Finite Group Databasebeta

← #10829767

Subgroups — #10829767

Order 768.

1106 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic, complemented
3#83260C211yesmaximal, characteristic, complemented
4C4×#949C211yesmaximal, characteristic, complemented
5#71879C212yesmaximal, complemented
6#71877C212yesmaximal, complemented
7#71818C211yesmaximal, characteristic, complemented
8#71790C211yesmaximal, characteristic, complemented
9#71667C212yesmaximal, complemented
10#71665C212yesmaximal, complemented
11#71646C211yesmaximal, characteristic, complemented
12#24171—31nomaximal, Sylow
13C3×#556—21no—
14C3×#555—21no—
15C3×#554—21no—
16C3×#553—22no—
17C3×#548—22no—
18C3×#547—21no—
19C3×#542—22no—
20C3×#537C2212yes—
21C3×#533—21no—
22C3×#532—21no—
23C3×#532C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#520C2212yes—
26C3×C4×#41C2211yescharacteristic
27C2×#949C2211yescharacteristic
28C2×#949C2212yes—
29C2×C4×D12C2211yescharacteristic
30C2×#837C2212yes—
31C2×#837C2212yes—
32C2×#837C2212yes—
33#5232C2211yescharacteristic
34#5232C2212yes—
35#5231—22no—
36#5083—21no—
37#5072—21no—
38#5057—22no—
39#5026—21no—
40C4×#215—21no—
41#5014—21no—
42C4×#213—21no—
43#5001C2211yescharacteristic
44#4995C2211yescharacteristic
45#4988C2212yes—
46#4987—22no—
47#4977C2212yes—
48C4×C4⋊Dic3C2211yescharacteristic
49#4924C2211yescharacteristic
50C4×#208C2211yescharacteristic
51#4532C2212yes—
52#4530C2212yes—
53#4530C2212yes—
54#3511D311yescharacteristic, complemented
55#3036—31no—
56C4×AGL(1,ℤ/8)—31no—
57#2563—31no—
58#2040—32no—
59#2006—32no—
60#1997—31no—
61#1980—31no—
62#1883—32no—
63#1860—32no—
64#1806—31no—
65C2×C3×#41—22no—
66C2×C3×#41C2311yescharacteristic
67C2×C3×#41C2312yes—
68C22×C3×Q8D412yes—
69C22×D12C2311yescharacteristic
70C22×C4×D3—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8D411yescharacteristic
76C3×#127D411yescharacteristic
77C3×#126—24no—
78C3×#125—22no—
79C3×#124—22no—
80C3×#124—24no—
81C3×#124D412yes—
82C3×#123—24no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—21no—
91C3×#121—21no—
92C3×#121—24no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8D411yescharacteristic
96C3×C4×Q8D411yescharacteristic
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—24no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×C4×D4D411yescharacteristic
105C3×C4×D4D411yescharacteristic
106C3×#117—22no—
107C3×#117—22no—
108C3×#117—21no—
109C3×#117C2311yescharacteristic
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4D412yes—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115#949—22no—
116#949—24no—
117#949—22no—
118#949—42no—
119C2×#215—21no—
120C2×#215—22no—
121C2×#213—21no—
122C2×#213—22no—
123C2×(C2×D3)⋊C4—21no—
124C2×(C2×D3)⋊C4—24no—
125C2×C4⋊Dic3—21no—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4×Dic3C2312yes—
128C2×#208C2311yescharacteristic
129C2×#208C2312yes—
130C2×#208C2311yescharacteristic
131C4×D12—21no—
132C4×D12—21no—
133C4×D12—21no—
134C4×D12—21no—
135#837—24no—
136#837—42no—
137#837—42no—
138#835—21no—
139#835—22no—
140#832—21no—
141#832—22no—
142C3⋊#95—21no—
143C3⋊#95—24no—
144#809—21no—
145#809—21no—
146#808—21no—
147#807—21no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)—21no—
150#803—21no—
151C4×C3⋊C8—21no—
152C2×AGL(1,ℤ/8)—31no—
153C2×AGL(1,ℤ/8)—32no—
154#556—21no—
155#555—21no—
156#554—21no—
157#553—22no—
158#548—22no—
159#547—21no—
160#542—22no—
161#537C2×D312yes—
162#533—21no—
163#532—21no—
164#532C2×D311yescharacteristic
165C2×C4⋊Q8C2×D311yescharacteristic
166#520C2×D312yes—
167C4×#41C2×D311yescharacteristic
168C2×C4×D4—31no—
169#463—62no—
170#461—61no—
171#459—61no—
172#442—61no—
173#440—61no—
174C4×SD16—61no—
175C4×D8—61no—
176#428—31no—
177#423—31no—
178C2×#104—32no—
179C2×#104—32no—
180C2×#104—32no—
181#418—31no—
182#417—32no—
183#417—31no—
184C4×M16—31no—
185#393—62no—
186#390—32no—
187#382—32no—
188#380—62no—
189C4×C4⋊C4—31no—
190#343—32no—
191#343—32no—
192#339—32no—
193C23×D3—21no—
194C3×#41—22no—
195C3×#41—22no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—22no—
200C3×#41—22no—
201C3×#41—21no—
202C3×#41—84no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—82no—
