Finite Group Databasebeta

← #11233303

Subgroups — #11233303

Order 768.

2282 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×#5721C211yesmaximal, characteristic, complemented
3#84093C212yesmaximal, complemented
4#84089C212yesmaximal, complemented
5#83227C211yesmaximal, characteristic, complemented
6#82876C214yesmaximal, complemented
7#82851C214yesmaximal, complemented
8#82758C212yesmaximal, complemented
9#82754C212yesmaximal, complemented
10#82695C214yesmaximal, complemented
11#82388C211yesmaximal, characteristic, complemented
12#82375C211yesmaximal, characteristic, complemented
13#82348C211yesmaximal, characteristic, complemented
14#82328C211yesmaximal, characteristic, complemented
15#82174C214yesmaximal, complemented
16#45335—31nomaximal, Sylow
17C22×C3×#41C2212yes—
18C3×#547—28no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#123C2214yes—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×#121C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C22×C3×C4⋊C4C2212yes—
26C2×Q8×Dic3C2212yes—
27C2×Q8×Dic3C2212yes—
28C2×#944C2212yes—
29C2×#936C2211yescharacteristic
30C2×#924C2212yes—
31#5721—28no—
32C2×#890C2212yes—
33C2×#889C2214yes—
34C2×#887C2212yes—
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37#5229C2212yes—
38#5222C2212yes—
39#5222C2212yes—
40#5222C2214yes—
41#5222C2214yes—
42#5221C2214yes—
43#5221C2214yes—
44#5221C2211yescharacteristic
45#5221C2212yes—
46#5212C2212yes—
47#5212C2214yes—
48#5210C2214yes—
49#5210C2214yes—
50#5210C2214yes—
51#5210C2214yes—
52C4×#217C2212yes—
53C4×#217C2212yes—
54C4×#217C2214yes—
55#4985C2212yes—
56#4983C2212yes—
57#4982C2214yes—
58#4981C2214yes—
59#4980C2214yes—
60#4980C2214yes—
61#4978C2214yes—
62#4976C2214yes—
63#4975C2211yescharacteristic
64#4973C2211yescharacteristic
65#4972C2212yes—
66#4953C2212yes—
67#4949C2214yes—
68#4949C2212yes—
69#4948C2214yes—
70#4947C2212yes—
71#4947C2212yes—
72#4945C2214yes—
73#4944C2214yes—
74#4944C2214yes—
75#4944C2214yes—
76#4936C2211yescharacteristic
77#4932C2211yescharacteristic
78C2×#557—31no—
79C2×#547D311yescharacteristic
80#2711—34no—
81#2694—34no—
82#2580—31no—
83#2536—31no—
84#2473—34no—
85#2458—34no—
86#2432—32no—
87#2429—32no—
88#2423—32no—
89#2398—31no—
90#2341—32no—
91#2327—31no—
92#2326—31no—
93C2×C3×#41—28no—
94C2×C3×#41—28no—
95C2×C3×#41—44no—
96C22×C3×Q8C2312yes—
97C22×C3×D4C2314yes—
98C22×C3×D4C2311yescharacteristic
99C22×C3×D4C2311yescharacteristic
100C23×C3×C4C2312yes—
101C23×C3×C4C2314yes—
102C23×C3×C4C2312yes—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2311yescharacteristic
105C3×C4⋊Q8D418yes—
106C3×#123—216no—
107C3×C22⋊Q8—28no—
108C3×#121—28no—
109C3×C4×D4—24no—
110C3×C4×D4—24no—
111C2×C3×C4⋊C4—28no—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2314yes—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2314yes—
120C2×C3×C42C2311yescharacteristic
121C2×#217C2312yes—
122C2×#217C2314yes—
123C2×#217C2312yes—
124C2×#217C2314yes—
125C2×#217C2314yes—
126C2×#217C2314yes—
127C2×#217C2314yes—
128C2×#217C2312yes—
129C2×#217C2312yes—
130C2×#217C2314yes—
131Q8×Dic3—28no—
132Q8×Dic3—44no—
133#944—28no—
134#936—24no—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2314yes—
138C2×C4⋊Dic3C2311yescharacteristic
139C2×C4⋊Dic3C2314yes—
140#924—44no—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2314yes—
148C2×C4×Dic3C2311yescharacteristic
149C2×C4×Dic3C2311yescharacteristic
150C2×C4×Dic3C2312yes—
151C2×C4×Dic3C2314yes—
152C2×C4×Dic3C2312yes—
153C2×C4×Dic3C2314yes—
154#890—28no—
155#889—216no—
156#887—28no—
157C4×Dic6—24no—
158C4×Dic6—24no—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2314yes—
170C3⋊#95C2312yes—
171C3⋊#95C2312yes—
172C22×#41C2×D312yes—
173#557—68no—
174#547—28no—
175C2×C4⋊Q8—32no—
176C2×C4⋊Q8C2×D311yescharacteristic
177C2×#125—34no—
178C2×#125—32no—
179C2×#124—31no—
180C2×#123C2×D314yes—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8C2×D312yes—
183C2×#121C2×D312yes—
184C2×C4×Q8—32no—
185C2×C4×Q8—32no—
