Finite Group Databasebeta

← #74620

Subgroups — #74620

Order 384.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 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×Dic3⋊C4C211yesmaximal, characteristic, complemented
2C2×#886C211yesmaximal, characteristic
3C3×#385C211yesmaximal, characteristic, complemented
4#5213C211yesmaximal, characteristic
5#4993C212yesmaximal, complemented
6#4982C212yesmaximal, complemented
7#4958C211yesmaximal, characteristic
8#4946C211yesmaximal, characteristic, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4660C212yesmaximal, complemented
12#2623—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)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2211yescharacteristic
27C2×Dic3⋊C4—24no—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31#886—24no—
32C3×#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#123—31no—
39C22×C4⋊C4—31no—
40#400—32no—
41#399—32no—
42#398—32no—
43#397—32no—
44#386—31no—
45#386—31no—
46#385—31no—
47#385D311yescharacteristic, complemented
48#381—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4—22no—
52C22×C3×C4—42no—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—22no—
60C22×Dic3—22no—
61C22×Dic3—22no—
62C22×Dic3—22no—
63C22×Dic3—42no—
64C22×Dic3D414yes—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C3×C4⋊C4—42no—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—22no—
73#217—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78C4⋊Dic3—22no—
79C4⋊Dic3—22no—
80Dic3⋊C4—24no—
81Dic3⋊C4—24no—
82Dic3⋊C4—42no—
83C22×D4—31no—
84C23×C4—31no—
85C23×C4—31no—
86C23×C4C2×D311yescharacteristic
87#123—64no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—64no—
94C2×C4⋊C4C2×D312yes—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4C2×D311yescharacteristic
101C2×C22⋊C4C2×D311yescharacteristic
102#95—32no—
103#95—32no—
104#95—32no—
105#95—32no—
106#95—32no—
107#95C2×D312yes—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3C2×D412yes—
111C23×C3#4112yes—
112C23×C3#4112yes—
113C23×C3C2411yescharacteristic
114C22×D3—21no—
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—22no—
118C22×D3—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—44no—
130#67—44no—
131C2×Dic3—28no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—44no—
146C2×Dic3C2×D412yes—
147C2×Dic3C2×D412yes—
148C24—31no—
149C24C22×D311yescharacteristic
150C2×D4—62no—
151C2×D4—62no—
152C2×D4—62no—
153C22×C4—22no—
154C22×C4—34no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—32no—
158C22×C4—31no—
159C22×C4—32no—
160C22×C4—42no—
161C22×C4—62no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4—122no—
165C22×C4C22×D312yes—
166C22×C4C22×D311yescharacteristic
167C22×C4C22×D312yes—
168C22×C4C22×D311yescharacteristic
169C4⋊C4—42no—
170C4⋊C4—64no—
171C4⋊C4—64no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—24no—
185C22×C3—24no—
186C22×C3#12314yes—
187C22×C3C22×D411yescharacteristic
188C22×C3C2×#4111yescharacteristic
189C22×C3C2×#4111yescharacteristic
190C22×C3H(5,𝔽2)12yes—
191C22×C32−1+412yes—
192C2×D3—41no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—42no—
196C2×D3—42no—
197C3×C4—41no—
198C3×C4—42no—
199C3×C4—41no—
200C3×C4—42no—
201Dic3—22no—
202Dic3—22no—
203Dic3—41no—
204Dic3—42no—
205Dic3—42no—
206C23—22no—
207C23—22no—
208C23—61no—
209C23—61no—
210C23—62no—
211C23—61no—
212C23—62no—
213C23#23512yes—
214C23D3×D412yes—
215C23#23912yes—
216C23C23×D311yescharacteristic
217D4—124no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—32no—
229C2×C4—32no—
230C2×C4—44no—
231C2×C4—68no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—124no—
246C2×C3—24no—
247C2×C3C2×#12311yescharacteristic
248C2×C3#53511yescharacteristic
249C2×C3#53611yescharacteristic
250C2×C3#53912yes—
251C2×C3#53912yes—
252D3—81no—
253C22—24no—
254C22—22no—
255C22—24no—
256C22—22no—
257C22—122no—
258C22—121no—
259C22—121no—
260C22—122no—
261C22—121no—
262C22#89414yes—
263C22C2×#23511yescharacteristic
264C22C2×D3×D411yescharacteristic
265C22#100412yes—
266C22C2×#23911yescharacteristic
267C22#101012yes—
268C4—41no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—62no—
273C4—62no—
274C4—122no—
275C4—121no—
276C4—122no—
277C3#262311yescharacteristic, complemented, Sylow
278C2—24no—
279C2—241no—
280C2C2×#89411yescharacteristic
281C2#556412yes—
282C2#559211yescharacteristic
283C2#564512yes—
284C2#564811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC22×#67C22
hypercenter#385D3
hypocenterC3#2623
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×Dic3⋊C4C2semidirect
3. C3×#385C2semidirect
5. #4993C2semidirect
6. #4982C2semidirect
8. #4946C2semidirect
9. #4943C2semidirect
10. #4675C2semidirect
11. #4660C2semidirect
47. #385D3semidirect
277. C3#2623semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3⋊C42
C2×#8862
C3×#3852
#52132
#49932
#49822
#49582
#49462
#49432
#46752
#46602
#26233