Finite Group Databasebeta

← #10845405

Subgroups — #10845405

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2563C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3#82417C211yesmaximal, characteristic
4C2×#5012C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6#69523C212yesmaximal, complemented
7#69520C212yesmaximal, complemented
8#69517C212yesmaximal, complemented
9#69513C212yesmaximal, complemented
10C4×#809C211yesmaximal, characteristic, complemented
11#69466C211yesmaximal, characteristic, complemented
12#25346—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#938—21no—
16C2×#930C2212yes—
17C22×#208C2212yes—
18C2×C4×D12C2211yescharacteristic
19C3×#393—22no—
20C3×#390—21no—
21C3×#382—21no—
22C3×#380C2212yes—
23C3×#380C2212yes—
24C3×C4×C4⋊C4C2211yescharacteristic
25#5212C2212yes—
26C2×#830C2212yes—
27#5084—24no—
28#5012—24no—
29#4987—21no—
30C2×#809C2212yes—
31C2×#809C2211yescharacteristic
32C2×#809C2211yescharacteristic
33C2×#808C2212yes—
34C2×#807C2212yes—
35#4961—22no—
36#4935C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4529C2212yes—
40#4529C2211yescharacteristic
41#4529C2211yescharacteristic
42#4529C2212yes—
43#4529C2212yes—
44C2×#476—31no—
45C2×#436—31no—
46#2563D311yescharacteristic, complemented
47#2537—31no—
48#2077—32no—
49#2033—32no—
50#1990—31no—
51#1947—31no—
52#1895—32no—
53#1809—32no—
54C4×#102—31no—
55C22×C3×Q8—21no—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67—22no—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4D412yes—
73C2×C3×C42—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76#938—44no—
77#930—24no—
78#930—42no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×#208—24no—
86C2×#208—24no—
87C2×#208—42no—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C4×D12—22no—
92C4×D12—22no—
93C3×#95—21no—
94C3×#95—22no—
95C3×#95—22no—
96C3⋊#95—22no—
97#830—24no—
98#809—22no—
99#809—24no—
100#809—22no—
101#808—24no—
102#807—42no—
103C4⋊(C3⋊C8)D414yescomplemented
104C4×C3⋊C8—22no—
105C22×M16—32no—
106C2×C4⋊Q8C2×D311yescharacteristic
107C2×#121—32no—
108C2×#121—61no—
109C2×C4×D4—31no—
110C2×C4×D4C2×D311yescharacteristic
111#476—64no—
112#436—64no—
113C2×#107—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#102—31no—
117C2×#102—32no—
118C2×#102—31no—
119C2×C22⋊C8—32no—
120C2×C4×C8—31no—
121#394—62no—
122#393—22no—
123#390—21no—
124#384—31no—
125#382—21no—
126#380—32no—
127#380—61no—
128#380C2×D312yes—
129#380C2×D312yes—
130C4×C4⋊C4C2×D311yescharacteristic
131#340—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340—32no—
136C24×C3C2×D412yes—
137C23×D3—21no—
138C2×C3×Q8—42no—
139C2×C3×Q8—81no—
140C2×C3×Q8—81no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C2×#67—44no—
158C2×#67—44no—
159C2×#67—82no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C2×D12—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12#4112yes—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—42no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—82no—
181C3×C22⋊C4—82no—
182C3×C42—42no—
183C3×C42C2×D412yes—
184C3×C42#4112yes—
185#217—44no—
186(C2×D3)⋊C4—44no—
187(C2×D3)⋊C4—42no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3—81no—
194#208—48no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8C2×D414yes—
203C2×C3⋊C8#4112yes—
204C22×Q8—21no—
205C22×D4—31no—
206C22×D4—62no—
207C22×D4#6711yescharacteristic
208C23×C4—61no—
209C23×C4C22×D312yes—
210C2×M16—64no—
211C2×M16—64no—
212C2×M16—122no—
213C22×C8—31no—
214C22×C8—32no—
215C22×C8—31no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218#121—64no—
219#121—122no—
220#121—124no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—62no—
224C4×D4—62no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4#6711yescharacteristic
231C2×C4⋊C4C22×D312yes—
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4#6712yes—
240C2×C42—21no—
241C2×C42C22×D311yescharacteristic
242#107—122no—
243#106—64no—
244C4⋊C8—34no—
245#102—62no—
246#102—64no—
247#102—62no—
248C22⋊C8—64no—
249C4×C8—62no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—62no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3C22×D411yescharacteristic
262C22×D3—22no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C3×Q8—84no—
270C3×D4—44no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4#12112yes—
