Finite Group Databasebeta

← #79634

Subgroups — #79634

Order 384.

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

normal subgroups only

This group has 340 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
1Q8×Dic6C211yesmaximal, characteristic, complemented
2#5543C211yesmaximal, characteristic
3C3×C4×SD16C211yesmaximal, characteristic
4#5164C212yesmaximal
5#5021C211yesmaximal, characteristic, complemented
6#5008C211yesmaximal, characteristic
7#4856C212yesmaximal, complemented
8#4796C212yesmaximal, complemented
9#4759C212yesmaximal
10#4682C211yesmaximal, characteristic, complemented
11#4679C211yesmaximal, characteristic, complemented
12#3385—31nomaximal, Sylow
13C2×C3×SD16C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16Q8×Dic3C2212yes—
17D4×Dic3C2212yes—
18C2×C4⋊Dic3—21no—
19#924—21no—
20#890C2212yes—
21#888—22no—
22#888C2212yes—
23#878—22no—
24#869—21no—
25#869C2211yescharacteristic
26#869C2211yescharacteristic
27C4×Dic6—21no—
28C4×Dic6C2211yescharacteristic
29C3×#106C2211yescharacteristic
30C3×#103C2211yescharacteristic
31C3×#102C2211yescharacteristic
32C3×C4×C8C2211yescharacteristic
33#835C2212yes—
34#832C2212yes—
35#818C2212yes—
36#818C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#808C2212yes—
40#807C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42Q82—31no—
43#554—31no—
44#501—32no—
45#500—31no—
46#479—32no—
47#477—31no—
48#476—31no—
49#474—32no—
50#446—31no—
51#440—32no—
52C4×SD16D311yescharacteristic
53C2×C3×Q8C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C22×Dic3—22no—
57C2×Dic6—21no—
58C2×Dic6—21no—
59C2×Dic6—22no—
60C2×Dic6D411yescharacteristic
61C3×SD16Q814yescomplemented
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4—21no—
65C3×C4⋊C4C2311yescharacteristic
66C3×C4⋊C4C2311yescharacteristic
67C3×C22⋊C4—21no—
68C3×C42—21no—
69C3×C42C2311yescharacteristic
70#217—22no—
71C4⋊Dic3—21no—
72C4⋊Dic3—21no—
73C4⋊Dic3—22no—
74C4⋊Dic3—22no—
75C4⋊Dic3D411yescharacteristic
76C4⋊Dic3C2312yes—
77C4⋊Dic3C2312yes—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4D412yes—
81C4×Dic3—22no—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2312yes—
84C2×SD16C2×D311yescharacteristic
85C4⋊Q8—32no—
86C4⋊Q8—31no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89C4⋊Q8—62no—
90#125—32no—
91C22⋊Q8—62no—
92C22⋊Q8—61no—
93C4×Q8—31no—
94C4×Q8—32no—
95C4×Q8—61no—
96C4×Q8C2×D311yescharacteristic
97C4×D4—32no—
98C4×D4C2×D311yescharacteristic
99C2×C4⋊C4—61no—
100#107—32no—
101#107—32no—
102#107—32no—
103#106—32no—
104#106C2×D311yescharacteristic
105C4⋊C8—32no—
106C4⋊C8—31no—
107#103—32no—
108#103C2×D311yescharacteristic
109#102—32no—
110#102C2×D311yescharacteristic
111#97—32no—
112C4×C8C2×D311yescharacteristic
113C23×C3—21no—
114C3×Q8—21no—
115C3×Q8C2×Q812yes—
116C3×D4—21no—
117C3×D4C2×Q812yes—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—42no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2411yescharacteristic
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—82no—
128C2×Dic3C2×D412yes—
129C2×Dic3C2×D412yes—
130Dic6—22no—
131Dic6—82no—
132Dic6—81no—
133Dic6—81no—
134C3×C8—21no—
135C3×C8C2×Q812yes—
136C3⋊C8—22no—
137C2×Q8—31no—
138C2×Q8—62no—
139C2×Q8—61no—
140C2×Q8—61no—
141C2×Q8C22×D311yescharacteristic
142C2×D4C22×D311yescharacteristic
143C22×C4—21no—
144C22×C4—62no—
145SD16Dic614yescomplemented
146C2×C8—32no—
147C2×C8C22×D311yescharacteristic
148C2×C8C22×D311yescharacteristic
149C4⋊C4—21no—
150C4⋊C4—31no—
151C4⋊C4—32no—
152C4⋊C4—32no—
153C4⋊C4—32no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4C22×D311yescharacteristic
161C4⋊C4C22×D311yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—62no—
164C42—21no—
165C42—32no—
166C42—62no—
167C42C22×D311yescharacteristic
168C22×C3—22no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3C22×D411yescharacteristic
172C3×C4—22no—
173C3×C4—21no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4#13712yes—
177C3×C4C22×Q811yescharacteristic
178C3×C42−1+411yescharacteristic
179Dic3—22no—
180Dic3—22no—
181Dic3—42no—
182Dic3—42no—
183C23—21no—
184Q8—21no—
185Q8—62no—
186Q8—241no—
187Q8—241no—
188Q8—242no—
189Q8C2×Dic612yes—
190D4—21no—
191D4C2×Dic612yes—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—32no—
196C2×C4—32no—
197C2×C4—42no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—242no—
201C2×C4D3×D411yescharacteristic
202C2×C4D3×D411yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—21no—
205C8—62no—
206C8C2×Dic612yes—
207C2×C3—42no—
208C2×C3D4×Q811yescharacteristic
209C2×C3C2×#13711yescharacteristic
210C2×C3#57611yescharacteristic
211C22—22no—
212C22—41no—
213C22—41no—
214C22C2×D3×D411yescharacteristic
215C4—21no—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—62no—
220C4—62no—
221C4—122no—
222C4—122no—
223C4#91512yes—
224C4C22×Dic611yescharacteristic
225C4#101011yescharacteristic
226C3#338511yescharacteristic, complemented, Sylow
227C2—42no—
228C2D4×Dic611yescharacteristic
229C2C2×#91511yescharacteristic
230C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×SD16C2
socleC22×C3C22×D4
prime radicalC2×C3×D4C23
hypercenterC4×SD16D3
hypocenterC3#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×Dic6C2semidirect
5. #5021C2semidirect
7. #4856C2semidirect
8. #4796C2semidirect
10. #4682C2semidirect
11. #4679C2semidirect
61. C3×SD16Q8semidirect
145. SD16Dic6semidirect
226. C3#3385semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic62
#55432
C3×C4×SD162
#51642
#50212
#50082
#48562
#47962
#47592
#46822
#46792
#33853