Finite Group Databasebeta

← #80025

Subgroups — #80025

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
1D4×#67C211yesmaximal, characteristic, complemented
2C22×#214C211yesmaximal, characteristic, complemented
3C2×#911C211yesmaximal, characteristic, complemented
4#5598C211yesmaximal, characteristic, complemented
5C3×#447C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5154C211yesmaximal, characteristic, complemented
8#5153C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4780C211yesmaximal, characteristic, complemented
11#4776C211yesmaximal, characteristic, complemented
12#4762C211yesmaximal, characteristic, complemented
13#4757C211yesmaximal, characteristic, complemented
14#4738C211yesmaximal, characteristic, complemented
15#4730C211yesmaximal, characteristic, complemented
16#3265—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67—21no—
19C2×#237C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic
21C22×Dic6C2211yescharacteristic, complemented
22C2×C3×D8C2211yescharacteristic
23C2×C3×D8C2211yescharacteristic, complemented
24C3×#121C2211yescharacteristic, complemented
25#953—21no—
26#940C2211yescharacteristic
27#939—21no—
28#939—21no—
29#938C2211yescharacteristic
30#937—21no—
31#936C2211yescharacteristic, complemented
32D4×Dic3—21no—
33D4×Dic3C2211yescharacteristic
34C2×#214—24no—
35C2×#214C2211yescharacteristic
36C2×#214C2211yescharacteristic
37C2×#214C2211yescharacteristic, complemented
38C2×#214C2211yescharacteristic, complemented
39C2×#213C2211yescharacteristic, complemented
40#930C2211yescharacteristic
41C4×#67C2211yescharacteristic
42C22×C3⋊C8C2211yescharacteristic, complemented
43#911—24no—
44C2×#204C2211yescharacteristic, complemented
45C2×#204C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47#897C2211yescharacteristic, complemented
48#887C2211yescharacteristic, complemented
49#885—21no—
50#878—21no—
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56#816C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#815C2211yescharacteristic, complemented
59Dic3⋊C8C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#807C2211yescharacteristic, 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#463—31no—
70#460—31no—
71#450—31no—
72#449—31no—
73#448—31no—
74#447D311yescharacteristic, complemented
75#440—31no—
76#435—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—44no—
94C2×#67D411yescharacteristic
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—21no—
98#237—22no—
99#237—41no—
100D3×D4—22no—
101D3×D4—41no—
102C2×D12C2311yescharacteristic
103C2×C4×D3C2311yescharacteristic
104C2×Dic6—22no—
105C2×Dic6—21no—
106C2×Dic6—41no—
107C2×Dic6C2311yescharacteristic
108C2×Dic6C2311yescharacteristic
109C3×D8—22no—
110C3×D8—41no—
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115#217—21no—
116#217—21no—
117#217—21no—
118#217—21no—
119#217D411yescharacteristic
120#214—22no—
121#214—22no—
122#214—22no—
123#214D414yescomplemented
124#213—22no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128C4⋊Dic3—21no—
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3—21no—
133C4×Dic3C2311yescharacteristic
134C2×C3⋊C8—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137#204—22no—
138#204—41no—
139ΓL(1,𝔽25)—22no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×D4—31no—
143C22×D4—61no—
144C22×D4C2×D311yescharacteristic
145AGL(1,ℤ/8)—64no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16—64no—
153C2×D8—31no—
154C2×D8C2×D311yescharacteristic
155C2×D8C2×D311yescharacteristic, complemented
156C2×M16—31no—
157C22×C8—31no—
158#127—31no—
159#124—31no—
160#124—61no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
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#107—31no—
175C4⋊C8—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8C2×D311yescharacteristic, complemented
184#97—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—42no—
194C22×D3—42no—
195C22×D3—81no—
196C22×D3C2×D411yescharacteristic
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—21no—
202C3×D4—42no—
203C3×D4—42no—
204C3×D4C2×D412yes—
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—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3C2×D411yescharacteristic
224C2×Dic3C2×D411yescharacteristic
225C2×Dic3C2×D411yescharacteristic
226D12—21no—
227D12—21no—
228C4×D3—21no—
229C4×D3—21no—
