Finite Group Databasebeta

← #81849

Subgroups — #81849

Order 384.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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#5716C211yesmaximal, characteristic, complemented
2D3×#124C211yesmaximal, characteristic, complemented
3C3×#496C211yesmaximal, characteristic, complemented
4#5170C212yesmaximal, complemented
5#5165C212yesmaximal, complemented
6#5156C212yesmaximal, complemented
7#5153C212yesmaximal, complemented
8#5074C211yesmaximal, characteristic, complemented
9#5055C211yesmaximal, characteristic, complemented
10#4714C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3213—31nomaximal, Sylow
13C2×D3×Q8C2211yescharacteristic
14C2×#237—21no—
15C2×D3×D4C2211yescharacteristic
16C2×C3×SD16C2212yescomplemented
17C2×C3×D8C2212yes—
18C3×#127C2211yescharacteristic, complemented
19C3×#124C2211yescharacteristic
20#947C2211yescharacteristic
21#938—21no—
22#938C2212yes—
23#936—21no—
24#936C2211yescharacteristic
25D4×Dic3—21no—
26D4×Dic3C2212yescomplemented
27C2×#214C2212yescomplemented
28C2×#213C2212yes—
29#885—21no—
30#885—21no—
31D3×C22⋊C4—21no—
32D3×C22⋊C4—21no—
33#875C2211yescharacteristic
34C42×D3C2211yescharacteristic
35#869C2211yescharacteristic, complemented
36C3×#97C2211yescharacteristic, complemented
37#835C2212yescomplemented
38#832C2212yes—
39#821C2212yes—
40(C2×D3)⋊C8C2212yes—
41#816C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#803C2211yescharacteristic, complemented
44#532—31no—
45C2×#124—31no—
46#496—31no—
47#496D311yescharacteristic, complemented
48#489—31no—
49#489—31no—
50#463—32no—
51#463—32no—
52#448—32no—
53#447—32no—
54#432—31no—
55C23×D3—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4C2311yescharacteristic
60C2×C3×D4C2312yes—
61C2×#67—22no—
62C2×#67—21no—
63C22×Dic3—22no—
64D3×Q8—42no—
65#237—44no—
66D3×D4—42no—
67C2×D12C2311yescharacteristic
68C2×C4×D3—21no—
69C2×C4×D3D411yescharacteristic
70C2×C4×D3D411yescharacteristic
71C2×C4×D3C2311yescharacteristic
72C2×Dic6—21no—
73C2×Dic6—21no—
74C2×Dic6C2311yescharacteristic
75C3×SD16—42no—
76C3×D8—42no—
77C2×C3×C8C2312yes—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—21no—
80C3×C42C2311yescharacteristic
81#217—21no—
82#217—21no—
83#217—22no—
84#217—22no—
85#214—42no—
86#213—42no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—21no—
89(C2×D3)⋊C4—21no—
90(C2×D3)⋊C4—21no—
91C4⋊Dic3—21no—
92C4⋊Dic3C2312yes—
93C4×Dic3D411yescharacteristic
94C4×Dic3D411yescharacteristic
95C4×Dic3C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97C2×#41—61no—
98C22×Q8—31no—
99C22×D4—31no—
100C2×SD16—32no—
101C2×SD16C2×D312yescomplemented
102C2×D8—32no—
103C2×D8C2×D312yes—
104C4⋊Q8—31no—
105#127C2×D311yescharacteristic, complemented
106#124—31no—
107#124—31no—
108#124—31no—
109#124—61no—
110#124—61no—
111#124—61no—
112#124C2×D311yescharacteristic
113#121—32no—
114#121—61no—
115C4×D4—32no—
116C4×D4—61no—
117C2×C22⋊C4—61no—
118C2×C22⋊C4—61no—
119C2×C42—31no—
120C4⋊C8—32no—
121#103—32no—
122#103—32no—
123#102—32no—
124#102—32no—
125C22⋊C8—32no—
126#97—31no—
127#97C2×D311yescharacteristic, complemented
128C23×C3—21no—
129C23×C3—22no—
130C22×D3—21no—
131C22×D3—42no—
132C22×D3—42no—
133C22×D3—42no—
134C22×D3C2×D411yescharacteristic
135C3×Q8—22no—
136C3×D4—22no—
137C3×D4—22no—
138C3×D4—22no—
139C3×D4—42no—
140C3×D4—42no—
141C2×C3×C4—21no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2411yescharacteristic
145#67—42no—
146#67—44no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—44no—