206C2×C3×Q8C2×D412yes—
207C2×C3×Q8C2×D411yescharacteristic
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—24no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4C2×D412yes—
214C2×C3×D4C2×D411yescharacteristic
215C22×C3×C4—24no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—22no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D414yes—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—21no—
225C22×Dic3—22no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12#4112yescomplemented
232C2×C4×D3—44no—
233C2×C4×D3—41no—
234C2×C4×D3—41no—
235C3×C4⋊C4—24no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4C2×D411yescharacteristic
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C42—24no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42C2×D411yescharacteristic
260C3×C42C2×D411yescharacteristic
261C3×C42C2×D411yescharacteristic
262C3×C42C2×D411yescharacteristic
263#215—24no—
264#215—44no—
265#213—24no—
266#213—44no—
267(C2×D3)⋊C4—41no—
268(C2×D3)⋊C4—41no—
269(C2×D3)⋊C4—84no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4×Dic3—22no—
274C4×Dic3—22no—
275#208—21no—
276#208—22no—
277#208—21no—
278#208—22no—
279#208—21no—
280#208#4112yescomplemented
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—22no—
284C2×#41—22no—
285C2×#41C22×D311yescharacteristic
286C2×#41C22×D312yes—
287C22×Q8#6712yes—
288C22×D4—31no—
289C23×C4—61no—
290AGL(1,ℤ/8)—62no—
291AGL(1,ℤ/8)—62no—
292AGL(1,ℤ/8)—64no—
293AGL(1,ℤ/8)—122no—
294C2×SD16—61no—
295C2×SD16—62no—
296C2×D8—61no—
297C2×D8—62no—
298C2×M16—31no—
299C2×M16—32no—
300C2×M16—31no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8#6711yescharacteristic
306#127#6711yescharacteristic
307#126—24no—
308#125—22no—
309#124—22no—
310#124—24no—
311#124#6712yes—
312#123—24no—
313C22⋊Q8—22no—
314C22⋊Q8—24no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317#121—22no—
318#121—22no—
319#121—22no—
320#121—21no—
321#121—21no—
322#121—24no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8#6711yescharacteristic
326C4×Q8#6711yescharacteristic
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—24no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—61no—
335C4×D4—61no—
336C4×D4—61no—
337C4×D4—61no—
338C4×D4#6711yescharacteristic
339C4×D4#6711yescharacteristic
340#117—22no—
341#117—22no—
342#117—21no—
343#117C22×D311yescharacteristic
344C2×C4⋊C4—22no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—61no—
347C2×C4⋊C4#6712yes—
348C2×C4⋊C4C22×D312yes—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—64no—
351C2×C42—21no—
352C2×C42—32no—
353C2×C42C22×D311yescharacteristic
354#107—61no—
355#106—61no—
356C4⋊C8—61no—
357C4⋊C8—61no—
358#104—64no—
359#104—122no—
360#104—122no—
361#103—61no—
362#103—62no—
363#102—61no—
364#102—62no—
365#102—61no—
366#102—61no—
367#97—61no—
368C4×C8—61no—
369#95—61no—
370#95—64no—
371C23×C3—21no—
372C23×C3—22no—
373C23×C3C22×D411yescharacteristic
374C22×D3—22no—
375C22×D3—42no—
376C22×D3—42no—
377C22×D3—41no—
378C22×D3—41no—
379C3×Q8—21no—
380C3×Q8—21no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8—44no—
384C3×Q8—82no—
385C3×D4—21no—
386C3×D4—21no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—42no—
390C3×D4—42no—
391C3×D4—41no—
392C3×D4—42no—
393C3×D4—41no—
394C3×D4—42no—
395C3×D4—84no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—24no—
402C2×C3×C4—41no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—42no—
409C2×C3×C4—44no—
410C2×C3×C4—84no—
411C2×C3×C4#12011yescharacteristic
412C2×C3×C4#12011yescharacteristic
413C2×C3×C4#12011yescharacteristic
414C2×C3×C4#12011yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4H(5,𝔽2)11yescharacteristic
420C2×C3×C42−1+411yescharacteristic
421C2×Dic3—42no—
422C2×Dic3—42no—
423C2×Dic3—42no—
424C2×Dic3—41no—
425C2×Dic3—42no—
426D12—22no—
427D12—22no—
428D12—41no—
429D12—41no—
430C4×D3—82no—
431C4×D3—82no—
432C3⋊C8—22no—
433C3⋊C8—41no—
434C3⋊C8—42no—
435C24—61no—
436#41—22no—
437#41—21no—
438#41—22no—
439#41—21no—
440#41—21no—
441#41—22no—
442#41—21no—
443#41—22no—
444#41—84no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—82no—
