Finite Group Databasebeta

← #11658822

Subgroups — #11658822

Order 768.

2666 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5745C211yesmaximal, characteristic
2C22×#901C211yesmaximal, characteristic
3#85731C211yesmaximal, characteristic
4C2×C3×#424C211yesmaximal, characteristic, complemented
5#83750C211yesmaximal, characteristic, complemented
6C2×#5102C211yesmaximal, characteristic, complemented
7#82204C211yesmaximal, characteristic, complemented
8#79081C218yesmaximal, complemented
9#79080C214yesmaximal, complemented
10D3×#424C214yesmaximal, complemented
11#79012C218yesmaximal, complemented
12#60100—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×#926C2211yescharacteristic
18C22×#208C2211yescharacteristic
19C22×#208C2211yescharacteristic
20#5745—28no—
21C2×#907C2214yes—
22C2×D3×M16C2214yes—
23C2×#901—28no—
24C2×#901C2214yes—
25C2×#901C2214yes—
26C2×#901C2214yes—
27C22×ΓL(1,𝔽25)C2212yes—
28C22×D3×C8C2212yes—
29#5523C414yes—
30#5522C2214yes—
31#5521C412yes—
32D3×#117C2214yes—
33#5516C412yes—
34#5504C414yes—
35#5486C414yes—
36C4×#235C2214yes—
37C3×#424C2212yes—
38C3×#424C2212yes—
39C3×#424C2214yescomplemented
40C2×C3×C4⋊C8C2214yescomplemented
41#5119C2214yes—
42#5118C2214yescomplemented
43#5102C2212yes—
44#5102C2212yes—
45#5102C2214yescomplemented
46#5102C2214yescomplemented
47C2×Dic3⋊C8C2214yescomplemented
48#4999C2214yes—
49#4999C2214yescomplemented
50#4924C2214yescomplemented
51#4839C22116yescomplemented
52#4839C22116yescomplemented
53#4837C22116yescomplemented
54D3×C4⋊C8C22116yescomplemented
55C2×#568—31no—
56C22×#131—31no—
57#3428—31no—
58#2906—38no—
59#2906—38no—
60#2905—34no—
61#2905—31no—
62#2905—31no—
63C2×#424—34no—
64C2×#424—31no—
65C2×#424D311yescharacteristic, complemented
66C23×C3×C4C2311yescharacteristic
67C22×#67C2×C412yes—
68C2×#235C2×C412yes—
69C2×#235C2×C412yes—
70C2×#235D418yes—
71C2×#235Q818yes—
72C2×#235C2312yes—
73C2×#235C2312yes—
74C22×D12C2×C411yescharacteristic
75C22×C4×D3C2312yes—
76C22×Dic6C2×C411yescharacteristic
77C2×C3×M16—24no—
78C2×C3×M16C2312yes—
79C2×C3×M16C2312yes—
80C22×C3×C8—24no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2312yes—
84C22×C3×C8C2312yes—
85C22×C3×C8C2312yes—
86C3×#117C2×C412yes—
87C3×#117C2×C412yes—
88C3×#117C2312yes—
89C3×#117C2312yes—
90C2×C3×C4⋊C4C2×C412yes—
91C2×C3×C22⋊C4C2×C412yes—
92C2×C3×C42C2312yes—
93C2×#217C2×C412yes—
94C4×#67C2×C4116yes—
95#926C2×C412yes—
96#926C2×C412yes—
97#926C2312yes—
98#926C2312yes—
99C2×C4⋊Dic3C2×C412yes—
100C2×C4×Dic3C2312yes—
101C2×#208—24no—
102C2×#208—24no—
103C2×#208C2312yes—
104C2×#208C2312yes—
105C2×#208C2312yes—
106C2×#208C2312yes—
107C22×C3⋊C8C2312yes—
108C22×C3⋊C8C2312yes—
109#907—216no—
110D3×M16—216no—
111#901—216no—
112#901—216no—
113#901—216no—
114C2×ΓL(1,𝔽25)—28no—
115C2×ΓL(1,𝔽25)C2318yes—
116C2×ΓL(1,𝔽25)C2318yes—
117C2×D3×C8—28no—
118C2×D3×C8C2318yes—
119C2×D3×C8C2318yes—
120#893C2×C418yes—
121#892C2×C418yes—
122D3×C4⋊C4C2×C418yes—
123#887C2×C418yes—
