Finite Group Databasebeta

← #78109

Subgroups — #78109

Order 384.

526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 133 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×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#883C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4#4989C212yesmaximal, complemented
5#4985C211yesmaximal, characteristic, complemented
6#4982C212yesmaximal, complemented
7#4956C212yesmaximal
8#4952C212yesmaximal, complemented
9#4949C211yesmaximal, characteristic, complemented
10#4664C211yesmaximal, characteristic, complemented
11#4659C211yesmaximal, characteristic, complemented
12#2623—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29D3×C4⋊C4—24no—
30#883—24no—
31C3×#95C2211yescharacteristic
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#123—31no—
39C22×C4⋊C4—31no—
40#400—31no—
41#400—31no—
42#399—31no—
43#399D311yescharacteristic, complemented
44#398—32no—
45#397—32no—
46#386—32no—
47#385—32no—
48#381—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C2×#67—22no—
56C2×#67—24no—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C2×C4×D3—24no—
63C2×C4×D3—24no—
64C2×C4×D3—42no—
65C2×C4×D3—42no—
66C2×C4×D3D414yes—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—42no—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—42no—
71#217—22no—
72#217—42no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—24no—
75C4⋊Dic3—22no—
76C4⋊Dic3—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—24no—
79Dic3⋊C4—42no—
80C22×D4—31no—
81C23×C4—31no—
82C23×C4—32no—
83#123—64no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—64no—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C4⋊C4C2×D312yes—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D311yescharacteristic
97C2×C22⋊C4C2×D312yes—
98#95—32no—
99#95—31no—
100#95—32no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95C2×D311yescharacteristic
105C23×C3—21no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—21no—
109C23×C3—21no—
110C23×C3C2411yescharacteristic
111C22×D3—22no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3C2×D412yes—
115C22×D3#4114yes—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4C2×D412yes—
126#67—44no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3C2×D412yes—
140C4×D3—28no—
141C4×D3—48no—
142C24—31no—
143C24C22×D311yescharacteristic
144C2×D4—62no—
145C2×D4—64no—
146C22×C4—31no—
147C22×C4—34no—
148C22×C4—31no—
149C22×C4—32no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—64no—
153C22×C4—64no—
154C22×C4—122no—
155C22×C4—122no—
156C22×C4C22×D312yes—
157C22×C4C22×D312yes—
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D311yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—64no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—122no—
167C22⋊C4—22no—
168C22⋊C4—42no—
169C22⋊C4—62no—
170C22⋊C4—64no—
171C22⋊C4—62no—
172C22⋊C4—122no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3C22×D411yescharacteristic
179C22×C3C2×#4112yes—
180C22×C3H(5,𝔽2)12yes—
181C22×C32−1+411yescharacteristic
182C22×C32−1+411yescharacteristic
183C2×D3—22no—
184C2×D3—22no—
185C2×D3—24no—
186C2×D3—24no—
187C2×D3#12314yes—
188C3×C4—22no—
189C3×C4—42no—
190C3×C4—41no—
191C3×C4—42no—
192Dic3—22no—
193Dic3—41no—
194Dic3—42no—
195Dic3—41no—
196Dic3—42no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23—21no—
202C23—32no—
203C23—34no—
204C23—62no—
205C23—61no—
206C23—61no—
207C23C23×D311yescharacteristic
208D4—124no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—32no—
219C2×C4—68no—
220C2×C4—62no—
221C2×C4—61no—
222C2×C4—62no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—61no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—61no—
232C2×C4—128no—
233C2×C4D3×D412yes—
234C2×C3—81no—
235C2×C3C2×#12311yescharacteristic
236C2×C3#53511yescharacteristic
237C2×C3#53611yescharacteristic
238C2×C3#53912yes—
239C2×C3#53912yes—
240D3—24no—
241C22—34no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—64no—
248C22—64no—
249C22—62no—
250C22—62no—
251C22C2×D3×D411yescharacteristic
252C22#100412yes—
253C22#100711yescharacteristic
254C22D3×#4112yes—
255C22#101011yescharacteristic
256C4—22no—
257C4—42no—
258C4—42no—
259C4—41no—
260C4—62no—
261C4—122no—
262C4—121no—
263C4—122no—
264C4—121no—
265C3#262311yescharacteristic, complemented, Sylow
266C2—64no—
267C2—81no—
268C2#562811yescharacteristic
269C2#563812yes—
270C2D3×#12311yescharacteristic
271C2#566012yes—
272C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC22×#67C22
hypercenter#399D3
hypocenterC3#2623
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊C4C2semidirect
3. C3×#399C2semidirect
4. #4989C2semidirect
5. #4985C2semidirect
6. #4982C2semidirect
8. #4952C2semidirect
9. #4949C2semidirect
10. #4664C2semidirect
11. #4659C2semidirect
43. #399D3semidirect
265. C3#2623semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊C42
C2×#8832
C3×#3992
#49892
#49852
#49822
#49562
#49522
#49492
#46642
#46592
#26233