Finite Group Databasebeta

← #68178

Subgroups — #68178

Order 384.

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

normal subgroups only

This group has 216 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#4837C211yesmaximal, 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#1495—31nomaximal, Sylow
9C3×C4×D4C2211yescharacteristic
10#938C411yescharacteristic
11#936C411yescharacteristic
12C4×#67—21no—
13#926—21no—
14C2×ΓL(1,𝔽25)—21no—
15#897C411yescharacteristic
16#890C411yescharacteristic
17#883—21no—
18#879—21no—
19C42×D3C2211yescharacteristic
20C4×Dic6C2211yescharacteristic
21C3×C4⋊C8C2211yescharacteristic
22C3×C4×C8C2211yescharacteristic
23(C2×D3)⋊C8—21no—
24Dic3⋊C8—21no—
25Dic3⋊C8—21no—
26C4⋊(C3⋊C8)C2211yescharacteristic
27C4×C3⋊C8C2211yescharacteristic
28#529—31no—
29#431—31no—
30#423—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—
62ΓL(1,𝔽25)—42no—
63C2×M16—61no—
64#126—61no—
65#125—31no—
66#124—31no—
67#123—61no—
68C22⋊Q8—31no—
69#121—31no—
70C4×Q8—31no—
71C4×D4—61no—
72C4×D4C2×D311yescharacteristic
73#117—61no—
74C2×C42—31no—
75C4⋊C8—31no—
76C4⋊C8—61no—
77C4⋊C8—61no—
78C4⋊C8C2×D311yescharacteristic
79C22⋊C8—61no—
80C4×C8—31no—
81C4×C8C2×D311yescharacteristic
82C23×C3—21no—
83C22×D3C22⋊C411yescharacteristic
84C3×D4—21no—
85C3×D4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4C22×C411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91C2×C3×C4C2×D411yescharacteristic
92#67—42no—
93C2×Dic3—21no—
94C2×Dic3—21no—
95C2×Dic3C22⋊C411yescharacteristic
96C2×Dic3C22⋊C411yescharacteristic
97C2×Dic3C22⋊C411yescharacteristic
98C4×D3—21no—
99C4×D3—21no—
100C4×D3—21no—
101C4×D3—21no—
102C4×D3—41no—
103Dic6—21no—
104Dic6—21no—
105C3×C8—41no—
106C3×C8—41no—
107C3⋊C8—41no—
108C3⋊C8—41no—
109C2×Q8—31no—
110C2×D4—61no—
111C2×D4C4×D311yescharacteristic
112C22×C4—21no—
113C22×C4—31no—
114C22×C4—31no—
115C22×C4—31no—
116M16—122no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—61no—
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—61no—
136C42C22×D311yescharacteristic
137C22×C3—41no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3C2×C22⋊C411yescharacteristic
141C2×D3—21no—
142C2×D3—21no—
143C2×D3—21no—
144C3×C4—21no—
145C3×C4—41no—
146C3×C4#13111yescharacteristic
147C3×C4#13111yescharacteristic
148C3×C4AGL(1,ℤ/8)11yescharacteristic
149C3×C4#13711yescharacteristic
150Dic3—21no—
151Dic3—21no—
152Dic3—21no—
153Dic3—41no—
154Dic3—41no—
155C23—21no—
156C23—31no—
157Q8—61no—
158Q8—61no—
159D4—21no—
160D4—21no—
161D4—122no—
162C2×C4—21no—
163C2×C4—31no—
164C2×C4—31no—
165C2×C4—31no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—61no—
169C2×C4—61no—
170C2×C4—61no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—121no—
175C2×C4C2×C4×D311yescharacteristic
176C2×C4D3×D411yescharacteristic
177C2×C4D3×D411yescharacteristic
178C8—41no—
179C8—41no—
180C8—121no—
181C8—121no—
182C2×C3—81no—
183C2×C3#40811yescharacteristic
184C2×C3#41711yescharacteristic
185C2×C3#42111yescharacteristic
186D3—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—61no—
191C22—61no—
192C22—61no—
193C22D3×C22⋊C411yescharacteristic
194C4—21no—
195C4—41no—
196C4—61no—
197C4—61no—
198C4—61no—
199C4—121no—
200C4—121no—
201C4#90111yescharacteristic
202C4#90711yescharacteristic
203C4#91111yescharacteristic
204C4#91511yescharacteristic
205C3#149511yescharacteristic, complemented, Sylow
206C2—81no—
207C2—121no—
208C2#472711yescharacteristic
209C2#477111yescharacteristic
210C2#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#1495
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. #4837C2semidirect
3. #4685C2semidirect
4. C3×D4⋊C8C2semidirect
5. D4⋊(C3⋊C8)C2semidirect
6. #4484C2semidirect
7. #4456C2semidirect
34. D4⋊C8D3semidirect
205. C3#1495semidirect

Maximal subgroups

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

subgroupindex
#55572
#48372
#46852
C3×D4⋊C82
D4⋊(C3⋊C8)2
#44842
#44562
#14953