Finite Group Databasebeta

← #10700583

Subgroups — #10700583

Order 768.

1192 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85731C211yesmaximal, characteristic
2C2×#5132C211yesmaximal, characteristic
3#83762C211yesmaximal, characteristic
4C2×C3×#344C211yesmaximal, characteristic, complemented
5C2×#4559C211yesmaximal, characteristic, complemented
6C2×#4559C211yesmaximal, characteristic, complemented
7C2×#4533C211yesmaximal, characteristic, complemented
8#10756—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5523—22no—
17#5522—22no—
18#5521—21no—
19D3×#117—22no—
20#5516—21no—
21#5504—22no—
22#5486C414yes—
23C4×#235—22no—
24C2×#824—21no—
25#5132—22no—
26#5132C414yescomplemented
27C2×#823—21no—
28#5129C414yescomplemented
29#5126—22no—
30#5110—22no—
31C3×#344—22no—
32#4559—22no—
33#4559—22no—
34#4533—22no—
35#3428—31no—
36C2×#413—31no—
37#2879—31no—
38C2×#344—31no—
39C2×#344—31no—
40C2×#344—31no—
41C2×#344D311yescharacteristic, complemented
42C23×C3×C4C2311yescharacteristic
43C22×#67C2×C412yes—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235—21no—
52C2×#235C2×C412yes—
53C2×#235C2×C412yes—
54C2×#235D412yes—
55C2×#235Q812yes—
56C22×D12D411yescharacteristic
57C22×C4×D3D412yes—
58C22×Dic6D411yescharacteristic
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C2×C3×M16—21no—
69C2×C3×M16C2×C412yescomplemented
70C2×C3×M16C2×C412yescomplemented
71C22×C3×C8—21no—
72C22×C3×C8—21no—
73C3×#117—21no—
74C3×#117—21no—
75C3×#117D412yes—
76C3×#117D412yes—
77C2×C3×C4⋊C4—21no—
78C2×C3×C22⋊C4C2×C412yes—
79C2×C3×C42—21no—
80C2×#217C2×C412yes—
81C4×#67—28no—
82#926—21no—
83#926—21no—
84#926D412yes—
85#926D412yes—
86C2×C4⋊Dic3—21no—
87C2×C4×Dic3—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208—21no—
99C2×#208C2×C412yescomplemented
100C22×C3⋊C8—21no—
101C22×C3⋊C8C2×C412yescomplemented
102#893—24no—
103#892—24no—
104D3×C4⋊C4—24no—
105#887—24no—
106#67⋊C4—28no—
107D3×C22⋊C4—24no—
108#877—24no—
109C4×D12—24no—
110#872—24no—
111C42×D3—24no—
112C4×Dic6—24no—
113#824—24no—
114#823—24no—
115(C2×D3)⋊C8—28no—
116C22×#41—31no—
117C22×M16—31no—
118C22×M16—31no—
119C22×M16C2×D311yescharacteristic
120C22×M16C2×D311yescharacteristic
121#520—62no—
122#520—62no—
123#519—61no—
124#518—34no—
125#518—61no—
126C4×#41—62no—
127C4×#41—62no—
128C2×#117—31no—
129C2×#117—62no—
130C2×#117C2×D311yescharacteristic
131#413—34no—
132#413—62no—
133C2×#101—61no—
134C2×#100—61no—
135#408—34no—
136#408—62no—
137#407—62no—
138#344—22no—
139#344—62no—
140#344—62no—
141#344—62no—
142C24×C3C22×C411yescharacteristic
143C23×D3C22⋊C412yes—
144C22×C3×C4—22no—
145C22×C3×C4C22×C411yescharacteristic
146C22×C3×C4C22×C411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×Q811yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×C3×C4#4111yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C2×#67—22no—
160C2×#67—22no—
161C2×#67—22no—
162C2×#67—22no—
163C2×#67C4214yes—
164C2×#67C4⋊C414yes—
165C22×Dic3—22no—
166C22×Dic3C22⋊C412yes—
167#235—22no—
168#235—22no—
169#235—22no—
170#235—22no—
171#235—22no—
172#235—22no—
173#235—22no—
174#235—22no—
175#235—22no—
176#235—22no—
177#235—22no—
178#235—22no—
