Finite Group Databasebeta

← #71001

Subgroups — #71001

Order 384.

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

normal subgroups only

This group has 260 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#5666C211yesmaximal, characteristic, complemented
2C3×#458C211yesmaximal, characteristic, complemented
3#5244C211yesmaximal, characteristic, complemented
4#5241C211yesmaximal, characteristic, complemented
5#4539C211yesmaximal, characteristic, complemented
6#4474C211yesmaximal, characteristic, complemented
7#4471C211yesmaximal, characteristic, complemented
8#2200—31nomaximal, Sylow
9C3×H(5,𝔽2)C2211yescharacteristic
10C3×#123—21no—
11C3×#123C2211yescharacteristic
12#949—21no—
13#940C2211yescharacteristic
14#937—21no—
15#937C2211yescharacteristic
16#930—21no—
17#898—21no—
18#895—21no—
19#884—21no—
20#883—21no—
21#882—21no—
22C3×#99—21no—
23C3×#99C2211yescharacteristic
24#837—21no—
25#834—21no—
26#834C2211yescharacteristic
27#833C2211yescharacteristic
28#560—31no—
29#458D311yescharacteristic, complemented
30B(4,𝔽2)—31no—
31#453—31no—
32#353—31no—
33#352—31no—
34AΣL(1,𝔽16)—31no—
35C23×D3D411yescharacteristic
36C3×#41—21no—
37C3×#41—41no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4—41no—
41C2×C3×D4C2311yescharacteristic
42C22×C3×C4D411yescharacteristic, complemented
43C2×#67—21no—
44C2×#67—21no—
45C2×#67—21no—
46C2×#67—41no—
47C2×D12—21no—
48C2×D12—21no—
49C2×D12D411yescharacteristic, complemented
50C2×C4×D3—21no—
51C3×C4⋊C4—21no—
52C3×C4⋊C4—41no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—41no—
56C3×C22⋊C4D411yescharacteristic, complemented
57#217—21no—
58#217D411yescharacteristic
59#215—41no—
60#213—41no—
61(C2×D3)⋊C4—21no—
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64(C2×D3)⋊C4—21no—
65(C2×D3)⋊C4—41no—
66C4⋊Dic3—21no—
67Dic3⋊C4—21no—
68C4×Dic3D411yescharacteristic, complemented
69#208—21no—
70H(5,𝔽2)C2×D311yescharacteristic
71AGL(1,ℤ/8)—61no—
72#127—31no—
73#126—61no—
74#123—21no—
75#123—61no—
76#123C2×D311yescharacteristic
77#121—61no—
78#121—61no—
79#121—61no—
80#120—31no—
81#120—61no—
82#120—61no—
83#104—61no—
84#100—31no—
85#99—21no—
86#99—31no—
87#99—61no—
88#99C2×D311yescharacteristic
89C23×C3—21no—
90C23×C3—21no—
91C23×C3C2×D411yescharacteristic
92C23×C3C2×D411yescharacteristic
93C22×D3—21no—
94C22×D3—21no—
95C22×D3—41no—
96C22×D3—81no—
97C3×Q8—21no—
98C3×D4—21no—
99C3×D4—41no—
100C3×D4—41no—
101C3×D4—41no—
102C3×D4—41no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—41no—
108C2×C3×C4—41no—
109C2×C3×C4—41no—
110C2×C3×C4C2×D411yescharacteristic
111#67—81no—
112#67—81no—
113#67—81no—
114#67—82no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—41no—
118D12—41no—
119D12—81no—
120D12—81no—
121C4×D3—81no—
122C3⋊C8—41no—
123C24—31no—
124#41—21no—
125#41—41no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—31no—
129C2×D4—41no—
130C2×D4—61no—
131C2×D4—61no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—121no—
136C2×D4C22×D311yescharacteristic
137C22×C4—61no—
138C22×C4#6711yescharacteristic, complemented
139SD16—121no—
140D8—121no—
141M16—61no—
142C4⋊C4—21no—
143C4⋊C4—41no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C22⋊C4—21no—
147C22⋊C4—21no—
148C22⋊C4—31no—
149C22⋊C4—41no—
150C22⋊C4—61no—
151C22⋊C4—61no—
152C22⋊C4—61no—
153C22⋊C4—61no—
154C22⋊C4—61no—
155C22⋊C4—121no—
156C22⋊C4#6711yescharacteristic, complemented
157C42—31no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—81no—
164C22×C3—81no—
165C22×C3#12011yescharacteristic
166C2×D3—41no—
167C2×D3—41no—
168C2×D3—41no—
169C2×D3—41no—
170C2×D3—42no—
171C2×D3—41no—
172C2×D3—41no—
173C2×D3—81no—
174C3×C4—21no—
175C3×C4—41no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—81no—
179Dic3—41no—
180Dic3—41no—
181Dic3—81no—
182C23—21no—
183C23—21no—
184C23—61no—
185C23—61no—
186C23—121no—
187C23—241no—
188C23D3×D411yescharacteristic
189C23C2×#6711yescharacteristic
190Q8—21no—
191D4—21no—
192D4—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—121no—
197D4—241no—
198D4—241no—
199D4—241no—
200D4—241no—
201D4—241no—
202D4—242no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—121no—
213C2×C4—241no—
214C2×C4D3×D411yescharacteristic
215C8—121no—
216C2×C3—21no—
217C2×C3—41no—
218C2×C3—41no—
219C2×C3—81no—
220C2×C3B(4,𝔽2)11yescharacteristic
221D3—81no—
222D3—81no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—81no—
229C22—81no—
230C22—121no—
231C22—121no—
232C22—121no—
233C22—121no—
234C22—122no—
235C22—121no—
236C22—121no—
237C22—241no—
238C22#93711yescharacteristic
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—81no—
244C4—121no—
245C4—121no—
246C4—241no—
247C3#220011yescharacteristic, complemented, Sylow
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—241no—
253C2—241no—
254C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×#458C2
socleC2×C3B(4,𝔽2)
hypercenter#458D3
hypocenterC3#2200
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #5666C2semidirect
2. C3×#458C2semidirect
3. #5244C2semidirect
4. #5241C2semidirect
5. #4539C2semidirect
6. #4474C2semidirect
7. #4471C2semidirect
29. #458D3semidirect
42. C22×C3×C4D4semidirect
49. C2×D12D4semidirect
56. C3×C22⋊C4D4semidirect
68. C4×Dic3D4semidirect
138. C22×C4#67semidirect
156. C22⋊C4#67semidirect
247. C3#2200semidirect

Maximal subgroups

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

subgroupindex
#56662
C3×#4582
#52442
#52412
#45392
#44742
#44712
#22003