Finite Group Databasebeta

← #70629

Subgroups — #70629

Order 384.

340 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×#204C211yesmaximal, characteristic
2C2×C3×#107C211yesmaximal, characteristic, complemented
3C2×#815C211yesmaximal, characteristic, complemented
4#4988C211yesmaximal, characteristic
5#4973C211yesmaximal, characteristic, complemented
6C2×#810C211yesmaximal, characteristic, complemented
7C2×#809C211yesmaximal, characteristic
8#1973—31nomaximal, Sylow
9C22×D12C2211yescharacteristic
10C22×Dic6C2211yescharacteristic
11C22×C3×C8C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×(C2×D3)⋊C4—22no—
15C2×C4×Dic3C2211yescharacteristic
16C22×C3⋊C8C2211yescharacteristic
17C2×#204—22no—
18C2×#204—22no—
19C2×#204C414yescomplemented
20C3×#107—22no—
21C3⋊#95—22no—
22#815—22no—
23#810—22no—
24#809—22no—
25C22×SD16—31no—
26C2×#107D311yescharacteristic, complemented
27C2×#103—31no—
28C2×#102—31no—
29C2×#97—31no—
30#391—31no—
31#390—31no—
32C23×D3—21no—
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4C2311yescharacteristic
36C22×Dic3—21no—
37C22×Dic3D412yes—
38C2×D12—21no—
39C2×D12—21no—
40C2×D12—21no—
41C2×D12—21no—
42C2×D12—21no—
43C2×D12C2×C412yes—
44C2×Dic6—21no—
45C2×Dic6—21no—
46C2×Dic6—21no—
47C2×Dic6—21no—
48C2×Dic6—21no—
49C2×Dic6C2×C412yes—
50C2×C3×C8C2×C412yes—
51C2×C3×C8D412yes—
52C2×C3×C8D412yes—
53C3×C4⋊C4—21no—
54C3×C4⋊C4—21no—
55C3×C4⋊C4—21no—
56C3×C4⋊C4—21no—
57(C2×D3)⋊C4—44no—
58C4×Dic3—21no—
59C4×Dic3—21no—
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62C2×C3⋊C8D412yes—
63#204—28no—
64C22×Q8—31no—
65C22×D4—31no—
66C2×SD16—34no—
67C2×SD16—62no—
68C2×SD16—62no—
69C22×C8—31no—
70C22×C8C2×D311yescharacteristic
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C4⋊C4C2×D311yescharacteristic
73C2×C22⋊C4—62no—
74C2×C42—31no—
75#107—22no—
76#103—62no—
77#102—62no—
78#97—62no—
79#95—62no—
80C23×C3C2×D411yescharacteristic
81C22×D3—22no—
82C22×D3—41no—
83C22×D3—41no—
84C22×D3—42no—
85C22×D3—42no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—42no—
89C2×C3×C4—42no—
90C2×C3×C4C22×C411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4#4111yescharacteristic
95C2×C3×C4#4111yescharacteristic
96C2×Dic3—22no—
97C2×Dic3—22no—
98C2×Dic3—22no—
99C2×Dic3—22no—
100C2×Dic3—41no—
101C2×Dic3—41no—
102D12—22no—
103D12—22no—
104D12—41no—
105D12—41no—
106Dic6—22no—
107Dic6—22no—
108Dic6—41no—
109Dic6—41no—
110C3×C8C22⋊C414yescomplemented
111C3⋊C8—22no—
112C24—61no—
113C2×Q8—32no—
114C2×Q8—61no—
115C2×Q8—61no—
116C2×Q8—61no—
117C2×Q8—61no—
118C2×Q8—61no—
119C2×D4—32no—
120C2×D4—61no—
121C2×D4—61no—
122C2×D4—61no—
123C2×D4—61no—
124C2×D4—61no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—32no—
128C22×C4—61no—
129C22×C4C22×D311yescharacteristic
130SD16—68no—
131C2×C8—32no—
132C2×C8—61no—
133C2×C8—61no—
134C2×C8C4×D312yes—
135C2×C8D1212yes—
136C2×C8#6712yes—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C22⋊C4—124no—
142C42—61no—
143C42—61no—
144C22×C3C4×D411yescharacteristic
145C22×C3#12111yescharacteristic
146C22×C3#12711yescharacteristic
147C22×C3AGL(1,ℤ/8)12yes—
148C22×C3#13712yes—
149C2×D3—42no—
150C2×D3—44no—
151C2×D3—82no—
152C2×D3—81no—
153C2×D3—81no—
154C3×C4—81no—
155C3×C4—81no—
156C3×C4C2×C22⋊C411yescharacteristic
157C3×C4C4×D411yescharacteristic
158C3×C4#12111yescharacteristic
159C3×C4#12411yescharacteristic
160Dic3—42no—
161Dic3—42no—
162C23—62no—
163C23—121no—
164C23—122no—
165C23—122no—
166C23—121no—
167C23D3×D411yescharacteristic
168Q8—62no—
169Q8—62no—
170Q8—121no—
171Q8—121no—
172D4—62no—
173D4—62no—
174D4—121no—
175D4—121no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—121no—
185C2×C4—121no—
186C2×C4C2×C4×D311yescharacteristic
187C2×C4C2×D1211yescharacteristic
188C2×C4D3×D411yescharacteristic
189C2×C4#23911yescharacteristic
190C2×C4#23911yescharacteristic
191C2×C4C2×#6711yescharacteristic
192C8—62no—
193C8(C2×D3)⋊C414yescomplemented
194C2×C3#39011yescharacteristic
195C2×C3#44012yes—
196C2×C3#46812yes—
197C2×C3#49611yescharacteristic
198C2×C3#49711yescharacteristic
199D3—82no—
200C22—124no—
201C22—122no—
202C22—242no—
203C22—241no—
204C22—241no—
205C22#89311yescharacteristic
206C22#89511yescharacteristic
207C22#91112yes—
208C22#91812yes—
209C22#94011yescharacteristic
210C4—81no—
211C4—81no—
212C4—122no—
213C4—122no—
214C4#89311yescharacteristic
215C4#89511yescharacteristic
216C4C2×(C2×D3)⋊C411yescharacteristic
217C4#94711yescharacteristic
218C3#197311yescharacteristic, complemented, Sylow
219C2—242no—
220C2#488612yes—
221C2#488912yes—
222C2#498811yescharacteristic
223C2#515411yescharacteristic
224C2#519011yescharacteristic
subgroupG/H
centerC23D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#107C2
socleC23×C3C2×D4
prime radicalC22×D12C22
cosocleC22×C3×C4C23
hypercenterC2×#107D3
hypocenterC3#1973
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×C3×#107C2semidirect
3. C2×#815C2semidirect
5. #4973C2semidirect
6. C2×#810C2semidirect
19. C2×#204C4semidirect
26. C2×#107D3semidirect
110. C3×C8C22⋊C4semidirect
193. C8(C2×D3)⋊C4semidirect
218. C3#1973semidirect

Maximal subgroups

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

subgroupindex
C22×#2042
C2×C3×#1072
C2×#8152
#49882
#49732
C2×#8102
C2×#8092
#19733