186C2×C4×Q8—32no—
187C2×C4×Q8—31no—
188C2×C4×Q8—31no—
189C2×C4×D4C2×D311yescharacteristic
190C2×C4×D4C2×D311yescharacteristic
191C22×C4⋊C4C2×D312yes—
192#400—32no—
193#399—32no—
194#399—34no—
195#398—34no—
196#398—34no—
197#398—32no—
198#397—32no—
199#396—34no—
200#394—32no—
201#394—34no—
202#394—34no—
203#394—32no—
204#393—32no—
205#393—32no—
206#391—32no—
207#391—31no—
208#390—32no—
209#390—31no—
210#390—34no—
211#390—34no—
212#389—32no—
213#389—34no—
214#388—34no—
215#388—34no—
216#388—34no—
217#388—34no—
218#387—34no—
219#387—34no—
220#387—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#384—31no—
226#382—31no—
227#380—32no—
228#380—34no—
229C4×C22⋊C4—32no—
230C4×C22⋊C4—34no—
231C4×C22⋊C4—32no—
232#376—31no—
233C24×C3C2412yes—
234C24×C3C2412yes—
235C3×#41—432no—
236C2×C3×Q8—24no—
237C2×C3×Q8—24no—
238C2×C3×Q8C2×D418yes—
239C2×C3×D4—28no—
240C2×C3×D4—22no—
241C2×C3×D4—22no—
242C2×C3×D4—28no—
243C2×C3×D4—22no—
244C2×C3×D4—42no—
245C2×C3×D4—44no—
246C2×C3×D4#4114yes—
247C22×C3×C4—28no—
248C22×C3×C4—24no—
249C22×C3×C4—28no—
250C22×C3×C4—24no—
251C22×C3×C4—24no—
252C22×C3×C4—24no—
253C22×C3×C4—44no—
254C22×C3×C4—44no—
255C22×C3×C4C2412yes—
256C22×C3×C4C2412yes—
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2411yescharacteristic
259C22×C3×C4C2414yes—
260C22×C3×C4C2411yescharacteristic
261C22×Dic3C2412yes—
262C22×Dic3C2412yes—
263C22×Dic3C2414yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C22×Dic3C2412yes—
267C22×Dic3C2412yes—
268C2×Dic6—22no—
269C2×Dic6—24no—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6—22no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—28no—
275C3×C4⋊C4—22no—
276C3×C4⋊C4C2×D418yes—
277C3×C4⋊C4C2×D418yes—
278C3×C22⋊C4—24no—
279C3×C22⋊C4—24no—
280C3×C22⋊C4—28no—
281C3×C42C2×D414yes—
282#217—28no—
283#217—28no—
284#217—24no—
285#217—24no—
286#217—44no—
287#217—42no—
288#217—42no—
289#217—44no—
290#217—44no—
291#217—44no—
292C4⋊Dic3—22no—
293C4⋊Dic3—22no—
294C4⋊Dic3—28no—
295C4⋊Dic3—24no—
296C4⋊Dic3—44no—
297Dic3⋊C4—24no—
298Dic3⋊C4—24no—
299Dic3⋊C4—28no—
300Dic3⋊C4—28no—
301Dic3⋊C4—44no—
302Dic3⋊C4—44no—
303Dic3⋊C4—44no—
304C4×Dic3—24no—
305C4×Dic3—24no—
306C4×Dic3—28no—
307C4×Dic3—22no—
308C4×Dic3—22no—
309C4×Dic3—44no—
310C2×#41—28no—
311C2×#41—28no—
312C2×#41—44no—
313C22×Q8—31no—
314C22×Q8—31no—
315C22×Q8C22×D312yes—
316C22×D4C22×D314yes—
317C22×D4C22×D311yescharacteristic
318C22×D4C22×D311yescharacteristic
319C23×C4C22×D312yes—
320C23×C4C22×D314yes—
321C23×C4C22×D312yes—
322C4⋊Q8—68no—
323C4⋊Q8#6718yes—
324#125—616no—
325#125—68no—
326#124—64no—
327#123—216no—
328C22⋊Q8—28no—
329C22⋊Q8—124no—
330#121—28no—
331C4×Q8—68no—
332C4×Q8—68no—
333C4×Q8—64no—
334C4×Q8—64no—
335C4×Q8—124no—
336C4×D4—24no—
337C4×D4—24no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C4⋊C4C22×D314yes—
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D312yes—
355C2×C4⋊C4C22×D312yes—
356C2×C22⋊C4—34no—
357C2×C22⋊C4—34no—
358C2×C22⋊C4—32no—
359C2×C22⋊C4—32no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4C22×D312yes—
367C2×C22⋊C4C22×D312yes—
368C2×C22⋊C4C22×D314yes—
369C2×C42—34no—
370C2×C42—32no—
371C2×C42—32no—
372C2×C42—31no—
373C2×C42—31no—
374C2×C42—34no—
375C2×C42C22×D311yescharacteristic
376#95—32no—
377#95—32no—
378#95—32no—
379#95—34no—
380#95—32no—
381#95—32no—
382#95—32no—
383#95—32no—
384#95—32no—
385#95—34no—
386#95—34no—
387#95—34no—
388#95—34no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—24no—
392C23×C3—24no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—22no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3C2×#4114yes—
401C23×C3C2511yescharacteristic
402C3×Q8—216no—
403C3×D4—28no—
404C3×D4—416no—
405C3×D4—44no—
406C2×C3×C4—24no—
407C2×C3×C4—24no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—24no—
415C2×C3×C4—416no—
416C2×C3×C4—48no—
417C2×C3×C4—48no—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D418yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C2×#4112yes—
