Finite Group Databasebeta

← #10829874

Subgroups — #10829874

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic, complemented
3#83297C211yesmaximal, characteristic, complemented
4#82329C211yesmaximal, characteristic, complemented
5#71856C212yesmaximal, complemented
6#71852C212yesmaximal, complemented
7#71840C211yesmaximal, characteristic, complemented
8#71816C211yesmaximal, characteristic, complemented
9#71667C212yesmaximal, complemented
10#71665C212yesmaximal, complemented
11#71617C211yesmaximal, characteristic, complemented
12#24180—31nomaximal, Sylow
13C3×#559—22no—
14C3×#556—21no—
15C3×#555—21no—
16C3×#553—22no—
17C3×#550—21no—
18C3×#547—21no—
19C3×#538—22no—
20C3×#535C2212yes—
21C3×#533C2211yescharacteristic
22C3×#532C2211yescharacteristic
23C3×#525—22no—
24C3×#520C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C2×#949C2212yes—
27C2×#940C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×#837C2212yes—
30C2×#837C2212yes—
31#5232C2212yes—
32#5221C2211yescharacteristic
33C2×#833C2212yes—
34#5213—22no—
35#5087—21no—
36#5084—21no—
37#5083—21no—
38#5072—21no—
39#5066—22no—
40#5057—22no—
41#5001C2212yes—
42#4995C2212yes—
43#4988C2211yescharacteristic
44#4977C2211yescharacteristic
45#4956—22no—
46#4932C2211yescharacteristic
47#4924C2211yescharacteristic
48#4924C2211yescharacteristic
49#4532C2212yes—
50#4532C2212yes—
51#4530C2212yes—
52#3506D311yescharacteristic, complemented
53#3077—31no—
54#3036—31no—
55#2328—31no—
56#2037—32no—
57#2002—32no—
58#1995—31no—
59#1963—31no—
60#1883—32no—
61#1860—32no—
62#1800—31no—
63C2×C3×#41—22no—
64C2×C3×#41C2312yes—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4D412yes—
67C22×#67—22no—
68C22×D12C2311yescharacteristic
69C22×C4×D3—21no—
70C3×C4⋊Q8D411yescharacteristic, complemented
71C3×C4⋊Q8D411yescharacteristic, complemented
72C3×C4⋊Q8D411yescharacteristic, complemented
73C3×#127—22no—
74C3×#127D411yescharacteristic, complemented
75C3×#126—24no—
76C3×#125D412yescomplemented
77C3×#124—24no—
78C3×#124—22no—
79C3×#124D412yescomplemented
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—21no—
85C3×#121—22no—
86C3×#121—24no—
87C3×#121—21no—
88C3×#121—21no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#120—24no—
92C3×C4×Q8—22no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×D4—24no—
96C3×C4×D4—22no—
97C3×C4×D4—24no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—21no—
104C3×#117—22no—
105C3×#117—22no—
106C3×#117C2312yes—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4D412yes—
110C2×C3×C42C2311yescharacteristic
111C2×#217—22no—
112#949—24no—
113#949—42no—
114C2×#215—22no—
115#940—22no—
116#940—22no—
117C2×#213—22no—
118C2×(C2×D3)⋊C4—21no—
119C2×(C2×D3)⋊C4—22no—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C2×#208C2312yes—
124C2×#208C2312yes—
125C4×D12—21no—
126C4×D12—21no—
127C4×D12—21no—
128C4×D12—21no—
129#837—24no—
130#837—42no—
131#835—22no—
132#833—42no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—22no—
137#809—22no—
138#809—22no—
139#808—22no—
140#807—22no—
141C4⋊(C3⋊C8)—21no—
142C4⋊(C3⋊C8)—21no—
143C4⋊(C3⋊C8)—21no—
144C4⋊(C3⋊C8)—21no—
145C2×AGL(1,ℤ/8)—32no—
146#559—22no—
147#556—21no—
148#555—21no—
149#553—22no—
150#550—21no—
151#547—21no—
152#538—22no—
153#535C2×D312yes—
154#533C2×D311yescharacteristic
155#532C2×D311yescharacteristic
156C2×#127—31no—
157#525—22no—
158#520C2×D312yes—
159C2×C4×D4—31no—
160C2×C4×D4C2×D311yescharacteristic
161#478—62no—
162#476—61no—
163#474—61no—
164#463—62no—
165#461—61no—
166#459—61no—
167#428—32no—
168#423—31no—
169#423—31no—
170C2×#104—32no—
171C2×#104—32no—
172#418—32no—
173#417—32no—
174C2×#100—32no—
175#390—31no—
176#390—31no—
177#386—62no—
178#386—62no—
179#382—31no—
180#376—31no—
181#343—32no—
182#339—32no—
183#339—32no—
184C24×C3C2×D412yes—