179#235—22no—
180#235—22no—
181#235—22no—
182#235—22no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12C22⋊C412yes—
188C2×D12C4⋊C412yes—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—28no—
194C2×C4×D3C22⋊C414yes—
195C2×C4×D3C4⋊C414yes—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6—21no—
200C2×Dic6C22⋊C412yes—
201C2×Dic6C4⋊C412yes—
202C3×M16—22no—
203C3×M16—22no—
204C3×M16—22no—
205C3×M16—22no—
206C3×M16—41no—
207C3×M16—41no—
208C3×M16—41no—
209C3×M16—41no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—41no—
213C2×C3×C8—41no—
214C2×C3×C8—41no—
215C2×C3×C8—41no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4C22⋊C414yes—
220C3×C42—22no—
221C3×C42—22no—
222#217—22no—
223#217C22⋊C414yes—
224(C2×D3)⋊C4—28no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227Dic3⋊C4—28no—
228C4×Dic3—28no—
229C4×Dic3—22no—
230C4×Dic3—22no—
231#208—22no—
232#208—22no—
233#208—41no—
234#208—41no—
235#208—41no—
236#208—41no—
237#208—41no—
238#208—41no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C2×C3⋊C8—41no—
244C2×C3⋊C8—41no—
245C2×#41—32no—
246C2×#41—32no—
247C2×#41—32no—
248C2×#41—32no—
249C2×#41—61no—
250C2×#41—61no—
251C2×#41—61no—
252C2×#41—61no—
253C2×#41—61no—
254C2×#41—61no—
255C2×#41—61no—
256C2×#41—61no—
257C22×Q8—31no—
258C22×D4—32no—
259C22×D4—31no—
260C23×C4—32no—
261C23×C4C22×D311yescharacteristic
262C2×M16—21no—
263C2×M16—21no—
264C2×M16—21no—
265C2×M16—21no—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—21no—
270C2×M16—21no—
271C2×M16—21no—
272C2×M16—32no—
273C2×M16—61no—
274C2×M16—61no—
275C2×M16—61no—
276C2×M16—61no—
277C2×M16—61no—
278C2×M16—61no—
279C2×M16—61no—
280C2×M16—61no—
281C2×M16—61no—
282C2×M16—61no—
283C2×M16—61no—
284C2×M16C4×D312yescomplemented
285C2×M16C4×D312yescomplemented
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—32no—
289C22×C8—61no—
290C4×Q8—64no—
291C4×Q8—64no—
292C4×D4—64no—
293C4×D4—68no—
294C4×D4—64no—
295C4×D4—68no—
296#117—21no—
297#117—21no—
298#117—32no—
299#117—32no—
300#117—64no—
301#117—64no—
302#117—64no—
303#117—61no—
304#117—61no—
305#117D1212yes—
306#117#6712yes—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—64no—
309C2×C4⋊C4—61no—
310C2×C22⋊C4—32no—
311C2×C22⋊C4—64no—
312C2×C22⋊C4C4×D312yes—
313C2×C42—21no—
314C2×C42—64no—
315C2×C42—61no—
316#101—64no—
317#100—64no—
318C22⋊C8—68no—
319C23×C3—81no—
320C23×C3C2×C4211yescharacteristic
321C23×C3C2×C22⋊C411yescharacteristic
322C23×C3C2×C22⋊C411yescharacteristic
323C23×C3C2×C22⋊C411yescharacteristic
324C23×C3C2×C4⋊C411yescharacteristic
325C23×C3#11711yescharacteristic
326C23×C3#11711yescharacteristic
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—82no—
330C22×D3#9514yes—
331C2×C3×C4—24no—
332C2×C3×C4—82no—
333C2×C3×C4C2×C22⋊C411yescharacteristic
334C2×C3×C4C2×C22⋊C411yescharacteristic
335C2×C3×C4C2×C4⋊C411yescharacteristic
336C2×C3×C4C2×C4⋊C411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4C4×D411yescharacteristic
339C2×C3×C4C4×D411yescharacteristic
340C2×C3×C4C4×D411yescharacteristic
341C2×C3×C4C4×D411yescharacteristic
342C2×C3×C4C4×D411yescharacteristic
343C2×C3×C4C4×D411yescharacteristic
344C2×C3×C4C4×D411yescharacteristic
