Finite Group Databasebeta

← #70972

Subgroups — #70972

Order 384.

308 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 51 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#5605C211yesmaximal, characteristic
2C3×#448C211yesmaximal, characteristic, complemented
3#5231C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4468C211yesmaximal, characteristic, complemented
8#1601—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic, complemented
10C2×D3×D4—21no—
11C2×C3×SD16—21no—
12C3×#121C2211yescharacteristic
13C2×#215—21no—
14#939—21no—
15#937—21no—
16#930C2211yescharacteristic
17C2×(C2×D3)⋊C4—21no—
18C2×(C2×D3)⋊C4C2211yescharacteristic
19#894—21no—
20#883—21no—
21#882—21no—
22C3×#103—21no—
23C3×C22⋊C8C2211yescharacteristic, complemented
24#835—21no—
25C3⋊#95—21no—
26C3⋊#95C2211yescharacteristic, complemented
27#830C2211yescharacteristic, complemented
28#534—31no—
29#448—31no—
30#448D311yescharacteristic, complemented
31#393—31no—
32#327—31no—
33#327—31no—
34#323—31no—
35C23×D3—21no—
36C23×D3D411yescharacteristic
37C2×C3×Q8—21no—
38C2×C3×Q8—21no—
39C2×C3×Q8—42no—
40C2×C3×D4—21no—
41C2×C3×D4D411yescharacteristic
42C22×C3×C4—21no—
43C22×C3×C4C2311yescharacteristic
44C2×#67—21no—
45C2×#67—21no—
46C2×#67—21no—
47C22×Dic3—21no—
48C22×Dic3D411yescharacteristic
49D3×D4—42no—
50D3×D4—42no—
51C2×D12—21no—
52C2×D12D411yescharacteristic
53C2×C4×D3—21no—
54C3×SD16—42no—
55C2×C3×C8—21no—
56C3×C4⋊C4D411yescharacteristic
57C3×C22⋊C4—21no—
58#217—21no—
59#215—42no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—21no—
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64(C2×D3)⋊C4—42no—
65C4⋊Dic3D411yescharacteristic
66Dic3⋊C4—21no—
67C2×C3⋊C8—21no—
68C22×Q8C2×D311yescharacteristic, complemented
69C22×D4—61no—
70C2×SD16—21no—
71C2×SD16—61no—
72#123—61no—
73#123—61no—
74#121—31no—
75#121—61no—
76#121C2×D311yescharacteristic
77#120—61no—
78#120—61no—
79C2×C22⋊C4—31no—
80C2×C22⋊C4—61no—
81#103—21no—
82#103—61no—
83C22⋊C8—31no—
84C22⋊C8C2×D311yescharacteristic, complemented
85#95—31no—
86#95—61no—
87C23×C3—21no—
88C23×C3C2×D411yescharacteristic
89C22×D3—21no—
90C22×D3—21no—
91C22×D3—21no—
92C22×D3—42no—
93C22×D3—41no—
94C22×D3—41no—
95C22×D3—81no—
96C22×D3—81no—
97C22×D3SD1612yescomplemented
98C3×Q8—22no—
99C3×Q8—22no—
100C3×Q8—41no—
101C3×Q8—41no—
102C3×D4—41no—
103C3×D4—41no—
104C3×D4—41no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4—41no—
108C2×C3×C4—41no—
109C2×C3×C4—41no—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112#67—42no—
113#67—41no—
114#67—41no—
115#67—81no—
116C2×Dic3—21no—
117C2×Dic3—21no—
118C2×Dic3—41no—
119C2×Dic3—41no—
120C2×Dic3—41no—
121C2×Dic3—41no—
122D12—41no—
123D12—42no—
124C4×D3—42no—
125C3×C8—41no—
126C3⋊C8—41no—
127C24—31no—
128C24—61no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—42no—
132C2×D4—21no—
133C2×D4—31no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4—122no—
139C2×D4—122no—
140C2×D4#6711yescharacteristic
141C22×C4—21no—
142C22×C4—31no—
143C22×C4—61no—
144C22×C4—61no—
145C22×C4C22×D311yescharacteristic
146SD16—42no—
147SD16—122no—
148C2×C8—21no—
149C2×C8—61no—
150C4⋊C4—31no—
151C4⋊C4—61no—
152C4⋊C4#6711yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—122no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3#12011yescharacteristic
166C22×C3C2×SD1611yescharacteristic
167C22×C3AGL(1,ℤ/8)11yescharacteristic
168C2×D3—22no—
169C2×D3—41no—
170C2×D3—41no—
171C2×D3—42no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3—41no—
175C2×D3—81no—
176C2×D3—84no—
177C3×C4—21no—
178C3×C4—21no—
179C3×C4—41no—
180C3×C4—42no—
181Dic3—41no—
182Dic3—41no—
183Dic3—81no—
184C23—21no—
185C23—32no—
186C23—61no—
187C23—61no—
188C23—61no—
189C23—121no—
190C23—121no—
191C23—122no—
192C23—241no—
193C23—241no—
194C23D3×D411yescharacteristic
195Q8—22no—
196Q8—22no—
197Q8—41no—
198Q8—41no—
199D4—41no—
200D4—41no—
201D4—41no—
202D4—121no—
203D4—122no—
204D4—122no—
205D4—121no—
206D4—121no—
207D4—241no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—122no—
216C2×C4—121no—
217C2×C4—121no—
218C2×C4—121no—
219C2×C4—121no—
220C2×C4D3×D411yescharacteristic
221C2×C4C2×#6711yescharacteristic
222C8—41no—
223C8—121no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—81no—
227C2×C3#45011yescharacteristic
228C2×C3#45511yescharacteristic
229C2×C3B(4,𝔽2)11yescharacteristic
230D3—42no—
231D3—81no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—62no—
238C22—122no—
239C22—121no—
240C22—121no—
241C22—121no—
242C22—121no—
243C22—121no—
244C22—241no—
245C22—244no—
246C22D3×SD1611yescharacteristic
247C22#91411yescharacteristic
248C22#93711yescharacteristic
249C4—21no—
250C4—21no—
251C4—41no—
252C4—42no—
253C4—121no—
254C4—121no—
255C4—241no—
256C3#160111yescharacteristic, complemented, Sylow
257C2—21no—
258C2—21no—
259C2—81no—
260C2—122no—
261C2—241no—
262C2#503211yescharacteristic
263C2#516911yescharacteristic
264C2#520211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#448C2
socleC22×C3#120
prime radical#5605C2
hypercenter#448D3
hypocenterC3#1601
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C3×#448C2semidirect
3. #5231C2semidirect
4. #5227C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4468C2semidirect
9. C22×C3×Q8C22semidirect
23. C3×C22⋊C8C22semidirect
26. C3⋊#95C22semidirect
27. #830C22semidirect
30. #448D3semidirect
68. C22×Q8C2×D3semidirect
84. C22⋊C8C2×D3semidirect
97. C22×D3SD16semidirect
256. C3#1601semidirect

Maximal subgroups

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

subgroupindex
#56052
C3×#4482
#52312
#52272
#45372
#44702
#44682
#16013