Finite Group Databasebeta

← #10770602

Subgroups — #10770602

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86051C211yesmaximal, characteristic
2C3×#3196C211yesmaximal, characteristic, complemented
3#82808C211yesmaximal, characteristic, complemented
4#82736C211yesmaximal, characteristic, complemented
5#71710C211yesmaximal, characteristic, complemented
6#71178C212yesmaximal, complemented
7#71171C212yesmaximal, complemented
8#70982C212yesmaximal, complemented
9#70975C212yesmaximal, complemented
10#69978C211yesmaximal, characteristic, complemented
11#69972C211yesmaximal, characteristic, complemented
12#18457—31nomaximal, Sylow
13C3×#525C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15#5833—21no—
16C2×#930C2211yescharacteristic
17C2×#924C2211yescharacteristic
18#5667—22no—
19#5645C2212yes—
20#5634—21no—
21#5617—21no—
22#5597C2212yes—
23#5591—22no—
24#5555—21no—
25#5552—21no—
26#5550—21no—
27#5543—21no—
28#5534—21no—
29C3×#485—21no—
30C3×#484C2211yescharacteristic, complemented
31C3×#481C2211yescharacteristic, complemented
32C3×#470—21no—
33C3×#468—21no—
34C3×#452—21no—
35C3×#451C2211yescharacteristic, complemented
36C3×#448C2211yescharacteristic, complemented
37C3×#407C2211yescharacteristic
38#5230C2212yescomplemented
39#5227C2212yes—
40#5211C2211yescharacteristic
41C2×#830C2211yescharacteristic
42#5048—22no—
43#5046—21no—
44#5043—21no—
45#4986—21no—
46#4985C2212yescomplemented
47#4979—21no—
48#4968C2212yes—
49#4952—22no—
50#4538C2212yescomplemented
51#4537C2212yescomplemented
52#4470C2212yes—
53#4470C2212yes—
54#4469C2212yescomplemented
55#4469C2212yescomplemented
56#3536—31no—
57#3196D311yescharacteristic, complemented
58#3187—31no—
59#2817—31no—
60#1616—32no—
61#1615—32no—
62#1604—32no—
63#1603—32no—
64#1516—31no—
65#1510—31no—
66#1472—31no—
67C2×C3×#41D411yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C22×#67—21no—
71C2×#237—22no—
72C2×#235—21no—
73C22×D12D411yescharacteristic
74C22×Dic6D411yescharacteristic
75C2×C3×Q16—21no—
76C2×C3×SD16—21no—
77C2×C3×M16—21no—
78C3×#123—21no—
79C3×C22⋊Q8—21no—
80C3×C22⋊Q8—21no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8D411yescharacteristic
84C3×C22⋊Q8C2311yescharacteristic
85C3×#121D411yescharacteristic
86C3×#121C2311yescharacteristic
87C3×#120—21no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×#117D411yescharacteristic
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×#217—21no—
95C2×#217D412yes—
96#946—22no—
97C2×#215—22no—
98#939—22no—
99#939—22no—
100#939—22no—
101#938—22no—
102#937—22no—
103D4×Dic3—22no—
104#930—21no—
105#930—21no—
106#930—21no—
107#930D412yes—
108#930C2312yes—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3D411yescharacteristic
113#924—21no—
114#924—21no—
115#924—21no—
116#924—21no—
117#924—21no—
118C2×Dic3⋊C4—22no—
119C2×Dic3⋊C4C2312yes—
120C22×C3⋊C8—21no—
121#898—22no—
122#897—22no—
123#896—22no—
124#896—22no—
125#894—22no—
126#890—22no—
127#886—22no—
128#886—22no—
129#885—22no—
130#885—22no—
131#884—22no—
132#883—22no—
133#67⋊C4—22no—
134D3×C22⋊C4—22no—
135#878—22no—
136#878—22no—
137#875—21no—
138C4×D12—21no—
139#869—21no—
140C4×Dic6—21no—
141C3×#107—21no—
142C3×#106—21no—
143C3×#103—21no—
144C3×#103—21no—
145C3×#103—21no—
146C3×#102—21no—
147C3×C22⋊C8—21no—
148C3×C22⋊C8C2311yescharacteristic
149C3×C22⋊C8C2311yescharacteristic
150#835—22no—
151C3⋊#95—21no—
152C3⋊#95—22no—
153C3⋊#95—22no—
154C3⋊#95—22no—
155C3⋊#95C2312yes—
156#830—21no—
157#830C2312yes—
158#809—22no—
159#808—22no—
160#562—62no—
161#554—61no—
162#548—61no—
163#547—61no—
164#546—62no—
165#546—61no—
166#543—61no—
167#542—61no—
168#539—32no—
169#538—61no—
170#537—32no—
171#537—61no—
172#525C2×D311yescharacteristic
173C2×C22⋊Q8—31no—
174C2×C22⋊Q8C2×D311yescharacteristic
175C2×#121—31no—
176#485—21no—
177#484C2×D311yescharacteristic, complemented
178#483—61no—
179#481—32no—
180#481C2×D311yescharacteristic, complemented
181#470—21no—
182#468—21no—
183#465—62no—
184#452—21no—
185#451C2×D311yescharacteristic, complemented
186#450—61no—
187#448—32no—
188#448C2×D311yescharacteristic, complemented
