Finite Group Databasebeta

← #10981932

Subgroups — #10981932

Order 768.

2262 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C211yesmaximal, characteristic, complemented
2C2×#5546C211yesmaximal, characteristic
3C3×#2547C211yesmaximal, characteristic, complemented
4#83261C211yesmaximal, characteristic, complemented
5#82959C212yesmaximal
6#82955C212yesmaximal, complemented
7#82388C211yesmaximal, characteristic, complemented
8#77543C212yesmaximal, complemented
9#77537C212yesmaximal, complemented
10#77533C212yesmaximal, complemented
11#77460C212yesmaximal, complemented
12#76615C212yesmaximal, complemented
13#76504C212yesmaximal, complemented
14#73908C214yesmaximal, complemented
15#73905C214yesmaximal, complemented
16#73668C211yesmaximal, characteristic, complemented
17#73643C211yesmaximal, characteristic, complemented
18#35712—31nomaximal, Sylow
19C22×#237C2211yescharacteristic
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#124C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×#946C2212yes—
24C2×Q8×Dic3C2212yes—
25C2×#944C2212yes—
26C2×D4×Dic3C2211yescharacteristic
27C2×C4×#67C2212yes—
28C22×Dic3⋊C4C2212yes—
29#5721—28no—
30#5546—28no—
31C2×D3×C4⋊C4C2211yescharacteristic
32C2×#890C2212yes—
33C2×#889C2214yes—
34C2×#887C2212yes—
35C2×#887C2211yescharacteristic
36C2×#67⋊C4C2212yes—
37C2×#877C2212yes—
38C2×#872C2211yescharacteristic
39C2×C4×Dic6C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#397C2212yes—
42C3×#397C2212yes—
43C3×#391C2212yes—
44C3×#380C2214yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46#5229C2212yes—
47#5221C2212yes—
48#5210C2214yes—
49#4994C2212yes—
50#4992C2212yes—
51#4992C2212yes—
52#4983C2212yes—
53#4983C2212yes—
54#4981C2212yes—
55#4978C2214yes—
56#4972C2212yes—
57#4959C2214yes—
58#4959C2214yes—
59#4951C2214yes—
60#4951C2214yes—
61#4950C2214yes—
62#4949C2212yes—
63#4947C2212yes—
64#4946C2214yes—
65#4946C2214yes—
66Dic3×C22⋊C4C2214yes—
67C4×(C2×D3)⋊C4C2212yes—
68#4936C2211yescharacteristic
69C4×C4⋊Dic3C2212yes—
70C4×C4⋊Dic3C2212yes—
71#4929C2212yes—
72#4677C2211yescharacteristic
73#4673C2211yescharacteristic
74#4668C2214yes—
75#4666C2214yes—
76#4659C2211yescharacteristic
77#4658C2214yes—
78#4657C2214yes—
79#4656C2211yescharacteristic
80#4654C2214yes—
81#4652C2214yes—
82#4652C2214yes—
83#4651C2212yes—
84#4648C2212yes—
85#4647C2214yes—
86C2×#557—31no—
87C2×#520—31no—
88#2633—32no—
89#2621—31no—
90#2611—32no—
91#2599—32no—
92#2580—31no—
93#2547D311yescharacteristic, complemented
