Finite Group Databasebeta

← #70971

Subgroups — #70971

Order 384.

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

normal subgroups only

This group has 310 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#5597C211yesmaximal, characteristic, complemented
2C3×#447C211yesmaximal, characteristic, complemented
3#5222C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4469C211yesmaximal, characteristic, complemented
8#1598—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×#237—21no—
11C2×C3×D8—21no—
12C3×#121C2211yescharacteristic
13C2×#217—21no—
14C2×#217—21no—
15#939—21no—
16#939—21no—
17C2×#213—21no—
18#930C2211yescharacteristic
19C2×Dic3⋊C4C2211yescharacteristic
20#896—21no—
21#884—21no—
22#878—21no—
23C3×#102—21no—
24C3×C22⋊C8C2211yescharacteristic
25#832—21no—
26C3⋊#95C2211yescharacteristic
27#830C2211yescharacteristic
28#537—31no—
29#447—31no—
30#447D311yescharacteristic, complemented
31#394—31no—
32#327—31no—
33#327—31no—
34#324—31no—
35C24×C3—21no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—42no—
40C2×C3×D4D411yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C2×#67—21no—
43C2×#67—21no—
44C2×#67—21no—
45C22×Dic3—21no—
46C22×Dic3—21no—
47C22×Dic3D411yescharacteristic
48C22×Dic3D411yescharacteristic
49#237—42no—
50#237—42no—
51C2×D12D411yescharacteristic
52C2×C4×D3—21no—
53C2×Dic6—21no—
54C3×D8—42no—
55C2×C3×C8—21no—
56C3×C4⋊C4D411yescharacteristic
57C3×C22⋊C4—21no—
58#217—21no—
59#217—42no—
60#217—42no—
61#213—42no—
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64C4⋊Dic3D411yescharacteristic
65Dic3⋊C4—21no—
66Dic3⋊C4—21no—
67Dic3⋊C4—21no—
68C2×C3⋊C8—21no—
69C2×#41—61no—
70C22×D4C2×D311yescharacteristic
71C2×D8—21no—
72C2×D8—61no—
73C22⋊Q8—61no—
74C22⋊Q8—61no—
75#121—31no—
76#121—61no—
77#121—61no—
78#121—61no—
79#121C2×D311yescharacteristic
80C2×C4⋊C4—31no—
81C2×C22⋊C4—61no—
82C2×C22⋊C4—61no—
83#102—21no—
84#102—61no—
85C22⋊C8—31no—
86C22⋊C8C2×D311yescharacteristic
87#95—31no—
88C23×C3—22no—
89C23×C3—21no—
90C23×C3—41no—
91C23×C3—41no—
92C23×C3—41no—
93C23×C3—41no—
94C23×C3—41no—
95C23×C3—41no—
96C23×C3C2×D411yescharacteristic
97C22×D3—21no—
98C3×D4—22no—
99C3×D4—22no—
100C3×D4—41no—
101C3×D4—41no—
102C3×D4—41no—
103C3×D4—41no—
104C3×D4—41no—
105C2×C3×C4—21no—
106C2×C3×C4—41no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109#67—42no—
110#67—41no—
111#67—41no—
112#67—81no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—42no—
118C2×Dic3—41no—
119C2×Dic3—41no—
120C2×Dic3—41no—
121C2×Dic3—41no—
122C2×Dic3D812yescomplemented
123D12—41no—
124C4×D3—42no—
125Dic6—42no—
126C3×C8—41no—
127C3⋊C8—41no—
128C24—21no—
129#41—122no—
130#41—122no—
131C2×Q8—61no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—31no—
136C2×D4—42no—
137C2×D4—61no—
138C2×D4—61no—
139C2×D4—61no—
140C2×D4#6711yescharacteristic
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—61no—
144C22×C4—61no—
145C22×C4—61no—
146C22×C4C22×D311yescharacteristic
147D8—42no—
148D8—122no—
149C2×C8—21no—
150C2×C8—61no—
151C4⋊C4—31no—
152C4⋊C4—61no—
153C4⋊C4—61no—
154C4⋊C4—61no—
155C4⋊C4#6711yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—61no—
160C22⋊C4—122no—
161C22⋊C4—122no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—42no—
170C22×C3—81no—
171C22×C3—81no—
172C22×C3—81no—
173C22×C3—81no—
174C22×C3#12011yescharacteristic
175C22×C3C2×D811yescharacteristic
176C22×C3AGL(1,ℤ/8)11yescharacteristic
177C2×D3—41no—
178C2×D3—41no—
179C2×D3—41no—
180C3×C4—21no—
181C3×C4—21no—
182C3×C4—41no—
183Dic3—22no—
184Dic3—41no—
185Dic3—41no—
186Dic3—81no—
187C23—22no—
188C23—21no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—61no—
196C23D3×D411yescharacteristic
197Q8—122no—
198D4—22no—
199D4—22no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—41no—
205D4—121no—
206D4—122no—
207D4—121no—
208D4—121no—
209D4—241no—
210C2×C4—21no—
211C2×C4—32no—
212C2×C4—41no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4D3×D411yescharacteristic
224C2×C4C2×#6711yescharacteristic
225C8—41no—
226C8—121no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—81no—
230C2×C3—82no—
231C2×C3#44711yescharacteristic
232C2×C3#45411yescharacteristic
233C2×C3B(4,𝔽2)11yescharacteristic
234D3—81no—
235C22—21no—
236C22—21no—
237C22—42no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—42no—
242C22—41no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—121no—
248C22—121no—
249C22—121no—
250C22D3×D811yescharacteristic
251C22#91111yescharacteristic
252C22#93711yescharacteristic
253C4—21no—
254C4—21no—
255C4—41no—
256C4—62no—
257C4—121no—
258C4—121no—
259C4—241no—
260C3#159811yescharacteristic, complemented, Sylow
261C2—21no—
262C2—21no—
263C2—82no—
264C2—81no—
265C2—241no—
266C2#503211yescharacteristic
267C2#515611yescharacteristic
268C2#520111yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#447C2
socleC22×C3#120
hypercenter#447D3
hypocenterC3#1598
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #5597C2semidirect
2. C3×#447C2semidirect
3. #5222C2semidirect
4. #5217C2semidirect
5. #4537C2semidirect
6. #4470C2semidirect
7. #4469C2semidirect
30. #447D3semidirect
122. C2×Dic3D8semidirect
260. C3#1598semidirect

Maximal subgroups

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

subgroupindex
#55972
C3×#4472
#52222
#52172
#45372
#44702
#44692
#15983