189#407C2×D311yescharacteristic
190C2×C22⋊C8—31no—
191#400—32no—
192#400—61no—
193#397—62no—
194#395—32no—
195#395—61no—
196#381—31no—
197#328—32no—
198#327—32no—
199#327—32no—
200#327—32no—
201#324—32no—
202#324—32no—
203C24×C3C2×D411yescharacteristic
204C23×D3—21no—
205C3×#41—81no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—81no—
209C2×C3×Q8C2×D411yescharacteristic
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4C2×D411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—81no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C2×#67—22no—
225C2×#67—22no—
226C2×#67—22no—
227C2×#67—22no—
228C2×#67—82no—
229C22×Dic3—22no—
230C22×Dic3—21no—
231C22×Dic3—22no—
232C22×Dic3C2×D412yes—
233C22×Dic3C2×D412yes—
234#237—84no—
235#235—82no—
236C2×D12—21no—
237C2×D12—81no—
238C2×D12C2×D412yes—
239C2×C4×D3—22no—
240C2×C4×D3—22no—
241C2×Dic6—21no—
242C2×Dic6—22no—
243C2×Dic6—21no—
244C2×Dic6—81no—
245C3×Q16—81no—
246C3×SD16—81no—
247C3×M16—81no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C22⋊C4—21no—
259C3×C22⋊C4—21no—
260C3×C22⋊C4—21no—
261C3×C22⋊C4—21no—
262C3×C22⋊C4—21no—
263C3×C22⋊C4—41no—
264C3×C22⋊C4—41no—
265C3×C42—21no—
266#217—22no—
267#217—22no—
268#217—22no—
269#217—81no—
270#215—82no—
271(C2×D3)⋊C4—22no—
272(C2×D3)⋊C4—22no—
273(C2×D3)⋊C4—22no—
274(C2×D3)⋊C4—22no—
275(C2×D3)⋊C4—22no—
276(C2×D3)⋊C4—22no—
277C4⋊Dic3—22no—
278C4⋊Dic3—22no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3C2×D412yes—
283Dic3⋊C4—22no—
284Dic3⋊C4—22no—
285Dic3⋊C4—22no—
286Dic3⋊C4—22no—
287Dic3⋊C4—82no—
288C4×Dic3—22no—
289C2×C3⋊C8—22no—
290C2×C3⋊C8—81no—
291C2×#41—61no—
292C2×#41—62no—
293C2×#41#6711yescharacteristic
294C22×Q8—31no—
295C22×Q8C22×D311yescharacteristic
296C22×D4—31no—
297C22×D4—61no—
298C23×C4C22×D311yescharacteristic
299C2×Q16—21no—
300C2×SD16—21no—
301C2×SD16—62no—
302C2×M16—21no—
303C22×C8—61no—
304C4⋊Q8—61no—
305#126—62no—
306#125—62no—
307#124—61no—
308#124—62no—
309#124—62no—
310#123—21no—
311#123—62no—
312#123—62no—
313#123—62no—
314#123—62no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8—62no—
320C22⋊Q8—62no—
321C22⋊Q8—62no—
322C22⋊Q8—62no—
323C22⋊Q8—61no—
324C22⋊Q8—62no—
325C22⋊Q8—61no—
326C22⋊Q8—61no—
327C22⋊Q8—61no—
328C22⋊Q8—61no—
329C22⋊Q8—62no—
330C22⋊Q8#6711yescharacteristic
331C22⋊Q8C22×D311yescharacteristic
332#121—32no—
333#121—32no—
334#121—62no—
335#121—61no—
336#121—62no—
337#121—62no—
338#121—62no—
339#121—62no—
340#121—61no—
341#121—61no—
342#121#6711yescharacteristic
343#121C22×D311yescharacteristic
344#120—21no—
345#120—62no—
346C4×Q8—61no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—61no—
350C4×D4—62no—
351C4×D4—62no—
352#117#6711yescharacteristic
353C2×C4⋊C4—21no—
354C2×C4⋊C4—31no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4—62no—
359C2×C4⋊C4C22×D311yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—62no—
363C2×C22⋊C4—61no—
364C2×C22⋊C4—61no—
365#107—21no—
366#106—21no—
367#106—62no—
368#103—21no—
369#103—21no—
370#103—21no—
371#103—62no—
372#102—21no—
373#102—62no—
374C22⋊C8—21no—
375C22⋊C8—32no—
376C22⋊C8—61no—
377C22⋊C8C22×D311yescharacteristic
378C22⋊C8C22×D311yescharacteristic
379#95—32no—
380#95—62no—
381#95—62no—
382#95—62no—
383#95—61no—
384C23×C3—22no—
385C23×C3—21no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3#12011yescharacteristic
389C23×C3#12011yescharacteristic
390C23×C3C22×D411yescharacteristic
391C22×D3—22no—
392C22×D3—42no—
393C22×D3—81no—
394C22×D3—81no—
395C3×Q8—41no—
396C3×Q8—42no—
397C3×Q8—41no—
398C3×Q8—41no—
399C3×D4—41no—
400C3×D4—42no—
401C3×D4—42no—
402C3×D4—81no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4#12011yescharacteristic
418C2×C3×C4#12011yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421#67—44no—
422#67—82no—
423#67—82no—
424#67—82no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—24no—
429C2×Dic3—42no—
430C2×Dic3—42no—
431C2×Dic3—42no—
432C2×Dic3—42no—
433C2×Dic3—81no—
434C2×Dic3—82no—
435D12—42no—
436D12—42no—
437C4×D3—82no—
438C4×D3—82no—
439Dic6—42no—
440Dic6—81no—
441Dic6—82no—
442C3×C8—41no—
443C3×C8—41no—
444C3⋊C8—42no—
445C24—61no—