150C2×Dic3C2×D411yescharacteristic
151C2×Dic3C2×D411yescharacteristic
152C2×Dic3C2×D411yescharacteristic
153D12—22no—
154C4×D3—22no—
155C4×D3—22no—
156C4×D3—41no—
157C4×D3—42no—
158Dic6—22no—
159Dic6—42no—
160Dic6—42no—
161C3×C8—22no—
162C3⋊C8—22no—
163C24—61no—
164#41—124no—
165C2×Q8—31no—
166C2×Q8—61no—
167C2×Q8—61no—
168C2×Q8—122no—
169C2×Q8C22×D311yescharacteristic
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—31no—
173C2×D4—61no—
174C2×D4—62no—
175C2×D4—122no—
176C2×D4C22×D311yescharacteristic
177C2×D4C22×D312yes—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—61no—
182C22×C4—62no—
183SD16—42no—
184SD16—122no—
185D8—42no—
186D8—122no—
187C2×C8—32no—
188C2×C8C22×D312yes—
189C4⋊C4—32no—
190C4⋊C4—61no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—61no—
194C22⋊C4—61no—
195C22⋊C4—61no—
196C22⋊C4—61no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C42—31no—
202C42—31no—
203C42—31no—
204C42C22×D311yescharacteristic
205C22×C3—41no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—41no—
209C22×C3—42no—
210C22×C3—41no—
211C22×C3C22×D411yescharacteristic
212C2×D3—21no—
213C2×D3—21no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—41no—
217C2×D3—84no—
218C2×D3AGL(1,ℤ/8)12yes—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—41no—
222C3×C4H(5,𝔽2)12yes—
223Dic3—21no—
224Dic3—21no—
225Dic3—21no—
226Dic3—41no—
227Dic3—42no—
228C23—21no—
229C23—22no—
230C23—31no—
231C23—61no—
232C23—122no—
233C23—122no—
234C23—122no—
235Q8—22no—
236Q8—62no—
237Q8—122no—
238Q8—122no—
239D4—22no—
240D4—22no—
241D4—22no—
242D4—42no—
243D4—42no—
244D4—62no—
245D4—122no—
246D4—124no—
247C2×C4—21no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—61no—
254C2×C4—62no—
255C2×C4—121no—
256C2×C4—122no—
257C2×C4—124no—
258C2×C4D3×D411yescharacteristic
259C2×C4D3×D411yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—22no—
262C8—62no—
263C2×C3—81no—
264C2×C3—82no—
265C2×C3#53511yescharacteristic
266C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
267C2×C3#57511yescharacteristic
268D3—22no—
269D3—81no—
270C22—32no—
271C22—41no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—42no—
276C22—41no—
277C22—61no—
278C22—61no—
279C22—121no—
280C22—121no—
281C22—121no—
282C22—244no—
283C22C2×D3×D411yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—61no—
288C4—61no—
289C4—61no—
290C4—121no—
291C4—122no—
292C4#100412yes—
293C3#321311yescharacteristic, complemented, Sylow
294C2—62no—
295C2—81no—
296C2—82no—
297C2—241no—
298C2#571511yescharacteristic
299C2D3×AGL(1,ℤ/8)11yescharacteristic
300C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#496C2
socleC22×C3C22×D4
hypercenter#496D3
hypocenterC3#3213
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #5716C2semidirect
2. D3×#124C2semidirect
3. C3×#496C2semidirect
4. #5170C2semidirect
5. #5165C2semidirect
6. #5156C2semidirect
7. #5153C2semidirect
8. #5074C2semidirect
9. #5055C2semidirect
10. #4714C2semidirect
11. #4705C2semidirect
16. C2×C3×SD16C22semidirect
18. C3×#127C22semidirect
26. D4×Dic3C22semidirect
27. C2×#214C22semidirect
35. #869C22semidirect
36. C3×#97C22semidirect
37. #835C22semidirect
41. #816C22semidirect
42. Dic3⋊C8C22semidirect
43. #803C22semidirect
47. #496D3semidirect
101. C2×SD16C2×D3semidirect
105. #127C2×D3semidirect
127. #97C2×D3semidirect
293. C3#3213semidirect

Maximal subgroups

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

subgroupindex
#57162
D3×#1242
C3×#4962
#51702
#51652
#51562
#51532
#50742
#50552
#47142
#47052
#32133