Finite Group Databasebeta

← #79205

Subgroups — #79205

Order 384.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#5519C211yesmaximal, characteristic
2#5500C211yesmaximal, characteristic
3C3×#428C211yesmaximal, characteristic
4#5122C211yesmaximal, characteristic
5#5121C211yesmaximal, characteristic
6#4998C211yesmaximal, characteristic
7C2×#808C211yesmaximal, characteristic
8#4878C212yesmaximal, complemented
9#4876C212yesmaximal, complemented
10#4856C212yesmaximal, complemented
11#4855C212yesmaximal, complemented
12#3163—31nomaximal, Sylow
13C2×C3×M16C2211yescharacteristic
14C3×#117C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16#926C2211yescharacteristic
17#926C2211yescharacteristic
18C2×C4⋊Dic3C2211yescharacteristic
19#924—22no—
20C22×C3⋊C8C2211yescharacteristic
21#890—21no—
22#890C2212yes—
23#890C2212yes—
24#888—21no—
25#888C2212yes—
26#888C2212yes—
27#878—22no—
28#870—21no—
29#869—21no—
30C3×#107C2212yes—
31C3×#106C2212yes—
32#818C2212yes—
33#817C2212yes—
34#815C2212yes—
35C8×Dic3C2212yes—
36#808C2212yes—
37#808C2211yescharacteristic
38#808C2212yes—
39#808C2211yescharacteristic
40#807C2211yescharacteristic
41#807C2211yescharacteristic
42#544—31no—
43#544—31no—
44#501—32no—
45#501—32no—
46#499—32no—
47#498—32no—
48#428—31no—
49#428D311yescharacteristic
50#427—31no—
51C2×#106—31no—
52#405—31no—
53C22×C3×C4—21no—
54C22×C3×C4C2311yescharacteristic
55C22×Dic3—21no—
56C2×Dic6—22no—
57C2×Dic6—22no—
58C3×M16Q814yescomplemented
59C2×C3×C8C2312yes—
60C3×C4⋊C4—21no—
61C3×C4⋊C4C2311yescharacteristic
62C3×C4⋊C4C2311yescharacteristic
63C3×C4⋊C4C2312yes—
64C3×C22⋊C4—21no—
65C3×C42—21no—
66#217—21no—
67#217D412yes—
68C4⋊Dic3—21no—
69C4⋊Dic3—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3D411yescharacteristic
72C4⋊Dic3D411yescharacteristic
73C4⋊Dic3C2311yescharacteristic
74C4⋊Dic3C2311yescharacteristic
75C4⋊Dic3C2312yes—
76Dic3⋊C4—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80C4×Dic3—21no—
81C4×Dic3C2312yes—
82C2×C3⋊C8Q814yescomplemented
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85C2×M16C2×D311yescharacteristic
86C22×C8—31no—
87C4⋊Q8—32no—
88C4⋊Q8—32no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91#125—32no—
92#125—32no—
93#125—61no—
94#125—61no—
95C22⋊Q8—62no—
96C22⋊Q8—62no—
97#117—31no—
98#117—31no—
99#117C2×D311yescharacteristic
100C2×C4⋊C4—31no—
101C2×C4⋊C4C2×D311yescharacteristic
102#107—32no—
103#107—31no—
104#107—31no—
105#107C2×D312yes—
106#106—32no—
107#106—32no—
108#106—32no—
109#106—31no—
110#106—31no—
111#106C2×D312yes—
112#97—32no—
113C4×C8—32no—
114C23×C3C2×D411yescharacteristic
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117C2×C3×C4—21no—
118C2×C3×C4—81no—
119C2×C3×C4C2×D411yescharacteristic
120C2×C3×C4C2×Q812yes—
121C2×C3×C4C2×Q812yes—
122C2×C3×C4C2411yescharacteristic
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—81no—
127C2×Dic3C2×D412yes—
128C2×Dic3C2×D412yes—
129Dic6—82no—
130Dic6—82no—
131C3×C8C2×Q814yes—
132C3⋊C8C2×Q812yes—
133C3⋊C8C2×Q812yes—
134C2×Q8—62no—
135C2×Q8—62no—
136C22×C4—21no—
137C22×C4—61no—
138C22×C4C22×D311yescharacteristic
139M16Dic614yescomplemented
140C2×C8—34no—
141C2×C8—31no—
142C2×C8—31no—
143C2×C8C22×D312yes—
144C4⋊C4—21no—
145C4⋊C4—31no—
146C4⋊C4—31no—
147C4⋊C4—32no—
148C4⋊C4—31no—
149C4⋊C4—31no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—61no—
157C4⋊C4C22×D312yes—
158C4⋊C4C22×D311yescharacteristic
159C4⋊C4C22×D311yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—32no—
162C22⋊C4—61no—
163C42—21no—
164C42—32no—
165C42—61no—
166C22×C3—21no—
167C22×C3—21no—
168C22×C3C4⋊Q812yes—
169C22×C3C22×D411yescharacteristic
170C3×C4—41no—
171C3×C4—42no—
172C3×C4—41no—
173C3×C4C4⋊Q812yes—
174C3×C4C22×Q811yescharacteristic
175C3×C4C22×Q811yescharacteristic
176Dic3—22no—
177Dic3—22no—
178Dic3—41no—
179Dic3—42no—
180Dic3—41no—
181C23D3×D411yescharacteristic
182Q8—242no—
183Q8—242no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—32no—
188C2×C4—32no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—62no—
192C2×C4—81no—
193C2×C4—241no—
194C2×C4C2×Dic612yes—
195C2×C4D3×D411yescharacteristic
196C2×C4D3×Q812yes—
197C2×C4C23×D311yescharacteristic
198C8—32no—
199C8—32no—
200C8C2×Dic614yes—
201C2×C3—22no—
202C2×C3C2×C4⋊Q811yescharacteristic
203C2×C3#57711yescharacteristic
204C2×C3#57711yescharacteristic
205C22—21no—
206C22—21no—
207C22#88812yes—
208C22C2×D3×D411yescharacteristic
209C4—42no—
210C4—41no—
211C4—41no—
212C4—62no—
213C4—62no—
214C4—121no—
215C4—122no—
216C4—121no—
217C4#88812yes—
218C4C22×Dic611yescharacteristic
219C4C2×D3×Q811yescharacteristic
220C3#316311yescharacteristic, complemented, Sylow
221C2—22no—
222C2C2×#88811yescharacteristic
223C2#577511yescharacteristic
224C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radicalC23×C3C2×D4
hypercenter#428D3
hypocenterC3#3163
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
8. #4878C2semidirect
9. #4876C2semidirect
10. #4856C2semidirect
11. #4855C2semidirect
58. C3×M16Q8semidirect
82. C2×C3⋊C8Q8semidirect
139. M16Dic6semidirect
220. C3#3163semidirect

Maximal subgroups

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

subgroupindex
#55192
#55002
C3×#4282
#51222
#51212
#49982
C2×#8082
#48782
#48762
#48562
#48552
#31633