Finite Group Databasebeta

← #81850

Subgroups — #81850

Order 384.

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

normal subgroups only

This group has 434 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
2#5668C211yesmaximal, characteristic, complemented
3C3×#496C211yesmaximal, characteristic, complemented
4#5172C212yesmaximal, complemented
5#5162C212yesmaximal, complemented
6#5158C212yesmaximal, complemented
7#5150C212yesmaximal, complemented
8#5074C211yesmaximal, characteristic, complemented
9#5055C211yesmaximal, characteristic, complemented
10#4714C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3214—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C2×#237—21no—
15C2×#237C2211yescharacteristic
16C2×C3×SD16C2212yes—
17C2×C3×D8C2212yes—
18C3×#127C2211yescharacteristic
19C3×#124C2211yescharacteristic
20#944C2211yescharacteristic
21#940C2211yescharacteristic
22#938—21no—
23#938C2212yes—
24#936—21no—
25D4×Dic3—21no—
26D4×Dic3C2212yes—
27C2×#214C2212yes—
28C2×#213C2212yes—
29#884—21no—
30#884—21no—
31#67⋊C4—21no—
32#67⋊C4—21no—
33#875C2211yescharacteristic
34C42×D3C2211yescharacteristic
35#869C2211yescharacteristic
36C3×#97C2211yescharacteristic
37#835C2212yes—
38#832C2212yes—
39#821C2212yes—
40(C2×D3)⋊C8C2212yes—
41#816C2212yes—
42Dic3⋊C8C2212yes—
43#803C2211yescharacteristic
44#532—31no—
45#532—31no—
46#496—31no—
47#496D311yescharacteristic, complemented
48#489—31no—
49#489—31no—
50#461—32no—
51#459—32no—
52#449—32no—
53#449—32no—
54#432—31no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4C2311yescharacteristic
59C2×C3×D4C2312yes—
60C2×#67—21no—
61C2×#67—22no—
62C2×#67—21no—
63C2×#67—21no—
64C22×Dic3—21no—
65C22×Dic3—22no—
66#239—42no—
67#237—42no—
68#237—44no—
69C2×D12—21no—
70C2×D12C2311yescharacteristic
71C2×C4×D3—21no—
72C2×C4×D3D411yescharacteristic
73C2×C4×D3D411yescharacteristic
74C2×C4×D3C2311yescharacteristic
75C2×Dic6—21no—
76C2×Dic6C2311yescharacteristic
77C3×SD16—42no—
78C3×D8—42no—
79C2×C3×C8C2312yes—
80C3×C22⋊C4—21no—
81C3×C22⋊C4—21no—
82C3×C42C2311yescharacteristic
83#217—22no—
84#217—22no—
85#214—42no—
86#213—42no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2312yes—
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C4×Dic3D411yescharacteristic
94C4×Dic3D411yescharacteristic
95C4×Dic3C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97C2×#41—31no—
98C2×#41—31no—
99C2×#41—61no—
100C2×SD16—32no—
101C2×SD16C2×D312yes—
102C2×D8—32no—
103C2×D8C2×D312yes—
104C4⋊Q8—31no—
105C4⋊Q8—31no—
106#127—31no—
107#127C2×D311yescharacteristic
108#124—31no—
109#124—61no—
110#124C2×D311yescharacteristic
111#121—32no—
112#121—61no—
113#121—61no—
114#121—61no—
115C4×D4—32no—
116C4×D4—61no—
117C4×D4—61no—
118C4×D4—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
128C23×C3—22no—
129C23×C3—21no—
130C22×D3—21no—
131C22×D3C2×D411yescharacteristic
132C3×Q8—22no—
133C3×D4—22no—
134C3×D4—22no—
135C3×D4—22no—
136C3×D4—42no—
137C3×D4—42no—
138C2×C3×C4—21no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2411yescharacteristic
142#67—42no—
143#67—42no—
144#67—44no—
145#67—42no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—42no—
149C2×Dic3—44no—
150C2×Dic3C2×D411yescharacteristic
151C2×Dic3C2×D411yescharacteristic
152C2×Dic3C2×D411yescharacteristic
153D12—22no—
154D12—42no—
155C4×D3—22no—
156C4×D3—22no—
157C4×D3—41no—
158C4×D3—42no—
159Dic6—22no—
160Dic6—42no—
161C3×C8—22no—
162C3⋊C8—22no—
163#41—122no—
164#41—124no—
165#41—122no—
166C2×Q8—31no—
167C2×Q8—61no—
168C2×Q8C22×D311yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—31no—
172C2×D4—61no—
173C2×D4—61no—
174C2×D4—61no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4C22×D311yescharacteristic
178C2×D4C22×D312yes—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—61no—
183C22×C4—62no—
184C22×C4—61no—
185SD16—42no—
186SD16—122no—
187D8—42no—
188D8—122no—
189C2×C8—32no—
190C2×C8C22×D312yes—
191C4⋊C4—32no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4—61no—
195C22⋊C4—21no—
196C22⋊C4—21no—
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—41no—
210C22×C3—42no—
211C22×C3C22×D411yescharacteristic
212C2×D3—21no—
213C2×D3—21no—
214C2×D3—21no—
215C2×D3—41no—
216C2×D3—41no—
217C2×D3—41no—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—41no—
221C3×C4H(5,𝔽2)12yes—
222Dic3—21no—
223Dic3—21no—
224Dic3—41no—
225Dic3—42no—
226Dic3AGL(1,ℤ/8)12yes—
227C23—21no—
228C23—22no—
229C23—31no—
230C23—61no—
231Q8—22no—
232Q8—62no—
233Q8—122no—
234D4—22no—
235D4—22no—
236D4—22no—
237D4—42no—
238D4—42no—
239D4—62no—
240D4—122no—
241D4—122no—
242D4—122no—
243D4—122no—
244D4—124no—
245C2×C4—21no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—121no—
254C2×C4—124no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4D3×D411yescharacteristic
258C2×C4D3×D411yescharacteristic
259C2×C4C23×D311yescharacteristic
260C8—22no—
261C8—62no—
262C2×C3—81no—
263C2×C3—82no—
264C2×C3#53511yescharacteristic
265C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
266C2×C3#57511yescharacteristic
267D3—41no—
268D3—81no—
269C22—41no—
270C22—42no—
271C22—42no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—61no—
276C22—61no—
277C22—61no—
278C22—121no—
279C22—121no—
280C22—121no—
281C22C2×D3×D411yescharacteristic
282C4—21no—
283C4—21no—
284C4—32no—
285C4—41no—
286C4—61no—
287C4—61no—
288C4—121no—
289C4—122no—
290C4#100412yes—
291C3#321411yescharacteristic, complemented, Sylow
292C2—81no—
293C2—82no—
294C2—121no—
295C2—241no—
296C2#571511yescharacteristic
297C2D3×AGL(1,ℤ/8)11yescharacteristic
298C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#496C2
socleC22×C3C22×D4
hypercenter#496D3
hypocenterC3#3214
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. #5668C2semidirect
3. C3×#496C2semidirect
4. #5172C2semidirect
5. #5162C2semidirect
6. #5158C2semidirect
7. #5150C2semidirect
8. #5074C2semidirect
9. #5055C2semidirect
10. #4714C2semidirect
11. #4705C2semidirect
47. #496D3semidirect
291. C3#3214semidirect

Maximal subgroups

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

subgroupindex
#57162
#56682
C3×#4962
#51722
#51622
#51582
#51502
#50742
#50552
#47142
#47052
#32143