Finite Group Databasebeta

← #78929

Subgroups — #78929

Order 384.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#1004C211yesmaximal, characteristic
2C22×#214C211yesmaximal, characteristic, complemented
3C2×#909C211yesmaximal, characteristic, complemented
4#5526C211yesmaximal, characteristic, complemented
5C3×#418C211yesmaximal, characteristic, complemented
6#5129C211yesmaximal, characteristic, complemented
7#5002C211yesmaximal, characteristic, complemented
8#4783C214yesmaximal, complemented
9#4776C214yesmaximal, complemented
10#3009—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×#67—22no—
13#1004—24no—
14#1004—24no—
15C2×#237—21no—
16C2×#237C2212yes—
17C2×D3×D4—21no—
18C2×D3×D4C2212yes—
19C2×#235C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C22×Dic6C2211yescharacteristic, complemented
22C2×C3×M16C2211yescharacteristic, complemented
23C3×#117C2211yescharacteristic, complemented
24C2×#214—24no—
25C2×#214C2214yescomplemented
26C22×C3⋊C8C2211yescharacteristic, complemented
27#909—24no—
28C2×Dic12C2212yescomplemented
29C2×#204C2212yescomplemented
30#897C2214yescomplemented
31#886—22no—
32#875—21no—
33#869—21no—
34C3×#102C2214yescomplemented
35(C2×D3)⋊C8C2214yescomplemented
36#810C2212yescomplemented
37#809C2212yescomplemented
38C2×H(5,𝔽2)—31no—
39C2×#137—31no—
40C22×SD16—31no—
41#536—31no—
42#452—34no—
43#450—34no—
44#418D311yescharacteristic, complemented
45#416—31no—
46#408—31no—
47C24×C3—21no—
48C23×D3—22no—
49C2×C3×D4—21no—
50C2×C3×D4—22no—
51C2×C3×D4D414yes—
52C2×C3×D4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C2×#67—216no—
55C2×#67—21no—
56C2×#67—22no—
57C2×#67—22no—
58C2×#67D412yes—
59C22×Dic3—22no—
60C22×Dic3—21no—
61#237—28no—
62#237—24no—
63#237—24no—
64D3×D4—28no—
65D3×D4—24no—
66D3×D4—24no—
67#235—22no—
68#235—22no—
69#235—22no—
70#235D414yes—
71C2×D12D411yescharacteristic
72C2×D12C2311yescharacteristic
73C2×C4×D3—21no—
74C2×C4×D3C2312yes—
75C2×Dic6—22no—
76C2×Dic6—22no—
77C2×Dic6—21no—
78C2×Dic6D411yescharacteristic
79C2×Dic6C2312yes—
80C2×Dic6C2311yescharacteristic
81C3×M16—22no—
82C2×C3×C8C2312yes—
83C3×C4⋊C4C2312yes—
84C3×C22⋊C4—21no—
85C3×C42—21no—
86#214—28no—
87(C2×D3)⋊C4—24no—
88C4⋊Dic3—24no—
89C2×C3⋊C8—22no—
90C2×C3⋊C8C2312yes—
91Dic12—24no—
92#204—24no—
93H(5,𝔽2)—64no—
94H(5,𝔽2)—64no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—31no—
98C2×#41—61no—
99C22×Q8—31no—
100C22×D4—32no—
101C22×D4—61no—
102C22×D4—62no—
103C22×D4C2×D311yescharacteristic
104#137—64no—
105C2×Q16—32no—
106C2×SD16—34no—
107C2×SD16—32no—
108C2×SD16—64no—
109C2×M16C2×D311yescharacteristic, complemented
110C22×C8—31no—
111C4⋊Q8—61no—
112#124—61no—
113#123—62no—
114C22⋊Q8—34no—
115#117C2×D311yescharacteristic, complemented
116#103—32no—
117#102—32no—
118#102C2×D314yescomplemented
119C22⋊C8—34no—
120C23×C3—22no—
121C23×C3—24no—
122C23×C3—82no—
123C23×C3C2×D411yescharacteristic
124C22×D3—21no—
125C22×D3—28no—
126C22×D3—84no—
127C22×D3C2×D412yes—
128C3×D4—24no—
129C3×D4—22no—
130C3×D4C2×D414yes—
131C2×C3×C4—22no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D412yes—
135C2×C3×C4C2411yescharacteristic
136#67—28no—
137#67—28no—
138#67—24no—
139#67—24no—
140C2×Dic3—28no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143C2×Dic3—42no—
144C2×Dic3C2×D412yes—
145D12—22no—
146D12—21no—
147D12C2×D412yes—
