Finite Group Databasebeta

← #80007

Subgroups — #80007

Order 384.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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×#938C211yesmaximal, characteristic, complemented
2#5606C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5158C212yesmaximal, complemented
6#5157C212yesmaximal, complemented
7#5034C211yesmaximal, characteristic, complemented
8#4784C212yesmaximal, complemented
9#4778C212yesmaximal, complemented
10D3×C22⋊C8C211yesmaximal, characteristic, complemented
11#4722C211yesmaximal, characteristic, complemented
12#3188—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—21no—
15C2×#237C2211yescharacteristic, complemented
16C22×C4×D3C2211yescharacteristic
17C2×C3×D8C2212yescomplemented
18C3×#121C2211yescharacteristic
19C2×#217—21no—
20#938—21no—
21#938—21no—
22#938C2212yes—
23#938C2212yes—
24#938C2211yescharacteristic, complemented
25#937—21no—
26D4×Dic3—21no—
27C2×#214C2212yescomplemented
28C2×C4⋊Dic3C2211yescharacteristic
29#924C2211yescharacteristic
30C2×D3×C8C2212yescomplemented
31#897C2211yescharacteristic, complemented
32#892C2211yescharacteristic, complemented
33#883—21no—
34#67⋊C4—21no—
35C3×#102C2212yescomplemented
36C3×C22⋊C8C2211yescharacteristic
37#832C2212yescomplemented
38#832C2212yescomplemented
39#830C2211yescharacteristic
40(C2×D3)⋊C8C2212yescomplemented
41#818C2212yescomplemented
42#816C2212yescomplemented
43#808C2212yescomplemented
44#525—31no—
45C2×#121—31no—
46#484—31no—
47#482—32no—
48#481—31no—
49#466—32no—
50#465—32no—
51#450—31no—
52#449—32no—
53#447D311yescharacteristic, complemented
54C2×C22⋊C8—31no—
55C24×C3—21no—
56C23×D3D411yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4—42no—
60C2×C3×D4C2311yescharacteristic
61C2×C3×D4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C2×#67—21no—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—44no—
68C22×Dic3—21no—
69C22×Dic3—21no—
70C22×Dic3D411yescharacteristic
71#237—42no—
72C2×C4×D3—42no—
73C2×C4×D3D411yescharacteristic
74C2×C4×D3D411yescharacteristic
75C2×C4×D3C2311yescharacteristic
76C2×C4×D3C2311yescharacteristic
77C2×Dic6C2311yescharacteristic
78C3×D8—42no—
79C2×C3×C8C2312yes—
80C3×C4⋊C4C2311yescharacteristic
81C3×C22⋊C4—21no—
82#217—21no—
83#217—21no—
84#217—22no—
85#217—22no—
86#214—42no—
87(C2×D3)⋊C4—21no—
88(C2×D3)⋊C4—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3—21no—
91C4⋊Dic3C2312yes—
92C4⋊Dic3C2311yescharacteristic
93Dic3⋊C4—21no—
94C4×Dic3—21no—
95C2×C3⋊C8C2312yes—
96D3×C8—42no—
97C2×#41—31no—
98C22×D4—61no—
99C22×D4C2×D311yescharacteristic
100C23×C4—31no—
101C2×SD16—32no—
102C2×D8C2×D312yescomplemented
103C22×C8—32no—
104#123—61no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107#121—32no—
108#121—32no—
109#121—31no—
110#121—61no—
111#121—61no—
112#121C2×D311yescharacteristic
113#120—61no—
114C4×D4—61no—
115C4×D4—61no—
116#117—31no—
117C2×C4⋊C4—31no—
118C2×C22⋊C4—61no—
119#107—32no—
120#106—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124#102C2×D312yescomplemented
125C22⋊C8—31no—
126C22⋊C8—32no—
127C22⋊C8C2×D311yescharacteristic
128C23×C3—22no—
129C23×C3—21no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—42no—
133C23×C3C2×D411yescharacteristic
134C22×D3—22no—
135C22×D3—22no—
136C22×D3—41no—
137C22×D3C2×D411yescharacteristic
138C22×D3C2×D411yescharacteristic
139C3×D4—22no—
140C3×D4—22no—
141C3×D4—22no—
142C3×D4—42no—
143C3×D4—42no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2411yescharacteristic
148#67—42no—
149#67—44no—
