Finite Group Databasebeta

← #80086

Subgroups — #80086

Order 384.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C211yesmaximal, characteristic, complemented
2C2×D3×SD16C211yesmaximal, characteristic, complemented
3C2×#911C211yesmaximal, characteristic, complemented
4#5607C211yesmaximal, characteristic, complemented
5C3×#449C211yesmaximal, characteristic, complemented
6#5238C211yesmaximal, characteristic, complemented
7#5170C211yesmaximal, characteristic, complemented
8#5158C211yesmaximal, characteristic, complemented
9#5040C211yesmaximal, characteristic, complemented
10#4806C211yesmaximal, characteristic, complemented
11#4802C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4772C211yesmaximal, characteristic, complemented
14#4738C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3010—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18#1010—24no—
19#1010—24no—
20C2×D3×Q8C2211yescharacteristic
21C2×D3×Q8C2211yescharacteristic, complemented
22C2×#237—21no—
23C2×#237—21no—
24C2×#237C2211yescharacteristic
25C2×#237C2211yescharacteristic, complemented
26C2×D3×D4C2211yescharacteristic
27C2×#235—21no—
28C2×#235C2211yescharacteristic
29C22×Dic6—21no—
30C22×Dic6C2211yescharacteristic, complemented
31C2×C3×SD16C2211yescharacteristic
32C2×C3×D8C2211yescharacteristic
33C3×#121C2211yescharacteristic
34C2×#215C2211yescharacteristic
35#940—21no—
36#938C2211yescharacteristic
37#937—21no—
38C2×#214C2211yescharacteristic
39C2×#214C2211yescharacteristic
40C2×#213C2211yescharacteristic
41#930C2211yescharacteristic
42D3×SD16—24no—
43#911—24no—
44C2×#204C2211yescharacteristic
45C2×#204C2211yescharacteristic
46C2×ΓL(1,𝔽25)C2211yescharacteristic
47C2×D3×C8C2211yescharacteristic
48#895C2211yescharacteristic
49#892C2211yescharacteristic
50#885—21no—
51#882—21no—
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×C22⋊C8C2211yescharacteristic, complemented
55#835C2211yescharacteristic, complemented
56#832C2211yescharacteristic, complemented
57#830C2211yescharacteristic, complemented
58(C2×D3)⋊C8C2211yescharacteristic, complemented
59(C2×D3)⋊C8C2211yescharacteristic, complemented
60#816C2211yescharacteristic, complemented
61#816C2211yescharacteristic, complemented
62#810C2211yescharacteristic, complemented
63#810C2211yescharacteristic, complemented
64C2×2−1+4—31no—
65C2×AGL(1,ℤ/8)—31no—
66C22×SD16—31no—
67#535—31no—
68#449—31no—
69#449—31no—
70#449—31no—
71#449D311yescharacteristic, complemented
72#448—31no—
73#448—31no—
74#448—31no—
75#448—31no—
76#419—31no—
77#416—31no—
78#408—31no—
79C23×D3—21no—
80C3×#41—22no—
81C3×#41—22no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4C2311yescharacteristic
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—21no—
93C2×#67—21no—
94C2×#67D411yescharacteristic
95C2×#67D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99D3×Q8—22no—
100D3×Q8—22no—
101D3×Q8—22no—
102D3×Q8D414yescomplemented
103#237—28no—
104#237—28no—
105#237—22no—
106#237—22no—
107#237—22no—
108#237D414yescomplemented
109D3×D4—22no—
110D3×D4—22no—
111#235—28no—
112#235—22no—
113#235—22no—
114C2×D12—21no—
115C2×D12D411yescharacteristic
116C2×D12C2311yescharacteristic
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3C2311yescharacteristic
120C2×C4×D3C2311yescharacteristic
121C2×Dic6—28no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—22no—
126C2×Dic6—22no—
