Finite Group Databasebeta

← #68129

Subgroups — #68129

Order 384.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×#898C211yesmaximal, characteristic
2#4719C211yesmaximal, characteristic, complemented
3#4718C211yesmaximal, characteristic, complemented
4C3×#324C211yesmaximal, characteristic, complemented
5#4528C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic
7#4468C211yesmaximal, characteristic
8#1473—31nomaximal, Sylow
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×(C2×D3)⋊C4C411yescharacteristic
11C2×(C2×D3)⋊C4C411yescharacteristic
12C2×(C2×D3)⋊C4C2211yescharacteristic
13C2×C4⋊Dic3C411yescharacteristic
14C2×Dic3⋊C4C411yescharacteristic
15C2×C4×Dic3C2211yescharacteristic
16#898—21no—
17#898—21no—
18#898—21no—
19#898—21no—
20C3×C22⋊C8C2211yescharacteristic
21C3×C22⋊C8C2211yescharacteristic
22#830C2211yescharacteristic
23#830C2211yescharacteristic
24#815—21no—
25Dic3⋊C8—21no—
26Dic3⋊C8—21no—
27C8×Dic3—21no—
28C2×#126—31no—
29#432—31no—
30#431—31no—
31#324—31no—
32#324D311yescharacteristic, complemented
33#323—31no—
34#323—31no—
35C23×D3C2×C411yescharacteristic
36C22×C3×C4C2×C411yescharacteristic
37C22×C3×C4C2×C411yescharacteristic
38C22×C3×C4C2311yescharacteristic
39C22×Dic3C2×C411yescharacteristic
40C22×Dic3C2×C411yescharacteristic
41C22×Dic3C2×C411yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C3×C4⋊C4—21no—
45C3×C4⋊C4—21no—
46(C2×D3)⋊C4—21no—
47(C2×D3)⋊C4—21no—
48(C2×D3)⋊C4—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3—21no—
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56C4×Dic3D411yescharacteristic
57C4×Dic3D411yescharacteristic
58C4×Dic3D411yescharacteristic
59C4×Dic3D411yescharacteristic
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62#126—61no—
63#126—61no—
64#126—61no—
65#126—61no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4C2×D311yescharacteristic
69C2×C22⋊C4—31no—
70C2×C22⋊C4—31no—
71C2×C22⋊C4—31no—
72C2×C42—31no—
73C4⋊C8—61no—
74C4⋊C8—61no—
75C22⋊C8—31no—
76C22⋊C8—31no—
77C22⋊C8C2×D311yescharacteristic
78C22⋊C8C2×D311yescharacteristic
79#97—61no—
80C4×C8—61no—
81C23×C3C22×C411yescharacteristic
82C22×D3—21no—
83C22×D3—21no—
84C22×D3—21no—
85C22×D3—41no—
86C22×D3—41no—
87C2×C3×C4—21no—
88C2×C3×C4—21no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95C2×Dic3—21no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—41no—
99C2×Dic3—41no—
100C2×Dic3C22⋊C411yescharacteristic
101C2×Dic3C22⋊C411yescharacteristic
102C2×Dic3C22⋊C411yescharacteristic
103C2×Dic3C22⋊C411yescharacteristic
104C3×C8—41no—
105C3×C8—41no—
106C3⋊C8—41no—
107C3⋊C8—41no—
108C24—31no—
109C22×C4—31no—
110C22×C4—31no—
111C22×C4—31no—
112C22×C4C4×D311yescharacteristic
113C22×C4C4×D311yescharacteristic
114C22×C4C22×D311yescharacteristic
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—61no—
118C2×C8—61no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C22⋊C4—61no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C42—31no—
132C42—31no—
133C42—31no—
134C42—31no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3C2×C22⋊C411yescharacteristic
138C22×C3#13111yescharacteristic
139C22×C3#13111yescharacteristic
140C2×D3—41no—
141C2×D3—41no—
142C2×D3—41no—
143C2×D3—81no—
144C2×D3—81no—
145C3×C4—21no—
146C3×C4—21no—
147C3×C4—41no—
148C3×C4—41no—
149Dic3—21no—
150Dic3—21no—
151Dic3—21no—
152Dic3—21no—
153Dic3—41no—
154C23—61no—
155C23—61no—
156C23—61no—
157C23—121no—
158C23—121no—
159C23C2×C4×D311yescharacteristic
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—31no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—31no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—121no—
174C2×C4—121no—
175C2×C4D3×D411yescharacteristic
176C2×C4D3×D411yescharacteristic
177C8—41no—
178C8—41no—
179C8—121no—
180C8—121no—
181C2×C3—21no—
182C2×C3—21no—
183C2×C3#40811yescharacteristic
184C2×C3#41011yescharacteristic
185C2×C3#41311yescharacteristic
186D3—81no—
187C22—21no—
188C22—21no—
189C22—121no—
190C22—121no—
191C22—121no—
192C22—241no—
193C22—241no—
194C22D3×C22⋊C411yescharacteristic
195C22#90111yescharacteristic
196C22#90711yescharacteristic
197C4—21no—
198C4—21no—
199C4—41no—
200C4—41no—
201C4—61no—
202C4—61no—
203C4—61no—
204C4—61no—
205C4—121no—
206C3#147311yescharacteristic, complemented, Sylow
207C2—21no—
208C2—21no—
209C2—241no—
210C2#472711yescharacteristic
211C2#474011yescharacteristic
212C2#475111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#324C2
socleC22×C3C2×C22⋊C4
prime radicalC23×D3C2×C4
cosocleC22×C3×C4C23
hypercenter#324D3
hypocenterC3#1473
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #4719C2semidirect
3. #4718C2semidirect
4. C3×#324C2semidirect
5. #4528C2semidirect
32. #324D3semidirect
206. C3#1473semidirect

Maximal subgroups

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

subgroupindex
C2×#8982
#47192
#47182
C3×#3242
#45282
#44682
#14733