Finite Group Databasebeta

← #74407

Subgroups — #74407

Order 384.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×C4⋊Dic3C211yesmaximal, characteristic, complemented
2C2×#886C211yesmaximal, characteristic
3C2×#886C211yesmaximal, characteristic
4C2×#877C211yesmaximal, characteristic, complemented
5C3×#385C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic
7#4994C212yesmaximal, complemented
8#4984C212yesmaximal, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4670C212yesmaximal, complemented
11#4661C212yesmaximal, complemented
12#2413—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×#217C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×C4⋊Dic3—24no—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2212yes—
28C2×C4⋊Dic3C2212yes—
29C2×C4⋊Dic3C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33#886—24no—
34#886—24no—
35#877—24no—
36C3×#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#123—31no—
41C2×#123—31no—
42C2×#117—31no—
43C22×C4⋊C4—31no—
44#398—32no—
45#396—32no—
46#389—32no—
47#388—32no—
48#386—31no—
49#385D311yescharacteristic, complemented
50#381—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C22×C3×C4—22no—
54C22×C3×C4—24no—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C22×C3×C4C2312yes—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67—22no—
62C22×Dic3—22no—
63C22×Dic3—22no—
64C22×Dic3—22no—
65C22×Dic3—22no—
66C22×Dic3—22no—
67C22×Dic3D414yes—
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2312yes—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C3×C4⋊C4—24no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4D414yes—
76#217—22no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3—24no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—24no—
85C4⋊Dic3—24no—
86Dic3⋊C4—24no—
87C4×Dic3—24no—
88C22×D4—31no—
89C23×C4—31no—
90C23×C4—31no—
91C23×C4C2×D311yescharacteristic
92#123—64no—
93#123—64no—
94#117—64no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—64no—
102C2×C4⋊C4C2×D312yes—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊C4C2×D311yescharacteristic
110C2×C42—32no—
111#95—32no—
112#95—32no—
113#95—32no—
114#95C2×D312yes—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3C2×D412yes—
118C23×C3C2×D412yes—
119C23×C3#4112yes—
120C23×C3C2411yescharacteristic
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—22no—
124C22×D3—21no—
125C22×D3—22no—
126C2×C3×C4—28no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D414yes—
136C2×C3×C4#4114yes—
137#67—44no—
138C2×Dic3—28no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—44no—
153C2×Dic3C2×D412yes—
154C2×Dic3C2×D412yes—
155C24—31no—
156C24C22×D311yescharacteristic
157C2×D4—62no—
158C2×D4—62no—
159C2×D4—62no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—34no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—31no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D312yes—
176C22×C4C22×D312yes—
177C4⋊C4—24no—
178C4⋊C4—64no—
179C4⋊C4—64no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—64no—
183C4⋊C4—64no—
184C22⋊C4—22no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—64no—
189C22⋊C4—64no—
190C22⋊C4D1214yes—
191C42—64no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3#12114yes—
197C22×C3C22×D411yescharacteristic
198C22×C3C22×D411yescharacteristic
199C22×C3C2×#4112yes—
200C22×C3C2×#4111yescharacteristic
201C22×C32−1+412yes—
202C2×D3—41no—
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—42no—
206C2×D3—42no—
207C3×C4—24no—
208C3×C4—24no—
209C3×C4—41no—
210C3×C4—41no—
211Dic3—22no—
212Dic3—22no—
213Dic3—42no—
214Dic3—41no—
215Dic3—42no—
216C23—22no—
217C23—22no—
218C23—61no—
219C23—61no—
220C23—62no—
221C23—61no—
222C23—62no—
223C23C2×D1212yes—
224C23D3×D412yes—
225C23#23912yes—
226C23C23×D311yescharacteristic
227D4—124no—
228C2×C4—28no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—68no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—61no—
247C2×C4—62no—
248C2×C4—61no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—61no—
252C2×C4—62no—
253C2×C4—124no—
254C2×C4C2×D1214yes—
255C2×C4#23714yes—
256C2×C3—24no—
257C2×C3C2×#12111yescharacteristic
258C2×C3#52511yescharacteristic
259C2×C3#53611yescharacteristic
260C2×C3#54712yes—
261C2×C3#55212yes—
262D3—81no—
263C22—24no—
264C22—22no—
265C22—24no—
266C22—22no—
267C22—122no—
268C22—121no—
269C22—121no—
270C22—122no—
271C22—121no—
272C22#89514yes—
273C22C22×D1211yescharacteristic
274C22C2×D3×D411yescharacteristic
275C22C2×#23712yes—
276C22C2×#23911yescharacteristic
277C22#101012yes—
278C4—24no—
279C4—24no—
280C4—41no—
281C4—41no—
282C4—62no—
283C4—62no—
284C4—122no—
285C4—121no—
286C4—122no—
287C3#241311yescharacteristic, complemented, Sylow
288C2—24no—
289C2—241no—
290C2C2×#89511yescharacteristic
291C2#555512yes—
292C2#558611yescharacteristic
293C2#564212yes—
294C2#564811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC22×#67C22
hypercenter#385D3
hypocenterC3#2413
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic3C2semidirect
4. C2×#877C2semidirect
5. C3×#385C2semidirect
7. #4994C2semidirect
8. #4984C2semidirect
9. #4943C2semidirect
10. #4670C2semidirect
11. #4661C2semidirect
49. #385D3semidirect
287. C3#2413semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic32
C2×#8862
C2×#8772
C3×#3852
#52132
#49942
#49842
#49432
#46702
#46612
#24133