185C23×D3—21no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—84no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8C2×D412yes—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—24no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—24no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—82no—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C2×#67—28no—
216C2×#67—82no—
217C2×#67—82no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12#4112yescomplemented
227C2×C4×D3—41no—
228C2×C4×D3—41no—
229C2×C4×D3—82no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4C2×D412yes—
240C3×C4⋊C4C2×D412yes—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C42—24no—
250C3×C42—22no—
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42C2×D411yescharacteristic
255#217—82no—
256#215—44no—
257#213—44no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—41no—
260(C2×D3)⋊C4—82no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3#4112yescomplemented
267#208—22no—
268#208—22no—
269#208—22no—
270#208—22no—
271C2×C3⋊C8—22no—
272C2×C3⋊C8—22no—
273C2×#41—22no—
274C2×#41C22×D312yes—
275C22×D4—31no—
276C22×D4—62no—
277C22×D4C22×D311yescharacteristic
278C23×C4—61no—
279C23×C4#6712yes—
280AGL(1,ℤ/8)—64no—
281AGL(1,ℤ/8)—122no—
282C2×SD16—62no—
283C2×D8—62no—
284C2×M16—32no—
285C2×M16—32no—
286C4⋊Q8#6711yescharacteristic, complemented
287C4⋊Q8#6711yescharacteristic, complemented
288C4⋊Q8#6711yescharacteristic, complemented
289#127—22no—
290#127—62no—
291#127—62no—
292#127#6711yescharacteristic, complemented
293#126—24no—
294#125#6712yescomplemented
295#124—24no—
296#124—22no—
297#124#6712yescomplemented
298#123—24no—
299#123—24no—
300C22⋊Q8—22no—
301C22⋊Q8—21no—
302C22⋊Q8—21no—
303#121—22no—
304#121—24no—
305#121—21no—
306#121—21no—
307#121—22no—
308#121—22no—
309#120—24no—
310C4×Q8—22no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4—24no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326#117—22no—
327#117—22no—
328#117C22×D312yes—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—61no—
333C2×C22⋊C4—62no—
334C2×C22⋊C4—62no—
335C2×C22⋊C4#6712yes—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341C4⋊C8—61no—
342C4⋊C8—61no—
343C4⋊C8—61no—
344C4⋊C8—61no—
345#104—64no—
346#104—122no—
347#103—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351#100—122no—
352#95—61no—
353#95—61no—
354#95—62no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—42no—
360C23×C3—42no—
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C22×D3—42no—
364C22×D3—42no—
365C22×D3—41no—
366C22×D3—41no—
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—42no—
370C3×Q8—81no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—42no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—81no—
381C3×D4—84no—
382C3×D4—84no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—44no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—42no—
393C2×C3×C4—82no—
394C2×C3×C4—84no—
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4#12011yescharacteristic
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4#12011yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4H(5,𝔽2)11yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405#67—88no—
406C2×Dic3—22no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—42no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414D12—22no—
415D12—22no—
416D12—41no—
417D12—41no—
418C4×D3—82no—
419C4×D3—82no—
420C3⋊C8—42no—
421C3⋊C8—42no—
422C24—61no—
423C24C2×#6712yes—
424#41—22no—
425#41—22no—
426#41—22no—
427#41—22no—
428#41—84no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×#6712yes—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—32no—
444C2×D4—68no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—242no—
451C2×D4—242no—
452C2×D4C2×#6712yes—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—82no—
461C22×C4—121no—
