Finite Group Databasebeta

← #80026

Subgroups — #80026

Order 384.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#67C211yesmaximal, characteristic, complemented
2C2×#932C211yesmaximal, characteristic, complemented
3C2×#911C211yesmaximal, characteristic, complemented
4#5599C211yesmaximal, characteristic, complemented
5C3×#447C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5151C211yesmaximal, characteristic, complemented
8#5150C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#4782C211yesmaximal, characteristic, complemented
11#4773C211yesmaximal, characteristic, complemented
12#4762C211yesmaximal, characteristic, complemented
13#4758C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3274—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67—21no—
19C2×#237C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic
21C2×#235C2211yescharacteristic, complemented
22C2×C3×D8C2211yescharacteristic
23C2×C3×D8C2211yescharacteristic, complemented
24C3×#121C2211yescharacteristic, complemented
25#953—21no—
26#940—21no—
27#940C2211yescharacteristic
28#940C2211yescharacteristic, complemented
29#939—21no—
30#939—21no—
31#938C2211yescharacteristic
32#937—21no—
33D4×Dic3C2211yescharacteristic
34C2×#214C2211yescharacteristic
35C2×#214C2211yescharacteristic, complemented
36#932—24no—
37C2×#213C2211yescharacteristic
38C2×#213C2211yescharacteristic, complemented
39C2×#213C2211yescharacteristic, complemented
40#930C2211yescharacteristic
41C4×#67C2211yescharacteristic
42C2×#208C2211yescharacteristic, complemented
43#911—24no—
44C2×D24C2211yescharacteristic, complemented
45C2×#204C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47#895C2211yescharacteristic, complemented
48#887C2211yescharacteristic, complemented
49#884—21no—
50#67⋊C4—21no—
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#808C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×AGL(1,ℤ/8)—31no—
64#550—31no—
65D42—31no—
66#493—31no—
67#469—31no—
68#469—31no—
69#460—31no—
70#459—31no—
71#449—31no—
72#449—31no—
73#447—31no—
74#447D311yescharacteristic, complemented
75C4×SD16—31no—
76#436—31no—
77C24×C3—21no—
78C23×D3—21no—
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—41no—
83C2×C3×D4C2311yescharacteristic
84C2×C3×D4C2311yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67—21no—
94C2×#67—21no—
95C2×#67—21no—
96C2×#67—44no—
97C2×#67D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100#237—22no—
101#237—41no—
102D3×D4—22no—
103D3×D4—41no—
104#235—22no—
105#235—41no—
106C2×D12—21no—
107C2×D12C2311yescharacteristic
108C2×D12C2311yescharacteristic
109C2×C4×D3—21no—
110C2×C4×D3C2311yescharacteristic
111C2×Dic6C2311yescharacteristic
112C3×D8—22no—
113C3×D8—41no—
114C2×C3×C8C2311yescharacteristic
115C2×C3×C8C2311yescharacteristic
116C3×C4⋊C4C2311yescharacteristic
117C3×C22⋊C4—21no—
118#217—21no—
119#217—21no—
120#217D411yescharacteristic
121#214—22no—
122#214D414yescomplemented
123#213—22no—
124#213—22no—
125#213—22no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4D411yescharacteristic
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3—21no—
133C4×Dic3C2311yescharacteristic
134#208—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137D24—41no—
138#204—22no—
139ΓL(1,𝔽25)—22no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4—31no—
143C22×D4—61no—
144C22×D4C2×D311yescharacteristic
145AGL(1,ℤ/8)—64no—
146AGL(1,ℤ/8)—64no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×D8C2×D311yescharacteristic
155C2×D8C2×D311yescharacteristic, complemented
156C2×M16—31no—
157C2×M16—31no—
158#127—31no—
159#127—31no—
160#127—61no—
161#121—31no—
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#121—61no—
167#121C2×D311yescharacteristic, complemented
168#120—61no—
169#120—61no—
170C4×Q8—31no—
171C4×D4—31no—
172C4×D4—31no—
173C4×D4—61no—
174#106—31no—
175C4⋊C8—31no—
176#103—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8C2×D311yescharacteristic, complemented
184C4×C8—31no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—81no—
191C23×C3C2×D411yescharacteristic
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—42no—
195C22×D3—42no—
196C22×D3—81no—
197C22×D3C2×D411yescharacteristic
198C3×D4—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—21no—
202C3×D4—21no—
203C3×D4—42no—
204C3×D4—81no—
205C3×D4C2×D412yes—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2411yescharacteristic
211#67—22no—
212#67—42no—
213#67—42no—
214#67—42no—
215#67—42no—
216#67—42no—
217#67—42no—
218#67—42no—
219#67—81no—