150#67—42no—
151#67—44no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—41no—
156C2×Dic3—42no—
157C2×Dic3—42no—
158C2×Dic3C2×D411yescharacteristic
159C2×Dic3C2×D411yescharacteristic
160C4×D3—22no—
161C4×D3—22no—
162C4×D3—41no—
163C4×D3—41no—
164Dic6—22no—
165C3×C8—22no—
166C3⋊C8—22no—
167C24—21no—
168C24—31no—
169#41—122no—
170C2×Q8—31no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—42no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—124no—
179C2×D4C22×D311yescharacteristic
180C2×D4C22×D312yes—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—61no—
187C22×C4—61no—
188C22×C4—122no—
189C22×C4C22×D311yescharacteristic
190SD16—122no—
191D8—42no—
192C2×C8—32no—
193C2×C8—122no—
194C2×C8C22×D312yes—
195C4⋊C4—31no—
196C4⋊C4—32no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4C22×D311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—61no—
203C22⋊C4—61no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C22⋊C4—61no—
207C22⋊C4—61no—
208C42—61no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—41no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—41no—
217C22×C3—84no—
218C22×C3#13512yes—
219C22×C3C22×D411yescharacteristic
220C2×D3—21no—
221C2×D3—21no—
222C2×D3—21no—
223C2×D3—21no—
224C2×D3—21no—
225C2×D3—44no—
226C2×D3AGL(1,ℤ/8)12yes—
227C3×C4—21no—
228C3×C4—41no—
229C3×C4H(5,𝔽2)12yes—
230Dic3—21no—
231Dic3—21no—
232Dic3—41no—
233Dic3—42no—
234Dic3—41no—
235C23—21no—
236C23—22no—
237C23—31no—
238C23—31no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—62no—
243C23—62no—
244C23—121no—
245C23D3×D411yescharacteristic
246Q8—62no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—124no—
253D4—122no—
254D4—124no—
255D4—122no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—121no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4D3×D411yescharacteristic
271C2×C4C23×D311yescharacteristic
272C8—22no—
273C8—62no—
274C2×C3—22no—
275C2×C3—81no—
276C2×C3—82no—
277C2×C3#53411yescharacteristic
278C2×C3C2×#13511yescharacteristic
279C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
280D3—22no—
281D3—41no—
282C22—21no—
283C22—21no—
284C22—32no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—61no—
292C22—61no—
293C22—61no—
294C22—61no—
295C22—61no—
296C22—84no—
297C22—124no—
298C22#91212yes—
299C22C2×D3×D411yescharacteristic
300C4—21no—
301C4—41no—
302C4—61no—
303C4—61no—
304C4—122no—
305C4—121no—
306C4—121no—
307C4#100412yes—
308C3#318811yescharacteristic, complemented, Sylow
309C2—22no—
310C2—62no—
311C2—82no—
312C2—81no—
313C2—121no—
314C2#558911yescharacteristic
315C2C2×#91211yescharacteristic
316C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3188
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×#938C2semidirect
2. #5606C2semidirect
3. C3×#447C2semidirect
4. #5218C2semidirect
5. #5158C2semidirect
6. #5157C2semidirect
7. #5034C2semidirect
8. #4784C2semidirect
9. #4778C2semidirect
10. D3×C22⋊C8C2semidirect
11. #4722C2semidirect
15. C2×#237C22semidirect
17. C2×C3×D8C22semidirect
24. #938C22semidirect
27. C2×#214C22semidirect
30. C2×D3×C8C22semidirect
31. #897C22semidirect
32. #892C22semidirect
35. C3×#102C22semidirect
37. #832C22semidirect
38. #832C22semidirect
40. (C2×D3)⋊C8C22semidirect
41. #818C22semidirect
42. #816C22semidirect
43. #808C22semidirect
53. #447D3semidirect
102. C2×D8C2×D3semidirect
124. #102C2×D3semidirect
308. C3#3188semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
#56062
C3×#4472
#52182
#51582
#51572
#50342
#47842
#47782
D3×C22⋊C82
#47222
#31883