Finite Group Databasebeta

← #80358

Subgroups — #80358

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#911C211yesmaximal, characteristic, complemented
2C2×D3×D8C211yesmaximal, characteristic, complemented
3#5717C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic, complemented
5C3×#459C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5074C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4782C211yesmaximal, characteristic, complemented
13#4778C211yesmaximal, characteristic, complemented
14#4776C211yesmaximal, characteristic, complemented
15#4773C211yesmaximal, characteristic, complemented
16#3264—31nomaximal, Sylow
17C2×#237C2211yescharacteristic, complemented
18C2×D3×D4—21no—
19C2×D3×D4C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic, complemented
21C2×D3×D4C2211yescharacteristic, complemented
22C2×C3×D8C2211yescharacteristic, complemented
23C2×C3×D8C2211yescharacteristic, complemented
24C3×#127C2211yescharacteristic, complemented
25C3×C4×D4C2211yescharacteristic, complemented
26#938—21no—
27#938C2211yescharacteristic
28#938C2211yescharacteristic, complemented
29#937—21no—
30#937—21no—
31C2×#214C2211yescharacteristic, complemented
32C2×#214C2211yescharacteristic, complemented
33C2×#213C2211yescharacteristic
34C2×#213C2211yescharacteristic, complemented
35C2×#213C2211yescharacteristic, complemented
36#930—21no—
37C2×(C2×D3)⋊C4—21no—
38#911—24no—
39D3×D8—24no—
40C2×D24C2211yescharacteristic
41C2×#204C2211yescharacteristic, complemented
42C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
43C2×D3×C8C2211yescharacteristic, complemented
44#897C2211yescharacteristic, complemented
45#895C2211yescharacteristic, complemented
46#886—21no—
47#882—21no—
48#875C2211yescharacteristic, complemented
49C4×D12C2211yescharacteristic
50C4×D12C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#808C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×D8—31no—
64#546—31no—
65D42—31no—
66#496—31no—
67#469—31no—
68#465—31no—
69#463—31no—
70#459D311yescharacteristic, complemented
71#450—31no—
72#449—31no—
73#447—31no—
74#447—31no—
75#442—31no—
76#435—31no—
77C23×D3—21no—
78C23×D3—21no—
79C23×D3—21no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C22×Dic3—21no—
93#237—22no—
94#237—41no—
95D3×D4—22no—
96D3×D4—22no—
97D3×D4—22no—
98D3×D4—41no—
99D3×D4—41no—
100D3×D4—41no—
101D3×D4—44no—
102C2×D12—21no—
103C2×D12D411yescharacteristic
104C2×D12C2311yescharacteristic
105C2×D12C2311yescharacteristic
106C2×C4×D3—21no—
107C2×C4×D3C2311yescharacteristic
108C2×C4×D3C2311yescharacteristic
109C2×Dic6C2311yescharacteristic
110C3×D8—22no—
111C3×D8—22no—
112C2×C3×C8C2311yescharacteristic
113C2×C3×C8C2311yescharacteristic
114C3×C4⋊C4C2311yescharacteristic
115C3×C22⋊C4—21no—
116C3×C42C2311yescharacteristic
117#217—21no—
118#217—21no—
119#214—22no—
120#214—41no—
121#213—22no—
122#213—41no—
123#213D414yescomplemented
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4D411yescharacteristic
130(C2×D3)⋊C4D411yescharacteristic
131C4⋊Dic3—21no—
132C4⋊Dic3D411yescharacteristic
133C4⋊Dic3C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—22no—
137#204—22no—
138ΓL(1,𝔽25)—22no—
139D3×C8—22no—
140C2×#41—31no—
141C22×D4—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×D8—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—64no—
154C2×D8C2×D311yescharacteristic, complemented
155C2×D8C2×D311yescharacteristic, complemented
156C2×M16—31no—
157C22×C8—31no—
158#127C2×D311yescharacteristic, complemented
159#124—31no—
160#123—61no—
161C22⋊Q8—31no—
162#121—31no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168#120—61no—
169#120—61no—
170C4×D4—31no—
171C4×D4—31no—
172C4×D4C2×D311yescharacteristic, complemented
173C2×C22⋊C4—61no—
174#106—31no—
175C4⋊C8C2×D311yescharacteristic, complemented
176#103—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8—31no—
184#97—31no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—81no—
197C22×D3—81no—
198C22×D3—81no—
199C22×D3C2×D411yescharacteristic
200C22×D3C2×D411yescharacteristic
201C3×D4—21no—
202C3×D4—21no—
203C3×D4—21no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4—42no—
207C3×D4—81no—
208C3×D4C2×D412yescomplemented
209C2×C3×C4—21no—
210C2×C3×C4—81no—
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2×D411yescharacteristic
213C2×C3×C4C2411yescharacteristic
214#67—42no—
215#67—42no—
216#67—42no—
217#67—42no—
218#67—42no—
219#67—42no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—42no—
223C2×Dic3C2×D411yescharacteristic
224C2×Dic3C2×D411yescharacteristic
225D12—22no—
226D12—21no—
227D12—21no—
228D12—21no—
229D12—81no—
230D12C2×D412yes—
231C4×D3—21no—
232C4×D3—21no—
233C4×D3—21no—