423C2×C3×C4H(5,𝔽2)12yes—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—24no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—24no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—24no—
440C2×Dic3—24no—
441C2×Dic3—22no—
442C2×Dic3—22no—
443C2×Dic32−1+414yes—
444Dic6—28no—
445Dic6—44no—
446C24C23×D312yes—
447C24C23×D312yes—
448#41—432no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—64no—
452C2×Q8—62no—
453C2×Q8—62no—
454C2×Q8—62no—
455C2×Q8—62no—
456C2×Q8C2×#6718yes—
457C2×D4—28no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—28no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4#23714yes—
465C22×C4—28no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—34no—
475C22×C4—32no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4C23×D312yes—
481C22×C4C23×D312yes—
482C22×C4C23×D311yescharacteristic
483C22×C4C23×D311yescharacteristic
484C22×C4C23×D314yes—
485C22×C4C23×D311yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—28no—
488C4⋊C4—22no—
489C4⋊C4—64no—
490C4⋊C4—62no—
491C4⋊C4—64no—
492C4⋊C4—64no—
493C4⋊C4—68no—
494C4⋊C4—62no—
495C4⋊C4—68no—
496C4⋊C4—68no—
497C4⋊C4—124no—
498C4⋊C4—124no—
499C4⋊C4—124no—
500C4⋊C4—124no—
501C4⋊C4C2×#6718yes—
502C4⋊C4C2×#6718yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—28no—
506C22⋊C4—68no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—68no—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—124no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—124no—
516C42—62no—
517C42—62no—
518C42—68no—
519C42—64no—
520C42—64no—
521C42—124no—
522C42C2×#6714yes—
523C22×C3—24no—
524C22×C3—28no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—44no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3C23×D411yescharacteristic
536C22×C3C22×#4111yescharacteristic
537C22×C3C2×H(5,𝔽2)11yescharacteristic
538C22×C3C2×2−1+412yes—
539C22×C3#58512yes—
540C3×C4—28no—
541C3×C4—22no—
542C3×C4—22no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4#54614yes—
546Dic3—24no—
547Dic3—42no—
548Dic3—42no—
549Dic3—42no—
550Dic3—42no—
551Dic3—42no—
552Dic3—44no—
553C23—22no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23C2×#23714yes—
565C23C24×D311yescharacteristic
566Q8—216no—
567Q8—68no—
568Q8—124no—
569D4—28no—
570D4—416no—
571D4—44no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—34no—
582C2×C4—416no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—64no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4C2×#23712yes—
605C2×C4#100412yes—
606C2×C4C22×#6712yes—
607C2×C4C22×#6718yes—
608C2×C4C22×#6712yes—
609C2×C4C22×#6712yes—
610C2×C3—44no—
611C2×C3—82no—
612C2×C3C2×#54611yescharacteristic
613C2×C3#348611yescharacteristic
614C2×C3#348711yescharacteristic
615C2×C3#351512yes—
616C2×C3#355412yes—
617C22—28no—
618C22—24no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22C22×#23711yescharacteristic
630C22C2×#100411yescharacteristic
631C22#596412yes—
632C22D3×2−1+412yes—
633C22C23×#6711yescharacteristic
634C4—28no—
635C4—22no—
636C4—22no—
637C4—42no—
638C4—42no—
639C4—64no—
640C4—122no—
641C4—122no—
642C4—124no—
643C4—122no—
644C4—122no—
645C4—122no—
646C4#580614yes—
647C3#4533511yescharacteristic, complemented, Sylow
648C2—44no—
649C2—82no—
650C2#8668212yes—
651C2#8669412yes—
652C2C2×#580611yescharacteristic
653C2#8727411yescharacteristic
654C2#8729411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC2×C3×#547C2
hypercenterC2×#547D3
hypocenterC3#45335
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#5721C2semidirect
3. #84093C2semidirect
4. #84089C2semidirect
5. #83227C2semidirect
6. #82876C2semidirect
7. #82851C2semidirect
8. #82758C2semidirect
9. #82754C2semidirect
10. #82695C2semidirect
11. #82388C2semidirect
12. #82375C2semidirect
13. #82348C2semidirect
14. #82328C2semidirect
15. #82174C2semidirect
647. C3#45335semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×#57212
#840932
#840892
#832272
#828762
#828512
#827582
#827542
#826952
#823882
#823752
#823482
#823282
#821742
#453353