Finite Group Databasebeta

← #80882

Subgroups — #80882

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
1C2×#911C211yesmaximal, characteristic, complemented
2C2×#908C211yesmaximal, characteristic, complemented
3#5609C211yesmaximal, characteristic, complemented
4#5599C211yesmaximal, characteristic, complemented
5C3×#469C211yesmaximal, characteristic, complemented
6#5151C211yesmaximal, characteristic, complemented
7#5128C211yesmaximal, characteristic, complemented
8#5037C211yesmaximal, characteristic, complemented
9#5036C211yesmaximal, characteristic, complemented
10#4865C211yesmaximal, characteristic, complemented
11#4852C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4781C211yesmaximal, characteristic, complemented
14#4773C211yesmaximal, characteristic, complemented
15#4762C211yesmaximal, characteristic, complemented
16#3274—31nomaximal, Sylow
17C2×#237C2211yescharacteristic, complemented
18C2×D3×D4—21no—
19C2×D3×D4C2211yescharacteristic
20C2×#235C2211yescharacteristic, complemented
21C22×D12C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic, complemented
23C2×C3×M16C2211yescharacteristic, complemented
24C3×#121C2211yescharacteristic
25C3×#121C2211yescharacteristic, complemented
26#940—21no—
27#940C2211yescharacteristic
28#940C2211yescharacteristic, complemented
29#939—21no—
30#937—21no—
31C2×#214C2211yescharacteristic, complemented
32C2×#213C2211yescharacteristic
33C2×#213C2211yescharacteristic, complemented
34C2×#213C2211yescharacteristic, complemented
35C4×#67C2211yescharacteristic
36#911—24no—
37#908—24no—
38C2×D24C2211yescharacteristic
39C2×D24C2211yescharacteristic, complemented
40C2×#204C2211yescharacteristic
41C2×#204C2211yescharacteristic, complemented
42C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
43#895C2211yescharacteristic
44#895C2211yescharacteristic, complemented
45#893C2211yescharacteristic
46#887C2211yescharacteristic, complemented
47#884—21no—
48#884—21no—
49#882—21no—
50#67⋊C4—21no—
51C3×#106C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#830C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic
60#809C2211yescharacteristic, complemented
61#809C2211yescharacteristic, 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#469D311yescharacteristic, complemented
69#460—31no—
70#459—31no—
71#449—31no—
72#449—31no—
73#447—31no—
74#447—31no—
75C4×SD16—31no—
76#436—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67D411yescharacteristic
92C22×Dic3—21no—
93#237—22no—
94#237—41no—
95D3×D4—22no—
96D3×D4—41no—
97D3×D4—44no—
98#235—22no—
99#235—41no—
100C2×D12—22no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—41no—
105C2×D12C2311yescharacteristic
106C2×D12C2311yescharacteristic
107C2×D12C2311yescharacteristic
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3C2311yescharacteristic
111C2×Dic6C2311yescharacteristic
112C3×D8—22no—
113C3×M16—22no—
114C2×C3×C8C2311yescharacteristic
115C2×C3×C8C2311yescharacteristic
116C3×C4⋊C4C2311yescharacteristic
117C3×C4⋊C4C2311yescharacteristic
118C3×C22⋊C4—21no—
119C3×C22⋊C4—21no—
120#217D411yescharacteristic
121#214—22no—
122#213—22no—
123#213—41no—
124#213—41no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4D411yescharacteristic
129Dic3⋊C4—21no—
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135D24—22no—
136D24—22no—
137#204—22no—
138#204D414yescomplemented
139ΓL(1,𝔽25)—22no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
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×D8—31no—
155C2×D8C2×D311yescharacteristic, complemented
156C2×M16—31no—
157C2×M16C2×D311yescharacteristic, complemented
158#127—31no—
159#127—31no—
160#127—61no—
161#121—31no—
162#121—31no—
163#121—61no—
164#121—61no—
165#121—61no—
166#121C2×D311yescharacteristic
167#121C2×D311yescharacteristic, complemented
168#120—61no—
169#120—61no—
170C4×Q8—31no—
171C4×D4—31no—
172C4×D4—31no—
173C4×D4—61no—
174#106C2×D311yescharacteristic, complemented
175C4⋊C8—31no—
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—
184C4×C8—31no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3C2×D411yescharacteristic
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—81no—
196C22×D3—81no—
197C22×D3C2×D411yescharacteristic
198C3×D4—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—21no—
202C3×D4—42no—
203C3×D4—81no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4C2×D411yescharacteristic
209C2×C3×C4C2411yescharacteristic
210#67—22no—
211#67—42no—
212#67—42no—
213#67—42no—
214#67—42no—
215#67—42no—
216#67—81no—
217#67—81no—
218C2×Dic3—21no—
219C2×Dic3—42no—
