Finite Group Databasebeta

← #83400

Subgroups — #83400

Order 384.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C3×#536C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic
3C2×#948C211yesmaximal, characteristic, complemented
4C2×#942C211yesmaximal, characteristic, complemented
5C2×#835C211yesmaximal, characteristic, complemented
6#5052C212yesmaximal, complemented
7#5051C212yesmaximal, complemented
8#5049C212yesmaximal, complemented
9#5048C212yesmaximal, complemented
10#5002C211yesmaximal, characteristic, complemented
11#4995C211yesmaximal, characteristic, complemented
12#3063—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic
15C2×#235C2211yescharacteristic
16C3×C4⋊Q8—21no—
17C3×#124—21no—
18C3×#123—22no—
19C3×C22⋊Q8C2212yes—
20C3×C22⋊Q8C2212yes—
21C3×#117C2211yescharacteristic
22#948—24no—
23#946—21no—
24C2×#216C2212yes—
25#942—24no—
26C2×#215C2212yes—
27#939—22no—
28#938—21no—
29#930C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31#924C2212yes—
32C2×#208C2212yes—
33C2×#208C2211yescharacteristic
34C22×C3⋊C8C2211yescharacteristic
35C22×C3⋊C8C2212yes—
36#835C2212yes—
37#835C2212yes—
38#810C2211yescharacteristic
39#810C2211yescharacteristic
40#809C2211yescharacteristic
41#809C2211yescharacteristic
42#808C2212yes—
43#807C2212yes—
44C2×#137—31no—
45C2×#131—31no—
46#537—31no—
47#536D311yescharacteristic, complemented
48#470—32no—
49#468—32no—
50#467—32no—
51#465—32no—
52#428—31no—
53#416—31no—
54C2×#103—31no—
55C3×#41—22no—
56C3×#41—22no—
57C2×C3×Q8—21no—
58C2×C3×Q8—22no—
59C2×C3×Q8—22no—
60C2×C3×Q8D411yescharacteristic
61C2×C3×Q8C2312yes—
62C2×C3×D4D411yescharacteristic
63C2×C3×D4D412yes—
64C22×C3×C4—21no—
65C22×C3×C4C2312yes—
66C22×C3×C4C2311yescharacteristic
67C2×#67—21no—
68C2×#67—22no—
69C22×Dic3—21no—
70#235—22no—
71#235—22no—
72C2×D12C2311yescharacteristic
73C2×C4×D3—21no—
74C2×Dic6C2311yescharacteristic
75C3×C4⋊C4—22no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4C2311yescharacteristic
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C22⋊C4—22no—
81C3×C22⋊C4—22no—
82C3×C42—21no—
83#217—22no—
84#216—24no—
85#215—24no—
86(C2×D3)⋊C4—22no—
87C4⋊Dic3—21no—
88C4⋊Dic3C2312yes—
89Dic3⋊C4—22no—
90#208—24no—
91#208D414yescomplemented
92C2×C3⋊C8—22no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8C2311yescharacteristic
95C2×C3⋊C8C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97C2×#41—31no—
98C2×#41C2×D311yescharacteristic
99C22×Q8C2×D311yescharacteristic
100#137—64no—
101C2×Q16—32no—
102C2×SD16—32no—
103#131—64no—
104C2×M16—31no—
105C2×M16—32no—
106C22×C8—31no—
107C22×C8—32no—
108C4⋊Q8—21no—
109#124—21no—
110#123—22no—
111C22⋊Q8—32no—
112C22⋊Q8—61no—
113C22⋊Q8C2×D312yes—
114C22⋊Q8C2×D312yes—
115#121—32no—
116#121—61no—
117#121—62no—
118#117C2×D311yescharacteristic
119C2×C4⋊C4—31no—
120#107—32no—
121#106—32no—
122#103—32no—
123#103—32no—
124#103—31no—
125#103—31no—
126#102—31no—
127#102—31no—
128C23×C3C2×D412yes—
129C23×C3C2×D411yescharacteristic
130C22×D3—21no—
131C3×Q8—22no—
132C3×Q8—22no—
133C3×Q8—22no—
134C3×Q8—42no—
135C3×D4—24no—
136C3×D4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—42no—
