Finite Group Databasebeta

← #10826845

Subgroups — #10826845

Order 768.

1802 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2#87274C211yesmaximal, characteristic, complemented
3#84081C212yesmaximal, complemented
4C2×#5095C211yesmaximal, characteristic, complemented
5C2×#5062C211yesmaximal, characteristic, complemented
6#72137C211yesmaximal, characteristic, complemented
7#71784C211yesmaximal, characteristic, complemented
8#71769C212yesmaximal, complemented
9#71731C212yesmaximal, complemented
10#71724C211yesmaximal, characteristic, complemented
11#71532C212yesmaximal, complemented
12#23909—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C2×#1007C2211yescharacteristic
16C3×#532—24no—
17C3×#532—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×#121—22no—
23C2×C3×C4×D4—22no—
24C22×C3×C42C2211yescharacteristic
25#5834—24no—
26#5831—24no—
27#5812C2211yescharacteristic
28C2×#942C2212yes—
29#5805—22no—
30#5805C2212yes—
31C2×#932C2212yes—
32#5791—22no—
33#5791C2212yes—
34#5239—24no—
35#5238—24no—
36C2×#836C2211yescharacteristic
37C2×#836C2211yescharacteristic
38#5225C2212yes—
39C2×#833C2212yes—
40#5215C2212yes—
41#5207C2212yes—
42#5095—24no—
43#5062—24no—
44C2×#805C2212yes—
45C2×#805C2212yes—
46#4523C2212yes—
47#4523C2212yes—
48#3487—31no—
49C2×#532D311yescharacteristic, complemented
50C2×#456—31no—
51C2×#454—31no—
52#3005—32no—
53#2004—32no—
54#2002—32no—
55#1896—32no—
56#1888—31no—
57#1886—31no—
58#1785—31no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—24no—
64C2×C3×#41—24no—
65C2×C3×#41—42no—
66C2×C3×#41—42no—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4—21no—
70C22×C3×D4—22no—
71C22×C3×D4D412yes—
72C22×C3×D4C2311yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C23×C3×C4—21no—
75C23×C3×C4—21no—
76C23×C3×C4D412yes—
77C23×C3×C4C2311yescharacteristic
78#1007—22no—
79#1007—22no—
80#1007—42no—
81C2×#239—22no—
82C2×D3×Q8—22no—
83C2×#235—22no—
84C2×#235C2312yes—
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8D414yescomplemented
87C3×#127—22no—
88C3×#127—22no—
89C3×#124—22no—
90C3×#124—22no—
91C3×#124—22no—
92C3×#124D414yescomplemented
93C3×#121—216no—
94C3×C4×D4—216no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C42—21no—
99C2×C3×C42—22no—
100C2×C3×C42—22no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103#953—24no—
104#953—22no—
105Q8×Dic3—24no—
106C2×#216—22no—
107#942—24no—
108#942—42no—
109C2×#215—22no—
110#940—24no—
111#940—22no—
112#938—28no—
113#938—24no—
114#936—24no—
115#936—22no—
116C2×#214—22no—
117#932—24no—
118#932—42no—
119C2×#213—22no—
120#930—24no—
121#930D414yescomplemented
122#929—22no—
123C4×#67—28no—
124C4×#67—24no—
125#926—22no—
126#926C2312yes—
127#924—24no—
128#924D414yescomplemented
129C2×#208C2312yes—
130C2×#208C2312yes—
131#836—22no—
132#836—22no—
133#835—24no—
134#833—42no—
135#832—24no—
136#830—24no—
137#805—24no—
138#805—24no—
139C2×2−1+4—31no—
140C22×#41C2×D311yescharacteristic
141C22×#41C2×D311yescharacteristic
142C2×#137—32no—
143C2×AGL(1,ℤ/8)—32no—
144#550—64no—
145#548—64no—
146#535—32no—
147#535—62no—
148#532—24no—
149#532—24no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#127C2×D311yescharacteristic
152C2×#124C2×D311yescharacteristic
153C2×#124C2×D311yescharacteristic
154#525—32no—
155#525—62no—
156C2×#121—22no—
157#519—31no—
158C2×C4×D4—22no—
159C22×C42C2×D311yescharacteristic
160#456—64no—
161#454—64no—
162#452—64no—
163#449—64no—
164C2×#104—32no—
165C2×#104—32no—
166#419—32no—
167#418—32no—
168C2×#101—31no—
169C2×#101—31no—
170C2×#100—32no—
171#407—32no—
172#339—32no—
173#339—32no—
174C24×C3—21no—
175C24×C3C2×D412yes—
176C24×C3C2×D411yescharacteristic
177C3×#41—48no—
178C3×#41—48no—
179C3×#41—48no—
180C3×#41—48no—
181C2×C3×Q8—22no—
