Finite Group Databasebeta

← #82423

Subgroups — #82423

Order 384.

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

normal subgroups only

This group has 562 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
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#876C211yesmaximal, characteristic, complemented
3#5223C211yesmaximal, characteristic
4#5222C211yesmaximal, characteristic, complemented
5#5213C212yesmaximal, complemented
6#5210C212yesmaximal, complemented
7#4993C211yesmaximal, characteristic, complemented
8#4985C211yesmaximal, characteristic, complemented
9#4961C212yesmaximal, complemented
10#4948C212yesmaximal, complemented
11#4936C211yesmaximal, characteristic, complemented
12#2636—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2212yes—
15C22×#67C2212yes—
16C3×C4×D4—24no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C42C2211yescharacteristic
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2212yes—
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32#876—24no—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×C4×D4D311yescharacteristic, complemented
40#400—31no—
41#399—31no—
42#396—32no—
43#394—31no—
44#394—32no—
45#392—31no—
46#386—32no—
47#385—32no—
48#382—31no—
49C24×C3C2312yes—
50C23×D3C2311yescharacteristic
51C2×C3×D4—24no—
52C2×C3×D4—22no—
53C22×C3×C4—24no—
54C22×C3×C4—42no—
55C22×C3×C4—42no—
56C22×C3×C4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—42no—
62C2×#67—42no—
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C3×C4⋊C4—22no—
69C3×C22⋊C4—24no—
70C3×C42D414yescomplemented
71#217—24no—
72#217—42no—
73#217—42no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—42no—
78C4⋊Dic3—42no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81Dic3⋊C4—22no—
82Dic3⋊C4—42no—
83C22×D4—32no—
84C22×D4C2×D311yescharacteristic
85C23×C4C2×D312yes—
86#126—64no—
87C4×D4—24no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C22⋊C4—32no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4C2×D312yes—
102C2×C42C2×D311yescharacteristic
103#95—32no—
104#95—31no—
105#95—31no—
106#95—32no—
107#95—32no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3#4114yes—
115C23×C3C2411yescharacteristic
116C22×D3—21no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—21no—
120C22×D3—21no—
121C3×D4—28no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—48no—
130C2×C3×C4C2×D412yes—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D412yes—
133#67—48no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C24—31no—
148C24C22×D312yes—
149C2×D4—24no—
150C2×D4—22no—
151C2×D4—64no—
152C2×D4—122no—
153C2×D4—122no—
154C22×C4—24no—
155C22×C4—32no—
156C22×C4—31no—
157C22×C4—32no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C22×D312yes—
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C22⋊C4—24no—
173C22⋊C4—64no—
174C22⋊C4—62no—
175C22⋊C4—62no—
176C22⋊C4—62no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C22⋊C4—122no—
180C42#6714yescomplemented
181C22×C3—24no—
182C22×C3—24no—
183C22×C3—24no—
184C22×C3—42no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3C22×D411yescharacteristic
189C22×C3C2×#4112yes—
190C22×C3H(5,𝔽2)12yes—
191C22×C3H(5,𝔽2)11yescharacteristic
192C22×C32−1+411yescharacteristic
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—41no—
196C2×D3—42no—
197C2×D3—42no—
198C3×C4—22no—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—42no—
202Dic3—41no—
203Dic3—41no—
204Dic3—41no—
205Dic3—42no—
206Dic3—42no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—62no—
214C23—62no—
215C23—61no—
216C23—61no—
217C23—61no—
218C23#23514yes—
219C23C23×D311yescharacteristic
220D4—28no—
221D4—128no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—48no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4C2×#6712yes—
244C2×C4C2×#6712yes—
245C2×C4C2×#6712yes—
246C2×C3—44no—
247C2×C3#52511yescharacteristic
248C2×C3#53511yescharacteristic
249C2×C3#53711yescharacteristic
250C2×C3#53812yes—
251C2×C3#53912yes—
252D3—81no—
253C22—24no—
254C22—24no—
255C22—24no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—121no—
261C22—121no—
262C22—122no—
263C22—122no—
264C22—121no—
265C22C2×#23512yes—
266C22#100412yes—
267C22#100911yescharacteristic
268C22#101011yescharacteristic
269C22C22×#6711yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—42no—
274C4—121no—
275C4—121no—
276C4—122no—
277C4—122no—
278C4—121no—
279C3#263611yescharacteristic, complemented, Sylow
280C2—44no—
281C2—241no—
282C2#556012yes—
283C2#556412yes—
284C2#579111yescharacteristic
285C2#580511yescharacteristic
286C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radical#5223C2
hypercenterC2×C4×D4D3
hypocenterC3#2636
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. C2×#876C2semidirect
4. #5222C2semidirect
5. #5213C2semidirect
6. #5210C2semidirect
7. #4993C2semidirect
8. #4985C2semidirect
9. #4961C2semidirect
10. #4948C2semidirect
11. #4936C2semidirect
39. C2×C4×D4D3semidirect
70. C3×C42D4semidirect
180. C42#67semidirect
279. C3#2636semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#8762
#52232
#52222
#52132
#52102
#49932
#49852
#49612
#49482
#49362
#26363