Finite Group Databasebeta

← #69141

Subgroups — #69141

Order 384.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#5651C211yesmaximal, characteristic, complemented
2#5053C211yesmaximal, characteristic
3#4746C211yesmaximal, characteristic
4#4742C211yesmaximal, characteristic
5C3×#352C211yesmaximal, characteristic, complemented
6#4476C211yesmaximal, characteristic, complemented
7#4473C211yesmaximal, characteristic, complemented
8#2201—31nomaximal, Sylow
9#1004C2211yescharacteristic
10C3×#123C2211yescharacteristic
11#936C2211yescharacteristic
12#935C2211yescharacteristic
13#929—21no—
14#924—21no—
15#909—21no—
16#889—21no—
17#888—21no—
18#886—21no—
19#879—21no—
20#879—21no—
21#878—21no—
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#825—21no—
26#806—21no—
27#806—21no—
28#563—31no—
29#458—31no—
30#458—31no—
31#455—31no—
32#354—31no—
33#352—31no—
34#352D311yescharacteristic, complemented
35C2×C3×D4C2311yescharacteristic
36C22×C3×C4D411yescharacteristic
37C2×#67—21no—
38C2×#67—21no—
39C22×Dic3D411yescharacteristic
40#237—41no—
41D3×D4—41no—
42#235—21no—
43C2×Dic6—21no—
44C2×Dic6D411yescharacteristic
45C3×M16—21no—
46C3×C4⋊C4—21no—
47C3×C22⋊C4—21no—
48C3×C22⋊C4—21no—
49C3×C22⋊C4D411yescharacteristic
50#217—21no—
51#217—21no—
52#217—21no—
53#217—21no—
54#217D411yescharacteristic
55(C2×D3)⋊C4—41no—
56(C2×D3)⋊C4—41no—
57C4⋊Dic3—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—41no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65Dic3⋊C4—41no—
66C4×Dic3—21no—
67C4×Dic3D411yescharacteristic
68Dic12—41no—
69#204—41no—
70H(5,𝔽2)—31no—
71#137—61no—
72C4⋊Q8—61no—
73#126—61no—
74#126—61no—
75#125—61no—
76#124—31no—
77#123—31no—
78#123—61no—
79#123—61no—
80#123C2×D311yescharacteristic
81C22⋊Q8—61no—
82C22⋊Q8—61no—
83#104—61no—
84#100C2×D311yescharacteristic
85#99—31no—
86#99—61no—
87#99—61no—
88#99C2×D311yescharacteristic
89C23×C3C2×D411yescharacteristic
90C23×C3C2×D411yescharacteristic
91C22×D3—21no—
92C22×D3—21no—
93C3×D4—41no—
94C2×C3×C4—21no—
95C2×C3×C4—21no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4C2×D411yescharacteristic
99#67—41no—
100#67—41no—
101#67—41no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110D12—21no—
111C4×D3—41no—
112Dic6—21no—
113Dic6—41no—
114Dic6—81no—
115C3×C8—41no—
116#41—61no—
117#41—121no—
118C2×Q8—31no—
119C2×Q8—61no—
120C2×D4—61no—
121C2×D4—61no—
122C2×D4—121no—
123C2×D4C22×D311yescharacteristic
124C22×C4—31no—
125C22×C4D1211yescharacteristic
126Q16—121no—
127SD16—121no—
128M16—21no—
129C4⋊C4—21no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C4⋊C4—61no—
136C4⋊C4—61no—
137C4⋊C4—121no—
138C4⋊C4—121no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—31no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—121no—
147C22⋊C4—121no—
148C22⋊C4D1211yescharacteristic
149C42—31no—
150C42—61no—
151C22×C3—21no—
152C22×C3—21no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3#12011yescharacteristic
156C2×D3—41no—
157C2×D3—81no—
158C2×D3—81no—
159C3×C4—21no—
160C3×C4—41no—
161C3×C4—41no—
162C3×C4—81no—
163Dic3—41no—
164Dic3—41no—
165Dic3—41no—
166Dic3—41no—
167Dic3—41no—
168Dic3—81no—
169C23—61no—
170C23—61no—
171C23C2×D1211yescharacteristic
172C23D3×D411yescharacteristic
173Q8—61no—
174Q8—121no—
175Q8—241no—
176D4—41no—
177D4—61no—
178D4—121no—
179D4—121no—
180D4—121no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4—121no—
194C2×C4D3×D411yescharacteristic
195C8—41no—
196C2×C3—21no—
197C2×C3—41no—
198C2×C3—41no—
199C2×C3B(4,𝔽2)11yescharacteristic
200D3—81no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—121no—
206C22—241no—
207C22—241no—
208C22#88211yescharacteristic
209C4—21no—
210C4—41no—
211C4—41no—
212C4—81no—
213C4—121no—
214C4—121no—
215C4—121no—
216C4—121no—
217C4—121no—
218C4—241no—
219C3#220111yescharacteristic, complemented, Sylow
220C2—21no—
221C2—41no—
222C2—41no—
223C2—241no—
224C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×#352C2
socleC2×C3B(4,𝔽2)
prime radical#1004C22
hypercenter#352D3
hypocenterC3#2201
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #5651C2semidirect
5. C3×#352C2semidirect
6. #4476C2semidirect
7. #4473C2semidirect
34. #352D3semidirect
219. C3#2201semidirect

Maximal subgroups

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

subgroupindex
#56512
#50532
#47462
#47422
C3×#3522
#44762
#44732
#22013