124#67⋊C4C2×C4116yes—
125D3×C22⋊C4C2×C418yes—
126#877C2×C418yes—
127C4×D12C2×C418yes—
128#872C2318yes—
129C42×D3C2318yes—
130C4×Dic6C2×C418yes—
131C3×C4⋊C8C2318yes—
132C3×C4⋊C8C2318yes—
133Dic3⋊C8C2318yes—
134Dic3⋊C8C23116yes—
135Dic3⋊C8C2318yes—
136C4⋊(C3⋊C8)C23116yes—
137C22×#41—31no—
138#568—68no—
139C2×#131—34no—
140C2×#131—34no—
141C2×#131—34no—
142C2×#131—34no—
143C2×#131—68no—
144C22×M16—31no—
145C22×M16—31no—
146C22×M16—34no—
147C22×M16—32no—
148C22×M16C2×D311yescharacteristic
149C23×C8—32no—
150C23×C8C2×D311yescharacteristic
151#520—34no—
152#520—34no—
153#519—32no—
154#518—32no—
155#518—34no—
156C4×#41—34no—
157C4×#41—34no—
158C2×#117—31no—
159C2×#117—34no—
160C2×#117C2×D311yescharacteristic
161#424—34no—
162#424—32no—
163#424—32no—
164#424—34no—
165#424—34no—
166#424—34no—
167#424—34no—
168#424—316no—
169#424—316no—
170#424C2×D312yes—
171#424C2×D312yes—
172#424C2×D314yescomplemented
173#423—34no—
174#423—34no—
175#423—316no—
176C2×C4⋊C8—34no—
177C2×C4⋊C8—316no—
178C2×C4⋊C8C2×D314yescomplemented
179C24×C3C22×C411yescharacteristic
180C23×D3C22×C412yes—
181C22×C3×C4C22×C414yes—
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×Q814yes—
187C22×C3×C4C2411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C2×#67C4⋊C4116yes—
191C2×#67C22×C414yes—
192C2×#67C22×C414yes—
193C22×Dic3C22×C414yes—
194C22×Dic3C22×C412yes—
195#235C4⋊C4116yes—
196#235C4⋊C4116yes—
197#235C2×D418yes—
198#235C2×D418yes—
199#235C2×Q818yes—
200#235C2×Q818yes—
201C2×D12C4⋊C418yes—
202C2×D12C22×C412yes—
203C2×D12C22×C412yes—
204C2×C4×D3C22×C4116yes—
205C2×C4×D3C22×C414yes—
206C2×C4×D3C2×D418yes—
207C2×C4×D3C2×Q818yes—
208C2×C4×D3C2414yes—
209C2×Dic6C4⋊C418yes—
210C2×Dic6C22×C412yes—
211C2×Dic6C22×C412yes—
212C3×M16—24no—
213C3×M16—24no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—24no—
217C2×C3×C8—24no—
218C2×C3×C8—24no—
219C2×C3×C8C2412yes—
220C2×C3×C8C2414yes—
221C2×C3×C8C2412yes—
222C3×C4⋊C4C22×C414yes—
223C3×C4⋊C4C22×C414yes—
224C3×C22⋊C4C22×C414yes—
225C3×C22⋊C4C22×C414yes—
226C3×C42C22×C414yes—
227C3×C42C2414yes—
228#217C22×C414yes—
229#217C22×C414yes—
230(C2×D3)⋊C4C22×C4116yes—
231C4⋊Dic3C22×C414yes—
232C4⋊Dic3C22×C414yes—
233Dic3⋊C4C22×C4116yes—
234C4×Dic3C22×C4116yes—
235C4×Dic3C22×C414yes—
236C4×Dic3C2414yes—
237#208—24no—
238#208—24no—
239#208—24no—
240#208—24no—
241C2×C3⋊C8—24no—
242C2×C3⋊C8—24no—
243C2×C3⋊C8C2414yes—
244C2×C3⋊C8C2414yes—
245ΓL(1,𝔽25)—216no—
246ΓL(1,𝔽25)—216no—
247D3×C8—216no—
248D3×C8—216no—
249C2×#41—32no—
250C2×#41—32no—
251C2×#41—38no—
252C2×#41—38no—
253C2×#41—32no—
254C2×#41—32no—
255C22×Q8—31no—
256C22×D4—32no—
257C22×D4—31no—
258C23×C4—32no—
259C23×C4C22×D311yescharacteristic
260#131—616no—
261#131—616no—
262#131—616no—
263#131—616no—
264C2×M16—24no—
265C2×M16—32no—
266C2×M16—32no—
267C2×M16—32no—
268C2×M16—32no—
269C2×M16—38no—
270C2×M16—38no—
271C2×M16—68no—
272C2×M16—616no—
273C2×M16—64no—
274C2×M16—64no—
275C2×M16C22×D312yes—
276C2×M16C22×D312yes—