148C4×D3—24no—
149C4×D3—22no—
150C4×D3—22no—
151Dic6—22no—
152Dic6—21no—
153Dic6—22no—
154Dic6—22no—
155Dic6—42no—
156Dic6C2×D412yes—
157C3×C8—22no—
158C3⋊C8—22no—
159C24—21no—
160C24—62no—
161#41—34no—
162#41—64no—
163#41—68no—
164#41—64no—
165#41—62no—
166#41—62no—
167#41—62no—
168C2×Q8—31no—
169C2×Q8—31no—
170C2×Q8—32no—
171C2×Q8—61no—
172C2×Q8—62no—
173C2×Q8—62no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—32no—
177C2×D4—31no—
178C2×D4—31no—
179C2×D4—62no—
180C2×D4—64no—
181C2×D4—68no—
182C2×D4—62no—
183C2×D4—616no—
184C2×D4—61no—
185C2×D4—64no—
186C2×D4D1214yes—
187C2×D4C22×D312yes—
188C22×C4—32no—
189C22×C4—62no—
190C22×C4—61no—
191C22×C4—61no—
192C22×C4C22×D311yescharacteristic
193Q16—64no—
194SD16—64no—
195SD16—68no—
196M16—22no—
197C2×C8—32no—
198C2×C8—62no—
199C2×C8C22×D312yes—
200C4⋊C4—64no—
201C4⋊C4C22×D312yes—
202C22⋊C4—21no—
203C22⋊C4—64no—
204C42—21no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—44no—
208C22×C3—84no—
209C22×C3#12012yes—
210C22×C3C22×D411yescharacteristic
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—24no—
214C2×D3—42no—
215C2×D3—88no—
216C3×C4—42no—
217C3×C4#12012yes—
218C3×C4C22×D411yescharacteristic
219C3×C4C22×D411yescharacteristic
220Dic3—22no—
221Dic3—22no—
222Dic3—42no—
223C23—22no—
224C23—24no—
225C23—32no—
226C23—68no—
227C23—61no—
228C23—82no—
229C23—244no—
230C23D3×D411yescharacteristic
231Q8—32no—
232Q8—62no—
233Q8—61no—
234Q8—62no—
235Q8—62no—
236Q8—122no—
237D4—24no—
238D4—22no—
239D4—32no—
240D4—68no—
241D4—68no—
242D4—64no—
243D4—64no—
244D4—62no—
245D4—61no—
246D4C2×D1214yes—
247C2×C4—22no—
248C2×C4—32no—
249C2×C4—68no—
250C2×C4—61no—
251C2×C4—64no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—122no—
256C2×C4C2×D1212yes—
257C2×C4D3×D411yescharacteristic
258C2×C4D3×D412yes—
259C2×C4C23×D311yescharacteristic
260C8—22no—
261C8—62no—
262C2×C3—22no—
263C2×C3—44no—
264C2×C3C2×#12011yescharacteristic
265C2×C3#57712yes—
266D3—42no—
267D3—42no—
268C22—22no—
269C22—24no—
270C22—44no—
271C22—62no—
272C22—62no—
273C22—64no—
274C22—84no—
275C22—122no—
276C22—248no—
277C22#88212yes—
278C22C2×D3×D411yescharacteristic
279C4—42no—
280C4—62no—
281C4—62no—
282C4—122no—
283C4#88212yes—
284C4C22×D1211yescharacteristic
285C4C2×D3×D411yescharacteristic
286C3#300911yescharacteristic, complemented, Sylow
287C2—22no—
288C2—44no—
289C2—122no—
290C2—122no—
291C2C2×#88211yescharacteristic
292C2#577512yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#418C2
socleC22×C3C22×D4
prime radicalC2×#1004C2
hypercenter#418D3
hypocenterC3#3009
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C22×#214C2semidirect
3. C2×#909C2semidirect
4. #5526C2semidirect
5. C3×#418C2semidirect
6. #5129C2semidirect
7. #5002C2semidirect
8. #4783C2semidirect
9. #4776C2semidirect
21. C22×Dic6C22semidirect
22. C2×C3×M16C22semidirect
23. C3×#117C22semidirect
25. C2×#214C22semidirect
26. C22×C3⋊C8C22semidirect
28. C2×Dic12C22semidirect
29. C2×#204C22semidirect
30. #897C22semidirect
34. C3×#102C22semidirect
35. (C2×D3)⋊C8C22semidirect
36. #810C22semidirect
37. #809C22semidirect
44. #418D3semidirect
109. C2×M16C2×D3semidirect
115. #117C2×D3semidirect
118. #102C2×D3semidirect
286. C3#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C22×#2142
C2×#9092
#55262
C3×#4182
#51292
#50022
#47832
#47762
#30093