Finite Group Databasebeta

← #68322

Subgroups — #68322

Order 384.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#5634C211yesmaximal, characteristic
2#5044C211yesmaximal, characteristic, complemented
3#4722C211yesmaximal, characteristic, complemented
4#4677C211yesmaximal, characteristic
5C3×#328C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic
7#4452C211yesmaximal, characteristic, complemented
8#1614—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10#946—21no—
11C2×(C2×D3)⋊C4—21no—
12C2×(C2×D3)⋊C4C2211yescharacteristic
13C2×C4⋊Dic3—21no—
14C2×C4⋊Dic3C2211yescharacteristic
15#924C2211yescharacteristic
16#897—21no—
17#896—21no—
18#896—21no—
19D3×C22⋊C4—21no—
20#878—21no—
21C3×C22⋊C8C2211yescharacteristic
22C3×#95C2211yescharacteristic
23#835—21no—
24#830C2211yescharacteristic
25#818—21no—
26#816—21no—
27#807—21no—
28#543—31no—
29#484—31no—
30#484—31no—
31#399—31no—
32#328—31no—
33#328D311yescharacteristic, complemented
34#323—31no—
35C23×D3D411yescharacteristic
36C2×C3×Q8D411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C22×Dic3—21no—
40C22×Dic3D411yescharacteristic
41C2×C4×D3—21no—
42C2×C4×D3—21no—
43C2×Dic6—21no—
44C2×Dic6D411yescharacteristic
45C2×C3×C8—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4D411yescharacteristic
48C3×C22⋊C4—21no—
49#217—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52(C2×D3)⋊C4—21no—
53(C2×D3)⋊C4—21no—
54(C2×D3)⋊C4—42no—
55C4⋊Dic3—21no—
56C4⋊Dic3—21no—
57C4⋊Dic3—21no—
58C4⋊Dic3—42no—
59C4⋊Dic3D411yescharacteristic
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63C2×C3⋊C8—21no—
64C22⋊Q8—31no—
65C22⋊Q8—61no—
66C22⋊Q8—61no—
67C22⋊Q8—61no—
68C22⋊Q8—61no—
69C22⋊Q8—61no—
70C22⋊Q8C2×D311yescharacteristic
71C2×C4⋊C4—31no—
72C2×C4⋊C4—61no—
73C2×C22⋊C4—31no—
74C2×C22⋊C4—61no—
75C2×C22⋊C4—61no—
76#107—61no—
77#107—61no—
78#103—61no—
79#103—61no—
80C22⋊C8—31no—
81C22⋊C8C2×D311yescharacteristic
82#95C2×D311yescharacteristic
83C23×C3C2×D411yescharacteristic
84C22×D3—21no—
85C22×D3—21no—
86C22×D3—41no—
87C22×D3—41no—
88C22×D3Q1612yes—
89C3×Q8—41no—
90C2×C3×C4—22no—
91C2×C3×C4—21no—
92C2×C3×C4—21no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4C2×D411yescharacteristic
97C2×C3×C4C2×D411yescharacteristic
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—21no—
101C2×Dic3—41no—
102C2×Dic3—41no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C4×D3—81no—
106C4×D3—81no—
107Dic6—41no—
108Dic6—81no—
109C3×C8—41no—
110C3⋊C8—41no—
111C24—31no—
112C2×Q8—31no—
113C2×Q8—61no—
114C2×Q8D1211yescharacteristic
115C22×C4—21no—
116C22×C4—31no—
117C22×C4—61no—
118C22×C4—61no—
119C22×C4—61no—
120C22×C4C22×D311yescharacteristic
121C2×C8—21no—
122C2×C8—61no—
123C4⋊C4—21no—
124C4⋊C4—31no—
125C4⋊C4—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—61no—
131C4⋊C4—122no—
132C4⋊C4D1211yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C22⋊C4—61no—
139C22⋊C4—122no—
140C22×C3—21no—
141C22×C3—21no—
142C22×C3#12011yescharacteristic
143C22×C3C2×Q1611yescharacteristic
144C22×C3#13711yescharacteristic
145C2×D3—22no—
146C2×D3—41no—
147C2×D3—41no—
148C2×D3—42no—
149C2×D3—81no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—42no—
153C3×C4—41no—
154C3×C4—41no—
155Dic3—41no—
156Dic3—41no—
157Dic3—41no—
158Dic3—81no—
159C23—32no—
160C23—61no—
161C23—61no—
162C23—121no—
163C23—121no—
164C23D3×D411yescharacteristic
165Q8—41no—
166Q8—121no—
167Q8—241no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—121no—
178C2×C4—121no—
179C2×C4—121no—
180C2×C4—121no—
181C2×C4—241no—
182C2×C4—241no—
183C2×C4C2×D1211yescharacteristic
184C2×C4D3×D411yescharacteristic
185C8—41no—
186C8—121no—
187C2×C3—21no—
188C2×C3—21no—
189C2×C3#45111yescharacteristic
190C2×C3#45311yescharacteristic
191C2×C3#45811yescharacteristic
192D3—42no—
193C22—21no—
194C22—21no—
195C22—62no—
196C22—122no—
197C22—121no—
198C22—121no—
199C22—241no—
200C22#88211yescharacteristic
201C22#91511yescharacteristic
202C22D3×Q1611yescharacteristic
203C4—21no—
204C4—21no—
205C4—42no—
206C4—41no—
207C4—41no—
208C4—121no—
209C4—121no—
210C4—121no—
211C4—241no—
212C3#161411yescharacteristic, complemented, Sylow
213C2—21no—
214C2—21no—
215C2—122no—
216C2#474211yescharacteristic
217C2#480911yescharacteristic
218C2#482211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radicalC23×D3D4
hypercenter#328D3
hypocenterC3#1614
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #5044C2semidirect
3. #4722C2semidirect
5. C3×#328C2semidirect
7. #4452C2semidirect
33. #328D3semidirect
212. C3#1614semidirect

Maximal subgroups

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

subgroupindex
#56342
#50442
#47222
#46772
C3×#3282
#44682
#44522
#16143