Finite Group Databasebeta

← #81056

Subgroups — #81056

Order 384.

340 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#5722C211yesmaximal, characteristic, complemented
2#5543C211yesmaximal, characteristic
3C3×#476C211yesmaximal, characteristic
4#5078C211yesmaximal, characteristic, complemented
5C4×#214C211yesmaximal, characteristic
6#4857C212yesmaximal, complemented
7#4856C212yesmaximal, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4759C212yesmaximal
11#4757C212yesmaximal
12#3385—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×D4C2211yescharacteristic
15#944—22no—
16D4×Dic3C2212yes—
17C2×#214C2211yescharacteristic
18C2×C4⋊Dic3—21no—
19#924—21no—
20#890C2212yes—
21#888—21no—
22#888C2212yes—
23#887—21no—
24#887C2212yes—
25#878—22no—
26#869C2211yescharacteristic
27C4×Dic6C2211yescharacteristic
28C4×Dic6C2211yescharacteristic
29C3×#106C2212yes—
30C3×C4⋊C8C2211yescharacteristic
31C3×#102C2212yes—
32#832C2211yescharacteristic
33#818C2212yes—
34#816C2212yes—
35#815C2212yes—
36Dic3⋊C8C2212yes—
37#810C2211yescharacteristic
38#808C2211yescharacteristic
39#807C2212yes—
40#807C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42Q82—31no—
43#554—31no—
44#501—32no—
45#500—31no—
46#479—32no—
47#477—31no—
48#476D311yescharacteristic
49#474—32no—
50#446—31no—
51#440—32no—
52C4×SD16—31no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×Dic3—22no—
57C2×Dic6—21no—
58C2×Dic6—22no—
59C2×Dic6D411yescharacteristic
60C2×Dic6C2311yescharacteristic
61C2×C3×C8C2312yes—
62C3×C4⋊C4—21no—
63C3×C4⋊C4C2312yes—
64C3×C4⋊C4C2311yescharacteristic
65C3×C22⋊C4—21no—
66C3×C42C2311yescharacteristic
67#217—22no—
68#214Q814yescomplemented
69C4⋊Dic3—21no—
70C4⋊Dic3—22no—
71C4⋊Dic3D411yescharacteristic
72C4⋊Dic3C2312yes—
73C4⋊Dic3C2311yescharacteristic
74Dic3⋊C4—21no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4D412yes—
79C4×Dic3—21no—
80C4×Dic3—22no—
81C4×Dic3C2312yes—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84C2×SD16—31no—
85C4⋊Q8—32no—
86C4⋊Q8—31no—
87C4⋊Q8—61no—
88C4⋊Q8—62no—
89C4⋊Q8C2×D311yescharacteristic
90#125—32no—
91C22⋊Q8—62no—
92C22⋊Q8—61no—
93C4×Q8—31no—
94C4×Q8—32no—
95C4×Q8—31no—
96C4×Q8—61no—
97C4×D4—32no—
98C4×D4C2×D311yescharacteristic
99C2×C4⋊C4—61no—
100#107—32no—
101#107—32no—
102#107—32no—
103#106—31no—
104#106C2×D312yes—
105C4⋊C8—32no—
106C4⋊C8C2×D311yescharacteristic
107#103—32no—
108#103—31no—
109#102—31no—
110#102C2×D312yes—
111#97—32no—
112C4×C8—31no—
113C23×C3—21no—
114C3×Q8—81no—
115C3×D4—21no—
116C3×D4C2×Q812yes—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—42no—
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2411yescharacteristic
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—82no—
127C2×Dic3C2×D412yes—
128C2×Dic3C2×D412yes—
129Dic6—22no—
130Dic6—21no—
131Dic6—82no—
132Dic6—81no—
133Dic6C2×Q812yes—
134C3×C8—22no—
135C3⋊C8—21no—
136C3⋊C8C2×Q812yes—
137C2×Q8—21no—
138C2×Q8—31no—
139C2×Q8—31no—
140C2×Q8—62no—
141C2×Q8—61no—
142C2×D4C22×D311yescharacteristic
143C22×C4—21no—
144C22×C4—62no—
145SD16—34no—
146C2×C8—31no—
147C2×C8—31no—
148C2×C8C22×D312yes—
149C4⋊C4—21no—
150C4⋊C4—31no—
151C4⋊C4—32no—
152C4⋊C4—31no—
153C4⋊C4—32no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4C22×D312yes—
161C4⋊C4C22×D311yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—62no—
164C42—32no—
165C42—62no—
166C42—61no—
167C42C22×D311yescharacteristic
168C22×C3—22no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3C22×D411yescharacteristic
172C3×C4—21no—
173C3×C4—41no—
174C3×C4—42no—
175C3×C4#13712yes—
176C3×C4C22×Q811yescharacteristic
177C3×C42−1+411yescharacteristic
178Dic3—22no—
179Dic3—22no—
180Dic3—22no—
181Dic3—41no—
182Dic3—42no—
183C23—21no—
184Q8—32no—
185Q8—62no—
186Q8—61no—
187Q8—81no—
188Q8—241no—
189Q8—242no—
190D4—21no—
191D4D3×Q812yes—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—32no—
195C2×C4—32no—
196C2×C4—42no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—62no—
200C2×C4—242no—
201C2×C4D3×D411yescharacteristic
202C2×C4D3×D411yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—22no—
205C8—32no—
206C8—61no—
207C2×C3—42no—
208C2×C3D4×Q811yescharacteristic
209C2×C3C2×#13711yescharacteristic
210C2×C3#57611yescharacteristic
211C22—22no—
212C22—41no—
213C22—41no—
214C22C2×D3×D411yescharacteristic
215C4—21no—
216C4—41no—
217C4—42no—
218C4—62no—
219C4—62no—
220C4—62no—
221C4—122no—
222C4—121no—
223C4#91512yes—
224C4C2×D3×Q811yescharacteristic
225C4#101011yescharacteristic
226C3#338511yescharacteristic, complemented, Sylow
227C2—42no—
228C2#563211yescharacteristic
229C2C2×#91511yescharacteristic
230C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#476C2
socleC22×C3C22×D4
prime radicalC2×C3×D4C23
hypercenter#476D3
hypocenterC3#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #5722C2semidirect
4. #5078C2semidirect
6. #4857C2semidirect
7. #4856C2semidirect
8. #4850C2semidirect
9. #4831C2semidirect
68. #214Q8semidirect
226. C3#3385semidirect

Maximal subgroups

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

subgroupindex
#57222
#55432
C3×#4762
#50782
C4×#2142
#48572
#48562
#48502
#48312
#47592
#47572
#33853