127C2×Dic6D411yescharacteristic
128C2×Dic6C2311yescharacteristic
129C2×Dic6C2311yescharacteristic
130C3×SD16—22no—
131C3×D8—22no—
132C2×C3×C8C2311yescharacteristic
133C2×C3×C8C2311yescharacteristic
134C3×C4⋊C4C2311yescharacteristic
135C3×C22⋊C4—21no—
136#217—21no—
137#215—22no—
138#214—22no—
139#214—22no—
140#213—22no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—21no—
144C4⋊Dic3C2311yescharacteristic
145C4×Dic3—21no—
146C2×C3⋊C8C2311yescharacteristic
147C2×C3⋊C8C2311yescharacteristic
148#204—22no—
149#204—22no—
150ΓL(1,𝔽25)—22no—
151D3×C8—22no—
1522−1+4—64no—
1532−1+4—64no—
154C2×#41—31no—
155C2×#41—31no—
156C2×#41—31no—
157C2×#41—61no—
158C2×#41—61no—
159C2×#41—61no—
160C2×#41C2×D311yescharacteristic, complemented
161C22×Q8—31no—
162C22×Q8—31no—
163C22×Q8—31no—
164C22×Q8—61no—
165C22×D4—31no—
166AGL(1,ℤ/8)—64no—
167C2×SD16—31no—
168C2×SD16—31no—
169C2×SD16—31no—
170C2×SD16—31no—
171C2×SD16—31no—
172C2×SD16—64no—
173C2×SD16C2×D311yescharacteristic
174C2×D8—31no—
175C2×D8C2×D311yescharacteristic
176C2×M16—31no—
177C22×C8—31no—
178#127—61no—
179#124—61no—
180#121—31no—
181#121—31no—
182#121—31no—
183#121C2×D311yescharacteristic
184#120—61no—
185#120—61no—
186#117—31no—
187#103—31no—
188#103—31no—
189#103—31no—
190#103—31no—
191#103—31no—
192#103C2×D311yescharacteristic, complemented
193#102—31no—
194#102C2×D311yescharacteristic, complemented
195C22⋊C8—31no—
196C22⋊C8—31no—
197C22⋊C8—31no—
198C22⋊C8C2×D311yescharacteristic, complemented
199C23×C3—21no—
200C23×C3—21no—
201C23×C3C2×D411yescharacteristic
202C22×D3—21no—
203C22×D3—42no—
204C22×D3—81no—
205C22×D3—81no—
206C22×D3C2×D411yescharacteristic
207C22×D3C2×D411yescharacteristic
208C3×Q8—21no—
209C3×Q8C2×D412yes—
210C3×D4—24no—
211C3×D4—21no—
212C3×D4—21no—
213C3×D4—21no—
214C3×D4—81no—
215C3×D4C2×D412yes—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4C2×D411yescharacteristic
222C2×C3×C4C2411yescharacteristic
223#67—24no—
224#67—22no—
225#67—24no—
226#67—22no—
227#67—42no—
228#67—81no—
229#67—81no—
230C2×Dic3—24no—
231C2×Dic3—24no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—24no—
235C2×Dic3—21no—
236C2×Dic3C2×D411yescharacteristic
237C2×Dic3C2×D411yescharacteristic
238D12—22no—
239D12—21no—
240D12—21no—
241D12—81no—
242C4×D3—24no—
243C4×D3—24no—
244C4×D3—21no—
245C4×D3—21no—
246C4×D3C2×D412yes—
247C4×D3C2×D412yes—
248Dic6—24no—
249Dic6—24no—
250Dic6—21no—
251Dic6—21no—
252Dic6—22no—
253Dic6—24no—
254Dic6C2×D412yes—
255Dic6C2×D412yes—
256C3×C8—21no—
257C3×C8—21no—
258C3⋊C8—21no—
259C3⋊C8—21no—
260C24—61no—
261#41—22no—
262#41—22no—
263#41—34no—
264#41—62no—
265#41—62no—
266#41—68no—
267#41—62no—
268#41—62no—
269#41—68no—
270#41—68no—
271#41—62no—
272C2×Q8—31no—
273C2×Q8—34no—
274C2×Q8—31no—
275C2×Q8—31no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—62no—
279C2×Q8—62no—
280C2×Q8—68no—
281C2×Q8—62no—
282C2×Q8—62no—
283C2×Q8—62no—
284C2×Q8—61no—
285C2×Q8C22×D311yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—31no—
289C2×D4—31no—
290C2×D4—31no—
291C2×D4—31no—
292C2×D4—61no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4C22×D311yescharacteristic
301C2×D4C22×D311yescharacteristic
302C22×C4—21no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4C22×D311yescharacteristic
311SD16—22no—
312SD16—62no—
313SD16—62no—
314SD16—62no—
315SD16—62no—
316SD16—62no—
317D8—22no—
318D8—62no—
319M16—62no—
