Finite Group Databasebeta

← #69705

Subgroups — #69705

Order 384.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#5515C211yesmaximal, characteristic
2C3×#395C211yesmaximal, characteristic, complemented
3#5228C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic
5#4528C211yesmaximal, characteristic, complemented
6#4452C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic
8#1603—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C2×C3×C4⋊C4—21no—
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×#216—21no—
13C2×#215—21no—
14#930C2211yescharacteristic
15#929—21no—
16C4×#67—21no—
17#924C2211yescharacteristic
18#898—21no—
19#896—21no—
20#894—21no—
21#889—21no—
22C3×#95—21no—
23C3×#95C2211yescharacteristic
24#835—21no—
25#835—21no—
26#830C2211yescharacteristic
27#830C2211yescharacteristic
28#539—31no—
29#451—31no—
30#448—31no—
31#395D311yescharacteristic, complemented
32#328—31no—
33#327—31no—
34#324—31no—
35C2×C3×Q8—21no—
36C2×C3×Q8—21no—
37C2×C3×Q8—42no—
38C22×C3×C4—21no—
39C22×C3×C4—21no—
40C22×C3×C4D411yescharacteristic
41C22×C3×C4D411yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C2×#67—21no—
44C2×D12D411yescharacteristic
45C2×C4×D3—21no—
46C2×Dic6D411yescharacteristic
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—42no—
50#217—21no—
51#216—42no—
52#215—42no—
53(C2×D3)⋊C4—21no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57C4⋊Dic3D411yescharacteristic
58C4⋊Dic3D411yescharacteristic
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63C4×Dic3—21no—
64C2×C3⋊C8—21no—
65C2×C3⋊C8—21no—
66C22×Q8C2×D311yescharacteristic
67C2×Q16—61no—
68C2×SD16—61no—
69#126—61no—
70#125—61no—
71#123—61no—
72#123—61no—
73C22⋊Q8—31no—
74C22⋊Q8—61no—
75#121—31no—
76C4×D4—61no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2×D311yescharacteristic
79#103—61no—
80#103—61no—
81C22⋊C8—31no—
82C22⋊C8—31no—
83#95—21no—
84#95C2×D311yescharacteristic
85C23×C3C2×D411yescharacteristic
86C22×D3—21no—
87C3×Q8—22no—
88C3×Q8—22no—
89C3×Q8—41no—
90C3×Q8—41no—
91C2×C3×C4—22no—
92C2×C3×C4—22no—
93C2×C3×C4—21no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4—41no—
103C2×C3×C4C2×D411yescharacteristic
104C2×C3×C4C2×D411yescharacteristic
105#67—41no—
106#67—41no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110D12—41no—
111C4×D3—81no—
112Dic6—41no—
113C3⋊C8—41no—
114C3⋊C8—41no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—31no—
118C2×Q8—42no—
119C2×D4—31no—
120C2×D4—61no—
121C22×C4—21no—
122C22×C4—21no—
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—31no—
134C4⋊C4—31no—
135C4⋊C4—42no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C22⋊C4—61no—
141C22⋊C4—61no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C42—61no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3#12011yescharacteristic
149C22×C3#13511yescharacteristic
150C22×C3#13711yescharacteristic
151C2×D3—41no—
152C2×D3—41no—
153C2×D3—41no—
154C3×C4—21no—
155C3×C4—21no—
156C3×C4—42no—
157C3×C4—41no—
158C3×C4—41no—
159C3×C4—42no—
160Dic3—41no—
161Dic3—41no—
162Dic3—41no—
163C23—61no—
164C23C2×#6711yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169Q8—121no—
170D4—121no—
171D4—121no—
172D4—121no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—241no—
189C2×C4D3×D411yescharacteristic
190C2×C4D3×D411yescharacteristic
191C8—121no—
192C8—121no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3#45211yescharacteristic
196C2×C3#45611yescharacteristic
197C2×C3B(4,𝔽2)11yescharacteristic
198D3—81no—
199C22—21no—
200C22—21no—
201C22—121no—
202C22—121no—
203C22—121no—
204C22#93711yescharacteristic
205C22#94211yescharacteristic
206C22#95011yescharacteristic
207C4—21no—
208C4—21no—
209C4—42no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—121no—
214C4—121no—
215C4—121no—
216C3#160311yescharacteristic, complemented, Sylow
217C2—21no—
218C2—21no—
219C2—241no—
220C2#503211yescharacteristic
221C2#504711yescharacteristic
222C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#395C2
socleC22×C3#120
prime radical#930C22
hypercenter#395D3
hypocenterC3#1603
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C3×#395C2semidirect
3. #5228C2semidirect
5. #4528C2semidirect
6. #4452C2semidirect
31. #395D3semidirect
216. C3#1603semidirect

Maximal subgroups

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

subgroupindex
#55152
C3×#3952
#52282
#52272
#45282
#44522
#44512
#16033