Finite Group Databasebeta

← #78671

Subgroups — #78671

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×#1009C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic, complemented
3C2×#909C211yesmaximal, characteristic, complemented
4C22×#204C211yesmaximal, characteristic, complemented
5C3×#408C211yesmaximal, characteristic, complemented
6#5131C211yesmaximal, characteristic, complemented
7#5113C211yesmaximal, characteristic, complemented
8#4733C214yesmaximal, complemented
9#4731C214yesmaximal, complemented
10#3009—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12#1009—24no—
13#1009—24no—
14C2×#239—21no—
15C2×D3×D4—22no—
16C2×D3×D4—21no—
17C2×#235C2212yes—
18C2×#235C2211yescharacteristic
19C22×D12C2212yes—
20C22×D12C2211yescharacteristic
21C22×Dic6C2211yescharacteristic, complemented
22C2×C3×M16C2211yescharacteristic, complemented
23C22×C3×C8C2211yescharacteristic, complemented
24#944—21no—
25#936—21no—
26#935—22no—
27#926C2211yescharacteristic, complemented
28#924C2214yescomplemented
29#909—24no—
30C2×Dic12C2212yescomplemented
31C2×#204—24no—
32C2×#204C2214yescomplemented
33C2×#204C2212yescomplemented
34C3×C22⋊C8C2214yescomplemented
35#821C2214yescomplemented
36#821C2212yescomplemented
37#816C2212yescomplemented
38C2×H(5,𝔽2)—31no—
39C2×#137—31no—
40C22×SD16—31no—
41#536—31no—
42#452—34no—
43#450—34no—
44#418—31no—
45#416—31no—
46#408D311yescharacteristic, complemented
47C23×D3—22no—
48C23×D3—21no—
49C3×#41—22no—
50C3×#41—22no—
51C2×C3×Q8D411yescharacteristic
52C2×C3×D4D412yes—
53C2×C3×D4D411yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67—22no—
57C2×#67—21no—
58C22×Dic3—21no—
59#239—28no—
60D3×D4—216no—
61D3×D4—28no—
62#235—22no—
63#235—24no—
64#235—24no—
65#235D414yes—
66C2×D12—22no—
67C2×D12—21no—
68C2×D12—22no—
69C2×D12—24no—
70C2×D12—24no—
71C2×D12D414yes—
72C2×D12C2311yescharacteristic
73C2×D12C2312yes—
74C2×C4×D3—22no—
75C2×C4×D3—21no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—21no—
79C2×Dic6C2312yes—
80C2×Dic6C2311yescharacteristic
81C3×M16—22no—
82C2×C3×C8—22no—
83C2×C3×C8C2312yes—
84C2×C3×C8C2312yes—
85#217—24no—
86#217—21no—
87C4⋊Dic3C2312yes—
88Dic3⋊C4—24no—
89C4×Dic3—21no—
90Dic12—24no—
91#204—28no—
92#204—24no—
93H(5,𝔽2)—64no—
94H(5,𝔽2)—64no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—61no—
98C2×#41C2×D311yescharacteristic
99C22×Q8—31no—
100C22×D4—31no—
101C22×D4—32no—
102C22×D4—62no—
103C22×D4—61no—
104#137—64no—
105C2×Q16—32no—
106C2×SD16—34no—
107C2×SD16—32no—
108C2×SD16—64no—
109C2×M16C2×D311yescharacteristic, complemented
110C22×C8C2×D311yescharacteristic, complemented
111C4⋊Q8—61no—
112#124—61no—
113#123—62no—
114C22⋊Q8—34no—
115#117—31no—
116#103—32no—
117#102—32no—
118#102—34no—
119C22⋊C8C2×D314yescomplemented
120C23×C3C2×D412yes—
121C23×C3C2×D411yescharacteristic
122C22×D3—22no—
123C22×D3—21no—
124C22×D3—28no—
125C22×D3—24no—
126C22×D3—82no—
127C22×D3—84no—
128C3×Q8—22no—
129C3×D4—24no—
130C3×D4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D412yes—
137C2×C3×C4C2411yescharacteristic
138#67—28no—
139#67—24no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—42no—
144D12—28no—
145D12—24no—
146D12—21no—
147D12—22no—
148D12C2×D412yes—
149D12C2×D414yes—
150C4×D3—28no—
151C4×D3—24no—
152Dic6—21no—
153Dic6—22no—
