Finite Group Databasebeta

← #69140

Subgroups — #69140

Order 384.

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

normal subgroups only

This group has 246 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#5664C211yesmaximal, characteristic, complemented
2#5053C211yesmaximal, characteristic, complemented
3#4747C211yesmaximal, characteristic
4#4742C211yesmaximal, characteristic, complemented
5C3×#352C211yesmaximal, characteristic, complemented
6#4476C211yesmaximal, characteristic, complemented
7#4474C211yesmaximal, characteristic, complemented
8#2198—31nomaximal, Sylow
9#1004C2211yescharacteristic
10C3×#123C2211yescharacteristic
11#940C2211yescharacteristic
12#935C2211yescharacteristic
13#929—21no—
14C4×#67—21no—
15#908—21no—
16#894—21no—
17#887—21no—
18#886—21no—
19#885—21no—
20#885—21no—
21#67⋊C4—21no—
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#825—21no—
26#806—21no—
27#806—21no—
28#559—31no—
29#458—31no—
30#458—31no—
31#453—31no—
32#353—31no—
33#352—31no—
34#352D311yescharacteristic, complemented
35C2×C3×D4C2311yescharacteristic
36C22×C3×C4D411yescharacteristic
37C2×#67—21no—
38C2×#67—21no—
39C2×#67—21no—
40C2×#67—21no—
41C22×Dic3D411yescharacteristic
42#237—41no—
43D3×D4—41no—
44#235—21no—
45C2×D12D411yescharacteristic
46C2×C4×D3—21no—
47C2×Dic6—21no—
48C3×M16—21no—
49C3×C4⋊C4—21no—
50C3×C22⋊C4—21no—
51C3×C22⋊C4—21no—
52C3×C22⋊C4D411yescharacteristic
53#217—21no—
54#217—21no—
55#217D411yescharacteristic
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—41no—
60(C2×D3)⋊C4—41no—
61C4⋊Dic3—41no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—41no—
65C4×Dic3—21no—
66C4×Dic3—21no—
67C4×Dic3D411yescharacteristic
68D24—41no—
69#204—41no—
70H(5,𝔽2)—31no—
71AGL(1,ℤ/8)—61no—
72#127—31no—
73#124—61no—
74#124—61no—
75#123—31no—
76#123—61no—
77#123—61no—
78#123—61no—
79#123C2×D311yescharacteristic
80C4×Q8—61no—
81C4×D4—61no—
82C4×D4—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—
93C22×D3—21no—
94C3×D4—41no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4C2×D411yescharacteristic
100#67—42no—
101#67—42no—
102#67—41no—
103#67—41no—
104#67—41no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—41no—
111C2×Dic3—41no—
112D12—21no—
113D12—41no—
114C4×D3—41no—
115C4×D3—81no—
116Dic6—21no—
117Dic6—42no—
118C3×C8—41no—
119#41—61no—
120#41—121no—
121C2×Q8—61no—
122C2×D4—31no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4—61no—
126C2×D4—61no—
127C2×D4—121no—
128C2×D4C22×D311yescharacteristic
129C22×C4—31no—
130C22×C4—61no—
131C22×C4D1211yescharacteristic
132SD16—121no—
133D8—121no—
134M16—21no—
135C4⋊C4—21no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—121no—
139C4⋊C4—121no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C22⋊C4—31no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—121no—
149C22⋊C4—121no—
150C22⋊C4D1211yescharacteristic
151C42—31no—
152C42—61no—
153C42—61no—
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3#12011yescharacteristic
159C2×D3—41no—
160C2×D3—41no—
161C2×D3—41no—
162C2×D3—41no—
163C2×D3—81no—
164C2×D3—81no—
165C3×C4—21no—
166C3×C4—41no—
167C3×C4—41no—
168C3×C4—81no—
169Dic3—22no—
170Dic3—41no—
171Dic3—41no—
172Dic3—41no—
173Dic3—81no—
174C23—61no—
175C23—61no—
176C23—61no—
177C23C2×D1211yescharacteristic
178C23D3×D411yescharacteristic
179Q8—61no—
180Q8—122no—
181D4—41no—
182D4—61no—
183D4—121no—
184D4—122no—
185D4—122no—
186D4—121no—
187D4—121no—
188D4—121no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—121no—
201C2×C4—241no—
202C2×C4D3×D411yescharacteristic
203C8—41no—
204C2×C3—21no—
205C2×C3—41no—
206C2×C3—41no—
207C2×C3B(4,𝔽2)11yescharacteristic
208D3—81no—
209D3—81no—
210C22—21no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—121no—
215C22—121no—
216C22—121no—
217C22—121no—
218C22—241no—
219C22—241no—
220C22#88211yescharacteristic
221C4—21no—
222C4—41no—
223C4—41no—
224C4—62no—
225C4—81no—
226C4—121no—
227C4—121no—
228C4—121no—
229C4—241no—
230C3#219811yescharacteristic, complemented, Sylow
231C2—21no—
232C2—41no—
233C2—41no—
234C2—241no—
235C2—241no—
236C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×#352C2
socleC2×C3B(4,𝔽2)
prime radical#4747C2
hypercenter#352D3
hypocenterC3#2198
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. #5664C2semidirect
2. #5053C2semidirect
4. #4742C2semidirect
5. C3×#352C2semidirect
6. #4476C2semidirect
7. #4474C2semidirect
34. #352D3semidirect
230. C3#2198semidirect

Maximal subgroups

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

subgroupindex
#56642
#50532
#47472
#47422
C3×#3522
#44762
#44742
#21983