182C2×C3×Q8—21no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8C2×D412yes—
187C2×C3×Q8C2×D412yes—
188C2×C3×Q8C2×D412yes—
189C2×C3×D4—28no—
190C2×C3×D4—24no—
191C2×C3×D4—24no—
192C2×C3×D4—24no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—42no—
201C2×C3×D4—48no—
202C2×C3×D4—42no—
203C2×C3×D4—42no—
204C2×C3×D4C2×D412yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—22no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—24no—
214C22×C3×C4—24no—
215C22×C3×C4—42no—
216C22×C3×C4—44no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4C2×D414yes—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C2×#67—22no—
224C2×#67—28no—
225C2×#67—24no—
226#239—44no—
227#239—44no—
228D3×Q8—44no—
229D3×Q8—44no—
230#235—22no—
231#235—22no—
232#235—22no—
233#235—22no—
234#235—44no—
235#235—44no—
236C2×D12—22no—
237C2×D12C2×D412yes—
238C2×C4×D3—24no—
239C2×C4×D3—22no—
240C2×Dic6—22no—
241C2×Dic6C2×D412yes—
242C3×C4⋊C4—28no—
243C3×C22⋊C4—28no—
244C3×C22⋊C4—28no—
245C3×C42—24no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42C2×D412yes—
249C3×C42C2×D412yes—
250#217—28no—
251#217—24no—
252#217—22no—
253#216—44no—
254#215—44no—
255#214—44no—
256#213—44no—
257(C2×D3)⋊C4—24no—
258C4⋊Dic3—24no—
259C4⋊Dic3C2×D414yes—
260Dic3⋊C4—24no—
261C4×Dic3—24no—
262C4×Dic3—22no—
263#208—22no—
264#208—22no—
265#208—22no—
266#208—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8—22no—
2692−1+4—62no—
2702−1+4—62no—
2712−1+4—122no—
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—24no—
277C2×#41—24no—
278C2×#41—32no—
279C2×#41—42no—
280C2×#41—42no—
281C2×#41—62no—
282C2×#41—62no—
283C22×Q8—62no—
284C22×Q8C22×D311yescharacteristic
285C22×Q8C22×D311yescharacteristic
286C22×D4—22no—
287C22×D4—21no—
288C22×D4#6712yes—
289C22×D4C22×D311yescharacteristic
290C22×D4C22×D311yescharacteristic
291C23×C4—21no—
292C23×C4—21no—
293C23×C4#6712yes—
294C23×C4C22×D311yescharacteristic
295#137—64no—
296#137—122no—
297AGL(1,ℤ/8)—64no—
298AGL(1,ℤ/8)—122no—
299C2×Q16—62no—
300C2×SD16—62no—
301C2×SD16—62no—
302C2×D8—62no—
303C2×M16—32no—
304C2×M16—32no—
305C4⋊Q8—22no—
306C4⋊Q8#6714yescomplemented
307#127—22no—
308#127—22no—
309#127—62no—
310#127—64no—
311#124—22no—
312#124—22no—
313#124—22no—
314#124—62no—
315#124—64no—
316#124#6714yescomplemented
317#123—62no—
318C22⋊Q8—34no—
319C22⋊Q8—64no—
320#121—216no—
321#121—34no—
322#121—64no—
323#121—64no—
324#121—68no—
325#120—62no—
326#120—64no—
327C4×Q8—64no—
328C4×D4—216no—
329C4×D4—68no—
330C4×D4—64no—
331#117—32no—
332#117—62no—
333C2×C4⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42C22×D311yescharacteristic
340C2×C42C22×D311yescharacteristic
341#104—64no—
342#104—64no—
343#103—64no—
344#102—64no—
345#101—62no—
346#101—62no—
347#100—122no—
348C22⋊C8—64no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—42no—
356C23×C3—44no—
357C23×C3—44no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3#12012yes—
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C3×Q8—24no—
364C3×Q8—24no—
365C3×Q8—24no—
366C3×Q8—42no—
367C3×D4—24no—
368C3×D4—48no—
369C3×D4—416no—
370C3×D4—48no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—42no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—48no—
385C2×C3×C4—48no—
386C2×C3×C4—48no—
387C2×C3×C4—42no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4#12012yes—
391C2×C3×C4#12014yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4H(5,𝔽2)12yes—
396#67—42no—
397#67—42no—
398#67—84no—
399#67—88no—
400C2×Dic3—24no—
401C2×Dic3—22no—
402D12—22no—
403D12—22no—
404D12—44no—
405C4×D3—48no—
406C4×D3—42no—
407C4×D3—42no—
408Dic6—22no—
409Dic6—22no—
410Dic6—44no—
411C3⋊C8—42no—
412C3⋊C8—42no—
413C24—21no—
414C24C2×#6712yes—
415C24C2×#6711yescharacteristic
416#41—48no—
417#41—48no—