154Dic6—22no—
155Dic6—42no—
156Dic6C2×D412yes—
157C3×C8—22no—
158C3×C8—22no—
159C24—62no—
160C24—61no—
161#41—22no—
162#41—22no—
163#41—34no—
164#41—64no—
165#41—64no—
166#41—68no—
167#41—62no—
168C2×Q8—31no—
169C2×Q8—32no—
170C2×Q8—61no—
171C2×Q8—62no—
172C2×Q8—62no—
173C2×Q8D1211yescharacteristic
174C2×D4—34no—
175C2×D4—32no—
176C2×D4—31no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—616no—
180C2×D4—68no—
181C2×D4—62no—
182C2×D4—61no—
183C2×D4—62no—
184C2×D4—61no—
185C2×D4—62no—
186C2×D4D1212yes—
187C2×D4D1211yescharacteristic
188C22×C4—62no—
189C22×C4—61no—
190C22×C4—61no—
191C22×C4C22×D312yes—
192C22×C4C22×D311yescharacteristic
193Q16—64no—
194SD16—68no—
195SD16—64no—
196M16—22no—
197C2×C8—22no—
198C2×C8C22×D312yes—
199C2×C8C22×D312yes—
200C4⋊C4—32no—
201C4⋊C4—64no—
202C22⋊C4—61no—
203C22⋊C4—64no—
204C42—61no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—24no—
208C22×C3#12012yes—
209C22×C3C22×D411yescharacteristic
210C2×D3—24no—
211C2×D3—22no—
212C2×D3—44no—
213C2×D3—42no—
214C2×D3—84no—
215C2×D3—88no—
216C3×C4—22no—
217C3×C4#12012yes—
218C3×C4C22×D411yescharacteristic
219C3×C4C22×D411yescharacteristic
220Dic3—22no—
221Dic3—42no—
222Dic3—42no—
223C23—68no—
224C23—62no—
225C23—61no—
226C23—64no—
227C23—242no—
228C23—244no—
229C23C2×D1212yes—
230C23C2×D1211yescharacteristic
231Q8—22no—
232Q8—32no—
233Q8—61no—
234Q8—62no—
235Q8—62no—
236Q8—122no—
237D4—24no—
238D4—22no—
239D4—34no—
240D4—32no—
241D4—68no—
242D4—68no—
243D4—64no—
244D4—64no—
245D4—61no—
246D4—62no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—68no—
250C2×C4—61no—
251C2×C4—64no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—122no—
255C2×C4C2×D1212yes—
256C2×C4C2×D1211yescharacteristic
257C2×C4D3×D412yes—
258C2×C4D3×D412yes—
259C2×C4C23×D311yescharacteristic
260C8—22no—
261C8—22no—
262C2×C3—22no—
263C2×C3—42no—
264C2×C3C2×#12011yescharacteristic
265C2×C3#57712yes—
266D3—44no—
267D3—42no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—64no—
272C22—62no—
273C22—124no—
274C22—122no—
275C22—244no—
276C22—248no—
277C22#88212yes—
278C22C22×D1211yescharacteristic
279C4—22no—
280C4—62no—
281C4—122no—
282C4—122no—
283C4#88212yes—
284C4C2×D3×D411yescharacteristic
285C4C2×D3×D411yescharacteristic
286C3#300911yescharacteristic, complemented, Sylow
287C2—22no—
288C2—42no—
289C2—124no—
290C2—122no—
291C2C2×#88211yescharacteristic
292C2#575112yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#408C2
socleC22×C3C22×D4
prime radicalC2×#1009C2
hypercenter#408D3
hypocenterC3#3009
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #5825C2semidirect
3. C2×#909C2semidirect
4. C22×#204C2semidirect
5. C3×#408C2semidirect
6. #5131C2semidirect
7. #5113C2semidirect
8. #4733C2semidirect
9. #4731C2semidirect
21. C22×Dic6C22semidirect
22. C2×C3×M16C22semidirect
23. C22×C3×C8C22semidirect
27. #926C22semidirect
28. #924C22semidirect
30. C2×Dic12C22semidirect
32. C2×#204C22semidirect
33. C2×#204C22semidirect
34. C3×C22⋊C8C22semidirect
35. #821C22semidirect
36. #821C22semidirect
37. #816C22semidirect
46. #408D3semidirect
109. C2×M16C2×D3semidirect
110. C22×C8C2×D3semidirect
119. C22⋊C8C2×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×#10092
#58252
C2×#9092
C22×#2042
C3×#4082
#51312
#51132
#47332
#47312
#30093