Finite Group Databasebeta

← #71883

Subgroups — #71883

Order 384.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C3×#538C211yesmaximal, characteristic
2#5044C211yesmaximal, characteristic, complemented
3#5034C211yesmaximal, characteristic, complemented
4#4949C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4537C211yesmaximal, characteristic, complemented
7#4525C211yesmaximal, characteristic, complemented
8#1618—31nomaximal, Sylow
9C3×#126—21no—
10C3×#124—21no—
11C3×#123—21no—
12C3×C22⋊Q8C2211yescharacteristic
13C3×#121—21no—
14C3×#121C2211yescharacteristic
15C3×#120—21no—
16C3×C4×D4—21no—
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×#217—21no—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4—21no—
21#835—21no—
22#832—21no—
23C3⋊#95C2211yescharacteristic
24#830C2211yescharacteristic
25#830C2211yescharacteristic
26#808—21no—
27#807—21no—
28#538D311yescharacteristic
29#484—31no—
30#481—31no—
31#399—31no—
32#328—31no—
33#327—31no—
34#323—31no—
35C24×C3D411yescharacteristic
36C2×C3×Q8D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4D411yescharacteristic
41C22×C3×C4—21no—
42C22×C3×C4D411yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C22×Dic3—21no—
45C22×Dic3—21no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4D411yescharacteristic
49C3×C4⋊C4D411yescharacteristic
50C3×C22⋊C4—21no—
51C3×C22⋊C4—21no—
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C42—21no—
58#217—42no—
59C4⋊Dic3—21no—
60C4⋊Dic3—21no—
61Dic3⋊C4—42no—
62C2×C3⋊C8—21no—
63C2×C3⋊C8—21no—
64#126—21no—
65#124—21no—
66#123—21no—
67C22⋊Q8C2×D311yescharacteristic
68#121—21no—
69#121C2×D311yescharacteristic
70#120—21no—
71C4×D4—21no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—61no—
74C2×C22⋊C4—61no—
75C2×C22⋊C4C2×D311yescharacteristic
76#107—61no—
77#106—61no—
78#103—61no—
79#102—61no—
80C22⋊C8—31no—
81C22⋊C8—31no—
82#95—31no—
83C23×C3—21no—
84C23×C3—21no—
85C23×C3—21no—
86C23×C3—21no—
87C23×C3—41no—
88C23×C3—41no—
89C23×C3C2×D411yescharacteristic
90C3×Q8—41no—
91C3×D4—41no—
92C3×D4—41no—
93C3×D4—41no—
94C3×D4—81no—
95C3×D4—81no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4—81no—
103C2×C3×C4C2×D411yescharacteristic
104C2×C3×C4C2×D411yescharacteristic
105C2×Dic3—22no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C3⋊C8—41no—
112C3⋊C8—41no—
113C24#6711yescharacteristic
114C2×Q8#6711yescharacteristic
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4#6711yescharacteristic
119C22×C4—21no—
120C22×C4—61no—
121C22×C4—61no—
122C22×C4#6711yescharacteristic
123C22×C4C22×D311yescharacteristic
124C2×C8—61no—
125C2×C8—61no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—122no—
131C4⋊C4#6711yescharacteristic
132C4⋊C4#6711yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—122no—
141C42—21no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3—42no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3—41no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3—81no—
152C22×C3#12011yescharacteristic
153C22×C3#13511yescharacteristic
154C22×C3#13711yescharacteristic
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—41no—
161Dic3—42no—
162Dic3—81no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—41no—
168C23—41no—
169C23C2×#6711yescharacteristic
170Q8—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—81no—
175D4—81no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—62no—
183C2×C4—81no—
184C2×C4—121no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4—121no—
188C2×C4—121no—
189C2×C4C2×#6711yescharacteristic
190C2×C4C2×#6711yescharacteristic
191C8—121no—
192C8—121no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3—81no—
196C2×C3—81no—
197C2×C3#45211yescharacteristic
198C2×C3#45311yescharacteristic
199C2×C3#45811yescharacteristic
200C22—21no—
201C22—21no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—81no—
210C22#95011yescharacteristic
211C22#95111yescharacteristic
212C22#95311yescharacteristic
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—122no—
220C4—241no—
221C3#161811yescharacteristic, complemented, Sylow
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#523911yescharacteristic
227C2#524111yescharacteristic
228C2#524311yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#538C2
socleC22×C3#120
hypercenter#538D3
hypocenterC3#1618
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
2. #5044C2semidirect
3. #5034C2semidirect
4. #4949C2semidirect
5. #4538C2semidirect
6. #4537C2semidirect
7. #4525C2semidirect
221. C3#1618semidirect

Maximal subgroups

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

subgroupindex
C3×#5382
#50442
#50342
#49492
#45382
#45372
#45252
#16183