Finite Group Databasebeta

← #77120

Subgroups — #77120

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×#938C211yesmaximal, characteristic, complemented
2C2×#893C211yesmaximal, characteristic, complemented
3C2×#884C212yesmaximal, complemented
4C3×#394C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#4981C211yesmaximal, characteristic, complemented
7#4956C212yesmaximal, complemented
8#4956C212yesmaximal, complemented
9#4945C212yesmaximal, complemented
10#4674C211yesmaximal, characteristic, complemented
11D3×#95C211yesmaximal, characteristic, complemented
12#2543—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2212yes—
16C22×D12C2211yescharacteristic
17C22×C4×D3C2212yes—
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×C3×C22⋊C4C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25#938—24no—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#893—24no—
33#884—28no—
34C3×#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#121—31no—
40C2×#121—32no—
41C2×C4×D4—31no—
42#398—31no—
43#397—31no—
44#394D311yescharacteristic, complemented
45#392—31no—
46#388—32no—
47#386—32no—
48#386—32no—
49C2×#95—31no—
50C24×C3C2312yes—
51C23×D3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—24no—
64C2×#67—42no—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C2×D12—22no—
70C2×D12—22no—
71C2×D12D414yescomplemented
72C2×C4×D3—24no—
73C2×C4×D3—22no—
74C2×C4×D3—24no—
75C2×C4×D3—42no—
76C2×C4×D3—42no—
77C2×C4×D3—42no—
78C2×C4×D3D414yescomplemented
79C3×C4⋊C4—22no—
80C3×C22⋊C4—24no—
81C3×C22⋊C4—42no—
82#217—24no—
83#217—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—22no—
87C4⋊Dic3—22no—
88Dic3⋊C4—24no—
89C4×Dic3—22no—
90C22×D4—32no—
91C22×D4—32no—
92C22×D4—31no—
93C22×D4C2×D311yescharacteristic
94C23×C4—31no—
95C23×C4—32no—
96C23×C4—31no—
97#121—68no—
98#121—64no—
99C4×D4—64no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D312yes—
109C2×C22⋊C4C2×D312yes—
110C2×C42—31no—
111#95—32no—
112#95—31no—
113#95—32no—
114#95—32no—
115#95C2×D311yescharacteristic
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3C2411yescharacteristic
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3C2×D412yes—
131C22×D3C2×D412yes—
132C22×D3C2×D412yes—
133C22×D3#4112yes—
134C3×D4—44no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4C2×D412yes—
144C2×C3×C4C2×D412yes—
145#67—48no—
146#67—48no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3C2×D412yes—
156C2×Dic3#4114yes—
157D12—28no—
158C4×D3—44no—
159C4×D3—48no—
160C4×D3—44no—
161C24—31no—
162C24—31no—
163C24C22×D312yes—
164C2×D4—22no—
165C2×D4—34no—
166C2×D4—42no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—64no—
172C2×D4—64no—
173C2×D4—64no—
174C2×D4—122no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—34no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—62no—
181C22×C4—64no—
182C22×C4—64no—
183C22×C4—122no—
184C22×C4—122no—
185C22×C4—122no—
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—64no—
192C4⋊C4—62no—
193C22⋊C4—24no—
194C22⋊C4—42no—
195C22⋊C4—62no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—62no—
200C42—62no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3C22×D411yescharacteristic
209C22×C3C22×D411yescharacteristic
210C22×C3C2×#4111yescharacteristic
211C22×C3C2×#4112yes—
212C22×C3H(5,𝔽2)12yes—
213C2×D3—22no—
214C2×D3—22no—
215C2×D3—24no—
216C2×D3—24no—
217C2×D3—22no—
218C2×D3—24no—
219C2×D3—41no—
220C2×D3—41no—
221C2×D3—42no—
222C2×D3#12114yes—
223C3×C4—22no—
224C3×C4—22no—
225C3×C4—41no—
226C3×C4—42no—
227Dic3—22no—
228Dic3—24no—
229Dic3—42no—
230Dic3—41no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—32no—
239C23—32no—
240C23—32no—
241C23—32no—
242C23—62no—
243C23—62no—
244C23—61no—
245C23—62no—
246C23—61no—
247C23—62no—
248C23C23×D311yescharacteristic
249D4—44no—
250D4—68no—
251D4—128no—
252D4—128no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—32no—
262C2×C4—34no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
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×C4D3×D412yes—
276C2×C3—82no—
277C2×C3C2×#12111yescharacteristic
278C2×C3#53211yescharacteristic
279C2×C3#53411yescharacteristic
280C2×C3#54612yes—
281C2×C3#55312yes—
282D3—24no—
283D3—42no—
284C22—34no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—64no—
293C22—62no—
294C22—62no—
295C22—64no—
296C22—62no—
297C22—64no—
298C22—122no—
299C22—121no—
300C22—121no—
301C22C2×D3×D411yescharacteristic
302C22C2×D3×D411yescharacteristic
303C22#100412yes—
304C22D3×#4111yescharacteristic
305C22D3×#4112yes—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—42no—
310C4—62no—
311C4—64no—
312C4—121no—
313C4—122no—
314C3#254311yescharacteristic, complemented, Sylow
315C2—64no—
316C2—82no—
317C2—122no—
318C2#558911yescharacteristic
319C2D3×#12111yescharacteristic
320C2#561212yes—
321C2#566112yes—
322C2#566811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2543
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#938C2semidirect
2. C2×#893C2semidirect
3. C2×#884C2semidirect
4. C3×#394C2semidirect
5. #5223C2semidirect
6. #4981C2semidirect
7. #4956C2semidirect
8. #4956C2semidirect
9. #4945C2semidirect
10. #4674C2semidirect
11. D3×#95C2semidirect
44. #394D3semidirect
71. C2×D12D4semidirect
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×#8932
C2×#8842
C3×#3942
#52232
#49812
#49562
#49452
#46742
D3×#952
#25433