Finite Group Databasebeta

← #80664

Subgroups — #80664

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5627C211yesmaximal, characteristic, complemented
2D3×#121C211yesmaximal, characteristic, complemented
3C3×#465C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5169C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
8#5048C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4866C211yesmaximal, characteristic, complemented
11#4863C211yesmaximal, characteristic, complemented
12#4811C211yesmaximal, characteristic, complemented
13#4810C211yesmaximal, characteristic, complemented
14#4784C211yesmaximal, characteristic, complemented
15#4773C211yesmaximal, characteristic, complemented
16#3188—31nomaximal, Sylow
17C2×#239C2211yescharacteristic, complemented
18C2×D3×D4—21no—
19C2×D3×D4C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic, complemented
22C22×C3×C8C2211yescharacteristic, complemented
23C3×C22⋊Q8C2211yescharacteristic, complemented
24C3×#121C2211yescharacteristic
25#946C2211yescharacteristic
26C2×#215C2211yescharacteristic, complemented
27#939—21no—
28#938C2211yescharacteristic
29C2×#213C2211yescharacteristic, complemented
30#930C2211yescharacteristic
31#930C2211yescharacteristic, complemented
32C22×C3⋊C8C2211yescharacteristic, complemented
33C2×D24C2211yescharacteristic, complemented
34#895C2211yescharacteristic
35#895C2211yescharacteristic
36#894—21no—
37#893—21no—
38#892C2211yescharacteristic, complemented
39D3×C4⋊C4C2211yescharacteristic
40#884—21no—
41#882—21no—
42#67⋊C4—21no—
43D3×C22⋊C4—21no—
44C3×#106C2211yescharacteristic, complemented
45C3×#103C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#821C2211yescharacteristic, complemented
50#821C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55#818C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#525—31no—
61C2×#121—31no—
62#484—31no—
63#482—31no—
64#482—31no—
65#481—31no—
66#466—31no—
67#466—31no—
68#465—31no—
69#465D311yescharacteristic, complemented
70#450—31no—
71#449—31no—
72#449—31no—
73#447—31no—
74C2×C22⋊C8—31no—
75C23×D3—21no—
76C23×D3D411yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3D411yescharacteristic
86#239—41no—
87#239—41no—
88D3×D4—41no—
89D3×D4—41no—
90D3×D4—44no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3—41no—
99C2×C4×D3—41no—
100C2×C4×D3D411yescharacteristic
101C2×C4×D3D411yescharacteristic
102C2×C4×D3C2311yescharacteristic
103C2×C4×D3C2311yescharacteristic
104C3×SD16—41no—
105C2×C3×C8—41no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4—21no—
109C3×C4⋊C4C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113#217—21no—
114#215—41no—
115#213—41no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122C4⋊Dic3C2311yescharacteristic
123C4⋊Dic3C2311yescharacteristic
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126C4×Dic3—21no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8C2311yescharacteristic
129C2×C3⋊C8C2311yescharacteristic
130D24—41no—
131C2×#41—31no—
132C22×D4—31no—
133C22×D4—61no—
134C23×C4—31no—
135C2×SD16—31no—
136C2×SD16C2×D311yescharacteristic, complemented
137C2×D8—31no—
138C2×D8—31no—
139C22×C8—31no—
140C22×C8C2×D311yescharacteristic, complemented
141#123—61no—
142C22⋊Q8—31no—
143C22⋊Q8C2×D311yescharacteristic, complemented
144#121—31no—
145#121—31no—
146#121—31no—
147#121—31no—
148#121—31no—
149#121—61no—
150#121—61no—
151#121C2×D311yescharacteristic
152#120—61no—
153C4×D4—61no—
154C4×D4—61no—
155#117—31no—
156C2×C4⋊C4—31no—
157C2×C22⋊C4—61no—
158#107—31no—
159#107—31no—
160#106—31no—
161#106C2×D311yescharacteristic, complemented
162#103—31no—
163#103C2×D311yescharacteristic, complemented
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102—31no—
169#102C2×D311yescharacteristic, complemented
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C22⋊C8—31no—
174C23×C3—21no—
175C23×C3C2×D411yescharacteristic
176C22×D3—22no—
177C22×D3—22no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—41no—
181C22×D3—42no—
182C22×D3—42no—
183C22×D3—42no—
184C22×D3C2×D411yescharacteristic
185C22×D3C2×D411yescharacteristic
186C3×Q8—21no—
187C3×Q8—21no—
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—42no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2411yescharacteristic
198#67—42no—