230Dic6—21no—
231Dic6—21no—
232Dic6—21no—
233Dic6—42no—
234Dic6C2×D412yescomplemented
235C3×C8—21no—
236C3×C8—21no—
237C3⋊C8—21no—
238C3⋊C8C2×D412yescomplemented
239C24—21no—
240C24—61no—
241#41—62no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8—31no—
245C2×Q8—62no—
246C2×Q8—61no—
247C2×Q8—121no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—31no—
252C2×D4—31no—
253C2×D4—41no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C2×D4—124no—
263C2×D4C22×D311yescharacteristic
264C2×D4C22×D311yescharacteristic
265C2×D4C22×D311yescharacteristic
266C22×C4—31no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4C22×D311yescharacteristic
271SD16—34no—
272SD16—62no—
273SD16—62no—
274SD16—62no—
275SD16—62no—
276SD16—121no—
277D8—22no—
278D8—41no—
279D8—62no—
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—31no—
293C22⋊C4—31no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C42—31no—
301C42—61no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—22no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—84no—
314C22×C3AGL(1,ℤ/8)12yescomplemented
315C22×C3C22×D411yescharacteristic
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—21no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—84no—
323C3×C4—21no—
324C3×C4—41no—
325C3×C4C22×D411yescharacteristic
326C3×C4H(5,𝔽2)11yescharacteristic
327Dic3—22no—
328Dic3—21no—
329Dic3—21no—
330Dic3—21no—
331Dic3—41no—
332Dic3—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—31no—
337C23—42no—
338C23—42no—
339C23—61no—
340C23—81no—
341C23—122no—
342C23—122no—
343C23—241no—
344C23D3×D411yescharacteristic
345Q8—32no—
346Q8—61no—
347Q8—61no—
348Q8—61no—
349Q8—122no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—42no—
357D4—62no—
358D4—61no—
359D4—61no—
360D4—122no—
361D4—122no—
362D4—122no—
363D4—122no—
364D4—122no—
365D4—241no—
366D4D3×D412yes—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4D3×D411yescharacteristic
382C2×C4C23×D311yescharacteristic
383C8—21no—
384C8—21no—
385C8—32no—
386C8—61no—
387C2×C3—22no—
388C2×C3—42no—
389C2×C3—81no—
390C2×C3—81no—
391C2×C3D4211yescharacteristic
392C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
393C2×C3#57711yescharacteristic
394D3—41no—
395D3—81no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—61no—
408C22—61no—
409C22—61no—
410C22—84no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—244no—
415C22#91112yescomplemented
416C22C2×D3×D411yescharacteristic
417C4—21no—
418C4—41no—
419C4—61no—
420C4—61no—
421C4—61no—
422C4—62no—
423C4—121no—
424C4—121no—
425C4C2×D3×D411yescharacteristic
426C4#100411yescharacteristic
427C3#326511yescharacteristic, complemented, Sylow
428C2—22no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—121no—
433C2—241no—
434C2#559011yescharacteristic
435C2C2×#91111yescharacteristic
436C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3265
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. C22×#214C2semidirect
3. C2×#911C2semidirect
4. #5598C2semidirect
5. C3×#447C2semidirect
6. #5158C2semidirect
7. #5154C2semidirect
8. #5153C2semidirect
9. #5038C2semidirect
10. #4780C2semidirect
11. #4776C2semidirect
12. #4762C2semidirect
13. #4757C2semidirect
14. #4738C2semidirect
15. #4730C2semidirect
19. C2×#237C22semidirect
21. C22×Dic6C22semidirect
23. C2×C3×D8C22semidirect
24. C3×#121C22semidirect
31. #936C22semidirect
37. C2×#214C22semidirect
38. C2×#214C22semidirect
39. C2×#213C22semidirect
42. C22×C3⋊C8C22semidirect
44. C2×#204C22semidirect
45. C2×#204C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
47. #897C22semidirect
48. #887C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
54. #832C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. #816C22semidirect
57. #816C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #809C22semidirect
61. #807C22semidirect
74. #447D3semidirect
123. #214D4semidirect
155. C2×D8C2×D3semidirect
167. #121C2×D3semidirect
181. #102C2×D3semidirect
183. C22⋊C8C2×D3semidirect
234. Dic6C2×D4semidirect
238. C3⋊C8C2×D4semidirect
314. C22×C3AGL(1,ℤ/8)semidirect
415. C22#911semidirect
427. C3#3265semidirect

Maximal subgroups

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

subgroupindex
D4×#672
C22×#2142
C2×#9112
#55982
C3×#4472
#51582
#51542
#51532
#50382
#47802
#47762
#47622
#47572
#47382
#47302
#32653