Finite Group Databasebeta

← #69920

Subgroups — #69920

Order 384.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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×#929C211yesmaximal, characteristic
2C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
3#5207C211yesmaximal, characteristic, complemented
4#4469C212yesmaximal, complemented
5#4468C212yesmaximal, complemented
6#1468—31nomaximal, Sylow
7C23×C3×C4C2211yescharacteristic
8C22×#67C411yescharacteristic
9C22×C3×C8—21no—
10C2×#217C411yescharacteristic
11#929—21no—
12#929—21no—
13#929—21no—
14#929—21no—
15C2×(C2×D3)⋊C4C411yescharacteristic
16C2×(C2×D3)⋊C4C2211yescharacteristic
17C2×Dic3⋊C4C411yescharacteristic
18C2×Dic3⋊C4C2211yescharacteristic
19C2×#208—21no—
20C3×C22⋊C8—21no—
21C3×C22⋊C8C2212yes—
22#830—21no—
23#830C2212yes—
24C2×#123—31no—
25#407—31no—
26C2×C22⋊C8D311yescharacteristic, complemented
27#324—32no—
28#323—32no—
29C24×C3C2×C411yescharacteristic
30C23×D3C2×C411yescharacteristic
31C22×C3×C4—42no—
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4C2311yescharacteristic
38C2×#67—21no—
39C2×#67—21no—
40C2×#67—42no—
41C22×Dic3C2×C411yescharacteristic
42C22×Dic3C2×C411yescharacteristic
43C22×Dic3C2×C411yescharacteristic
44C2×C3×C8—22no—
45C2×C3×C8—42no—
46#217—21no—
47#217—21no—
48(C2×D3)⋊C4—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52Dic3⋊C4—21no—
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56#208—42no—
57C2×C3⋊C8—22no—
58C22×D4—31no—
59C23×C4C2×D311yescharacteristic
60C2×M16—61no—
61C22×C8—21no—
62#123—61no—
63#123—61no—
64#123—61no—
65#123—61no—
66C2×C4⋊C4—31no—
67C2×C4⋊C4—31no—
68C2×C22⋊C4—31no—
69C2×C22⋊C4—31no—
70C2×C22⋊C4—31no—
71C22⋊C8—21no—
72C22⋊C8—32no—
73C22⋊C8—61no—
74C22⋊C8C2×D312yes—
75C23×C3—22no—
76C23×C3—22no—
77C23×C3—41no—
78C23×C3C22⋊C411yescharacteristic
79C23×C3C22⋊C411yescharacteristic
80C23×C3C22×C411yescharacteristic
81C22×D3—21no—
82C22×D3—22no—
83C22×D3—41no—
84C22×D3—41no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4C22⋊C411yescharacteristic
92C2×C3×C4C22⋊C411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95#67—42no—
96#67—42no—
97C2×Dic3—21no—
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—22no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C3×C8—42no—
104C3⋊C8—42no—
105C24—31no—
106C24C4×D311yescharacteristic
107C2×D4—61no—
108C2×D4—61no—
109C2×D4—122no—
110C22×C4—31no—
111C22×C4—31no—
112C22×C4—31no—
113C22×C4—42no—
114C22×C4C4×D311yescharacteristic
115C22×C4D1211yescharacteristic
116C22×C4D1211yescharacteristic
117C22×C4#6711yescharacteristic
118C22×C4#6711yescharacteristic
119C22×C4C22×D311yescharacteristic
120M16—122no—
121C2×C8—22no—
122C2×C8—42no—
123C2×C8—62no—
124C4⋊C4—61no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C22⋊C4—61no—
129C22⋊C4—61no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C22×C3—21no—
135C22×C3—21no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—44no—
142C22×C3#9912yes—
143C22×C3C2×C22⋊C411yescharacteristic
144C22×C3#13112yes—
145C2×D3—41no—
146C2×D3—41no—
147C2×D3—41no—
148C2×D3—82no—
149C3×C4—21no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—21no—
153Dic3—41no—
154Dic3—41no—
155Dic3—41no—
156C23—22no—
157C23—22no—
158C23—41no—
159C23—61no—
160C23—62no—
161C23—121no—
162C23—121no—
163C23(C2×D3)⋊C411yescharacteristic
164C23(C2×D3)⋊C411yescharacteristic
165C23C2×C4×D311yescharacteristic
166D4—122no—
167D4—122no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—62no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4—121no—
179C2×C4—121no—
180C2×C4(C2×D3)⋊C411yescharacteristic
181C2×C4(C2×D3)⋊C411yescharacteristic
182C2×C4C2×D1211yescharacteristic
183C2×C4C2×#6711yescharacteristic
184C8—42no—
185C8—122no—
186C2×C3—22no—
187C2×C3—22no—
188C2×C3—41no—
189C2×C3#40811yescharacteristic
190C2×C3C2×#9911yescharacteristic
191C2×C3#41311yescharacteristic
192D3—81no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—44no—
201C22—121no—
202C22—121no—
203C22—121no—
204C22—242no—
205C22#80612yes—
206C22#90112yes—
207C22C2×(C2×D3)⋊C411yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—121no—
213C4—121no—
214C4—121no—
215C3#146811yescharacteristic, complemented, Sylow
216C2—22no—
217C2—22no—
218C2—41no—
219C2—241no—
220C2C2×#80611yescharacteristic
221C2#511011yescharacteristic
222C2#513211yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊C4
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×C22⋊C8C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#67C4
cosocleC22×C3×C4C23
hypercenterC2×C22⋊C8D3
hypocenterC3#1468
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊C8C2semidirect
3. #5207C2semidirect
4. #4469C2semidirect
5. #4468C2semidirect
26. C2×C22⋊C8D3semidirect
215. C3#1468semidirect

Maximal subgroups

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

subgroupindex
C2×#9292
C2×C3×C22⋊C82
#52072
#44692
#44682
#14683