Finite Group Databasebeta

← #80227

Subgroups — #80227

Order 384.

758 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#41C211yesmaximal, characteristic, complemented
2C2×#918C211yesmaximal, characteristic, complemented
3C2×#914C211yesmaximal, characteristic, complemented
4#5627C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5238C211yesmaximal, characteristic, complemented
7#5187C211yesmaximal, characteristic, complemented
8#5173C211yesmaximal, characteristic, complemented
9#5047C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4802C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4770C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4726C211yesmaximal, characteristic, complemented
16#3005—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18D3×#41—24no—
19D3×#41—24no—
20C2×#239C2211yescharacteristic
21C2×#239C2211yescharacteristic
22C2×D3×Q8C2211yescharacteristic
23C2×#237—21no—
24C2×#237C2211yescharacteristic
25C2×D3×D4—21no—
26C2×D3×D4C2211yescharacteristic
27C2×#235—21no—
28C2×#235C2211yescharacteristic
29C22×C4×D3—21no—
30C22×C4×D3C2211yescharacteristic
31C2×C3×Q16C2211yescharacteristic, complemented
32C2×C3×SD16C2211yescharacteristic, complemented
33C3×C22⋊Q8C2211yescharacteristic, complemented
34#946C2211yescharacteristic, complemented
35C2×#216C2211yescharacteristic, complemented
36C2×#215C2211yescharacteristic, complemented
37C2×#215C2211yescharacteristic, complemented
38C2×#213C2211yescharacteristic, complemented
39#930C2211yescharacteristic, complemented
40#918—24no—
41#914—24no—
42C2×D24C2211yescharacteristic, complemented
43C2×#204C2211yescharacteristic, complemented
44C2×ΓL(1,𝔽25)C2211yescharacteristic
45C2×ΓL(1,𝔽25)C2211yescharacteristic
46#895C2211yescharacteristic, complemented
47#894—21no—
48#893—21no—
49#892C2211yescharacteristic
50#882—21no—
51#67⋊C4—21no—
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#835C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58#821C2211yescharacteristic, complemented
59(C2×D3)⋊C8C2211yescharacteristic, complemented
60(C2×D3)⋊C8C2211yescharacteristic, complemented
61#816C2211yescharacteristic, complemented
62#810C2211yescharacteristic, complemented
63#809C2211yescharacteristic, complemented
64C22×#41—31no—
65C2×#137—31no—
66C2×AGL(1,ℤ/8)—31no—
67#525—31no—
68#452—31no—
69#452—31no—
70#452—31no—
71#452D311yescharacteristic, complemented
72#449—31no—
73#449—31no—
74#449—31no—
75#449—31no—
76#419—31no—
77#418—31no—
78#407—31no—
79C23×D3—21no—
80C23×D3D411yescharacteristic
81C3×#41—22no—
82C3×#41—22no—
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4—21no—
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4C2311yescharacteristic
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C22×Dic3—21no—
94C22×Dic3D411yescharacteristic
95#239—22no—
96#239—22no—
97#239—22no—
98#239—22no—
99D3×Q8—22no—
100D3×Q8D414yescomplemented
101#237—28no—
102#237—22no—
103#237—22no—
104D3×D4—28no—
105D3×D4—22no—
106D3×D4D414yescomplemented
107#235—28no—
108#235—22no—
109#235—22no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12C2311yescharacteristic
114C2×D12C2311yescharacteristic
115C2×C4×D3—28no—
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—21no—
122C2×C4×D3D411yescharacteristic
123C2×C4×D3D411yescharacteristic
124C2×C4×D3C2311yescharacteristic
125C2×C4×D3C2311yescharacteristic
126C2×Dic6—21no—
127C2×Dic6C2311yescharacteristic
128C3×Q16—22no—
129C3×SD16—22no—
130C2×C3×C8C2311yescharacteristic