290C2×C3×C4#12112yes—
291C2×C3×C4AGL(1,ℤ/8)12yes—
292C2×C3×C4#13712yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)11yescharacteristic
298C2×C3×C42−1+411yescharacteristic
299#67—88no—
300C2×Dic3—41no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—41no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306D12—22no—
307D12—22no—
308D12—41no—
309D12—41no—
310C4×D3—84no—
311C3⋊C8—24no—
312C3⋊C8—22no—
313C3⋊C8—41no—
314C24—61no—
315C24C2×#6712yes—
316C2×Q8—42no—
317C2×Q8—81no—
318C2×Q8—81no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—32no—
323C2×D4—61no—
324C2×D4—61no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—61no—
328C2×D4—124no—
329C2×D4—124no—
330C2×D4—242no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—61no—
341C22×C4—62no—
342C22×C4—122no—
343C22×C4—122no—
344C22×C4—122no—
345C22×C4C2×#6712yes—
346C22×C4C2×#6711yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C23×D311yescharacteristic
349M16—128no—
350C2×C8—34no—
351C2×C8—32no—
352C2×C8—61no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—42no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—122no—
372C4⋊C4—241no—
373C22⋊C4—24no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C22⋊C4—122no—
379C42—42no—
380C42D3×D412yes—
381C42#23712yes—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3C2×#12111yescharacteristic
390C22×C3#54611yescharacteristic
391C22×C3#54711yescharacteristic
392C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
393C22×C3C2×#13711yescharacteristic
394C22×C3#57611yescharacteristic
395C22×C3#57711yescharacteristic
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—42no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3—81no—
403C3×C4—22no—
404C3×C4—42no—
405C3×C4—81no—
406C3×C4—82no—
407C3×C4—81no—
408C3×C4#46814yes—
409C3×C4#52911yescharacteristic
410C3×C4#54211yescharacteristic
411C3×C4#54611yescharacteristic
412C3×C4#54811yescharacteristic
413Dic3—81no—
414Dic3—82no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—62no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23C22×#6711yescharacteristic
430Q8—84no—
431D4—44no—
432D4—62no—
433D4—62no—
434D4—121no—
435D4—121no—
436D4—248no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—244no—
462C2×C4#93912yes—
463C2×C4#93912yes—
464C2×C4#94212yes—
465C2×C4#94912yes—
466C2×C4C2×D3×D411yescharacteristic
467C2×C4C2×#23711yescharacteristic
468C2×C4D3×#4111yescharacteristic
469C2×C4D3×#4111yescharacteristic
470C2×C4#100911yescharacteristic
471C2×C4#101011yescharacteristic
472C8—62no—
473C8—64no—
474C8—121no—
475C2×C3—82no—
476C2×C3#242711yescharacteristic
477C2×C3C2×#46811yescharacteristic
478C2×C3#306211yescharacteristic
479C2×C3#332511yescharacteristic
480C2×C3#333111yescharacteristic
481C2×C3#335411yescharacteristic
482C2×C3#335711yescharacteristic
483D3—82no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—241no—
498C22C2×#93911yescharacteristic
499C22C2×#94211yescharacteristic
500C22C2×#94911yescharacteristic
501C22#582411yescharacteristic
502C22#582911yescharacteristic
503C22#583511yescharacteristic
504C22#583611yescharacteristic
505C4—22no—
506C4—42no—
507C4—82no—
508C4—81no—
509C4—81no—
510C4—241no—
511C4—242no—
512C4#504914yes—
513C4#567611yescharacteristic
514C4#567711yescharacteristic
515C4#568011yescharacteristic
516C4#568211yescharacteristic
517C3#2534611yescharacteristic, complemented, Sylow
518C2—82no—
519C2—242no—
520C2C2×#504911yescharacteristic
521C2#8299611yescharacteristic
522C2#8337911yescharacteristic
523C2#8362711yescharacteristic
524C2#8363611yescharacteristic
525C2#8365011yescharacteristic
526C2#8365411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2563C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenter#2563D3
hypocenterC3#25346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
1. C3×#2563C2semidirect
2. C2×#5084C2semidirect
4. C2×#5012C2semidirect
5. #71787C2semidirect
6. #69523C2semidirect
7. #69520C2semidirect
8. #69517C2semidirect
9. #69513C2semidirect
10. C4×#809C2semidirect
11. #69466C2semidirect
46. #2563D3semidirect
103. C4⋊(C3⋊C8)D4semidirect
517. C3#25346semidirect

Maximal subgroups

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

subgroupindex
C3×#25632
C2×#50842
#824172
C2×#50122
#717872
#695232
#695202
#695172
#695132
C4×#8092
#694662
#253463