234C4×D3—21no—
235C4×D3—42no—
236Dic6—21no—
237Dic6—21no—
238C3×C8—21no—
239C3×C8—21no—
240C3⋊C8—21no—
241C3⋊C8C2×D412yescomplemented
242C24—61no—
243C24—61no—
244C24—61no—
245#41—62no—
246#41—121no—
247C2×Q8—31no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—31no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—124no—
267C2×D4C22×D311yescharacteristic
268C2×D4C22×D311yescharacteristic
269C2×D4C22×D311yescharacteristic
270C22×C4—21no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—61no—
274C22×C4—61no—
275SD16—62no—
276SD16—62no—
277SD16—121no—
278D8—22no—
279D8—22no—
280D8—34no—
281D8—62no—
282D8—62no—
283D8—121no—
284M16—62no—
285C2×C8—31no—
286C2×C8—31no—
287C2×C8—62no—
288C2×C8C22×D311yescharacteristic
289C2×C8C22×D311yescharacteristic
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—61no—
293C4⋊C4C22×D311yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—31no—
296C22⋊C4—31no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C42C22×D311yescharacteristic
305C22×C3—22no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3C22×D411yescharacteristic
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—22no—
319C2×D3—21no—
320C2×D3—21no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—84no—
327C2×D3—84no—
328C2×D3—84no—
329C2×D3AGL(1,ℤ/8)12yescomplemented
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—41no—
333C3×C4C22×D411yescharacteristic
334C3×C4H(5,𝔽2)11yescharacteristic
335Dic3—21no—
336Dic3—21no—
337Dic3—41no—
338Dic3—41no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—31no—
343C23—31no—
344C23—61no—
345C23—61no—
346C23—122no—
347C23—122no—
348C23—122no—
349C23—122no—
350C23—122no—
351C23—122no—
352C23—241no—
353C23—241no—
354C23—241no—
355Q8—61no—
356Q8—61no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—32no—
363D4—42no—
364D4—62no—
365D4—61no—
366D4—61no—
367D4—61no—
368D4—81no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—122no—
373D4—122no—
374D4—122no—
375D4—241no—
376D4D3×D412yescomplemented
377C2×C4—21no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—81no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4D3×D411yescharacteristic
390C2×C4D3×D411yescharacteristic
391C2×C4C23×D311yescharacteristic
392C8—21no—
393C8—21no—
394C8—32no—
395C8—61no—
396C2×C3—42no—
397C2×C3—81no—
398C2×C3—81no—
399C2×C3D4211yescharacteristic
400C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C3#57611yescharacteristic
402D3—22no—
403D3—41no—
404D3—42no—
405D3—81no—
406C22—22no—
407C22—32no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—61no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—61no—
421C22—62no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—244no—
428C22—244no—
429C22—244no—
430C22C2×D3×D411yescharacteristic
431C4—21no—
432C4—21no—
433C4—41no—
434C4—61no—
435C4—61no—
436C4—121no—
437C4—121no—
438C4C2×D3×D411yescharacteristic
439C4#100911yescharacteristic
440C3#326411yescharacteristic, complemented, Sylow
441C2—42no—
442C2—62no—
443C2—81no—
444C2—81no—
445C2—121no—
446C2—122no—
447C2—241no—
448C2#560911yescharacteristic
449C2#575711yescharacteristic
450C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3264
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. C2×D3×D8C2semidirect
3. #5717C2semidirect
4. #5554C2semidirect
5. C3×#459C2semidirect
6. #5158C2semidirect
7. #5156C2semidirect
8. #5074C2semidirect
9. #5014C2semidirect
10. #4844C2semidirect
11. #4836C2semidirect
12. #4782C2semidirect
13. #4778C2semidirect
14. #4776C2semidirect
15. #4773C2semidirect
17. C2×#237C22semidirect
20. C2×D3×D4C22semidirect
21. C2×D3×D4C22semidirect
22. C2×C3×D8C22semidirect
23. C2×C3×D8C22semidirect
24. C3×#127C22semidirect
25. C3×C4×D4C22semidirect
28. #938C22semidirect
31. C2×#214C22semidirect
32. C2×#214C22semidirect
34. C2×#213C22semidirect
35. C2×#213C22semidirect
41. C2×#204C22semidirect
42. C2×ΓL(1,𝔽25)C22semidirect
43. C2×D3×C8C22semidirect
44. #897C22semidirect
45. #895C22semidirect
48. #875C22semidirect
50. C4×D12C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #816C22semidirect
59. #809C22semidirect
60. #808C22semidirect
61. #803C22semidirect
70. #459D3semidirect
123. #213D4semidirect
154. C2×D8C2×D3semidirect
155. C2×D8C2×D3semidirect
158. #127C2×D3semidirect
172. C4×D4C2×D3semidirect
175. C4⋊C8C2×D3semidirect
180. #102C2×D3semidirect
181. #102C2×D3semidirect
208. C3×D4C2×D4semidirect
241. C3⋊C8C2×D4semidirect
329. C2×D3AGL(1,ℤ/8)semidirect
376. D4D3×D4semidirect
440. C3#3264semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
C2×D3×D82
#57172
#55542
C3×#4592
#51582
#51562
#50742
#50142
#48442
#48362
#47822
#47782
#47762
#47732
#32643