Finite Group Databasebeta

← #77538

Subgroups — #77538

Order 384.

528 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 143 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×#888C211yesmaximal, characteristic, complemented
2C2×#878C212yesmaximal
3C2×#877C211yesmaximal, characteristic
4C3×#397C211yesmaximal, characteristic
5#4983C211yesmaximal, characteristic, complemented
6#4953C211yesmaximal, characteristic
7#4952C212yesmaximal
8#4661C211yesmaximal, characteristic, complemented
9#4657C212yesmaximal, complemented
10#4650C211yesmaximal, characteristic, complemented
11#4646C212yesmaximal, complemented
12#2621—31nomaximal, Sylow
13C23×Dic3C2211yescharacteristic
14C22×Dic6C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×#217C2211yescharacteristic
19C2×#217C2212yes—
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29#888—24no—
30#878—28no—
31#877—24no—
32C3×#95C2212yes—
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C2×C4⋊Q8—31no—
40C2×C22⋊Q8—32no—
41C2×#117—31no—
42#398—31no—
43#398—32no—
44#398—31no—
45#397—32no—
46#397—31no—
47#397D311yescharacteristic
48#391—32no—
49#382—31no—
50C24×C3C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×Dic3—22no—
56C22×Dic3—22no—
57C22×Dic3D414yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C2×Dic6—24no—
65C2×Dic6—24no—
66C2×Dic6—24no—
67C3×C4⋊C4—22no—
68C3×C22⋊C4—24no—
69C3×C22⋊C4—22no—
70#217—24no—
71#217—22no—
72C4⋊Dic3—22no—
73C4⋊Dic3—24no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76Dic3⋊C4—24no—
77Dic3⋊C4—24no—
78Dic3⋊C4—24no—
79C4×Dic3—22no—
80C4×Dic3D414yes—
81C22×Q8—32no—
82C23×C4—31no—
83C4⋊Q8—64no—
84C22⋊Q8—68no—
85#117—64no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D311yescharacteristic
97C2×C22⋊C4C2×D312yes—
98C2×C42—31no—
99C2×C42—31no—
100#95—32no—
101#95—31no—
102#95—32no—
103#95—32no—
104#95—32no—
105#95C2×D311yescharacteristic
106#95C2×D312yes—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3C2×D412yes—
112C23×C3C2411yescharacteristic
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4C2×D412yes—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—28no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3C2×D412yes—
136C2×Dic3C2×D412yes—
137C2×Dic3C2×D412yes—
138C2×Dic3C2×D412yes—
139C2×Dic3#4112yes—
140Dic6—48no—
141C24C22×D311yescharacteristic
142C2×Q8—64no—
143C2×Q8—64no—
144C2×Q8—64no—
145C22×C4—31no—
146C22×C4—34no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—62no—
153C22×C4—62no—
154C22×C4C22×D311yescharacteristic
155C22×C4C22×D312yes—
156C22×C4C22×D312yes—
157C22×C4C22×D311yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—62no—
160C4⋊C4—64no—
161C4⋊C4—64no—
162C4⋊C4—64no—
163C4⋊C4—64no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—62no—
169C22⋊C4—64no—
170C42—34no—
171C42—62no—
172C22×C3—22no—
173C22×C3—24no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—42no—
177C22×C3C22×D411yescharacteristic
178C22×C3C22×D411yescharacteristic
179C22×C3C2×#4111yescharacteristic
180C22×C32−1+412yes—
181C22×C32−1+412yes—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4—41no—
186Dic3—22no—
187Dic3—22no—
188Dic3—42no—
189Dic3—41no—
190Dic3—42no—
191Dic3#12114yes—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23D3×D412yes—
197C23C23×D311yescharacteristic
198Q8—128no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—68no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4D3×D412yes—
226C2×C3—42no—
227C2×C3C2×#12111yescharacteristic
228C2×C3#53611yescharacteristic
229C2×C3#53612yes—
230C2×C3#53611yescharacteristic
231C2×C3#54112yes—
232C22—24no—
233C22—22no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22C2×D3×D411yescharacteristic
238C22C2×D3×D411yescharacteristic
239C22D3×#4111yescharacteristic
240C22#101012yes—
241C22#101012yes—
242C4—22no—
243C4—34no—
244C4—42no—
245C4—41no—
246C4—42no—
247C4—62no—
248C4—62no—
249C4—121no—
250C4—122no—
251C4—122no—
252C3#262111yescharacteristic, complemented, Sylow
253C2—42no—
254C2D3×#12111yescharacteristic
255C2#564811yescharacteristic
256C2#564812yes—
257C2#567511yescharacteristic
258C2#570112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC24×C3C23
hypercenter#397D3
hypocenterC3#2621
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#888C2semidirect
5. #4983C2semidirect
8. #4661C2semidirect
9. #4657C2semidirect
10. #4650C2semidirect
11. #4646C2semidirect
252. C3#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#8882
C2×#8782
C2×#8772
C3×#3972
#49832
#49532
#49522
#46612
#46572
#46502
#46462
#26213