143C2×C3×C4C2×D412yes—
144C2×C3×C4C2×D412yes—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4#4112yes—
147C2×C3×C4C2411yescharacteristic
148#67—42no—
149#67—82no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—81no—
153D12—21no—
154D12—21no—
155C4×D3—42no—
156Dic6—21no—
157Dic6—21no—
158C3⋊C8—21no—
159C3⋊C8—21no—
160C3⋊C8C2×D414yes—
161#41—22no—
162#41—22no—
163#41—62no—
164#41—62no—
165C2×Q8—21no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—31no—
169C2×Q8#6711yescharacteristic
170C2×Q8C22×D312yes—
171C2×D4—31no—
172C2×D4—61no—
173C2×D4—62no—
174C2×D4#6712yes—
175C2×D4#6711yescharacteristic
176C22×C4—21no—
177C22×C4—61no—
178C22×C4—61no—
179C22×C4C22×D312yes—
180C22×C4C22×D311yescharacteristic
181Q16—64no—
182SD16—64no—
183M16—34no—
184M16—64no—
185C2×C8—32no—
186C2×C8—31no—
187C2×C8—31no—
188C2×C8—64no—
189C2×C8—62no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—32no—
193C4⋊C4—61no—
194C4⋊C4—62no—
195C4⋊C4C22×D311yescharacteristic
196C4⋊C4C22×D311yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—62no—
201C22⋊C4—62no—
202C42—21no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3#12112yes—
209C22×C3C22×D411yescharacteristic
210C2×D3—41no—
211C2×D3—41no—
212C2×D3—41no—
213C3×C4—22no—
214C3×C4—42no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4#12112yes—
218C3×C4C22×D411yescharacteristic
219C3×C4C2×#4111yescharacteristic
220Dic3—41no—
221Dic3—42no—
222C23—61no—
223C23C2×#6712yes—
224C23C2×#6711yescharacteristic
225Q8—22no—
226Q8—22no—
227Q8—22no—
228Q8—42no—
229Q8—61no—
230Q8—61no—
231D4—24no—
232D4—22no—
233D4—61no—
234D4—61no—
235D4—122no—
236D4—242no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—62no—
244C2×C4—61no—
245C2×C4—122no—
246C2×C4—241no—
247C2×C4D3×D412yes—
248C2×C4#23712yes—
249C2×C4C2×#6712yes—
250C2×C4C2×#6711yescharacteristic
251C2×C4C23×D311yescharacteristic
252C8—34no—
253C8—61no—
254C8—61no—
255C2×C3—22no—
256C2×C3—42no—
257C2×C3C2×#12111yescharacteristic
258C2×C3#57711yescharacteristic
259C2×C3#57811yescharacteristic
260D3—81no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—121no—
267C22—121no—
268C22—121no—
269C22#93912yes—
270C22C22×#6711yescharacteristic
271C4—22no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—121no—
276C4—122no—
277C4#93912yes—
278C4C2×D3×D411yescharacteristic
279C4C2×#23711yescharacteristic
280C3#306311yescharacteristic, complemented, Sylow
281C2—22no—
282C2—42no—
283C2—241no—
284C2C2×#93911yescharacteristic
285C2#583711yescharacteristic
286C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radical#5833C2
hypercenter#536D3
hypocenterC3#3063
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
3. C2×#948C2semidirect
4. C2×#942C2semidirect
5. C2×#835C2semidirect
6. #5052C2semidirect
7. #5051C2semidirect
8. #5049C2semidirect
9. #5048C2semidirect
10. #5002C2semidirect
11. #4995C2semidirect
47. #536D3semidirect
91. #208D4semidirect
280. C3#3063semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#58332
C2×#9482
C2×#9422
C2×#8352
#50522
#50512
#50492
#50482
#50022
#49952
#30633