131C2×C3×C8C2311yescharacteristic
132C3×C4⋊C4—21no—
133C3×C4⋊C4C2311yescharacteristic
134C3×C22⋊C4—21no—
135#216—22no—
136#215—22no—
137#215—22no—
138#213—22no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143C4⋊Dic3C2311yescharacteristic
144Dic3⋊C4—21no—
145C4×Dic3—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148D24—22no—
149#204—22no—
150ΓL(1,𝔽25)—22no—
151ΓL(1,𝔽25)—22no—
152C2×#41—31no—
153C2×#41—31no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—61no—
157C2×#41—61no—
158C2×#41—64no—
159C2×#41—64no—
160C2×#41C2×D311yescharacteristic
161C22×Q8—31no—
162C22×D4—31no—
163C22×D4—61no—
164C23×C4—31no—
165C23×C4—61no—
166#137—64no—
167AGL(1,ℤ/8)—64no—
168C2×Q16—31no—
169C2×Q16C2×D311yescharacteristic, complemented
170C2×SD16—31no—
171C2×SD16—31no—
172C2×SD16—31no—
173C2×SD16C2×D311yescharacteristic, complemented
174C2×D8—31no—
175C2×D8—31no—
176C2×M16—31no—
177C2×M16—31no—
178#123—61no—
179C22⋊Q8—31no—
180C22⋊Q8C2×D311yescharacteristic, complemented
181#121—31no—
182#121—31no—
183#120—61no—
184C4×D4—61no—
185C4×D4—61no—
186#117—31no—
187#103—31no—
188#103—31no—
189#103—31no—
190#103C2×D311yescharacteristic, complemented
191#102—31no—
192#102—31no—
193#102—31no—
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—21no—
200C23×C3C2×D411yescharacteristic
201C22×D3—21no—
202C22×D3—22no—
203C22×D3—24no—
204C22×D3—21no—
205C22×D3—41no—
206C22×D3—41no—
207C22×D3—42no—
208C22×D3—42no—
209C22×D3C2×D411yescharacteristic
210C22×D3C2×D411yescharacteristic
211C3×Q8—21no—
212C3×Q8—21no—
213C3×Q8—21no—
214C3×Q8C2×D412yes—
215C3×D4—24no—
216C3×D4—21no—
217C3×D4C2×D412yes—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4C2×D411yescharacteristic
225C2×C3×C4C2411yescharacteristic
226#67—24no—
227#67—24no—
228#67—24no—
229#67—42no—
230C2×Dic3—24no—
231C2×Dic3—21no—
232C2×Dic3—22no—
233C2×Dic3—21no—
234C2×Dic3C2×D411yescharacteristic
235C2×Dic3C2×D411yescharacteristic
236D12—24no—
237D12—21no—
238D12—21no—
239D12—21no—
240D12—42no—
241D12—42no—
242D12C2×D412yes—
243C4×D3—24no—
244C4×D3—24no—
245C4×D3—22no—
246C4×D3—24no—
247C4×D3—21no—
248C4×D3—21no—
249C4×D3—22no—
250C4×D3—42no—
251C4×D3C2×D412yes—
252C4×D3C2×D412yes—
253Dic6—21no—
254Dic6—24no—
255Dic6C2×D412yes—
256C3×C8—21no—
257C3×C8—21no—
258C3⋊C8—21no—
259C3⋊C8—21no—
260C24—31no—
261C24—61no—
262#41—22no—
263#41—22no—
264#41—62no—
265#41—62no—
266#41—68no—
267#41—62no—
268#41—62no—
269#41—62no—
270#41—68no—
271#41—62no—
272#41—62no—
273#41—62no—
274C2×Q8—34no—
275C2×Q8—31no—
276C2×Q8—61no—
277C2×Q8—62no—
278C2×Q8C22×D311yescharacteristic
279C2×Q8C22×D311yescharacteristic
280C2×D4—21no—
281C2×D4—34no—
282C2×D4—31no—
283C2×D4—31no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—62no—
288C2×D4—61no—
289C2×D4—68no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4C22×D311yescharacteristic
294C22×C4—21no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—68no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4C22×D311yescharacteristic
309Q16—22no—
310Q16—62no—
311SD16—22no—
312SD16—62no—
313SD16—62no—
314SD16—62no—
315D8—62no—
316D8—62no—
317M16—62no—
318M16—62no—
319C2×C8—31no—
320C2×C8—31no—
321C2×C8C22×D311yescharacteristic
322C2×C8C22×D311yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—31no—
325C4⋊C4—61no—
326C4⋊C4C22×D311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C42—61no—
333C22×C3—21no—
334C22×C3—22no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3C22×D411yescharacteristic
340C2×D3—21no—
