Finite Group Databasebeta

← #79088

Subgroups — #79088

Order 384.

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

normal subgroups only

This group has 504 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
1C2×#908C211yesmaximal, characteristic, complemented
2C2×#904C211yesmaximal, characteristic, complemented
3#5525C211yesmaximal, characteristic, complemented
4#5523C211yesmaximal, characteristic, complemented
5C3×#424C211yesmaximal, characteristic, complemented
6#5133C211yesmaximal, characteristic, complemented
7C2×#821C211yesmaximal, characteristic, complemented
8#4848C212yesmaximal, complemented
9#4845C212yesmaximal, complemented
10#4844C212yesmaximal, complemented
11#4843C212yesmaximal, complemented
12#3045—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C2×C3×M16C2211yescharacteristic
17C22×C3×C8C2211yescharacteristic
18C3×#117C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20#908—24no—
21C2×Dic12C2211yescharacteristic, complemented
22#904—24no—
23C2×D24C2212yes—
24C2×D24C2211yescharacteristic, complemented
25C2×#204C2212yes—
26C2×#204C2211yescharacteristic, complemented
27C2×#204C2211yescharacteristic, complemented
28#895—21no—
29#895—21no—
30#892—21no—
31D3×C4⋊C4—21no—
32#882—22no—
33#67⋊C4—22no—
34#875C2212yes—
35#874C2212yes—
36C4×D12C2212yescomplemented
37C4×Dic6C2212yescomplemented
38C3×C4⋊C8C2212yescomplemented
39C3×C4⋊C8C2212yescomplemented
40#821C2212yes—
41#821C2212yes—
42#821C2212yescomplemented
43#816C2212yescomplemented
44C2×AGL(1,ℤ/8)—31no—
45C2×#135—31no—
46#535—31no—
47#520—31no—
48#464—32no—
49#463—32no—
50#460—32no—
51#459—32no—
52#424D311yescharacteristic, complemented
53#417—31no—
54C2×#102—31no—
55C23×D3—21no—
56C22×C3×C4C2311yescharacteristic
57C2×#67—21no—
58C2×#67—21no—
59C22×Dic3—21no—
60#235—22no—
61#235—22no—
62#235—22no—
63#235D414yescomplemented
64C2×D12—22no—
65C2×D12—22no—
66C2×D12—21no—
67C2×D12—22no—
68C2×D12—22no—
69C2×D12C2312yes—
70C2×D12C2311yescharacteristic
71C2×D12C2311yescharacteristic
72C2×C4×D3—21no—
73C2×C4×D3—21no—
74C2×C4×D3—22no—
75C2×Dic6C2311yescharacteristic
76C2×Dic6C2311yescharacteristic
77C3×M16—22no—
78C2×C3×C8—22no—
79C2×C3×C8C2312yes—
80C2×C3×C8C2311yescharacteristic
81C2×C3×C8C2311yescharacteristic
82C3×C4⋊C4D411yescharacteristic
83C3×C4⋊C4D411yescharacteristic
84C3×C22⋊C4D412yes—
85C3×C42C2312yes—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89C4⋊Dic3—21no—
90C4⋊Dic3C2312yes—
91Dic3⋊C4—22no—
92C4×Dic3—22no—
93Dic12—22no—
94D24—24no—
95D24—22no—
96#204—22no—
97#204—24no—
98#204—22no—
99C2×#41—31no—
100C2×#41—31no—
101C22×D4—31no—
102AGL(1,ℤ/8)—64no—
103#135—64no—
104C2×Q16—31no—
105C2×SD16—31no—
106C2×SD16—31no—
107C2×SD16—32no—
108C2×D8—31no—
109C2×D8—32no—
110C2×M16C2×D311yescharacteristic
111C22×C8C2×D311yescharacteristic
112#127—32no—
113#124—32no—
114#121—61no—
115#121—61no—
116#120—62no—
117C4×Q8—32no—
118C4×D4—32no—
119C4×D4—62no—
120#117—61no—
121#117C2×D311yescharacteristic
122C2×C4⋊C4—31no—
123C2×C4⋊C4—61no—
124C4⋊C8C2×D312yescomplemented
125C4⋊C8C2×D312yescomplemented
126#103—32no—
127#102—32no—
128#102—32no—
129#102—32no—
130C23×C3C2×D411yescharacteristic
131C22×D3—21no—
132C22×D3—22no—
133C22×D3—21no—
134C22×D3—42no—
135C22×D3—81no—
136C22×D3—81no—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D412yes—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D412yes—
141C2×C3×C4#4112yes—
142C2×C3×C4C2411yescharacteristic
143#67—24no—
144#67—42no—
145C2×Dic3—21no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—81no—
149D12—22no—
150D12—21no—
151D12—21no—
152D12—22no—
153D12—21no—
154D12—42no—
155D12—82no—