277C22×C8—24no—
278C22×C8—32no—
279C22×C8—32no—
280C22×C8—38no—
281C22×C8—38no—
282C22×C8—68no—
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D312yes—
286C22×C8C22×D312yes—
287C22×C8C22×D312yes—
288C4×Q8—38no—
289C4×Q8—38no—
290C4×D4—316no—
291C4×D4—38no—
292C4×D4—316no—
293C4×D4—38no—
294#117—32no—
295#117—32no—
296#117—32no—
297#117—32no—
298#117—38no—
299#117—38no—
300#117—38no—
301#117C4×D312yes—
302#117C4×D312yes—
303#117C22×D312yes—
304#117C22×D312yes—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—38no—
307C2×C4⋊C4C4×D312yes—
308C2×C22⋊C4—32no—
309C2×C22⋊C4—38no—
310C2×C22⋊C4C4×D312yes—
311C2×C42—32no—
312C2×C42—38no—
313C2×C42C22×D312yes—
314C4⋊C8—38no—
315C4⋊C8—316no—
316C4⋊C8—38no—
317C4⋊C8—316no—
318C4⋊C8C22×D318yes—
319C4⋊C8C22×D318yes—
320C23×C3—42no—
321C23×C3C2×C4⋊C414yes—
322C23×C3C23×C411yescharacteristic
323C23×C3C23×C411yescharacteristic
324C23×C3C23×C411yescharacteristic
325C22×D3—44no—
326C22×D3C2×C4⋊C418yes—
327C22×D3C23×C414yes—
328C2×C3×C4—44no—
329C2×C3×C4C2×C4⋊C414yes—
330C2×C3×C4C2×C4⋊C414yes—
331C2×C3×C4C2×C4⋊C414yes—
332C2×C3×C4C23×C418yes—
333C2×C3×C4C23×C411yescharacteristic
334C2×C3×C4C23×C411yescharacteristic
335C2×C3×C4C23×C411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C22×D412yes—
339C2×C3×C4C22×Q812yes—
340C2×C3×C4C22×Q812yes—
341C2×C3×C4C22×Q812yes—
342C2×C3×C4C2511yescharacteristic
343#67C2×C4⋊C4116yes—
344#67C2×C4⋊C4116yes—
345C2×Dic3—44no—
346C2×Dic3C2×C4⋊C418yes—
347C2×Dic3C23×C418yes—
348C2×Dic3C23×C414yes—
349D12C2×C4⋊C418yes—
350D12C2×C4⋊C418yes—
351C4×D3—416no—
352C4×D3C2×C4⋊C4116yes—
353C4×D3C22×D418yes—
354C4×D3C22×Q818yes—
355Dic6C2×C4⋊C418yes—
356Dic6C2×C4⋊C418yes—
357C3×C8—22no—
358C3×C8—24no—
359C3×C8—22no—
360C3⋊C8—24no—
361C3⋊C8—24no—
362C24—32no—
363C24C2×C4×D311yescharacteristic
364#41—316no—
365#41—316no—
366#41—38no—
367#41—38no—
368#41—38no—
369#41—38no—
370C2×Q8—38no—
371C2×Q8—32no—
372C2×Q8—32no—
373C2×D4—316no—
374C2×D4—34no—
375C2×D4—34no—
376C2×D4—38no—
377C2×D4—32no—
378C2×D4—32no—
379C22×C4—32no—
380C22×C4—34no—
381C22×C4—38no—
382C22×C4—38no—
383C22×C4—34no—
384C22×C4—34no—
385C22×C4—316no—
386C22×C4C2×C4×D314yes—
387C22×C4C2×C4×D311yescharacteristic
388C22×C4C2×C4×D311yescharacteristic
389C22×C4C2×C4×D311yescharacteristic
390C22×C4D3×D414yes—
391C22×C4D3×Q814yes—
392C22×C4C23×D311yescharacteristic
393C22×C4C23×D311yescharacteristic
394C22×C4C23×D311yescharacteristic
395M16—24no—
396M16—24no—
397M16—616no—
398M16—616no—
399M16—64no—
400M16—64no—
401M16—64no—
402M16—64no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—34no—
409C2×C8—34no—
410C2×C8—616no—
411C2×C8—616no—
412C2×C8—64no—
413C2×C8—64no—
414C2×C8C23×D312yes—
415C2×C8C23×D314yes—
416C2×C8C23×D312yes—
417C4⋊C4—34no—
418C4⋊C4—316no—
419C4⋊C4—34no—
420C4⋊C4C2×C4×D314yes—
421C4⋊C4C2×C4×D314yes—
422C22⋊C4—34no—
423C22⋊C4—34no—
424C22⋊C4—316no—
425C22⋊C4C2×C4×D314yes—
426C22⋊C4C2×C4×D314yes—
427C42—316no—
428C42—34no—