94#2473—34no—
95#2460—34no—
96#2446—32no—
97#2432—32no—
98#2398—32no—
99#2396—31no—
100#2367—32no—
101#2360—32no—
102#2356—31no—
103C22×C3×Q8C2312yes—
104C22×C3×D4C2311yescharacteristic
105C23×C3×C4C2312yes—
106C22×#67C2312yes—
107C23×Dic3C2312yes—
108C23×Dic3C2312yes—
109C2×#237—24no—
110C2×#237—24no—
111C2×#237—24no—
112C22×C4×D3C2311yescharacteristic
113C22×C4×D3C2312yes—
114C22×Dic6C2311yescharacteristic
115C22×Dic6C2311yescharacteristic
116C3×C4⋊Q8—24no—
117C3×#124—28no—
118C3×C4×D4—24no—
119C2×C3×C4⋊C4C2311yescharacteristic
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2312yes—
122C2×C3×C4⋊C4C2312yes—
123C2×C3×C22⋊C4C2314yes—
124C2×C3×C22⋊C4C2314yes—
125C2×C3×C22⋊C4C2312yes—
126C2×C3×C42C2312yes—
127C2×C3×C42C2311yescharacteristic
128C2×#217C2314yes—
129C2×#217C2312yes—
130C2×#217C2314yes—
131#946—28no—
132Q8×Dic3—28no—
133#944—28no—
134D4×Dic3—24no—
135C4×#67—28no—
136C2×(C2×D3)⋊C4C2312yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×(C2×D3)⋊C4C2311yescharacteristic
139C2×(C2×D3)⋊C4C2311yescharacteristic
140C2×(C2×D3)⋊C4C2312yes—
141C2×C4⋊Dic3C2314yes—
142C2×C4⋊Dic3C2312yes—
143C2×C4⋊Dic3C2311yescharacteristic
144C2×C4⋊Dic3C2312yes—
145C2×C4⋊Dic3C2312yes—
146C2×C4⋊Dic3C2311yescharacteristic
147C2×Dic3⋊C4—28no—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×Dic3⋊C4C2314yes—
152C2×Dic3⋊C4C2311yescharacteristic
153C2×Dic3⋊C4C2312yes—
154C2×Dic3⋊C4C2312yes—
155C2×Dic3⋊C4C2311yescharacteristic
156C2×Dic3⋊C4C2314yes—
157C2×Dic3⋊C4C2312yes—
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2314yes—
160C2×C4×Dic3C2311yescharacteristic
161C2×C4×Dic3C2312yes—
162C2×C4×Dic3C2314yes—
163C2×C4×Dic3C2312yes—
164D3×C4⋊C4—24no—
165#890—28no—
166#889—216no—
167#887—24no—
168#887—28no—
169#67⋊C4—28no—
170#877—28no—
171#872—24no—
172C4×Dic6—24no—
173C4×Dic6D418yes—
174C3×#95C2314yes—
175C3×#95C2311yescharacteristic
176C3×#95C2311yescharacteristic
177C3×#95C2314yes—
178C3⋊#95C2312yes—
179C3⋊#95C2312yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2312yes—
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2312yes—
189C3⋊#95C2312yes—
190C22×#41—31no—
191#557—68no—
192C2×C4⋊Q8—32no—
193C2×C4⋊Q8C2×D311yescharacteristic
194C2×#125—32no—
195C2×#125—34no—
196C2×#124C2×D312yes—
197C2×C22⋊Q8—32no—
198#520—68no—
199C2×C4×Q8—32no—
200C2×C4×Q8—32no—
201C2×C4×Q8—31no—
202C2×C4×Q8—31no—
203C2×C4×Q8—31no—
204C2×C4×D4—31no—
205C2×C4×D4—32no—
206C2×C4×D4—32no—
207C2×C4×D4C2×D311yescharacteristic
208C2×#117—32no—
209C2×#117—31no—
