Finite Group Databasebeta

← #68276

Subgroups — #68276

Order 384.

238 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 63 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#5600C211yesmaximal, characteristic
2#4718C211yesmaximal, characteristic, complemented
3#4648C211yesmaximal, characteristic, complemented
4C3×#327C211yesmaximal, characteristic
5#4537C211yesmaximal, characteristic
6#4467C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1500—31nomaximal, Sylow
9C3×#121C2211yescharacteristic
10#936C411yescharacteristic, complemented
11#935—21no—
12D4×Dic3—21no—
13D4×Dic3C411yescharacteristic, complemented
14#924C2211yescharacteristic
15C2×C4×Dic3—21no—
16C2×C4×Dic3C2211yescharacteristic
17#890C411yescharacteristic, complemented
18#887C411yescharacteristic, complemented
19#879—21no—
20#877—21no—
21C3×C22⋊C8C2211yescharacteristic
22C3×#95C2211yescharacteristic
23C3⋊#95—21no—
24C3⋊#95C2211yescharacteristic
25#830C2211yescharacteristic
26Dic3⋊C8—21no—
27C8×Dic3—21no—
28#529—31no—
29#431—31no—
30#376—31no—
31#328—31no—
32#328—31no—
33#327—31no—
34#327D311yescharacteristic
35C2×C3×D4—21no—
36C2×C3×D4C2×C411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C22×Dic3—21no—
40C22×Dic3—21no—
41C22×Dic3D411yescharacteristic
42C22×Dic3D411yescharacteristic
43C2×Dic6C2×C411yescharacteristic
44C2×C3×C8—21no—
45C3×C4⋊C4C2×C411yescharacteristic
46C3×C22⋊C4—21no—
47#217—21no—
48#217—21no—
49#217—21no—
50#217—21no—
51C4⋊Dic3—21no—
52C4⋊Dic3C2×C411yescharacteristic
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56C4×Dic3—24no—
57C4×Dic3—21no—
58C4×Dic3C2×C411yescharacteristic
59C4×Dic3C2×C411yescharacteristic
60C4×Dic3D411yescharacteristic
61C4×Dic3D411yescharacteristic
62C2×C3⋊C8—21no—
63#126—61no—
64#125—31no—
65#124—31no—
66#123—61no—
67C22⋊Q8—31no—
68#121C2×D311yescharacteristic
69C4×Q8—31no—
70C4×D4—31no—
71C4×D4—61no—
72#117—61no—
73C2×C42—31no—
74C2×C42—61no—
75C4⋊C8—61no—
76C22⋊C8—31no—
77C22⋊C8C2×D311yescharacteristic
78C4×C8—61no—
79#95—31no—
80#95—61no—
81#95C2×D311yescharacteristic
82C23×C3—21no—
83C23×C3C2×D411yescharacteristic
84C3×D4—41no—
85C3×D4—41no—
86C3×D4—41no—
87C2×C3×C4—22no—
88C2×C3×C4—21no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4C22×C411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×Dic3—21no—
95C2×Dic3—21no—
96C2×Dic3—22no—
97C2×Dic3—22no—
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—41no—
101C2×Dic3—41no—
102C2×Dic3—42no—
103C2×Dic3C22⋊C411yescharacteristic
104C2×Dic3C22⋊C411yescharacteristic
105C2×Dic3C22⋊C411yescharacteristic
106C2×Dic3C22⋊C411yescharacteristic
107Dic6—41no—
108C3×C8—41no—
109C3⋊C8—41no—
110C2×Q8—31no—
111C2×D4—21no—
112C2×D4C4×D311yescharacteristic
113C22×C4—21no—
114C22×C4—31no—
115C22×C4—31no—
116C22×C4—61no—
117C22×C4—61no—
118C22×C4C22×D311yescharacteristic
119C2×C8—21no—
120C2×C8—61no—
121C4⋊C4—31no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4C4×D311yescharacteristic
127C22⋊C4—21no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C42—31no—
133C42—31no—
134C42—31no—
135C42—31no—
136C42—61no—
137C42—64no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3—41no—
141C22×C3—41no—
142C22×C3—41no—
143C22×C3C2×C22⋊C411yescharacteristic
144C22×C3#13511yescharacteristic
145C22×C3#13511yescharacteristic
146C3×C4—21no—
147C3×C4—21no—
148C3×C4—42no—
149C3×C4—41no—
150Dic3—21no—
151Dic3—21no—
152Dic3—21no—
153Dic3—42no—
154Dic3—41no—
155Dic3—41no—
156Dic3#10412yes—
157C23—21no—
158C23D3×D411yescharacteristic
159Q8—121no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—31no—
166C2×C4—31no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—62no—
173C2×C4—61no—
174C2×C4—61no—
175C2×C4—62no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4—122no—
179C2×C4—121no—
180C2×C4—121no—
181C2×C4C2×C4×D311yescharacteristic
182C2×C4D3×D411yescharacteristic
183C8—41no—
184C8—121no—
185C2×C3—21no—
186C2×C3—21no—
187C2×C3—81no—
188C2×C3#41011yescharacteristic
189C2×C3#41611yescharacteristic
190C2×C3C2×#10411yescharacteristic
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22D3×C22⋊C411yescharacteristic
197C22#91211yescharacteristic
198C22#91611yescharacteristic
199C4—21no—
200C4—21no—
201C4—32no—
202C4—42no—
203C4—41no—
204C4—61no—
205C4—61no—
206C4—61no—
207C4—122no—
208C4—121no—
209C4—121no—
210C3#150011yescharacteristic, complemented, Sylow
211C2—21no—
212C2—21no—
213C2—81no—
214C2#474011yescharacteristic
215C2#477211yescharacteristic
216C2D3×#10411yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3C2×C22⋊C4
prime radicalC3×#121C22
cosocleC22×C3×C4C23
hypercenter#327D3
hypocenterC3#1500
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #4718C2semidirect
3. #4648C2semidirect
6. #4467C2semidirect
7. #4452C2semidirect
10. #936C4semidirect
13. D4×Dic3C4semidirect
17. #890C4semidirect
18. #887C4semidirect
210. C3#1500semidirect

Maximal subgroups

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

subgroupindex
#56002
#47182
#46482
C3×#3272
#45372
#44672
#44522
#15003