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12111yescharacteristic
350C2×C3×C4#12111yescharacteristic
351C2×C3×C4#12111yescharacteristic
352C2×C3×C4#12111yescharacteristic
353C2×C3×C4C22⋊Q811yescharacteristic
354C2×C3×C4C22⋊Q811yescharacteristic
355C2×C3×C4C22⋊Q811yescharacteristic
356C2×C3×C4C22⋊Q811yescharacteristic
357C2×C3×C4#12311yescharacteristic
358C2×C3×C4#12311yescharacteristic
359C2×C3×C4#12311yescharacteristic
360C2×C3×C4#12311yescharacteristic
361#67—24no—
362#67—24no—
363#67—24no—
364#67—24no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—24no—
368C2×Dic3—82no—
369C2×Dic3#9514yes—
370D12—22no—
371D12—22no—
372D12—22no—
373D12—22no—
374C4×D3—24no—
375C4×D3—24no—
376C4×D3—24no—
377C4×D3—24no—
378C4×D3—88no—
379Dic6—22no—
380Dic6—22no—
381Dic6—22no—
382Dic6—22no—
383C3×C8—42no—
384C3×C8—42no—
385C3⋊C8—42no—
386C3⋊C8—42no—
387C24—32no—
388C24C2×C4×D311yescharacteristic
389#41—62no—
390#41—62no—
391#41—62no—
392#41—62no—
393#41—62no—
394#41—62no—
395#41—62no—
396#41—62no—
397#41—62no—
398#41—62no—
399#41—62no—
400#41—62no—
401#41—62no—
402#41—62no—
403#41—62no—
404#41—62no—
405C2×Q8—32no—
406C2×Q8—32no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×D4—34no—
412C2×D4—34no—
413C2×D4—32no—
414C2×D4—32no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C22×C4—22no—
424C22×C4—32no—
425C22×C4—34no—
426C22×C4—34no—
427C22×C4—68no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4C2×C4×D311yescharacteristic
434C22×C4C2×C4×D311yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4#23511yescharacteristic
437C22×C4#23511yescharacteristic
438C22×C4D3×D411yescharacteristic
439C22×C4D3×D411yescharacteristic
440C22×C4D3×D411yescharacteristic
441C22×C4D3×D411yescharacteristic
442C22×C4D3×D411yescharacteristic
443C22×C4#23711yescharacteristic
444C22×C4D3×Q811yescharacteristic
445C22×C4#23911yescharacteristic
446C22×C4C2×#6711yescharacteristic
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—62no—
456M16—62no—
457M16—121no—
458M16—121no—
459M16—121no—
460M16—121no—
461M16—121no—
462M16—121no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—62no—
470C2×C8—62no—
471C2×C8—62no—
472C2×C8—62no—
473C2×C8—121no—
474C2×C8—121no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—68no—
478C4⋊C4—62no—
479C4⋊C4—62no—
480C22⋊C4—22no—
481C22⋊C4—34no—
482C22⋊C4—68no—
483C22⋊C4—62no—
484C22⋊C4(C2×D3)⋊C414yes—
485C42—22no—
486C42—22no—
487C42—68no—
488C42—62no—
489C42—62no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3C2×#9511yescharacteristic
498C22×C3C4×C22⋊C411yescharacteristic
499C22×C3C4×C22⋊C411yescharacteristic
500C22×C3#37911yescharacteristic
501C22×C3#38011yescharacteristic
502C22×C3#38011yescharacteristic
503C22×C3#38111yescharacteristic
504C2×D3—24no—
505C2×D3—44no—
506C2×D3—44no—
507C2×D3—82no—
508C3×C4—44no—
509C3×C4#38511yescharacteristic
510C3×C4#38511yescharacteristic
511C3×C4#38511yescharacteristic
512C3×C4#38511yescharacteristic
513C3×C4#38611yescharacteristic
514C3×C4#38611yescharacteristic
515C3×C4#38611yescharacteristic
516C3×C4#38611yescharacteristic
517Dic3—24no—
518Dic3—44no—
519C23—34no—
520C23—62no—
