Finite Group Databasebeta

← #71889

Subgroups — #71889

Order 384.

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

normal subgroups only

This group has 244 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×#539C211yesmaximal, characteristic
2#5050C211yesmaximal, characteristic, complemented
3#5040C211yesmaximal, characteristic
4#4979C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4537C211yesmaximal, characteristic
7#4528C211yesmaximal, characteristic, complemented
8#1603—31nomaximal, Sylow
9C22×Dic6C2211yescharacteristic
10C3×#126—21no—
11C3×#125—21no—
12C3×#123—21no—
13C3×#123—21no—
14C3×C22⋊Q8—21no—
15C3×C22⋊Q8C2211yescharacteristic
16C3×#121C2211yescharacteristic
17C3×C4×D4—21no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×#216—21no—
20C2×#214—21no—
21C2×Dic3⋊C4—21no—
22C3⋊#95—21no—
23C3⋊#95C2211yescharacteristic
24#830C2211yescharacteristic
25#830C2211yescharacteristic
26#810—21no—
27#810—21no—
28#539D311yescharacteristic
29#451—31no—
30#448—31no—
31#395—31no—
32#328—31no—
33#327—31no—
34#324—31no—
35C2×C3×Q8D411yescharacteristic
36C2×C3×D4—21no—
37C2×C3×D4D411yescharacteristic
38C22×C3×C4—21no—
39C22×C3×C4D411yescharacteristic
40C22×C3×C4D411yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C22×Dic3—21no—
43C22×Dic3—21no—
44C2×Dic6—21no—
45C2×Dic6—21no—
46C2×Dic6—42no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4—21no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4D411yescharacteristic
54C3×C4⋊C4D411yescharacteristic
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4—21no—
58C3×C22⋊C4—21no—
59C3×C22⋊C4—21no—
60C3×C42—21no—
61#216—42no—
62#214—42no—
63Dic3⋊C4—42no—
64C2×C3⋊C8—21no—
65C2×C3⋊C8—21no—
66C22×Q8—31no—
67C2×Q16—61no—
68C2×SD16—61no—
69#126—21no—
70#125—21no—
71#123—21no—
72#123—21no—
73C22⋊Q8—21no—
74C22⋊Q8C2×D311yescharacteristic
75#121C2×D311yescharacteristic
76C4×D4—21no—
77C2×C4⋊C4—61no—
78C2×C4⋊C4C2×D311yescharacteristic
79#103—61no—
80#103—61no—
81C22⋊C8—31no—
82C22⋊C8—31no—
83#95—31no—
84#95—61no—
85C23×C3—21no—
86C23×C3C2×D411yescharacteristic
87C3×Q8—41no—
88C3×D4—41no—
89C3×D4—41no—
90C3×D4—41no—
91C2×C3×C4—21no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—21no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4—81no—
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4C2×D411yescharacteristic
103C2×Dic3—22no—
104C2×Dic3—22no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109Dic6—22no—
110Dic6—22no—
111Dic6—41no—
112Dic6—41no—
113C3⋊C8—41no—
114C3⋊C8—41no—
115C2×Q8—61no—
116C2×Q8—61no—
117C2×Q8—122no—
118C2×Q8#6711yescharacteristic
119C2×D4—21no—
120C2×D4#6711yescharacteristic
121C22×C4—21no—
122C22×C4—61no—
123C22×C4—61no—
124C22×C4#6711yescharacteristic
125C22×C4#6711yescharacteristic
126C22×C4C22×D311yescharacteristic
127Q16—122no—
128SD16—122no—
129C2×C8—61no—
130C2×C8—61no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—122no—
138C4⋊C4#6711yescharacteristic
139C4⋊C4#6711yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C42—21no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3#12011yescharacteristic
152C22×C3#13511yescharacteristic
153C22×C3#13711yescharacteristic
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—41no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—41no—
160C3×C4—41no—
161Dic3—42no—
162Dic3—42no—
163C23—21no—
164C23C2×#6711yescharacteristic
165Q8—41no—
166Q8—62no—
167Q8—62no—
168Q8—121no—
169Q8—121no—
170D4—41no—
171D4—41no—
172D4—41no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—81no—
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#45211yescharacteristic
197C2×C3#45611yescharacteristic
198C2×C3B(4,𝔽2)11yescharacteristic
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22#95011yescharacteristic
205C22#95111yescharacteristic
206C22#95311yescharacteristic
207C4—21no—
208C4—21no—
209C4—41no—
210C4—41no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—122no—
215C4—122no—
216C3#160311yescharacteristic, complemented, Sylow
217C2—21no—
218C2—21no—
219C2—81no—
220C2#523911yescharacteristic
221C2#524411yescharacteristic
222C2#524611yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#539C2
socleC22×C3#120
prime radicalC3×#121C22
hypercenter#539D3
hypocenterC3#1603
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
2. #5050C2semidirect
4. #4979C2semidirect
5. #4538C2semidirect
7. #4528C2semidirect
216. C3#1603semidirect

Maximal subgroups

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

subgroupindex
C3×#5392
#50502
#50402
#49792
#45382
#45372
#45282
#16033