96C23×C4—32no—
97#121—68no—
98#121—64no—
99C4×D4—64no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C22⋊C4—31no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
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#95C2×D311yescharacteristic
115#95C2×D312yes—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3C2×D412yes—
121C23×C3C2411yescharacteristic
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3C2×D412yes—
132C22×D3C2×D412yes—
133C22×D3#4112yes—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4C2×D412yes—
144#67—28no—
145#67—48no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—44no—
154C2×Dic3C2×D412yes—
155C2×Dic3C2×D412yes—
156C2×Dic3#4114yes—
157D12—44no—
158D12—48no—
159C4×D3—44no—
160C4×D3—48no—
161C24—31no—
162C24—32no—
163C24C22×D311yescharacteristic
164C2×D4—34no—
165C2×D4—64no—
166C2×D4—64no—
167C2×D4—64no—
168C2×D4—64no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—64no—
172C2×D4—62no—
173C2×D4—122no—
174C2×D4—122no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—34no—
179C22×C4—31no—
180C22×C4—64no—
181C22×C4—64no—
182C22×C4—62no—
183C22×C4—122no—
184C22×C4—122no—
185C22×C4—122no—
186C22×C4C22×D312yes—
187C22×C4C22×D312yes—
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—64no—
192C4⋊C4—62no—
193C22⋊C4—24no—
194C22⋊C4—22no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—62no—
199C22⋊C4—122no—
200C42—62no—
201C22×C3—22no—
202C22×C3—24no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3C22×D411yescharacteristic
207C22×C3C22×D411yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4112yes—
210C22×C3H(5,𝔽2)12yes—
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—24no—
214C2×D3—24no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3#12114yes—
223C3×C4—22no—
224C3×C4—42no—
225C3×C4—41no—
226C3×C4—42no—
227Dic3—22no—
228Dic3—22no—
229Dic3—24no—
230Dic3—41no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—32no—
236C23—32no—
237C23—32no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23—62no—
242C23—62no—
243C23—62no—
244C23—62no—
245C23—62no—
246C23—62no—
247C23D3×D412yes—
248C23C23×D311yescharacteristic
249D4—68no—
250D4—124no—
251D4—128no—
252D4—128no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—34no—
265C2×C4—61no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—61no—
271C2×C4—62no—
272C2×C4—124no—
273C2×C4—128no—
274C2×C4—124no—
275C2×C4D3×D412yes—
276C2×C3—42no—
277C2×C3C2×#12111yescharacteristic
278C2×C3#53211yescharacteristic
279C2×C3#53411yescharacteristic
280C2×C3#54612yes—
281C2×C3#55312yes—
282D3—24no—
283D3—82no—
284C22—24no—
285C22—22no—
286C22—34no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—62no—
291C22—64no—
292C22—62no—
293C22—64no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—122no—
299C22—122no—
300C22—122no—
301C22C2×D3×D411yescharacteristic
302C22C2×D3×D411yescharacteristic
303C22D3×#4112yes—
304C22D3×#4111yescharacteristic
305C22#100912yes—
306C4—22no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—62no—
311C4—64no—
312C4—62no—
313C4—121no—
314C3#254311yescharacteristic, complemented, Sylow
315C2—42no—
316C2—64no—
317C2—242no—
318C2D3×#12111yescharacteristic
319C2#565311yescharacteristic
320C2#565412yes—
321C2#566811yescharacteristic
322C2#570912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
hypercenter#397D3
hypocenterC3#2543
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#895C2semidirect
2. C2×#884C2semidirect
3. C2×#67⋊C4C2semidirect
4. C3×#397C2semidirect
5. #4983C2semidirect
6. #4960C2semidirect
7. #4958C2semidirect
8. #4672C2semidirect
9. #4669C2semidirect
10. #4665C2semidirect
11. D3×#95C2semidirect
43. #397D3semidirect
62. 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×#8952
C2×#8842
C2×#67⋊C42
C3×#3972
#49832
#49602
#49582
#46722
#46692
#46652
D3×#952
#25433