220#67—81no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3C2×D411yescharacteristic
226C2×Dic3C2×D411yescharacteristic
227C2×Dic3C2×D411yescharacteristic
228D12—21no—
229D12—21no—
230D12—21no—
231D12—21no—
232D12—81no—
233C4×D3—21no—
234C4×D3—21no—
235C4×D3—42no—
236Dic6—21no—
237Dic6C2×D412yescomplemented
238C3×C8—21no—
239C3×C8—21no—
240C3⋊C8—21no—
241C3⋊C8C2×D412yescomplemented
242C24—21no—
243C24—61no—
244#41—62no—
245#41—62no—
246#41—121no—
247#41—121no—
248C2×Q8—31no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—31no—
253C2×D4—31no—
254C2×D4—31no—
255C2×D4—41no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—62no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—121no—
268C2×D4—124no—
269C2×D4C22×D311yescharacteristic
270C2×D4C22×D311yescharacteristic
271C2×D4C22×D311yescharacteristic
272C22×C4—31no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4C22×D311yescharacteristic
277SD16—34no—
278SD16—62no—
279SD16—62no—
280D8—22no—
281D8—41no—
282D8—62no—
283D8—62no—
284D8—62no—
285D8—121no—
286M16—62no—
287M16—62no—
288C2×C8—31no—
289C2×C8—31no—
290C2×C8C22×D311yescharacteristic
291C2×C8C22×D311yescharacteristic
292C4⋊C4—31no—
293C4⋊C4—31no—
294C4⋊C4—61no—
295C4⋊C4C22×D311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C42—31no—
304C42—61no—
305C22×C3—21no—
306C22×C3—22no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—84no—
318C22×C3C22×D411yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3—21no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—84no—
329C3×C4—21no—
330C3×C4—41no—
331C3×C4C22×D411yescharacteristic
332C3×C4H(5,𝔽2)11yescharacteristic
333Dic3—22no—
334Dic3—21no—
335Dic3—21no—
336Dic3—41no—
337Dic3AGL(1,ℤ/8)12yescomplemented
338C23—21no—
339C23—21no—
340C23—21no—
341C23—31no—
342C23—42no—
343C23—42no—
344C23—61no—
345C23—61no—
346C23—81no—
347C23—122no—
348C23—122no—
349C23—241no—
350C23D3×D411yescharacteristic
351Q8—32no—
352Q8—61no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—62no—
360D4—61no—
361D4—61no—
362D4—61no—
363D4—61no—
364D4—81no—
365D4—122no—
366D4—122no—
367D4—122no—
368D4—122no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—241no—
373D4—241no—
374D4—241no—
375D4D3×D412yes—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4D3×D411yescharacteristic
390C2×C4C23×D311yescharacteristic
391C8—21no—
392C8—21no—
393C8—32no—
394C8—61no—
395C2×C3—42no—
396C2×C3—41no—
397C2×C3—81no—
398C2×C3—81no—
399C2×C3D4211yescharacteristic
400C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C3#57611yescharacteristic
402D3—41no—
403D3—81no—
404D3—81no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—84no—
421C22—121no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—244no—
428C22C2×D3×D411yescharacteristic
429C4—21no—
430C4—32no—
431C4—41no—
432C4—61no—
433C4—61no—
434C4—62no—
435C4—121no—
436C4C2×D3×D411yescharacteristic
437C4#100411yescharacteristic
438C3#327411yescharacteristic, complemented, Sylow
439C2—41no—
440C2—42no—
441C2—81no—
442C2—81no—
443C2—121no—
444C2—241no—
445C2—241no—
446C2#559011yescharacteristic
447C2#575711yescharacteristic
448C2D3×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#3274
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. D4×#67C2semidirect
2. C2×#932C2semidirect
3. C2×#911C2semidirect
4. #5599C2semidirect
5. C3×#447C2semidirect
6. #5158C2semidirect
7. #5151C2semidirect
8. #5150C2semidirect
9. #5039C2semidirect
10. #4782C2semidirect
11. #4773C2semidirect
12. #4762C2semidirect
13. #4758C2semidirect
14. #4734C2semidirect
15. #4729C2semidirect
19. C2×#237C22semidirect
21. C2×#235C22semidirect
23. C2×C3×D8C22semidirect
24. C3×#121C22semidirect
28. #940C22semidirect
35. C2×#214C22semidirect
38. C2×#213C22semidirect
39. C2×#213C22semidirect
42. C2×#208C22semidirect
44. C2×D24C22semidirect
45. C2×#204C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
47. #895C22semidirect
48. #887C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #816C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #809C22semidirect
61. #808C22semidirect
74. #447D3semidirect
122. #214D4semidirect
155. C2×D8C2×D3semidirect
167. #121C2×D3semidirect
181. #102C2×D3semidirect
183. C22⋊C8C2×D3semidirect
237. Dic6C2×D4semidirect
241. C3⋊C8C2×D4semidirect
337. Dic3AGL(1,ℤ/8)semidirect
438. C3#3274semidirect

Maximal subgroups

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

subgroupindex
D4×#672
C2×#9322
C2×#9112
#55992
C3×#4472
#51582
#51512
#51502
#50392
#47822
#47732
#47622
#47582
#47342
#47292
#32743