199#67—42no—
200#67—42no—
201#67—42no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—41no—
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207D12—21no—
208D12—21no—
209D12—21no—
210D12—21no—
211D12—42no—
212D12—42no—
213D12—42no—
214C4×D3—21no—
215C4×D3—21no—
216C4×D3—21no—
217C4×D3—21no—
218C4×D3—41no—
219C4×D3—41no—
220C4×D3—42no—
221C4×D3—42no—
222C3×C8—21no—
223C3×C8—21no—
224C3⋊C8—21no—
225C3⋊C8—21no—
226C24—31no—
227C24—61no—
228#41—121no—
229#41—121no—
230C2×Q8C22×D311yescharacteristic
231C2×D4—21no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—124no—
244C2×D4C22×D311yescharacteristic
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—121no—
253C22×C4—121no—
254C22×C4C22×D311yescharacteristic
255SD16—41no—
256SD16—121no—
257D8—121no—
258D8—121no—
259C2×C8—31no—
260C2×C8—31no—
261C2×C8—41no—
262C2×C8—121no—
263C2×C8C22×D311yescharacteristic
264C2×C8C22×D311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4C22×D311yescharacteristic
271C4⋊C4C22×D311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C42—61no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3AGL(1,ℤ/8)12yes—
288C22×C3C22×D411yescharacteristic
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—44no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—84no—
302C2×D3#13512yes—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4H(5,𝔽2)11yescharacteristic
308C3×C4H(5,𝔽2)11yescharacteristic
309Dic3—21no—
310Dic3—21no—
311Dic3—41no—
312Dic3—41no—
313C23—21no—
314C23—31no—
315C23—31no—
316C23—62no—
317C23—62no—
318C23—61no—
319C23—61no—
320C23—121no—
321C23—122no—
322C23—122no—
323C23—122no—
324C23D3×D411yescharacteristic
325Q8—21no—
326Q8—21no—
327D4—21no—
328D4—21no—
329D4—42no—
330D4—61no—
331D4—61no—
332D4—61no—
333D4—61no—
334D4—122no—
335D4—122no—
336D4—122no—
337D4—122no—
338D4—122no—
339D4—122no—
340D4—122no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4D3×D411yescharacteristic
360C2×C4C23×D311yescharacteristic
361C8—21no—
362C8—21no—
363C8—61no—
364C8—61no—
365C2×C3—22no—
366C2×C3—81no—
367C2×C3#53411yescharacteristic
368C2×C3C2×#13511yescharacteristic
369C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
370D3—22no—
371D3—41no—
372D3—81no—
373D3—81no—
374C22—21no—
375C22—21no—
376C22—32no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—61no—
381C22—61no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—124no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—244no—
393C22#91412yes—
394C22C2×D3×D411yescharacteristic
395C4—21no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—61no—
400C4—61no—
401C4—121no—
402C4—121no—
403C4#100411yescharacteristic
404C4#100911yescharacteristic
405C3#318811yescharacteristic, complemented, Sylow
406C2—22no—
407C2—62no—
408C2—81no—
409C2—121no—
410C2—241no—
411C2—241no—
412C2#560511yescharacteristic
413C2C2×#91411yescharacteristic
414C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3188
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5627C2semidirect
2. D3×#121C2semidirect
3. C3×#465C2semidirect
4. #5172C2semidirect
5. #5169C2semidirect
6. #5115C2semidirect
7. C2×(C2×D3)⋊C8C2semidirect
8. #5048C2semidirect
9. #5038C2semidirect
10. #4866C2semidirect
11. #4863C2semidirect
12. #4811C2semidirect
13. #4810C2semidirect
14. #4784C2semidirect
15. #4773C2semidirect
17. C2×#239C22semidirect
21. C2×C3×SD16C22semidirect
22. C22×C3×C8C22semidirect
23. C3×C22⋊Q8C22semidirect
26. C2×#215C22semidirect
29. C2×#213C22semidirect
31. #930C22semidirect
32. C22×C3⋊C8C22semidirect
33. C2×D24C22semidirect
38. #892C22semidirect
44. C3×#106C22semidirect
45. C3×#103C22semidirect
46. C3×#102C22semidirect
47. #835C22semidirect
48. #832C22semidirect
49. #821C22semidirect
50. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. #818C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #807C22semidirect
69. #465D3semidirect
136. C2×SD16C2×D3semidirect
140. C22×C8C2×D3semidirect
143. C22⋊Q8C2×D3semidirect
161. #106C2×D3semidirect
163. #103C2×D3semidirect
169. #102C2×D3semidirect
405. C3#3188semidirect

Maximal subgroups

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

subgroupindex
#56272
D3×#1212
C3×#4652
#51722
#51692
#51152
C2×(C2×D3)⋊C82
#50482
#50382
#48662
#48632
#48112
#48102
#47842
#47732
#31883