Finite Group Databasebeta

← #80553

Subgroups — #80553

Order 384.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×SD16C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3#5677C211yesmaximal, characteristic, complemented
4D4×D12C211yesmaximal, characteristic, complemented
5C3×#463C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5058C211yesmaximal, characteristic, complemented
9#5018C211yesmaximal, characteristic, complemented
10#4848C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4812C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4781C211yesmaximal, characteristic, complemented
15#4775C211yesmaximal, characteristic, complemented
16#3265—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic, complemented
18C2×#237C2211yescharacteristic, complemented
19C2×D3×D4C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic
21C22×D12—21no—
22C2×C3×SD16C2211yescharacteristic
23C2×C3×D8C2211yescharacteristic
24C3×#124C2211yescharacteristic, complemented
25C3×C4×D4C2211yescharacteristic
26#946C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic, complemented
29#938C2211yescharacteristic, complemented
30C2×#214C2211yescharacteristic
31C2×#213C2211yescharacteristic, complemented
32C2×#213C2211yescharacteristic, complemented
33#930—21no—
34D3×SD16—24no—
35#911—24no—
36C2×#204C2211yescharacteristic, complemented
37C2×#204C2211yescharacteristic, complemented
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39C2×D3×C8C2211yescharacteristic, complemented
40#895C2211yescharacteristic
41#895C2211yescharacteristic
42#885—21no—
43#884—21no—
44#882—21no—
45#882—21no—
46#67⋊C4—21no—
47#878—21no—
48#874C2211yescharacteristic
49C4×D12C2211yescharacteristic
50C4×Dic6C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic, complemented
56#821C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58(C2×D3)⋊C8C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#807C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×SD16—31no—
64#548—31no—
65D42—31no—
66#496—31no—
67#468—31no—
68#466—31no—
69#463D311yescharacteristic, complemented
70#460—31no—
71#450—31no—
72#449—31no—
73#448—31no—
74#447—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C22×Dic3—21no—
88D3×Q8—22no—
89D3×Q8—41no—
90#237—22no—
91#237—41no—
92D3×D4—22no—
93D3×D4—22no—
94D3×D4—41no—
95D3×D4—41no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—44no—
101C2×D12D411yescharacteristic
102C2×D12C2311yescharacteristic
103C2×D12C2311yescharacteristic
104C2×C4×D3—21no—
105C2×C4×D3C2311yescharacteristic
106C2×C4×D3C2311yescharacteristic
107C2×Dic6—21no—
108C2×Dic6C2311yescharacteristic
109C3×SD16—22no—
110C3×D8—22no—
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115C3×C22⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C42C2311yescharacteristic
118#217—21no—
119#215—22no—
120#215—41no—
121#214D414yescomplemented
122#213—22no—
123#213—41no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128(C2×D3)⋊C4D411yescharacteristic
129C4⋊Dic3D411yescharacteristic
130C4⋊Dic3C2311yescharacteristic
131Dic3⋊C4—21no—
132Dic3⋊C4—21no—
133C4×Dic3—21no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136#204—22no—
137#204—22no—
138ΓL(1,𝔽25)—22no—
139D3×C8—22no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145AGL(1,ℤ/8)—64no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—64no—
152C2×SD16C2×D311yescharacteristic
153C2×D8—31no—
154C2×D8—31no—
155C2×D8C2×D311yescharacteristic
156C2×M16—31no—
157C22×C8—31no—
158#127—31no—
159#124—61no—
160#124C2×D311yescharacteristic, complemented
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
165#121—31no—
166#121—61no—
167#121—61no—
168#120—61no—
169#120—61no—
170C4×Q8—31no—
171C4×D4—31no—
172C4×D4—61no—
173C4×D4C2×D311yescharacteristic
174#107—31no—
175C4⋊C8C2×D311yescharacteristic, complemented
176#103—31no—
177#103C2×D311yescharacteristic, complemented
178#102—31no—
179#102—31no—
180#102—31no—
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8—31no—
184#97—31no—
185C23×C3—21no—
186C23×C3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3C2×D411yescharacteristic
196C22×D3C2×D411yescharacteristic
197C3×Q8—21no—
198C3×Q8—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—21no—
202C3×D4C2×D412yes—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—42no—
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2×D411yescharacteristic
208C2×C3×C4C2411yescharacteristic
209#67—42no—
210#67—42no—
211#67—42no—
212#67—42no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—42no—
216C2×Dic3C2×D411yescharacteristic
217C2×Dic3C2×D411yescharacteristic
218D12—22no—
219D12—21no—
220D12—21no—
221D12—21no—
