Finite Group Databasebeta

← #68911

Subgroups — #68911

Order 384.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#890C211yesmaximal, characteristic
2C2×#816C211yesmaximal, characteristic
3#4973C211yesmaximal, characteristic
4C2×#810C211yesmaximal, characteristic
5C3×#340C211yesmaximal, characteristic
6#4550C211yesmaximal, characteristic
7#4529C211yesmaximal, characteristic
8#2026—31nomaximal, Sylow
9C22×Dic6C2211yescharacteristic
10C22×C3×C8C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C4⋊Dic3—21no—
14C2×C4⋊Dic3C2211yescharacteristic
15C2×Dic3⋊C4—21no—
16C2×C4×Dic3C2211yescharacteristic
17C22×C3⋊C8C2211yescharacteristic
18#890—22no—
19#890—22no—
20C3⋊#95—21no—
21C3⋊#95—21no—
22#816—22no—
23#810—22no—
24C2×#125—31no—
25C2×#103—31no—
26C2×#103—31no—
27#391—31no—
28#340—31no—
29#340D311yescharacteristic
30#336—31no—
31C22×C3×C4—21no—
32C22×C3×C4—21no—
33C22×C3×C4C2311yescharacteristic
34C22×Dic3—21no—
35C22×Dic3—21no—
36C22×Dic3D411yescharacteristic
37C22×Dic3D411yescharacteristic
38C2×Dic6—21no—
39C2×Dic6—21no—
40C2×Dic6—21no—
41C2×Dic6—21no—
42C2×Dic6—21no—
43C2×Dic6Q812yes—
44C2×C3×C8—21no—
45C2×C3×C8—21no—
46C2×C3×C8—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4D412yes—
51C4⋊Dic3—24no—
52C4⋊Dic3—21no—
53C4⋊Dic3D412yes—
54Dic3⋊C4—24no—
55C4×Dic3—21no—
56C4×Dic3—21no—
57C2×C3⋊C8—21no—
58C2×C3⋊C8—21no—
59C2×C3⋊C8—21no—
60C22×Q8—31no—
61C22×C8—31no—
62C22×C8C2×D311yescharacteristic
63#125—62no—
64#125—62no—
65C2×C4⋊C4—31no—
66C2×C4⋊C4—61no—
67C2×C4⋊C4—61no—
68C2×C4⋊C4C2×D311yescharacteristic
69C2×C4⋊C4C2×D311yescharacteristic
70C2×C42—31no—
71#103—62no—
72#103—62no—
73#95—61no—
74#95—61no—
75C23×C3C2×D411yescharacteristic
76C2×C3×C4—22no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4—41no—
81C2×C3×C4—41no—
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C2×C3×C4C2×Q811yescharacteristic
85C2×C3×C4#4111yescharacteristic
86C2×C3×C4#4111yescharacteristic
87C2×C3×C4#4111yescharacteristic
88C2×Dic3—21no—
89C2×Dic3—21no—
90C2×Dic3—22no—
91C2×Dic3—22no—
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3—21no—
96C2×Dic3—41no—
97C2×Dic3—41no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100Dic6—22no—
101Dic6—22no—
102Dic6—41no—
103Dic6—41no—
104C3×C8—41no—
105C3⋊C8—41no—
106C2×Q8—32no—
107C2×Q8—61no—
108C2×Q8—61no—
109C2×Q8—61no—
110C2×Q8—61no—
111C2×Q8—61no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—31no—
115C22×C4—31no—
116C22×C4—61no—
117C22×C4—61no—
118C22×C4C22×D311yescharacteristic
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—61no—
123C2×C8—61no—
124C2×C8—61no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—32no—
129C4⋊C4—64no—
130C4⋊C4—64no—
131C4⋊C4—61no—
132C4⋊C4D1212yes—
133C42—61no—
134C42—61no—
135C22×C3#12011yescharacteristic
136C22×C3C22⋊Q811yescharacteristic
137C22×C3#12411yescharacteristic
138C22×C3#13511yescharacteristic
139C22×C3#13511yescharacteristic
140C22×C3#13711yescharacteristic
141C22×C3#13711yescharacteristic
142C3×C4—42no—
143C3×C4—81no—
144C3×C4C22⋊Q811yescharacteristic
145C3×C4C22⋊Q811yescharacteristic
146C3×C4#12411yescharacteristic
147C3×C4#12411yescharacteristic
148Dic3—42no—
149Dic3—41no—
150Dic3—41no—
151Dic3—42no—
152C23D3×D411yescharacteristic
153Q8—62no—
154Q8—62no—
155Q8—121no—
156Q8—121no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—62no—
166C2×C4—62no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—121no—
172C2×C4—121no—
173C2×C4—121no—
174C2×C4—121no—
175C2×C4C2×D1211yescharacteristic
176C2×C4#23511yescharacteristic
177C2×C4#23511yescharacteristic
178C2×C4D3×D411yescharacteristic
179C2×C4#23711yescharacteristic
180C2×C4D3×Q811yescharacteristic
181C8—41no—
182C8—121no—
183C2×C3#39311yescharacteristic
184C2×C3#45211yescharacteristic
185C2×C3#45211yescharacteristic
186C2×C3#47911yescharacteristic
187C2×C3#47911yescharacteristic
188C2×C3#48811yescharacteristic
189C2×C3#49111yescharacteristic
190C22#88211yescharacteristic
191C22#88511yescharacteristic
192C22#89611yescharacteristic
193C22#91211yescharacteristic
194C22#91511yescharacteristic
195C22#91611yescharacteristic
196C22#91811yescharacteristic
197C4—42no—
198C4—81no—
199C4—122no—
200C4—121no—
201C4—121no—
202C4—122no—
203C4#87511yescharacteristic
204C4#88511yescharacteristic
205C4#89611yescharacteristic
206C4#89711yescharacteristic
207C3#202611yescharacteristic, complemented, Sylow
208C2#467311yescharacteristic
209C2#476811yescharacteristic
210C2#478311yescharacteristic
211C2#479711yescharacteristic
212C2#480811yescharacteristic
213C2#485711yescharacteristic
214C2#487711yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#340C2
socle, prime radicalC23×C3C2×D4
hypercenter#340D3
hypocenterC3#2026
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
207. C3#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×#8162
#49732
C2×#8102
C3×#3402
#45502
#45292
#20263