Finite Group Databasebeta

← #79996

Subgroups — #79996

Order 384.

444 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 113 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×#936C211yesmaximal, characteristic, complemented
2#5596C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5154C212yesmaximal, complemented
6#5153C212yesmaximal, complemented
7#5040C211yesmaximal, characteristic, complemented
8#4787C212yesmaximal, complemented
9#4765C212yesmaximal, complemented
10#4738C211yesmaximal, characteristic, complemented
11#4719C211yesmaximal, characteristic, complemented
12#3213—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C2×#237—21no—
15C22×Dic6C2211yescharacteristic
16C2×C3×D8C2212yes—
17C3×#121C2211yescharacteristic
18C2×#217—21no—
19C2×#217—21no—
20#940C2211yescharacteristic, complemented
21#939—21no—
22#936—21no—
23#936—21no—
24#936C2212yes—
25#936C2212yes—
26D4×Dic3C2211yescharacteristic, complemented
27C2×#214C2212yescomplemented
28C2×#213C2212yes—
29#930C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31C2×#204C2212yescomplemented
32#893C2211yescharacteristic, complemented
33#888C2211yescharacteristic, complemented
34#885—21no—
35#67⋊C4—21no—
36C3×#102C2212yes—
37C3×C22⋊C8C2211yescharacteristic
38#832C2212yes—
39#830C2211yescharacteristic
40#816C2212yescomplemented
41#815C2212yescomplemented
42Dic3⋊C8C2212yescomplemented
43#810C2212yescomplemented
44#532—31no—
45C2×#124—31no—
46#496—32no—
47#489—32no—
48#463—32no—
49#463—32no—
50#448—31no—
51#448—31no—
52#447—31no—
53#447D311yescharacteristic, complemented
54#432—31no—
55C24×C3—21no—
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4—42no—
59C2×C3×D4C2312yes—
60C2×C3×D4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C2×#67—21no—
63C2×#67—21no—
64C2×#67—21no—
65C22×Dic3—21no—
66C22×Dic3—21no—
67C22×Dic3D411yescharacteristic
68C22×Dic3D411yescharacteristic
69#237—44no—
70C2×D12C2311yescharacteristic
71C2×C4×D3—21no—
72C2×Dic6—21no—
73C2×Dic6—21no—
74C2×Dic6—42no—
75C2×Dic6C2312yes—
76C3×D8—42no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80#217—21no—
81#217—22no—
82#217—22no—
83#217—22no—
84#217—22no—
85#214—42no—
86#213—42no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—21no—
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91C4×Dic3D411yescharacteristic
92C4×Dic3D411yescharacteristic
93C4×Dic3C2311yescharacteristic
94C4×Dic3C2311yescharacteristic
95C2×C3⋊C8C2312yes—
96#204—42no—
97C2×#41—61no—
98C22×Q8—31no—
99C22×D4C2×D311yescharacteristic
100C2×SD16—32no—
101C2×SD16—32no—
102C2×D8—32no—
103C2×D8C2×D312yes—
104C4⋊Q8—31no—
105#127—31no—
106#124—32no—
107#124—32no—
108#124—61no—
109#124—61no—
110#124—61no—
111#121—31no—
112#121—61no—
113#121C2×D311yescharacteristic
114C4×D4—31no—
115C4×D4—31no—
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#102C2×D312yes—
125C22⋊C8—31no—
126C22⋊C8C2×D311yescharacteristic
127#97—32no—
128C23×C3—22no—
129C23×C3—21no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—42no—
133C23×C3C2×D411yescharacteristic
134C22×D3—21no—
135C3×D4—22no—
136C3×D4—22no—
137C3×D4—22no—
138C3×D4—42no—
139C3×D4—42no—
140C2×C3×C4—22no—
141C2×C3×C4—21no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2411yescharacteristic
144#67—42no—
145#67—42no—
146#67—42no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—41no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3C2×D411yescharacteristic
154C2×Dic3C2×D411yescharacteristic
155C2×Dic3C2×D411yescharacteristic
156C2×Dic3C2×D411yescharacteristic
157D12—22no—
158C4×D3—42no—
159Dic6—22no—
160Dic6—22no—
161Dic6—42no—
162Dic6—42no—
163C3×C8—22no—
164C3⋊C8—22no—
165C24—21no—
166#41—124no—
167C2×Q8—32no—
168C2×Q8—61no—
169C2×Q8—61no—
170C2×Q8—122no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—31no—
174C2×D4—42no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4C22×D312yes—
179C2×D4C22×D311yescharacteristic
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4C22×D311yescharacteristic
186SD16—122no—
187SD16—122no—
188D8—42no—
189D8—122no—
190C2×C8—32no—
191C2×C8C22×D312yes—
192C4⋊C4—31no—
193C4⋊C4—61no—
194C4⋊C4C22×D311yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—61no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22⋊C4—61no—
202C22⋊C4—61no—
203C42—31no—
204C42—31no—
205C42—31no—
206C42—31no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—42no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—41no—
214C22×C3—42no—
215C22×C3—84no—
216C22×C3AGL(1,ℤ/8)12yes—
217C22×C3C22×D411yescharacteristic
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—41no—
221C3×C4—21no—
222C3×C4—41no—
223C3×C4H(5,𝔽2)12yes—
224Dic3—21no—
225Dic3—21no—
226Dic3—21no—
227Dic3—21no—
228Dic3—42no—
229Dic3—41no—
230C23—21no—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—61no—
236C23D3×D411yescharacteristic
237Q8—62no—
238Q8—62no—
239Q8—122no—
240Q8—122no—
241D4—22no—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—42no—
246D4—62no—
247D4—122no—
248D4—122no—
249D4—122no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—122no—
260C2×C4—121no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4D3×D411yescharacteristic
264C2×C4C23×D311yescharacteristic
265C8—22no—
266C8—62no—
267C2×C3—22no—
268C2×C3—81no—
269C2×C3—82no—
270C2×C3#53511yescharacteristic
271C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
272C2×C3#57511yescharacteristic
273D3—81no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—84no—
283C22—121no—
284C22—121no—
285C22—121no—
286C22#91112yes—
287C22C2×D3×D411yescharacteristic
288C4—21no—
289C4—41no—
290C4—61no—
291C4—61no—
292C4—61no—
293C4—61no—
294C4—121no—
295C4—122no—
296C4#100412yes—
297C3#321311yescharacteristic, complemented, Sylow
298C2—22no—
299C2—82no—
300C2—81no—
301C2—241no—
302C2#559211yescharacteristic
303C2C2×#91111yescharacteristic
304C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3213
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#936C2semidirect
2. #5596C2semidirect
3. C3×#447C2semidirect
4. #5217C2semidirect
5. #5154C2semidirect
6. #5153C2semidirect
7. #5040C2semidirect
8. #4787C2semidirect
9. #4765C2semidirect
10. #4738C2semidirect
11. #4719C2semidirect
20. #940C22semidirect
26. D4×Dic3C22semidirect
27. C2×#214C22semidirect
31. C2×#204C22semidirect
32. #893C22semidirect
33. #888C22semidirect
40. #816C22semidirect
41. #815C22semidirect
42. Dic3⋊C8C22semidirect
43. #810C22semidirect
53. #447D3semidirect
297. C3#3213semidirect

Maximal subgroups

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

subgroupindex
C2×#9362
#55962
C3×#4472
#52172
#51542
#51532
#50402
#47872
#47652
#47382
#47192
#32133