210C22×C4⋊C4—32no—
211C22×C4⋊C4—31no—
212#400—34no—
213#400—31no—
214#399—32no—
215#399—31no—
216#398—32no—
217#398—32no—
218#398—32no—
219#398—34no—
220#397—32no—
221#397—32no—
222#397C2×D312yes—
223#397C2×D312yes—
224#396—32no—
225#395—31no—
226#394—34no—
227#394—34no—
228#393—32no—
229#393—31no—
230#391—32no—
231#391—32no—
232#391C2×D312yes—
233#390—32no—
234#389—32no—
235#389—34no—
236#388—34no—
237#388—34no—
238#387—34no—
239#387—34no—
240#387—34no—
241#387—34no—
242#385—34no—
243#385—34no—
244#385—34no—
245#385—34no—
246#385—34no—
247#383—32no—
248#382—31no—
249#380C2×D314yes—
250C4×C4⋊C4—32no—
251C4×C4⋊C4—32no—
252C4×C4⋊C4C2×D311yescharacteristic
253C4×C22⋊C4—34no—
254C4×C22⋊C4—32no—
255C4×C22⋊C4—34no—
256#376—32no—
257C24×C3C2412yes—
258C23×D3C2411yescharacteristic
259C2×C3×Q8—24no—
260C2×C3×Q8—24no—
261C2×C3×Q8—24no—
262C2×C3×D4—22no—
263C2×C3×D4—22no—
264C2×C3×D4#4114yes—
265C22×C3×C4—24no—
266C22×C3×C4—44no—
267C22×C3×C4C2412yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2414yes—
271C22×C3×C4C2411yescharacteristic
272C22×C3×C4C2411yescharacteristic
273C22×C3×C4C2411yescharacteristic
274C2×#67—24no—
275C2×#67—24no—
276C2×#67—24no—
277C22×Dic3—24no—
278C22×Dic3—24no—
279C22×Dic3—24no—
280C22×Dic3—24no—
281C22×Dic3—44no—
282C22×Dic3C2414yes—
283C22×Dic3C2411yescharacteristic
284C22×Dic3C2411yescharacteristic
285C22×Dic3C2412yes—
286C22×Dic3C2412yes—
287C22×Dic3C2412yes—
288C22×Dic3C2411yescharacteristic
289C22×Dic3C2412yes—
290#237—232no—
291C2×C4×D3—22no—
292C2×C4×D3—22no—
293C2×C4×D3—24no—
294C2×C4×D3—44no—
295C2×C4×D3#4114yes—
296C2×Dic6—22no—
297C2×Dic6—22no—
298C2×Dic6—22no—
299C2×Dic6—24no—
300C2×Dic6C2×D414yes—
301C3×C4⋊C4—24no—
302C3×C4⋊C4—22no—
303C3×C4⋊C4—24no—
304C3×C4⋊C4—42no—
305C3×C22⋊C4—28no—
306C3×C22⋊C4—28no—
307C3×C22⋊C4—24no—
308C3×C42—24no—
309C3×C42C2×D414yes—
310#217—24no—
311#217—28no—
312#217—44no—
313(C2×D3)⋊C4—28no—
314(C2×D3)⋊C4—22no—
315(C2×D3)⋊C4—22no—
316(C2×D3)⋊C4—24no—
317(C2×D3)⋊C4—42no—
318C4⋊Dic3—22no—
319C4⋊Dic3—24no—
320C4⋊Dic3—28no—
321C4⋊Dic3—24no—
322C4⋊Dic3—42no—
323C4⋊Dic3C2×D414yes—
324Dic3⋊C4—24no—
325Dic3⋊C4—22no—
326Dic3⋊C4—22no—
327Dic3⋊C4—24no—
328Dic3⋊C4—24no—
329Dic3⋊C4—28no—
330Dic3⋊C4—28no—
331Dic3⋊C4—28no—
332Dic3⋊C4—44no—
333Dic3⋊C4C2×D418yes—
334C4×Dic3—24no—
335C4×Dic3—22no—
336C4×Dic3—24no—
337C4×Dic3—28no—
338C4×Dic3—44no—
339C4×Dic3C2×D418yes—
340C2×#41—64no—
341C2×#41—64no—
342C2×#41—64no—
343C22×Q8—31no—
344C22×Q8—31no—
345C22×Q8C22×D312yes—