448C2×Q8C2×#6712yes—
449C2×Q8C2×#6711yescharacteristic
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—32no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4C2×#6712yes—
462C2×D4C2×#6711yescharacteristic
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—61no—
469C22×C4—62no—
470C22×C4—124no—
471C22×C4—121no—
472C22×C4—121no—
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6714yes—
475C22×C4C2×#6711yescharacteristic
476C22×C4C23×D311yescharacteristic
477SD16—64no—
478SD16—124no—
479D8—64no—
480D8—124no—
481M16—32no—
482M16—61no—
483M16—62no—
484M16—61no—
485M16—62no—
486M16—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—62no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—122no—
503C4⋊C4C2×#6711yescharacteristic
504C4⋊C4C2×#6711yescharacteristic
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—121no—
514C22⋊C4—121no—
515C22⋊C4—244no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—62no—
521C42—62no—
522C42C2×#6711yescharacteristic
523C42C2×#6711yescharacteristic
524C42C2×#6711yescharacteristic
525C42C2×#6711yescharacteristic
526C22×C3—21no—
527C22×C3—21no—
528C22×C3—42no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—41no—
533C22×C3—42no—
534C22×C3C2×#12011yescharacteristic
535C22×C3#54811yescharacteristic
536C22×C3#55011yescharacteristic
537C2×D3—44no—
538C2×D3—42no—
539C2×D3—82no—
540C2×D3—81no—
541C2×D3—81no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—42no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—44no—
550C3×C4#45412yes—
551C3×C4#52511yescharacteristic
552C3×C4#53711yescharacteristic
553C3×C4#54611yescharacteristic
554C3×C4#54711yescharacteristic
555Dic3—82no—
556Dic3—81no—
557C23—21no—
558C23—22no—
559C23—62no—
560C23—122no—
561C23—122no—
562C23—121no—
563C23—121no—
564C23C22×#6711yescharacteristic
565Q8—22no—
566Q8—21no—
567Q8—22no—
568Q8—21no—
569Q8—44no—
570Q8—82no—
571D4—22no—
572D4—21no—
573D4—22no—
574D4—21no—
575D4—42no—
576D4—41no—
577D4—42no—
578D4—42no—
579D4—41no—
580D4—42no—
581D4—62no—
582D4—62no—
583D4—84no—
584D4—121no—
585D4—121no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—84no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—121no—
606C2×C4—242no—
607C2×C4—242no—
608C2×C4#95311yescharacteristic
609C2×C4#95311yescharacteristic
610C2×C4#95311yescharacteristic
611C2×C4#95311yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4#100911yescharacteristic
615C2×C4#101011yescharacteristic
616C2×C4C22×#6711yescharacteristic
617C2×C4C22×#6711yescharacteristic
618C8—62no—
619C8—122no—
620C8—121no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—81no—
624C2×C3—82no—
625C2×C3#240711yescharacteristic
626C2×C3C2×#45411yescharacteristic
627C2×C3#301911yescharacteristic
628D3—82no—
629C22—21no—
630C22—21no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—124no—
638C22—122no—
639C22—242no—
640C22—241no—
641C22—241no—
642C22#583111yescharacteristic
643C22#583411yescharacteristic
644C22C2×#95311yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—44no—
652C4—41no—
653C4—242no—
654C4—241no—
655C4#524512yes—
656C4#582411yescharacteristic
657C4#582811yescharacteristic
658C4#582911yescharacteristic
659C4#583311yescharacteristic
660C3#2417111yescharacteristic, complemented, Sylow
661C2—21no—
662C2—21no—
663C2—81no—
664C2—82no—
665C2—242no—
666C2#8409711yescharacteristic
667C2#8410411yescharacteristic
668C2C2×#524511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3511C2
socleC22×C3C2×#120
hypercenter#3511D3
hypocenterC3#24171
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. C3×#3511C2semidirect
2. #83297C2semidirect
3. #83260C2semidirect
4. C4×#949C2semidirect
5. #71879C2semidirect
6. #71877C2semidirect
7. #71818C2semidirect
8. #71790C2semidirect
9. #71667C2semidirect
10. #71665C2semidirect
11. #71646C2semidirect
54. #3511D3semidirect
231. C2×D12#41semidirect
280. #208#41semidirect
660. C3#24171semidirect

Maximal subgroups

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

subgroupindex
C3×#35112
#832972
#832602
C4×#9492
#718792
#718772
#718182
#717902
#716672
#716652
#716462
#241713