341C2×D3—21no—
342C2×D3—21no—
343C2×D3—22no—
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—44no—
347C2×D3—48no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3#12012yes—
354C3×C4—22no—
355C3×C4—21no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4C22×D411yescharacteristic
359C3×C4C22×D411yescharacteristic
360Dic3—22no—
361Dic3—21no—
362Dic3—41no—
363Dic3#12012yes—
364C23—21no—
365C23—31no—
366C23—31no—
367C23—64no—
368C23—62no—
369C23—61no—
370C23—61no—
371C23—121no—
372C23—121no—
373C23—122no—
374C23—122no—
375C23D3×D411yescharacteristic
376Q8—21no—
377Q8—21no—
378Q8—21no—
379Q8—32no—
380Q8—64no—
381Q8—61no—
382Q8D3×D412yes—
383D4—24no—
384D4—21no—
385D4—32no—
386D4—64no—
387D4—64no—
388D4—64no—
389D4—64no—
390D4—61no—
391D4—61no—
392D4—61no—
393D4—122no—
394D4—122no—
395D4—122no—
396D4D3×D412yes—
397C2×C4—24no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—31no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—31no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—64no—
412C2×C4—62no—
413C2×C4—64no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—122no—
419C2×C4D3×D411yescharacteristic
420C2×C4C23×D311yescharacteristic
421C8—21no—
422C8—21no—
423C8—61no—
424C8—61no—
425C2×C3—41no—
426C2×C3—42no—
427C2×C3C2×#12011yescharacteristic
428C2×C3#57511yescharacteristic
429C2×C3#57511yescharacteristic
430D3—22no—
431D3—41no—
432D3—42no—
433D3—81no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—32no—
439C22—41no—
440C22—41no—
441C22—61no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—62no—
446C22—61no—
447C22—124no—
448C22—128no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22—121no—
454C22C2×D3×D411yescharacteristic
455C4—21no—
456C4—22no—
457C4—32no—
458C4—41no—
459C4—41no—
460C4—61no—
461C4—62no—
462C4—121no—
463C4C2×D3×D411yescharacteristic
464C4C2×D3×D411yescharacteristic
465C3#300511yescharacteristic, complemented, Sylow
466C2—41no—
467C2—42no—
468C2—62no—
469C2—121no—
470C2—122no—
471C2—241no—
472C2D3×#12011yescharacteristic
473C2#576811yescharacteristic
474C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
hypercenter#452D3
hypocenterC3#3005
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×D3×#41C2semidirect
2. C2×#918C2semidirect
3. C2×#914C2semidirect
4. #5627C2semidirect
5. C3×#452C2semidirect
6. #5238C2semidirect
7. #5187C2semidirect
8. #5173C2semidirect
9. #5047C2semidirect
10. #4810C2semidirect
11. #4802C2semidirect
12. #4781C2semidirect
13. #4770C2semidirect
14. #4734C2semidirect
15. #4726C2semidirect
31. C2×C3×Q16C22semidirect
32. C2×C3×SD16C22semidirect
33. C3×C22⋊Q8C22semidirect
34. #946C22semidirect
35. C2×#216C22semidirect
36. C2×#215C22semidirect
37. C2×#215C22semidirect
38. C2×#213C22semidirect
39. #930C22semidirect
42. C2×D24C22semidirect
43. C2×#204C22semidirect
46. #895C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #835C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. #821C22semidirect
59. (C2×D3)⋊C8C22semidirect
60. (C2×D3)⋊C8C22semidirect
61. #816C22semidirect
62. #810C22semidirect
63. #809C22semidirect
71. #452D3semidirect
100. D3×Q8D4semidirect
106. D3×D4D4semidirect
169. C2×Q16C2×D3semidirect
173. C2×SD16C2×D3semidirect
180. C22⋊Q8C2×D3semidirect
190. #103C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
465. C3#3005semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#412
C2×#9182
C2×#9142
#56272
C3×#4522
#52382
#51872
#51732
#50472
#48102
#48022
#47812
#47702
#47342
#47262
#30053