156D12—82no—
157D12C2×D412yes—
158C4×D3—24no—
159C4×D3—42no—
160C4×D3—82no—
161Dic6—21no—
162Dic6—21no—
163Dic6—21no—
164Dic6C2×D412yes—
165C3×C8—21no—
166C3×C8—22no—
167C3×C8—21no—
168C24—61no—
169#41—34no—
170#41—62no—
171#41—62no—
172#41—62no—
173C2×Q8—31no—
174C2×Q8—31no—
175C2×D4—31no—
176C2×D4—32no—
177C2×D4—31no—
178C2×D4—61no—
179C2×D4—61no—
180C2×D4—61no—
181C2×D4—62no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—62no—
185C22×C4—61no—
186C22×C4—61no—
187C22×C4—61no—
188C22×C4—62no—
189C22×C4C22×D311yescharacteristic
190Q16—62no—
191SD16—62no—
192SD16—62no—
193SD16—64no—
194D8—62no—
195D8—64no—
196M16—22no—
197C2×C8—22no—
198C2×C8C22×D311yescharacteristic
199C2×C8C22×D311yescharacteristic
200C2×C8C22×D312yes—
201C4⋊C4—32no—
202C4⋊C4—61no—
203C4⋊C4—62no—
204C4⋊C4D1211yescharacteristic
205C4⋊C4D1211yescharacteristic
206C22⋊C4—62no—
207C22⋊C4—62no—
208C22⋊C4—62no—
209C22⋊C4D1212yes—
210C42—62no—
211C42C22×D312yes—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3#12112yes—
215C22×C3C22×D411yescharacteristic
216C2×D3—22no—
217C2×D3—41no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—42no—
221C2×D3—41no—
222C2×D3—42no—
223C2×D3—41no—
224C2×D3—42no—
225C2×D3—84no—
226C3×C4—22no—
227C3×C4—22no—
228C3×C4#12112yes—
229C3×C4C22×D411yescharacteristic
230C3×C4C2×#4111yescharacteristic
231Dic3—22no—
232Dic3—41no—
233Dic3—42no—
234C23—61no—
235C23—62no—
236C23—61no—
237C23—122no—
238C23—241no—
239C23—241no—
240C23C2×D1211yescharacteristic
241Q8—32no—
242Q8—61no—
243Q8—61no—
244Q8—61no—
245D4—32no—
246D4—64no—
247D4—62no—
248D4—61no—
249D4—61no—
250D4—62no—
251D4—61no—
252D4—122no—
253D4—122no—
254D4—242no—
255D4—242no—
256C2×C4—61no—
257C2×C4—64no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—122no—
261C2×C4—242no—
262C2×C4—241no—
263C2×C4C2×D1212yes—
264C2×C4C2×D1212yes—
265C2×C4C2×D1211yescharacteristic
266C2×C4D3×D412yes—
267C2×C4#23912yes—
268C2×C4C23×D311yescharacteristic
269C8—21no—
270C8—21no—
271C8—22no—
272C2×C3—22no—
273C2×C3C2×#12111yescharacteristic
274C2×C3#57611yescharacteristic
275C2×C3#57711yescharacteristic
276D3—42no—
277D3—81no—
278D3—82no—
279C22—21no—
280C22—21no—
281C22—62no—
282C22—121no—
283C22—121no—
284C22—121no—
285C22—122no—
286C22—121no—
287C22—122no—
288C22—121no—
289C22—122no—
290C22—244no—
291C22#89512yes—
292C22C22×D1211yescharacteristic
293C4—22no—
294C4—22no—
295C4—62no—
296C4—122no—
297C4—121no—
298C4#89512yes—
299C4C2×D3×D411yescharacteristic
300C4C2×#23911yescharacteristic
301C3#304511yescharacteristic, complemented, Sylow
302C2—22no—
303C2—122no—
304C2—242no—
305C2—241no—
306C2C2×#89511yescharacteristic
307C2#575111yescharacteristic
308C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#424C2
socleC22×C3C22×D4
hypercenter#424D3
hypocenterC3#3045
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. C2×#908C2semidirect
2. C2×#904C2semidirect
3. #5525C2semidirect
4. #5523C2semidirect
5. C3×#424C2semidirect
6. #5133C2semidirect
7. C2×#821C2semidirect
8. #4848C2semidirect
9. #4845C2semidirect
10. #4844C2semidirect
11. #4843C2semidirect
21. C2×Dic12C22semidirect
24. C2×D24C22semidirect
26. C2×#204C22semidirect
27. C2×#204C22semidirect
36. C4×D12C22semidirect
37. C4×Dic6C22semidirect
38. C3×C4⋊C8C22semidirect
39. C3×C4⋊C8C22semidirect
42. #821C22semidirect
43. #816C22semidirect
52. #424D3semidirect
63. #235D4semidirect
124. C4⋊C8C2×D3semidirect
125. C4⋊C8C2×D3semidirect
301. C3#3045semidirect

Maximal subgroups

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

subgroupindex
C2×#9082
C2×#9042
#55252
#55232
C3×#4242
#51332
C2×#8212
#48482
#48452
#48442
#48432
#30453