222D12—21no—
223D12—42no—
224D12—42no—
225D12—42no—
226D12—81no—
227C4×D3—21no—
228C4×D3—21no—
229C4×D3—21no—
230C4×D3—21no—
231C4×D3—42no—
232Dic6—21no—
233Dic6—42no—
234Dic6C2×D412yescomplemented
235C3×C8—21no—
236C3×C8—21no—
237C3⋊C8—21no—
238C3⋊C8C2×D412yescomplemented
239C24—61no—
240C24—61no—
241#41—62no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8—61no—
245C2×Q8—62no—
246C2×Q8—121no—
247C2×Q8C22×D311yescharacteristic
248C2×D4—31no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—124no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4C22×D311yescharacteristic
265C2×D4C22×D311yescharacteristic
266C22×C4—21no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—61no—
270C22×C4—61no—
271SD16—22no—
272SD16—34no—
273SD16—62no—
274SD16—62no—
275SD16—62no—
276SD16—121no—
277D8—22no—
278D8—62no—
279D8—121no—
280M16—62no—
281C2×C8—31no—
282C2×C8—31no—
283C2×C8—62no—
284C2×C8C22×D311yescharacteristic
285C2×C8C22×D311yescharacteristic
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4C22×D311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—31no—
295C22⋊C4—31no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C42—61no—
301C42C22×D311yescharacteristic
302C22×C3—22no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3C22×D411yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—21no—
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—84no—
321C2×D3—84no—
322C2×D3AGL(1,ℤ/8)12yescomplemented
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—41no—
326C3×C4—41no—
327C3×C4C22×D411yescharacteristic
328C3×C4H(5,𝔽2)11yescharacteristic
329Dic3—22no—
330Dic3—21no—
331Dic3—21no—
332Dic3—41no—
333C23—21no—
334C23—21no—
335C23—31no—
336C23—31no—
337C23—61no—
338C23—61no—
339C23—122no—
340C23—122no—
341C23—122no—
342C23—122no—
343C23—241no—
344C23—241no—
345Q8—21no—
346Q8—21no—
347Q8—32no—
348Q8—61no—
349Q8—122no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—62no—
354D4—61no—
355D4—61no—
356D4—61no—
357D4—61no—
358D4—122no—
359D4—122no—
360D4—122no—
361D4—122no—
362D4—122no—
363D4—122no—
364D4—122no—
365D4—241no—
366D4D3×D412yes—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—42no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4D3×D411yescharacteristic
381C2×C4D3×D411yescharacteristic
382C2×C4C23×D311yescharacteristic
383C8—21no—
384C8—21no—
385C8—32no—
386C8—61no—
387C2×C3—42no—
388C2×C3—81no—
389C2×C3D4211yescharacteristic
390C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
391C2×C3#57711yescharacteristic
392D3—22no—
393D3—41no—
394D3—81no—
395D3—81no—
396C22—22no—
397C22—32no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—61no—
404C22—61no—
405C22—61no—
406C22—61no—
407C22—61no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—244no—
415C22—244no—
416C22C2×D3×D411yescharacteristic
417C4—21no—
418C4—21no—
419C4—41no—
420C4—41no—
421C4—61no—
422C4—61no—
423C4—62no—
424C4—121no—
425C4C2×D3×D411yescharacteristic
426C4#100911yescharacteristic
427C3#326511yescharacteristic, complemented, Sylow
428C2—42no—
429C2—62no—
430C2—81no—
431C2—121no—
432C2—241no—
433C2—241no—
434C2#560911yescharacteristic
435C2#577011yescharacteristic
436C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3265
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×D3×SD16C2semidirect
2. C2×#911C2semidirect
3. #5677C2semidirect
4. D4×D12C2semidirect
5. C3×#463C2semidirect
6. #5169C2semidirect
7. #5156C2semidirect
8. #5058C2semidirect
9. #5018C2semidirect
10. #4848C2semidirect
11. #4836C2semidirect
12. #4812C2semidirect
13. #4806C2semidirect
14. #4781C2semidirect
15. #4775C2semidirect
17. C2×D3×Q8C22semidirect
18. C2×#237C22semidirect
24. C3×#124C22semidirect
26. #946C22semidirect
27. C2×#215C22semidirect
28. C2×#215C22semidirect
29. #938C22semidirect
31. C2×#213C22semidirect
32. C2×#213C22semidirect
36. C2×#204C22semidirect
37. C2×#204C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
39. C2×D3×C8C22semidirect
50. C4×Dic6C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
55. #821C22semidirect
56. #821C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. (C2×D3)⋊C8C22semidirect
59. #810C22semidirect
60. #807C22semidirect
61. #803C22semidirect
69. #463D3semidirect
121. #214D4semidirect
160. #124C2×D3semidirect
175. C4⋊C8C2×D3semidirect
177. #103C2×D3semidirect
181. #102C2×D3semidirect
234. Dic6C2×D4semidirect
238. C3⋊C8C2×D4semidirect
322. C2×D3AGL(1,ℤ/8)semidirect
427. C3#3265semidirect

Maximal subgroups

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

subgroupindex
C2×D3×SD162
C2×#9112
#56772
D4×D122
C3×#4632
#51692
#51562
#50582
#50182
#48482
#48362
#48122
#48062
#47812
#47752
#32653