320C2×C8—31no—
321C2×C8—31no—
322C2×C8—62no—
323C2×C8C22×D311yescharacteristic
324C2×C8C22×D311yescharacteristic
325C4⋊C4—31no—
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×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3C22×D411yescharacteristic
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—21no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—84no—
352C2×D3#12012yes—
353C3×C4—21no—
354C3×C4—22no—
355C3×C4—41no—
356C3×C4C22×D411yescharacteristic
357C3×C4C22×D411yescharacteristic
358Dic3—21no—
359Dic3—22no—
360Dic3—22no—
361Dic3—41no—
362Dic3#12012yes—
363C23—21no—
364C23—21no—
365C23—31no—
366C23—31no—
367C23—61no—
368C23—122no—
369C23—241no—
370C23—241no—
371C23D3×D411yescharacteristic
372Q8—21no—
373Q8—32no—
374Q8—32no—
375Q8—62no—
376Q8—64no—
377Q8—64no—
378Q8—64no—
379Q8—61no—
380Q8—61no—
381Q8D3×D412yes—
382D4—24no—
383D4—21no—
384D4—21no—
385D4—21no—
386D4—62no—
387D4—62no—
388D4—62no—
389D4—64no—
390D4—64no—
391D4—61no—
392D4—61no—
393D4—81no—
394D4—122no—
395D4—241no—
396D4—241no—
397D4—241no—
398D4D3×D412yes—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—31no—
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—31no—
408C2×C4—64no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—64no—
412C2×C4—64no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4D3×D411yescharacteristic
419C2×C4C23×D311yescharacteristic
420C8—21no—
421C8—21no—
422C8—61no—
423C8—61no—
424C2×C3—41no—
425C2×C3—42no—
426C2×C3—81no—
427C2×C3C2×#12011yescharacteristic
428C2×C3#57711yescharacteristic
429C2×C3#57711yescharacteristic
430D3—22no—
431D3—41no—
432D3—81no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—32no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—61no—
447C22—61no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—244no—
452C22C2×D3×D411yescharacteristic
453C4—21no—
454C4—22no—
455C4—32no—
456C4—41no—
457C4—61no—
458C4—62no—
459C4—62no—
460C4—121no—
461C4C2×D3×D411yescharacteristic
462C4C2×D3×D411yescharacteristic
463C3#301011yescharacteristic, complemented, Sylow
464C2—41no—
465C2—42no—
466C2—62no—
467C2—81no—
468C2—121no—
469C2—241no—
470C2D3×#12011yescharacteristic
471C2#577011yescharacteristic
472C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3010
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C2×D3×SD16C2semidirect
3. C2×#911C2semidirect
4. #5607C2semidirect
5. C3×#449C2semidirect
6. #5238C2semidirect
7. #5170C2semidirect
8. #5158C2semidirect
9. #5040C2semidirect
10. #4806C2semidirect
11. #4802C2semidirect
12. #4781C2semidirect
13. #4772C2semidirect
14. #4738C2semidirect
15. #4727C2semidirect
17. C2×C3×#41C22semidirect
21. C2×D3×Q8C22semidirect
25. C2×#237C22semidirect
30. C22×Dic6C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. C3×C22⋊C8C22semidirect
55. #835C22semidirect
56. #832C22semidirect
57. #830C22semidirect
58. (C2×D3)⋊C8C22semidirect
59. (C2×D3)⋊C8C22semidirect
60. #816C22semidirect
61. #816C22semidirect
62. #810C22semidirect
63. #810C22semidirect
71. #449D3semidirect
102. D3×Q8D4semidirect
108. #237D4semidirect
160. C2×#41C2×D3semidirect
192. #103C2×D3semidirect
194. #102C2×D3semidirect
198. C22⋊C8C2×D3semidirect
463. C3#3010semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C2×D3×SD162
C2×#9112
#56072
C3×#4492
#52382
#51702
#51582
#50402
#48062
#48022
#47812
#47722
#47382
#47272
#30103