446C24D3×D411yescharacteristic
447#41—81no—
448#41—242no—
449#41—244no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—62no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—81no—
456C2×Q8—241no—
457C2×Q8C2×#6711yescharacteristic
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—32no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—62no—
465C2×D4—62no—
466C2×D4—61no—
467C2×D4—241no—
468C2×D4—242no—
469C2×D4C2×#6711yescharacteristic
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—61no—
479C22×C4—62no—
480C22×C4—81no—
481C22×C4D3×D411yescharacteristic
482C22×C4D3×D411yescharacteristic
483C22×C4D3×D411yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C2×#6711yescharacteristic
486C22×C4C23×D311yescharacteristic
487Q16—81no—
488SD16—81no—
489SD16—242no—
490M16—81no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—62no—
494C2×C8—241no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—62no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—62no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—242no—
512C4⋊C4C2×#6711yescharacteristic
513C4⋊C4C2×#6711yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—62no—
522C22⋊C4—62no—
523C22⋊C4—62no—
524C22⋊C4—62no—
525C22⋊C4—62no—
526C22⋊C4—62no—
527C22⋊C4—62no—
528C22⋊C4—62no—
529C22⋊C4—62no—
530C22⋊C4—241no—
531C42—21no—
532C42—62no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—82no—
545C22×C3C2×#12011yescharacteristic
546C22×C3#57712yes—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3—162no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—81no—
559C3×C4—81no—
560Dic3—42no—
561Dic3—42no—
562Dic3—42no—
563Dic3—82no—
564C23—22no—
565C23—21no—
566C23—41no—
567C23—41no—
568C23—62no—
569C23—122no—
570C23—241no—
571C23—241no—
572C23#93711yescharacteristic
573C23#93711yescharacteristic
574C23C2×D3×D411yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—42no—
578Q8—41no—
579Q8—122no—
580Q8—241no—
581Q8—242no—
582D4—41no—
583D4—42no—
584D4—42no—
585D4—81no—
586D4—122no—
587D4—122no—
588D4—124no—
589D4—242no—
590D4—242no—
591D4—242no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—122no—
613C2×C4—122no—
614C2×C4—242no—
615C2×C4—242no—
616C2×C4—241no—
617C2×C4—242no—
618C2×C4#93711yescharacteristic
619C2×C4#93711yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C22×#6711yescharacteristic
622C8—41no—
623C8—41no—
624C8—122no—
625C2×C3—22no—
626C2×C3—41no—
627C2×C3—41no—
628C2×C3—81no—
629C2×C3#300911yescharacteristic
630C2×C3#302011yescharacteristic
631C2×C3#302611yescharacteristic
632D3—82no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—82no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—482no—
649C22#577812yes—
650C22C2×#93711yescharacteristic
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—122no—
661C4—122no—
662C4—122no—
663C4—242no—
664C3#1845711yescharacteristic, complemented, Sylow
665C2—22no—
666C2—41no—
667C2—41no—
668C2—81no—
669C2—242no—
670C2#8337511yescharacteristic
671C2#8398211yescharacteristic
672C2#8400611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3196C2
socleC22×C3C2×#120
prime radical#86051C2
hypercenter#3196D3
hypocenterC3#18457
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3196

Complements

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

NH ≅ G/Ntype
2. C3×#3196C2semidirect
3. #82808C2semidirect
4. #82736C2semidirect
5. #71710C2semidirect
6. #71178C2semidirect
7. #71171C2semidirect
8. #70982C2semidirect
9. #70975C2semidirect
10. #69978C2semidirect
11. #69972C2semidirect
30. C3×#484C22semidirect
31. C3×#481C22semidirect
35. C3×#451C22semidirect
36. C3×#448C22semidirect
38. #5230C22semidirect
46. #4985C22semidirect
50. #4538C22semidirect
51. #4537C22semidirect
54. #4469C22semidirect
55. #4469C22semidirect
57. #3196D3semidirect
177. #484C2×D3semidirect
180. #481C2×D3semidirect
185. #451C2×D3semidirect
188. #448C2×D3semidirect
664. C3#18457semidirect

Maximal subgroups

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

subgroupindex
#860512
C3×#31962
#828082
#827362
#717102
#711782
#711712
#709822
#709752
#699782
#699722
#184573