Finite Group Databasebeta

← #68169

Subgroups — #68169

Order 384.

212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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#5557C211yesmaximal, characteristic
2#4833C211yesmaximal, characteristic, complemented
3#4685C211yesmaximal, characteristic, complemented
4C3×D4⋊C8C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#4484C211yesmaximal, characteristic, complemented
7#4456C211yesmaximal, characteristic, complemented
8#1482—31nomaximal, Sylow
9C3×C4×D4C2211yescharacteristic
10#938C411yescharacteristic
11#936C411yescharacteristic
12C4×#67—21no—
13#926—21no—
14#897C411yescharacteristic
15#890C411yescharacteristic
16#883—21no—
17#879—21no—
18C42×D3C2211yescharacteristic
19C4×Dic6C2211yescharacteristic
20C3×C4⋊C8C2211yescharacteristic
21C3×C4×C8C2211yescharacteristic
22(C2×D3)⋊C8—21no—
23(C2×D3)⋊C8—21no—
24Dic3⋊C8—21no—
25C8×Dic3—21no—
26C4⋊(C3⋊C8)C2211yescharacteristic
27C4×C3⋊C8C2211yescharacteristic
28#529—31no—
29#431—31no—
30#431—31no—
31Q8⋊C8—31no—
32Q8⋊C8—31no—
33D4⋊C8—31no—
34D4⋊C8D311yescharacteristic, complemented
35C2×C3×D4C2×C411yescharacteristic
36C22×C3×C4—21no—
37C2×#67—21no—
38C2×C4×D3C2×C411yescharacteristic
39C2×C4×D3D411yescharacteristic
40C2×C4×D3D411yescharacteristic
41C2×Dic6C2×C411yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C3×C4⋊C4C2×C411yescharacteristic
45C3×C22⋊C4—21no—
46C3×C42C2311yescharacteristic
47#217—21no—
48#217—21no—
49#217—21no—
50(C2×D3)⋊C4—21no—
51C4⋊Dic3—21no—
52C4⋊Dic3—21no—
53C4⋊Dic3C2×C411yescharacteristic
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56C4×Dic3—21no—
57C4×Dic3C2×C411yescharacteristic
58C4×Dic3D411yescharacteristic
59C4×Dic3D411yescharacteristic
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62#126—61no—
63#125—31no—
64#124—31no—
65#123—61no—
66C22⋊Q8—31no—
67#121—31no—
68C4×Q8—31no—
69C4×D4—61no—
70C4×D4C2×D311yescharacteristic
71#117—61no—
72C2×C42—31no—
73C4⋊C8—31no—
74C4⋊C8—61no—
75C4⋊C8C2×D311yescharacteristic
76C22⋊C8—61no—
77C22⋊C8—61no—
78C4×C8—31no—
79C4×C8—61no—
80C4×C8C2×D311yescharacteristic
81C23×C3—21no—
82C22×D3C22⋊C411yescharacteristic
83C3×D4—21no—
84C3×D4—21no—
85C2×C3×C4—21no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4C22×C411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91#67—42no—
92C2×Dic3—21no—
93C2×Dic3—21no—
94C2×Dic3C22⋊C411yescharacteristic
95C2×Dic3C22⋊C411yescharacteristic
96C2×Dic3C22⋊C411yescharacteristic
97C4×D3—21no—
98C4×D3—21no—
99C4×D3—21no—
100C4×D3—21no—
101C4×D3—41no—
102Dic6—21no—
103Dic6—21no—
104C3×C8—41no—
105C3×C8—41no—
106C3⋊C8—41no—
107C3⋊C8—41no—
108C2×Q8—31no—
109C2×D4—61no—
110C2×D4C4×D311yescharacteristic
111C22×C4—21no—
112C22×C4—31no—
113C22×C4—31no—
114C22×C4—31no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—61no—
118C2×C8—61no—
119C4⋊C4—31no—
120C4⋊C4—61no—
121C4⋊C4—61no—
122C4⋊C4—61no—
123C4⋊C4—61no—
124C4⋊C4C4×D311yescharacteristic
125C22⋊C4—21no—
126C22⋊C4—61no—
127C22⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C42—31no—
131C42—31no—
132C42—31no—
133C42—61no—
134C42C22×D311yescharacteristic
135C22×C3—41no—
136C22×C3—41no—
137C22×C3—41no—
138C22×C3C2×C22⋊C411yescharacteristic
139C2×D3—21no—
140C2×D3—21no—
141C2×D3—21no—
142C3×C4—21no—
143C3×C4—41no—
144C3×C4#13111yescharacteristic
145C3×C4#13111yescharacteristic
146C3×C4#13511yescharacteristic
147C3×C4#13511yescharacteristic
148Dic3—21no—
149Dic3—21no—
150Dic3—21no—
151Dic3—41no—
152Dic3—41no—
153C23—21no—
154C23—31no—
155Q8—61no—
156Q8—61no—
157D4—21no—
158D4—21no—
159D4—122no—
160C2×C4—21no—
161C2×C4—31no—
162C2×C4—31no—
163C2×C4—31no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—61no—
167C2×C4—61no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—121no—
173C2×C4C2×C4×D311yescharacteristic
174C2×C4D3×D411yescharacteristic
175C2×C4D3×D411yescharacteristic
176C8—41no—
177C8—41no—
178C8—121no—
179C8—121no—
180C2×C3—81no—
181C2×C3#40811yescharacteristic
182C2×C3#41611yescharacteristic
183C2×C3#42111yescharacteristic
184D3—41no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—61no—
189C22—61no—
190C22—61no—
191C22D3×C22⋊C411yescharacteristic
192C4—21no—
193C4—41no—
194C4—61no—
195C4—61no—
196C4—61no—
197C4—121no—
198C4—121no—
199C4#90111yescharacteristic
200C4#90711yescharacteristic
201C4#91211yescharacteristic
202C4#91611yescharacteristic
203C3#148211yescharacteristic, complemented, Sylow
204C2—81no—
205C2—121no—
206C2#472711yescharacteristic
207C2#477211yescharacteristic
208C2#482111yescharacteristic
subgroupG/H
centerC2×C4D3×D4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×D4⋊C8C2
socleC22×C3C2×C22⋊C4
prime radical#938C4
cosocleC3×C42C23
hypercenterD4⋊C8D3
hypocenterC3#1482
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #4833C2semidirect
3. #4685C2semidirect
4. C3×D4⋊C8C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4484C2semidirect
7. #4456C2semidirect
34. D4⋊C8D3semidirect
203. C3#1482semidirect

Maximal subgroups

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

subgroupindex
#55572
#48332
#46852
C3×D4⋊C82
D4⋊(C3⋊C8)2
#44842
#44562
#14823