220C2×Dic3C2×D411yescharacteristic
221C2×Dic3C2×D411yescharacteristic
222C2×Dic3C2×D411yescharacteristic
223D12—21no—
224D12—21no—
225D12—21no—
226D12—21no—
227D12—21no—
228D12—42no—
229D12—42no—
230D12—81no—
231D12C2×D412yes—
232C4×D3—21no—
233C4×D3—21no—
234C4×D3—42no—
235C4×D3—42no—
236Dic6—21no—
237Dic6C2×D412yescomplemented
238C3×C8—21no—
239C3×C8C2×D412yescomplemented
240C3⋊C8—21no—
241C3⋊C8—21no—
242C24—61no—
243C24—61no—
244#41—62no—
245#41—62no—
246#41—121no—
247#41—121no—
248C2×Q8—31no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—31no—
252C2×D4—31no—
253C2×D4—31no—
254C2×D4—31no—
255C2×D4—62no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—62no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—124no—
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—62no—
282D8—62no—
283D8—62no—
284D8—121no—
285D8—121no—
286M16—22no—
287M16—62no—
288C2×C8—31no—
289C2×C8—31no—
290C2×C8C22×D311yescharacteristic
291C2×C8C22×D311yescharacteristic
292C4⋊C4—31no—
293C4⋊C4—61no—
294C4⋊C4C22×D311yescharacteristic
295C4⋊C4C22×D311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—31no—
299C22⋊C4—31no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C42—31no—
304C42—61no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
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—22no—
317C2×D3—21no—
318C2×D3—21no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—84no—
328C2×D3—84no—
329C3×C4—21no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4C22×D411yescharacteristic
333C3×C4H(5,𝔽2)11yescharacteristic
334Dic3—22no—
335Dic3—21no—
336Dic3—21no—
337Dic3AGL(1,ℤ/8)12yescomplemented
338C23—21no—
339C23—21no—
340C23—31no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—122no—
345C23—122no—
346C23—122no—
347C23—122no—
348C23—241no—
349C23—241no—
350C23D3×D411yescharacteristic
351Q8—32no—
352Q8—61no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—32no—
358D4—42no—
359D4—62no—
360D4—61no—
361D4—61no—
362D4—61no—
363D4—61no—
364D4—61no—
365D4—81no—
366D4—122no—
367D4—122no—
368D4—122no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—122no—
373D4—241no—
374D4—241no—
375D4—241no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—31no—
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—61no—
393C8—61no—
394C8D3×D412yescomplemented
395C2×C3—41no—
396C2×C3—81no—
397C2×C3—81no—
398C2×C3D4211yescharacteristic
399C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
400C2×C3#57611yescharacteristic
401D3—41no—
402D3—42no—
403D3—81no—
404D3—81no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—61no—
415C22—61no—
416C22—61no—
417C22—62no—
418C22—121no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22—121no—
426C22—244no—
427C22—244no—
428C22C2×D3×D411yescharacteristic
429C4—21no—
430C4—32no—
431C4—41no—
432C4—41no—
433C4—61no—
434C4—61no—
435C4—62no—
436C4C2×D3×D411yescharacteristic
437C4#100911yescharacteristic
438C3#327411yescharacteristic, complemented, Sylow
439C2—41no—
440C2—81no—
441C2—81no—
442C2—121no—
443C2—122no—
444C2—241no—
445C2—241no—
446C2#560911yescharacteristic
447C2D3×AGL(1,ℤ/8)11yescharacteristic
448C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3274
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. C2×#908C2semidirect
3. #5609C2semidirect
4. #5599C2semidirect
5. C3×#469C2semidirect
6. #5151C2semidirect
7. #5128C2semidirect
8. #5037C2semidirect
9. #5036C2semidirect
10. #4865C2semidirect
11. #4852C2semidirect
12. #4786C2semidirect
13. #4781C2semidirect
14. #4773C2semidirect
15. #4762C2semidirect
17. C2×#237C22semidirect
20. C2×#235C22semidirect
22. C2×C3×D8C22semidirect
23. C2×C3×M16C22semidirect
25. C3×#121C22semidirect
28. #940C22semidirect
31. C2×#214C22semidirect
33. C2×#213C22semidirect
34. C2×#213C22semidirect
39. C2×D24C22semidirect
41. C2×#204C22semidirect
42. C2×ΓL(1,𝔽25)C22semidirect
44. #895C22semidirect
46. #887C22semidirect
51. C3×#106C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
54. #830C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #810C22semidirect
60. #809C22semidirect
61. #809C22semidirect
68. #469D3semidirect
138. #204D4semidirect
155. C2×D8C2×D3semidirect
157. C2×M16C2×D3semidirect
167. #121C2×D3semidirect
174. #106C2×D3semidirect
180. #102C2×D3semidirect
181. #102C2×D3semidirect
237. Dic6C2×D4semidirect
239. C3×C8C2×D4semidirect
337. Dic3AGL(1,ℤ/8)semidirect
394. C8D3×D4semidirect
438. C3#3274semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
C2×#9082
#56092
#55992
C3×#4692
#51512
#51282
#50372
#50362
#48652
#48522
#47862
#47812
#47732
#47622
#32743