429C42—34no—
430C42C2×C4×D314yes—
431C42C23×D314yes—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—44no—
436C22×C3C22×C4⋊C412yes—
437C22×C3C22×C4⋊C412yes—
438C22×C3C22×C4⋊C412yes—
439C22×C3C24×C411yescharacteristic
440C2×D3—28no—
441C2×D3—48no—
442C2×D3C22×C4⋊C418yes—
443C3×C4—28no—
444C3×C4C22×C4⋊C412yes—
445C3×C4C22×C4⋊C412yes—
446C3×C4C22×C4⋊C412yes—
447C3×C4C23×D411yescharacteristic
448C3×C4C23×Q811yescharacteristic
449Dic3—28no—
450Dic3C22×C4⋊C418yes—
451C23—34no—
452C23—38no—
453C23—42no—
454C23—124no—
455C23D3×C4⋊C414yes—
456C23C22×C4×D311yescharacteristic
457C23C22×C4×D311yescharacteristic
458C23C22×C4×D311yescharacteristic
459Q8—38no—
460Q8—38no—
461D4—316no—
462D4—316no—
463D4—38no—
464D4—38no—
465C2×C4—34no—
466C2×C4—38no—
467C2×C4—38no—
468C2×C4—38no—
469C2×C4—316no—
470C2×C4—38no—
471C2×C4—44no—
472C2×C4—1216no—
473C2×C4—124no—
474C2×C4D3×C4⋊C414yes—
475C2×C4D3×C4⋊C414yes—
476C2×C4D3×C4⋊C414yes—
477C2×C4C22×C4×D318yes—
478C2×C4C22×C4×D311yescharacteristic
479C2×C4C22×C4×D311yescharacteristic
480C2×C4C22×C4×D311yescharacteristic
481C2×C4C2×D3×D412yes—
482C2×C4C2×D3×D412yes—
483C2×C4C2×D3×D412yes—
484C2×C4C2×D3×Q812yes—
485C2×C4C2×D3×Q812yes—
486C2×C4C2×D3×Q812yes—
487C2×C4C24×D311yescharacteristic
488C8—22no—
489C8—22no—
490C8—24no—
491C8—64no—
492C8—64no—
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3C23×C4⋊C411yescharacteristic
497C2×C3#357312yes—
498D3—28no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—38no—
503C22—44no—
504C22—68no—
505C22—128no—
506C22C2×D3×C4⋊C412yes—
507C22C2×D3×C4⋊C412yes—
508C22C2×D3×C4⋊C412yes—
509C22C23×C4×D311yescharacteristic
510C4—28no—
511C4—38no—
512C4—68no—
513C4C2×D3×C4⋊C412yes—
514C4C2×D3×C4⋊C412yes—
515C4C2×D3×C4⋊C412yes—
516C4C22×D3×D411yescharacteristic
517C4C22×D3×Q811yescharacteristic
518C3#6010011yescharacteristic, complemented, Sylow
519C2—22no—
520C2—22no—
521C2—22no—
522C2—68no—
523C2C22×D3×C4⋊C411yescharacteristic
524C2#8716012yes—
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC2×C3×#424C2
prime radicalC22×#235C22
cosocleC2×C3×C4C25
hypercenterC2×#424D3
hypocenterC3#60100
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
4. C2×C3×#424C2semidirect
5. #83750C2semidirect
6. C2×#5102C2semidirect
7. #82204C2semidirect
8. #79081C2semidirect
9. #79080C2semidirect
10. D3×#424C2semidirect
11. #79012C2semidirect
39. C3×#424C22semidirect
40. C2×C3×C4⋊C8C22semidirect
42. #5118C22semidirect
45. #5102C22semidirect
46. #5102C22semidirect
47. C2×Dic3⋊C8C22semidirect
49. #4999C22semidirect
50. #4924C22semidirect
51. #4839C22semidirect
52. #4839C22semidirect
53. #4837C22semidirect
54. D3×C4⋊C8C22semidirect
65. C2×#424D3semidirect
172. #424C2×D3semidirect
178. C2×C4⋊C8C2×D3semidirect
518. C3#60100semidirect

Maximal subgroups

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

subgroupindex
C2×#57452
C22×#9012
#857312
C2×C3×#4242
#837502
C2×#51022
#822042
#790812
#790802
D3×#4242
#790122
#601003