346C22×D4—32no—
347C22×D4C22×D311yescharacteristic
348C23×C4—31no—
349C23×C4—32no—
350C23×C4—32no—
351C23×C4—32no—
352C23×C4C22×D312yes—
353C4⋊Q8—24no—
354C4⋊Q8—68no—
355#125—68no—
356#125—616no—
357#124—28no—
358C22⋊Q8—68no—
359C4×Q8—38no—
360C4×Q8—68no—
361C4×Q8—64no—
362C4×Q8—68no—
363C4×Q8—64no—
364C4×D4—24no—
365C4×D4—68no—
366C4×D4—68no—
367C4×D4—64no—
368#117—68no—
369#117—64no—
370C2×C4⋊C4—31no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—32no—
373C2×C4⋊C4—31no—
374C2×C4⋊C4—32no—
375C2×C4⋊C4—31no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—31no—
379C2×C4⋊C4—34no—
380C2×C4⋊C4—32no—
381C2×C4⋊C4—34no—
382C2×C4⋊C4—34no—
383C2×C4⋊C4—32no—
384C2×C4⋊C4—32no—
385C2×C4⋊C4—34no—
386C2×C4⋊C4—64no—
387C2×C4⋊C4—68no—
388C2×C4⋊C4C22×D311yescharacteristic
389C2×C4⋊C4C22×D312yes—
390C2×C4⋊C4C22×D312yes—
391C2×C4⋊C4C22×D312yes—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—34no—
394C2×C22⋊C4—31no—
395C2×C22⋊C4—31no—
396C2×C22⋊C4—34no—
397C2×C22⋊C4—32no—
398C2×C22⋊C4—34no—
399C2×C22⋊C4—32no—
400C2×C22⋊C4C22×D312yes—
401C2×C22⋊C4C22×D314yes—
402C2×C22⋊C4C22×D314yes—
403C2×C42—32no—
404C2×C42—31no—
405C2×C42—32no—
406C2×C42—34no—
407C2×C42—32no—
408C2×C42—34no—
409C2×C42C22×D312yes—
410C2×C42C22×D311yescharacteristic
411#95—32no—
412#95—34no—
413#95—32no—
414#95—32no—
415#95—32no—
416#95—34no—
417#95—34no—
418#95—34no—
419#95—32no—
420#95—32no—
421#95—34no—
422#95—34no—
423#95C22×D311yescharacteristic
424#95C22×D314yes—
425#95C22×D311yescharacteristic
426#95C22×D314yes—
427C23×C3—22no—
428C23×C3—22no—
429C23×C3—24no—
430C23×C3—24no—
431C23×C3C2×#4114yes—
432C23×C3C2511yescharacteristic
433C22×D3—21no—
434C22×D3—21no—
435C22×D3—22no—
436C22×D3—22no—
437C22×D3C2×#4112yes—
438C3×Q8—48no—
439C3×D4—28no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—24no—
443C2×C3×C4—22no—
444C2×C3×C4—24no—
445C2×C3×C4—22no—
446C2×C3×C4—22no—
447C2×C3×C4—24no—
448C2×C3×C4—22no—
449C2×C3×C4—22no—
450C2×C3×C4—22no—
451C2×C3×C4—24no—
452C2×C3×C4—22no—
453C2×C3×C4—84no—
454C2×C3×C4C22×D412yes—
455C2×C3×C4C22×D412yes—
456C2×C3×C4C22×D412yes—
457C2×C3×C4C2×#4112yes—
458C2×C3×C4C2×#4112yes—
459#67—216no—
460C2×Dic3—216no—
461C2×Dic3—22no—
462C2×Dic3—21no—
463C2×Dic3—22no—
464C2×Dic3—22no—
465C2×Dic3—21no—
466C2×Dic3—22no—
467C2×Dic3—24no—
468C2×Dic3—21no—
469C2×Dic3—22no—
470C2×Dic3—21no—
471C2×Dic3—24no—
472C2×Dic3—22no—
473C2×Dic3—24no—
474C2×Dic3—24no—
475C2×Dic3—22no—
476C2×Dic3—22no—
477C2×Dic3—22no—
478C2×Dic3—84no—