418#41—48no—
419#41—48no—
420#41—62no—
421#41—62no—
422#41—62no—
423#41—62no—
424#41—124no—
425#41—124no—
426#41—124no—
427#41—124no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—32no—
434C2×Q8—62no—
435C2×Q8—124no—
436C2×Q8—124no—
437C2×Q8C2×#6712yes—
438C2×Q8C2×#6712yes—
439C2×Q8C2×#6712yes—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—32no—
452C2×D4—42no—
453C2×D4—48no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—62no—
457C2×D4—62no—
458C2×D4—68no—
459C2×D4—64no—
460C2×D4C2×#6712yes—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—64no—
476C22×C4—62no—
477C22×C4D3×D414yes—
478C22×C4C2×#6711yescharacteristic
479C22×C4C2×#6712yes—
480C22×C4C23×D311yescharacteristic
481Q16—124no—
482SD16—124no—
483SD16—124no—
484D8—124no—
485M16—62no—
486M16—62no—
487M16—62no—
488M16—62no—
489C2×C8—62no—
490C2×C8—62no—
491C4⋊C4—28no—
492C4⋊C4—34no—
493C4⋊C4—64no—
494C4⋊C4—64no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—68no—
498C22⋊C4—64no—
499C22⋊C4—62no—
500C22⋊C4—64no—
501C42—24no—
502C42—21no—
503C42—21no—
504C42—64no—
505C42—62no—
506C42C2×#6712yes—
507C42C2×#6712yes—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—24no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—44no—
514C22×C3—44no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—44no—
518C22×C3—82no—
519C22×C3—82no—
520C22×C3—82no—
521C22×C3C2×#12011yescharacteristic
522C22×C3C2×#12011yescharacteristic
523C22×C3#53411yescharacteristic
524C2×D3—42no—
525C2×D3—44no—
526C3×C4—24no—
527C3×C4—24no—
528C3×C4—22no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4D4212yes—
532C3×C4D4212yes—
533Dic3—42no—
534Dic3—44no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—41no—
545C23—41no—
546C23—62no—
547C23#93712yes—
548C23C22×#6711yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—24no—
552Q8—42no—
553Q8—62no—
554Q8—62no—
555Q8—124no—
556D4—24no—
557D4—48no—
558D4—416no—
559D4—48no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—62no—
564D4—62no—
565D4—124no—
566D4—122no—
567D4—122no—
568D4—244no—
569D4—248no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—64no—
587C2×C4—62no—
588C2×C4—128no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4#93712yes—
592C2×C4#95314yes—
593C2×C4C2×D3×D412yes—
594C2×C4#100412yes—
595C2×C4C22×#6711yescharacteristic
596C2×C4C22×#6711yescharacteristic
597C8—122no—
598C8—122no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—42no—
602C2×C3—82no—
603C2×C3—82no—
604C2×C3#240311yescharacteristic
605C2×C3#301812yes—
606D3—82no—
607C22—22no—
608C22—22no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—122no—
621C22—124no—
622C22C2×#93711yescharacteristic
623C22#580411yescharacteristic
624C22C2×#95311yescharacteristic
625C4—24no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—122no—
631C4—124no—
632C4D4×#6712yes—
633C4D4×#6712yes—
634C3#2390911yescharacteristic, complemented, Sylow
635C2—21no—
636C2—21no—
637C2—42no—
638C2—82no—
639C2—82no—
640C2—242no—
641C2#8332212yes—
642C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#532C2
socleC22×C3C2×#120
hypercenterC2×#532D3
hypocenterC3#23909
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. #87274C2semidirect
3. #84081C2semidirect
4. C2×#5095C2semidirect
5. C2×#5062C2semidirect
6. #72137C2semidirect
7. #71784C2semidirect
8. #71769C2semidirect
9. #71731C2semidirect
10. #71724C2semidirect
11. #71532C2semidirect
49. C2×#532D3semidirect
86. C3×C4⋊Q8D4semidirect
92. C3×#124D4semidirect
121. #930D4semidirect
128. #924D4semidirect
306. C4⋊Q8#67semidirect
316. #124#67semidirect
634. C3#23909semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
#872742
#840812
C2×#50952
C2×#50622
#721372
#717842
#717692
#717312
#717242
#715322
#239093