Finite Group Databasebeta

← #83483

Subgroups — #83483

Order 384.

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

normal subgroups only

This group has 462 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×D42C211yesmaximal, characteristic
2#5543C211yesmaximal, characteristic, complemented
3#5218C212yesmaximal, complemented
4C2×#832C212yesmaximal, complemented
5#5075C211yesmaximal, characteristic, complemented
6#5070C211yesmaximal, characteristic, complemented
7#5041C212yesmaximal, complemented
8#5034C212yesmaximal, complemented
9C4×#214C211yesmaximal, characteristic, complemented
10D4×C3⋊C8C211yesmaximal, characteristic, complemented
11#5008C211yesmaximal, characteristic, complemented
12#3295—31nomaximal, Sylow
13C22×C3×D4—21no—
14C22×C3×D4C2212yes—
15C3×#127C2211yescharacteristic
16C3×#121—21no—
17C3×#121C2212yes—
18C3×#120—22no—
19C3×C4×D4C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21D4×Dic3—21no—
22C2×#214C2211yescharacteristic
23C2×#214C2212yescomplemented
24C2×C4⋊Dic3C2212yescomplemented
25#924C2212yescomplemented
26C22×C3⋊C8C2212yescomplemented
27#890—21no—
28#878—22no—
29#869C2211yescharacteristic, complemented
30C4×Dic6C2211yescharacteristic, complemented
31#832C2211yescharacteristic
32#832C2212yes—
33#832C2212yescomplemented
34#832C2212yescomplemented
35#832C2212yescomplemented
36#830C2212yescomplemented
37#810C2211yescharacteristic, complemented
38#808C2212yescomplemented
39#808C2211yescharacteristic, complemented
40C4⋊(C3⋊C8)C2211yescharacteristic, complemented
41C4×C3⋊C8C2211yescharacteristic, complemented
42#554—31no—
43D42D311yescharacteristic
44#486—31no—
45#481—32no—
46#476—31no—
47#465—32no—
48#460—31no—
49#450—32no—
50C4×SD16—31no—
51C8×D4—31no—
52C2×#102—32no—
53C24×C3—22no—
54C2×C3×D4—24no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4—42no—
60C2×C3×D4—44no—
61C2×C3×D4D411yescharacteristic
62C2×C3×D4C2312yes—
63C2×C3×D4C2311yescharacteristic
64C22×C3×C4—21no—
65C22×C3×C4C2312yes—
66C22×Dic3—22no—
67C2×Dic6—21no—
68C2×Dic6C2311yescharacteristic
69C3×C4⋊C4D411yescharacteristic
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—21no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4D412yes—
74C3×C42C2311yescharacteristic
75#217—22no—
76#214—22no—
77#214—42no—
78C4⋊Dic3—21no—
79C4⋊Dic3—22no—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2312yes—
82Dic3⋊C4—21no—
83Dic3⋊C4—22no—
84C4×Dic3—21no—
85C2×C3⋊C8—42no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88C2×C3⋊C8C2312yes—
89C22×D4—21no—
90C22×D4C2×D312yes—
91C2×SD16—31no—
92C2×SD16—32no—
93C22×C8—32no—
94C4⋊Q8—31no—
95#127C2×D311yescharacteristic
96#125—61no—
97C22⋊Q8—32no—
98C22⋊Q8—62no—
99#121—21no—
100#121C2×D312yes—
101#120—22no—
102C4×Q8—31no—
103C4×D4—61no—
104C4×D4C2×D311yescharacteristic
105C4×D4C2×D311yescharacteristic
106C2×C4⋊C4—32no—
107#106—31no—
108#106—32no—
109C4⋊C8—31no—
110#103—31no—
111#102—31no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102—32no—
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—21no—
119C23×C3—22no—
120C23×C3—44no—
121C23×C3—44no—
122C23×C3—82no—
123C23×C3C2×D412yes—
124C3×D4—22no—
125C3×D4—21no—
126C3×D4—22no—
127C3×D4—44no—
128C3×D4—44no—
129C3×D4—42no—
130C3×D4—81no—
131C3×D4SD1614yescomplemented