521C23—62no—
522C23—81no—
523C23—242no—
524C23C42×D311yescharacteristic
525C23#87211yescharacteristic
526C23D3×C22⋊C411yescharacteristic
527C23D3×C22⋊C411yescharacteristic
528C23D3×C4⋊C411yescharacteristic
529C23#89211yescharacteristic
530C23C2×(C2×D3)⋊C411yescharacteristic
531Q8—62no—
532Q8—62no—
533Q8—62no—
534Q8—62no—
535D4—64no—
536D4—64no—
537D4—64no—
538D4—64no—
539D4—62no—
540D4—62no—
541D4—62no—
542D4—62no—
543C2×C4—24no—
544C2×C4—34no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—82no—
553C2×C4—248no—
554C2×C4—242no—
555C2×C4C4×D1211yescharacteristic
556C2×C4C4×D1211yescharacteristic
557C2×C4D3×C22⋊C411yescharacteristic
558C2×C4D3×C22⋊C411yescharacteristic
559C2×C4#67⋊C411yescharacteristic
560C2×C4#67⋊C411yescharacteristic
561C2×C4#88211yescharacteristic
562C2×C4#88211yescharacteristic
563C2×C4#88311yescharacteristic
564C2×C4#88311yescharacteristic
565C2×C4#88411yescharacteristic
566C2×C4#88411yescharacteristic
567C2×C4D3×C4⋊C411yescharacteristic
568C2×C4D3×C4⋊C411yescharacteristic
569C2×C4#89311yescharacteristic
570C2×C4#89311yescharacteristic
571C2×C4#89411yescharacteristic
572C2×C4#89411yescharacteristic
573C2×C4#89511yescharacteristic
574C2×C4#89611yescharacteristic
575C2×C4#89611yescharacteristic
576C2×C4#89711yescharacteristic
577C2×C4C4×#6711yescharacteristic
578C2×C4C4×#6711yescharacteristic
579C2×C4#93711yescharacteristic
580C2×C4#93711yescharacteristic
581C2×C4#93811yescharacteristic
582C2×C4#94611yescharacteristic
583C8—42no—
584C8—42no—
585C8—122no—
586C8—122no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3C22⋊#9511yescharacteristic
595D3—44no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—64no—
604C22—124no—
605C22—124no—
606C22—242no—
607C22D3×#9511yescharacteristic
608C22#466411yescharacteristic
609C22#466611yescharacteristic
610C22#466711yescharacteristic
611C22C4×(C2×D3)⋊C411yescharacteristic
612C22#495511yescharacteristic
613C22#498711yescharacteristic
614C4—44no—
615C4—64no—
616C4—124no—
617C4#466511yescharacteristic
618C4#466511yescharacteristic
619C4#466811yescharacteristic
620C4#466811yescharacteristic
621C4#495611yescharacteristic
622C4#495611yescharacteristic
623C4#498911yescharacteristic
624C4#498911yescharacteristic
625C3#1075611yescharacteristic, complemented, Sylow
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—124no—
634C2(C2×D3)⋊#9511yescharacteristic
subgroupG/H
centerC4#4665
commutatorC23×C3C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#344C2
socleC2×C3C22⋊#95
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#344D3
hypocenterC3#10756
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×C3×#344C2semidirect
5. C2×#4559C2semidirect
6. C2×#4559C2semidirect
7. C2×#4533C2semidirect
26. #5132C4semidirect
28. #5129C4semidirect
41. C2×#344D3semidirect
69. C2×C3×M16C2×C4semidirect
70. C2×C3×M16C2×C4semidirect
99. C2×#208C2×C4semidirect
101. C22×C3⋊C8C2×C4semidirect
284. C2×M16C4×D3semidirect
285. C2×M16C4×D3semidirect
625. C3#10756semidirect

Maximal subgroups

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

subgroupindex
#857312
C2×#51322
#837622
C2×C3×#3442
C2×#45592
C2×#45332
#107563