462C22×C4—121no—
463C22×C4—242no—
464C22×C4C2×#6712yes—
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—124no—
469D8—124no—
470M16—62no—
471M16—62no—
472M16—62no—
473M16—62no—
474C2×C8—62no—
475C2×C8—62no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4C2×#6712yes—
488C4⋊C4C2×#6712yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—121no—
498C22⋊C4—121no—
499C22⋊C4—242no—
500C22⋊C4—242no—
501C42—24no—
502C42—22no—
503C42—32no—
504C42—61no—
505C42—61no—
506C42—61no—
507C42C2×#6711yescharacteristic
508C42C2×#6711yescharacteristic
509C42C2×#6711yescharacteristic
510C42C2×#6711yescharacteristic
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—44no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—82no—
521C22×C3C2×#12011yescharacteristic
522C22×C3#52511yescharacteristic
523C22×C3#53711yescharacteristic
524C2×D3—44no—
525C2×D3—42no—
526C2×D3—82no—
527C2×D3—81no—
528C2×D3—81no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4#54611yescharacteristic
538C3×C4#54711yescharacteristic
539C3×C4#54811yescharacteristic
540C3×C4#55011yescharacteristic
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—42no—
550C23—62no—
551C23—122no—
552C23—122no—
553C23—121no—
554C23—121no—
555C23C22×#6711yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—42no—
559Q8—81no—
560D4—22no—
561D4—22no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—62no—
570D4—62no—
571D4—81no—
572D4—84no—
573D4—84no—
574D4—121no—
575D4—121no—
576D4—248no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—62no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—122no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—242no—
598C2×C4—242no—
599C2×C4#95311yescharacteristic
600C2×C4#95311yescharacteristic
601C2×C4#95311yescharacteristic
602C2×C4#95311yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C2×C4#100911yescharacteristic
606C2×C4#101011yescharacteristic
607C2×C4C22×#6711yescharacteristic
608C2×C4C22×#6711yescharacteristic
609C8—122no—
610C8—122no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—82no—
614C2×C3—82no—
615C2×C3#240711yescharacteristic
616C2×C3#301911yescharacteristic
617C2×C3#302111yescharacteristic
618D3—82no—
619C22—21no—
620C22—21no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—82no—
629C22—124no—
630C22—122no—
631C22—242no—
632C22—241no—
633C22—241no—
634C22#582811yescharacteristic
635C22#583311yescharacteristic
636C22C2×#95311yescharacteristic
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—122no—
646C4—241no—
647C4—241no—
648C4#582411yescharacteristic
649C4#582911yescharacteristic
650C4#583111yescharacteristic
651C4#583411yescharacteristic
652C3#2418011yescharacteristic, complemented, Sylow
653C2—21no—
654C2—21no—
655C2—82no—
656C2—82no—
657C2—242no—
658C2#8409711yescharacteristic
659C2#8410411yescharacteristic
660C2#8412211yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3506C2
socleC22×C3C2×#120
hypercenter#3506D3
hypocenterC3#24180
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. C3×#3506C2semidirect
2. #83350C2semidirect
3. #83297C2semidirect
4. #82329C2semidirect
5. #71856C2semidirect
6. #71852C2semidirect
7. #71840C2semidirect
8. #71816C2semidirect
9. #71667C2semidirect
10. #71665C2semidirect
11. #71617C2semidirect
52. #3506D3semidirect
70. C3×C4⋊Q8D4semidirect
71. C3×C4⋊Q8D4semidirect
72. C3×C4⋊Q8D4semidirect
74. C3×#127D4semidirect
76. C3×#125D4semidirect
79. C3×#124D4semidirect
226. C2×D12#41semidirect
266. C4×Dic3#41semidirect
286. C4⋊Q8#67semidirect
287. C4⋊Q8#67semidirect
288. C4⋊Q8#67semidirect
292. #127#67semidirect
294. #125#67semidirect
297. #124#67semidirect
652. C3#24180semidirect

Maximal subgroups

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

subgroupindex
C3×#35062
#833502
#832972
#823292
#718562
#718522
#718402
#718162
#716672
#716652
#716172
#241803