Finite Group Databasebeta

← #68772

Subgroups — #68772

Order 384.

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

normal subgroups only

This group has 286 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
1#5523C211yesmaximal, characteristic
2#5133C211yesmaximal, characteristic
3#4995C211yesmaximal, characteristic, complemented
4#4936C211yesmaximal, characteristic, complemented
5C2×#805C211yesmaximal, characteristic
6C3×#339C211yesmaximal, characteristic, complemented
7#4554C211yesmaximal, characteristic, complemented
8#1860—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11C3×#117C2211yescharacteristic
12C2×C3×C42C2211yescharacteristic
13C2×C4⋊Dic3C2211yescharacteristic
14C2×C4⋊Dic3C2211yescharacteristic
15C2×#208C2211yescharacteristic
16#892—21no—
17D3×C4⋊C4—21no—
18#67⋊C4—22no—
19C4×D12—21no—
20C4×Dic6—21no—
21C3⋊#95—22no—
22#821—21no—
23#816—21no—
24#808—21no—
25#807—21no—
26#805C414yescomplemented
27#520—31no—
28#428—31no—
29C2×#104—31no—
30#417—31no—
31#382—31no—
32#343—31no—
33#339D311yescharacteristic, complemented
34C22×C3×C4—21no—
35C22×C3×C4C2311yescharacteristic
36C2×#67—21no—
37C22×Dic3—21no—
38C22×Dic3D412yes—
39#235—21no—
40#235—21no—
41#235C2×C412yes—
42#235Q812yes—
43C2×D12D411yescharacteristic
44C2×C4×D3—21no—
45C2×C4×D3—22no—
46C2×Dic6D411yescharacteristic
47C3×M16—21no—
48C3×M16—21no—
49C2×C3×C8—21no—
50C3×C4⋊C4D411yescharacteristic
51C3×C4⋊C4D411yescharacteristic
52C3×C22⋊C4—21no—
53C3×C42—21no—
54C3×C42—21no—
55C3×C42C2×C412yescomplemented
56(C2×D3)⋊C4—22no—
57C4⋊Dic3—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—21no—
60C4⋊Dic3—21no—
61Dic3⋊C4—22no—
62C4×Dic3—22no—
63#208—21no—
64#208C2×C412yescomplemented
65C2×C3⋊C8—21no—
66C2×#41—31no—
67C2×M16—31no—
68C2×M16C2×D311yescharacteristic
69C4×Q8—61no—
70C4×D4—61no—
71C4×D4—62no—
72#117—61no—
73#117C2×D311yescharacteristic
74C2×C4⋊C4—31no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4—61no—
77C2×C42C2×D311yescharacteristic
78#107—61no—
79#106—61no—
80#104—34no—
81#103—61no—
82#102—61no—
83#95—62no—
84C23×C3C2×D411yescharacteristic
85C22×D3—21no—
86C2×C3×C4—22no—
87C2×C3×C4—21no—
88C2×C3×C4—21no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4C22×C411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×Q811yescharacteristic
95C2×C3×C4#4111yescharacteristic
96C2×C3×C4#4111yescharacteristic
97#67—22no—
98#67—22no—
99C2×Dic3—21no—
100C2×Dic3—22no—
101C2×Dic3—42no—
102C2×Dic3—42no—
103C2×Dic3—41no—
104D12—21no—
105D12C4⋊C412yes—
106C4×D3—22no—
107C4×D3—22no—
108C4×D3—82no—
109Dic6—21no—
110Dic6C4⋊C412yes—
111C3×C8—41no—
112C3⋊C8—22no—
113#41—32no—
114#41—32no—
115#41—61no—
116#41—61no—
117C2×Q8—31no—
118C2×D4—31no—
119C2×D4—61no—
120C22×C4—21no—
121C22×C4—32no—
122C22×C4—62no—
123C22×C4—61no—
124C22×C4—61no—
125C22×C4C22×D311yescharacteristic
126M16—21no—
127M16—21no—
128M16—32no—
129M16—61no—
130C2×C8—21no—
131C2×C8—61no—
132C4⋊C4—62no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C4⋊C4—61no—
136C4⋊C4—61no—
137C4⋊C4D1211yescharacteristic
138C4⋊C4D1211yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—62no—
141C42—21no—
142C42—21no—
143C42—62no—
144C42C4×D312yescomplemented
145C22×C3—21no—
146C22×C3—21no—
147C22×C3C4×D411yescharacteristic
148C22×C3#12011yescharacteristic
149C22×C3C22⋊Q811yescharacteristic
150C2×D3—22no—
151C2×D3—41no—
152C2×D3—41no—
153C3×C4—42no—
154C3×C4—41no—
155C3×C4—41no—
156C3×C4C2×C4⋊C411yescharacteristic
157C3×C4C4×D411yescharacteristic
158C3×C4C22⋊Q811yescharacteristic
159C3×C4#12311yescharacteristic
160Dic3—22no—
161Dic3—42no—
162Dic3—81no—
163C23—61no—
164C23D3×D411yescharacteristic
165Q8—32no—
166Q8—61no—
167D4—32no—
168D4—62no—
169D4—62no—
170D4—61no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—61no—
177C2×C4—62no—
178C2×C4—62no—
179C2×C4—62no—
180C2×C4—122no—
181C2×C4—122no—
182C2×C4—121no—
183C2×C4—242no—
184C2×C4C2×C4×D311yescharacteristic
185C2×C4C2×D1211yescharacteristic
186C2×C4#23511yescharacteristic
187C2×C4D3×D411yescharacteristic
188C2×C4#23711yescharacteristic
189C2×C4D3×Q811yescharacteristic
190C8—41no—
191C8—62no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3#38511yescharacteristic
195C2×C3#45411yescharacteristic
196C2×C3#45511yescharacteristic
197D3—42no—
198C22—21no—
199C22—21no—
200C22—62no—
201C22—121no—
202C22—121no—
203C22#67⋊C411yescharacteristic
204C22#88211yescharacteristic
205C22#89611yescharacteristic
206C4—42no—
207C4—41no—
208C4—41no—
209C4—62no—
210C4—122no—
211C4—241no—
212C4C4×D1211yescharacteristic
213C4#88311yescharacteristic
214C4D3×C4⋊C411yescharacteristic
215C4#89711yescharacteristic
216C3#186011yescharacteristic, complemented, Sylow
217C2—21no—
218C2—21no—
219C2—122no—
220C2#466811yescharacteristic
221C2#482511yescharacteristic
222C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3#120
prime radicalC2×#235C22
cosocleC22×C3×C4C23
hypercenter#339D3
hypocenterC3#1860
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
3. #4995C2semidirect
4. #4936C2semidirect
6. C3×#339C2semidirect
7. #4554C2semidirect
26. #805C4semidirect
33. #339D3semidirect
55. C3×C42C2×C4semidirect
64. #208C2×C4semidirect
144. C42C4×D3semidirect
216. C3#1860semidirect

Maximal subgroups

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

subgroupindex
#55232
#51332
#49952
#49362
C2×#8052
C3×#3392
#45542
#18603