132C3×D4#4112yes—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—42no—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—82no—
145Dic6—21no—
146Dic6—21no—
147Dic6—81no—
148C3⋊C8—21no—
149C3⋊C8—21no—
150C3⋊C8—22no—
151C24—22no—
152C2×Q8—31no—
153C2×Q8—61no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4#6711yescharacteristic
162C2×D4C22×D312yes—
163C2×D4C22×D311yescharacteristic
164C22×C4—21no—
165C22×C4—62no—
166C22×C4C22×D312yes—
167SD16—62no—
168SD16—122no—
169C2×C8—31no—
170C2×C8—31no—
171C2×C8—32no—
172C2×C8—122no—
173C4⋊C4—31no—
174C4⋊C4—32no—
175C4⋊C4—62no—
176C4⋊C4—61no—
177C4⋊C4—61no—
178C4⋊C4—62no—
179C4⋊C4#6711yescharacteristic
180C4⋊C4C22×D311yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—22no—
183C22⋊C4—62no—
184C22⋊C4#6712yes—
185C42—61no—
186C42C22×D311yescharacteristic
187C22×C3—22no—
188C22×C3—22no—
189C22×C3—22no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—88no—
196C22×C3C2×SD1614yes—
197C22×C3C22×D411yescharacteristic
198C3×C4—22no—
199C3×C4—21no—
200C3×C4—41no—
201C3×C4C2×SD1612yes—
202C3×C4C2×#4111yescharacteristic
203C3×C4H(5,𝔽2)11yescharacteristic
204Dic3—41no—
205Dic3—42no—
206Dic3—41no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—44no—
211C23—82no—
212C23C2×#6712yes—
213Q8—61no—
214Q8—61no—
215Q8—241no—
216D4—22no—
217D4—21no—
218D4—22no—
219D4—44no—
220D4—44no—
221D4—42no—
222D4—81no—
223D4#21414yescomplemented
224D4#23712yes—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—62no—
232C2×C4—242no—
233C2×C4C2×#6711yescharacteristic
234C2×C4C2×#6712yes—
235C2×C4C2×#6711yescharacteristic
236C2×C4C23×D311yescharacteristic
237C8—61no—
238C8—61no—
239C8—62no—
240C2×C3—24no—
241C2×C3—42no—
242C2×C3—82no—
243C2×C3#54611yescharacteristic
244C2×C3C22×SD1611yescharacteristic
245C2×C3#57611yescharacteristic
246C22—22no—
247C22—22no—
248C22—22no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—88no—
255C22C2×#21414yes—
256C22C22×#6711yescharacteristic
257C4—22no—
258C4—21no—
259C4—41no—
260C4—121no—
261C4—121no—
262C4—122no—
263C4C2×#21412yes—
264C4C2×#23711yescharacteristic
265C4#100411yescharacteristic
266C3#329511yescharacteristic, complemented, Sylow
267C2—24no—
268C2—42no—
269C2—82no—
270C2C22×#21411yescharacteristic
271C2#580611yescharacteristic
272C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×D42C2
socleC22×C3C22×D4
hypercenterD42D3
hypocenterC3#3295
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
2. #5543C2semidirect
3. #5218C2semidirect
4. C2×#832C2semidirect
5. #5075C2semidirect
6. #5070C2semidirect
7. #5041C2semidirect
8. #5034C2semidirect
9. C4×#214C2semidirect
10. D4×C3⋊C8C2semidirect
11. #5008C2semidirect
23. C2×#214C22semidirect
24. C2×C4⋊Dic3C22semidirect
25. #924C22semidirect
26. C22×C3⋊C8C22semidirect
29. #869C22semidirect
30. C4×Dic6C22semidirect
33. #832C22semidirect
34. #832C22semidirect
35. #832C22semidirect
36. #830C22semidirect
37. #810C22semidirect
38. #808C22semidirect
39. #808C22semidirect
40. C4⋊(C3⋊C8)C22semidirect
41. C4×C3⋊C8C22semidirect
131. C3×D4SD16semidirect
223. D4#214semidirect
266. C3#3295semidirect

Maximal subgroups

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

subgroupindex
C3×D422
#55432
#52182
C2×#8322
#50752
#50702
#50412
#50342
C4×#2142
D4×C3⋊C82
#50082
#32953