Finite Group Databasebeta

← #70361

Subgroups — #70361

Order 384.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#5833C211yesmaximal, characteristic
2C2×C3×#104C211yesmaximal, characteristic, complemented
3#5232C211yesmaximal, characteristic, complemented
4#5133C211yesmaximal, characteristic, complemented
5#4935C211yesmaximal, characteristic, complemented
6C2×#805C211yesmaximal, characteristic, complemented
7#4554C211yesmaximal, characteristic, complemented
8#2039—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C2×C3×M16C2211yescharacteristic
12C2×C3×C42C2211yescharacteristic
13#946—21no—
14#939—22no—
15#938—21no—
16#930—21no—
17C2×C4⋊Dic3—22no—
18C2×C4⋊Dic3C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20#924—21no—
21C2×#208C2211yescharacteristic
22C3×#104—22no—
23#835—21no—
24#832—21no—
25#821—21no—
26#816—21no—
27#805—22no—
28#537—31no—
29C2×#104—31no—
30C2×#104D311yescharacteristic, complemented
31#417—31no—
32#417—31no—
33#384—31no—
34#343—31no—
35C3×#41—21no—
36C3×#41—21no—
37C3×#41—21no—
38C3×#41—21no—
39C2×C3×Q8D411yescharacteristic
40C2×C3×D4—21no—
41C2×C3×D4D411yescharacteristic
42C22×C3×C4—21no—
43C22×C3×C4—21no—
44C22×C3×C4C2311yescharacteristic
45C2×#67—21no—
46C2×#67—22no—
47C22×Dic3—21no—
48C22×Dic3D412yes—
49#235—21no—
50#235—21no—
51#235—21no—
52#235—21no—
53C2×D12D411yescharacteristic
54C2×C4×D3—21no—
55C2×Dic6D411yescharacteristic
56C3×M16—21no—
57C3×M16—21no—
58C2×C3×C8—21no—
59C3×C42—21no—
60C3×C42D412yescomplemented
61#217—22no—
62(C2×D3)⋊C4—22no—
63C4⋊Dic3—21no—
64C4⋊Dic3—21no—
65C4⋊Dic3—21no—
66C4⋊Dic3—21no—
67C4⋊Dic3—44no—
68Dic3⋊C4—22no—
69#208—21no—
70#208—21no—
71C2×C3⋊C8—21no—
72C2×#41—31no—
73C2×#41C2×D311yescharacteristic
74C2×M16—31no—
75C2×M16C2×D311yescharacteristic
76C22⋊Q8—61no—
77C22⋊Q8—61no—
78#121—61no—
79#121—62no—
80#121—61no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—62no—
84C2×C42C2×D311yescharacteristic
85#104—22no—
86#104—62no—
87#103—61no—
88#103—61no—
89#102—61no—
90#102—61no—
91C23×C3—21no—
92C23×C3C2×D411yescharacteristic
93C22×D3—21no—
94C3×Q8—21no—
95C3×Q8—21no—
96C3×D4—21no—
97C3×D4—21no—
98C3×D4—41no—
99C3×D4—41no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4#4111yescharacteristic
110C2×C3×C4#4111yescharacteristic
111C2×C3×C4#4111yescharacteristic
112#67—41no—
113#67—41no—
114#67—82no—
115C2×Dic3—21no—
116C2×Dic3—22no—
117C2×Dic3—42no—
118C2×Dic3—42no—
119C2×Dic3—41no—
120D12—21no—
121D12—21no—
122C4×D3—41no—
123C4×D3—41no—
124Dic6—21no—
125Dic6—21no—
126C3×C8—41no—
127C3⋊C8—41no—
128#41—21no—
129#41—21no—
130#41—21no—
131#41—21no—
132#41—61no—
133#41—61no—
134#41—61no—
135#41—61no—
136C2×Q8—31no—
137C2×Q8D1211yescharacteristic
138C2×D4—21no—
139C2×D4—31no—
140C2×D4—61no—
141C2×D4—62no—
142C2×D4D1211yescharacteristic
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—32no—
146C22×C4—61no—
147C22×C4—61no—
148C22×C4C22×D311yescharacteristic
149M16—21no—
150M16—21no—
151M16—61no—
152M16—61no—
153C2×C8—21no—
154C2×C8—61no—
155C4⋊C4—62no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4—124no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C42—21no—
164C42#6712yescomplemented
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3#12011yescharacteristic
171C22×C3#12111yescharacteristic
172C22×C3#12411yescharacteristic
173C2×D3—41no—
174C2×D3—41no—
175C2×D3—41no—
176C3×C4—24no—
177C3×C4—41no—
178C3×C4#12111yescharacteristic
179C3×C4#12111yescharacteristic
180C3×C4#12311yescharacteristic
181C3×C4#12311yescharacteristic
182Dic3—41no—
183Dic3—42no—
184Dic3—81no—
185C23—21no—
186C23—61no—
187C23D3×D411yescharacteristic
188Q8—21no—
189Q8—21no—
190Q8—61no—
191Q8—61no—
192D4—21no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—61no—
197D4—61no—
198D4—121no—
199D4—121no—
200D4—242no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4—121no—
210C2×C4—121no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4—121no—
214C2×C4C2×D1211yescharacteristic
215C2×C4#23511yescharacteristic
216C2×C4D3×D411yescharacteristic
217C2×C4#23711yescharacteristic
218C2×C4#23711yescharacteristic
219C2×C4C2×#6711yescharacteristic
220C8—41no—
221C8—121no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—81no—
225C2×C3#39411yescharacteristic
226C2×C3#45311yescharacteristic
227C2×C3#45611yescharacteristic
228D3—81no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—121no—
235C22—121no—
236C22—121no—
237C22#88211yescharacteristic
238C22#88511yescharacteristic
239C22#93911yescharacteristic
240C4—24no—
241C4—41no—
242C4—121no—
243C4—122no—
244C4—241no—
245C4#88311yescharacteristic
246C4#88611yescharacteristic
247C4#93011yescharacteristic
248C4#93811yescharacteristic
249C3#203911yescharacteristic, complemented, Sylow
250C2—21no—
251C2—21no—
252C2—81no—
253C2—241no—
254C2#482211yescharacteristic
255C2#482611yescharacteristic
256C2#496111yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC2×C3×#104C2
socleC22×C3#120
prime radical#5833C2
hypercenterC2×#104D3
hypocenterC3#2039
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×C3×#104C2semidirect
3. #5232C2semidirect
4. #5133C2semidirect
5. #4935C2semidirect
6. C2×#805C2semidirect
7. #4554C2semidirect
30. C2×#104D3semidirect
60. C3×C42D4semidirect
164. C42#67semidirect
249. C3#2039semidirect

Maximal subgroups

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

subgroupindex
#58332
C2×C3×#1042
#52322
#51332
#49352
C2×#8052
#45542
#20393