Finite Group Databasebeta

← #68293

Subgroups — #68293

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
2#5036C211yesmaximal, characteristic, complemented
3#4732C211yesmaximal, characteristic, complemented
4#4672C211yesmaximal, characteristic, complemented
5C3×#327C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1598—31nomaximal, Sylow
9C2×#237—21no—
10C22×D12C2211yescharacteristic
11C3×#121C2211yescharacteristic
12#939—21no—
13#939—21no—
14C2×#213—21no—
15#930C2211yescharacteristic
16C2×(C2×D3)⋊C4—21no—
17C2×(C2×D3)⋊C4—21no—
18C2×Dic3⋊C4C2211yescharacteristic
19C2×D24—21no—
20#896—21no—
21#884—21no—
22#878—21no—
23C3×C22⋊C8C2211yescharacteristic
24C3×#95C2211yescharacteristic
25#830C2211yescharacteristic
26#821—21no—
27#809—21no—
28#537—31no—
29#447—31no—
30#447—31no—
31#394—31no—
32#327—31no—
33#327D311yescharacteristic, complemented
34#324—31no—
35C23×D3—21no—
36C2×C3×D4—21no—
37C2×C3×D4D411yescharacteristic
38C22×C3×C4—21no—
39C22×C3×C4C2311yescharacteristic
40C2×#67—21no—
41C2×#67—21no—
42C2×#67—21no—
43C22×Dic3—21no—
44C22×Dic3D411yescharacteristic
45C22×Dic3D411yescharacteristic
46#237—42no—
47#237—42no—
48C2×D12—21no—
49C2×D12—21no—
50C2×D12—42no—
51C2×D12D411yescharacteristic
52C2×C4×D3—21no—
53C2×Dic6—21no—
54C2×C3×C8—21no—
55C3×C4⋊C4D411yescharacteristic
56C3×C22⋊C4—21no—
57#217—21no—
58#213—42no—
59(C2×D3)⋊C4—21no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—42no—
62(C2×D3)⋊C4—42no—
63C4⋊Dic3D411yescharacteristic
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66Dic3⋊C4—21no—
67C2×C3⋊C8—21no—
68D24—42no—
69C2×#41—61no—
70C22×D4—31no—
71C2×D8—61no—
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—61no—
84#102—61no—
85C22⋊C8—31no—
86C22⋊C8C2×D311yescharacteristic
87#95C2×D311yescharacteristic
88C23×C3—21no—
89C23×C3C2×D411yescharacteristic
90C22×D3—22no—
91C22×D3—21no—
92C22×D3—41no—
93C22×D3—41no—
94C22×D3—41no—
95C22×D3—41no—
96C22×D3—41no—
97C22×D3—41no—
98C3×D4—41no—
99C3×D4—41no—
100C3×D4—41no—
101C2×C3×C4—21no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108#67—42no—
109#67—41no—
110#67—41no—
111#67—81no—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—42no—
117C2×Dic3—41no—
118C2×Dic3D812yescomplemented
119D12—22no—
120D12—22no—
121D12—41no—
122D12—41no—
123D12—41no—
124C4×D3—42no—
125Dic6—42no—
126C3×C8—41no—
127C3⋊C8—41no—
128C24—61no—
129#41—122no—
130#41—122no—
131C2×Q8—61no—
132C2×D4—21no—
133C2×D4—31no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4—61no—
139C2×D4—122no—
140C2×D4D1211yescharacteristic
141C22×C4—21no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—61no—
145C22×C4—61no—
146C22×C4C22×D311yescharacteristic
147D8—122no—
148D8—122no—
149C2×C8—21no—
150C2×C8—61no—
151C4⋊C4—31no—
152C4⋊C4—61no—
153C4⋊C4—61no—
154C4⋊C4—61no—
155C4⋊C4D1211yescharacteristic
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—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3#12011yescharacteristic
168C22×C3C2×D811yescharacteristic
169C22×C3AGL(1,ℤ/8)11yescharacteristic
170C2×D3—42no—
171C2×D3—42no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3—42no—
175C2×D3—41no—
176C2×D3—81no—
177C2×D3—81no—
178C2×D3—81no—
179C2×D3—81no—
180C3×C4—21no—
181C3×C4—21no—
182C3×C4—41no—
183C3×C4—81no—
184Dic3—22no—
185Dic3—41no—
186Dic3—41no—
187C23—21no—
188C23—62no—
189C23—61no—
190C23—121no—
191C23—121no—
192C23—121no—
193C23—121no—
194C23—121no—
195C23—121no—
196C23D3×D411yescharacteristic
197Q8—122no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—62no—
202D4—62no—
203D4—121no—
204D4—121no—
205D4—121no—
206D4—122no—
207D4—121no—
208D4—121no—
209D4—241no—
210C2×C4—21no—
211C2×C4—32no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—121no—
223C2×C4C2×D1211yescharacteristic
224C2×C4D3×D411yescharacteristic
225C8—41no—
226C8—121no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—81no—
230C2×C3#44711yescharacteristic
231C2×C3#45411yescharacteristic
232C2×C3B(4,𝔽2)11yescharacteristic
233D3—81no—
234D3—82no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—122no—
241C22—122no—
242C22—122no—
243C22—121no—
244C22—121no—
245C22—121no—
246C22—241no—
247C22—241no—
248C22—241no—
249C22—241no—
250C22#88211yescharacteristic
251C22D3×D811yescharacteristic
252C22#91411yescharacteristic
253C4—21no—
254C4—21no—
255C4—41no—
256C4—62no—
257C4—81no—
258C4—121no—
259C4—121no—
260C3#159811yescharacteristic, complemented, Sylow
261C2—21no—
262C2—21no—
263C2—81no—
264C2—242no—
265C2—241no—
266C2#474111yescharacteristic
267C2#477311yescharacteristic
268C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3#120
hypercenter#327D3
hypocenterC3#1598
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #5597C2semidirect
2. #5036C2semidirect
3. #4732C2semidirect
4. #4672C2semidirect
5. C3×#327C2semidirect
6. #4469C2semidirect
7. #4451C2semidirect
33. #327D3semidirect
118. 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
#50362
#47322
#46722
C3×#3272
#44692
#44512
#15983