479C2×Dic3C22×D414yes—
480C2×Dic3C22×D414yes—
481C2×Dic3C2×#4112yes—
482C2×Dic32−1+414yes—
483C4×D3—28no—
484C4×D3—84no—
485Dic6—28no—
486Dic6—28no—
487C24—31no—
488C24C23×D312yes—
489#41—632no—
490C2×Q8—24no—
491C2×Q8—24no—
492C2×Q8—24no—
493C2×Q8—34no—
494C2×Q8—64no—
495C2×Q8—62no—
496C2×Q8—62no—
497C2×Q8—62no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—64no—
501C2×D4—64no—
502C2×D4—64no—
503C2×D4#23714yes—
504C22×C4—24no—
505C22×C4—32no—
506C22×C4—32no—
507C22×C4—31no—
508C22×C4—32no—
509C22×C4—34no—
510C22×C4—34no—
511C22×C4—31no—
512C22×C4—31no—
513C22×C4—32no—
514C22×C4—44no—
515C22×C4—64no—
516C22×C4—64no—
517C22×C4—64no—
518C22×C4—64no—
519C22×C4—64no—
520C22×C4—62no—
521C22×C4—62no—
522C22×C4—124no—
523C22×C4—124no—
524C22×C4C23×D312yes—
525C22×C4C23×D312yes—
526C22×C4C23×D312yes—
527C22×C4C23×D314yes—
528C22×C4C23×D311yescharacteristic
529C22×C4C23×D311yescharacteristic
530C22×C4C23×D311yescharacteristic
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—34no—
535C4⋊C4—38no—
536C4⋊C4—42no—
537C4⋊C4—64no—
538C4⋊C4—64no—
539C4⋊C4—68no—
540C4⋊C4—68no—
541C4⋊C4—68no—
542C4⋊C4—68no—
543C4⋊C4—64no—
544C4⋊C4—62no—
545C4⋊C4—62no—
546C4⋊C4—64no—
547C4⋊C4—62no—
548C4⋊C4—64no—
549C4⋊C4—124no—
550C4⋊C4—122no—
551C22⋊C4—28no—
552C22⋊C4—28no—
553C22⋊C4—24no—
554C22⋊C4—64no—
555C22⋊C4—68no—
556C22⋊C4—64no—
557C22⋊C4—68no—
558C22⋊C4—62no—
559C22⋊C4—62no—
560C22⋊C4—122no—
561C22⋊C4—124no—
562C42—24no—
563C42—38no—
564C42—64no—
565C42—68no—
566C42—62no—
567C42—64no—
568C42—124no—
569C42D3×D414yes—
570C22×C3—24no—
571C22×C3—28no—
572C22×C3—42no—
573C22×C3—42no—
574C22×C3—44no—
575C22×C3C23×D411yescharacteristic
576C22×C3C22×#4111yescharacteristic
577C22×C3C22×#4111yescharacteristic
578C22×C3C2×2−1+412yes—
579C22×C3#58512yes—
580C2×D3—22no—
581C2×D3—24no—
582C2×D3—41no—
583C2×D3—41no—
584C2×D3—42no—
585C3×C4—22no—
586C3×C4—22no—
587C3×C4—44no—
588C3×C4—42no—
589C3×C4—42no—
590C3×C4—42no—
591C3×C4#52514yes—
592Dic3—24no—
593Dic3—22no—
594Dic3—24no—
595Dic3—24no—
596Dic3—41no—
597Dic3—42no—
598Dic3—41no—
599Dic3—44no—
600C23—22no—
601C23—22no—
602C23—24no—
603C23—24no—
604C23—32no—
605C23—61no—
606C23—61no—
607C23—62no—
608C23—62no—
609C23C2×#23714yes—
610C23C24×D311yescharacteristic
611Q8—48no—
612Q8—68no—
613Q8—68no—
614D4—28no—
615D4—616no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—34no—
630C2×C4—34no—
631C2×C4—32no—
632C2×C4—34no—
633C2×C4—616no—
634C2×C4—62no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—62no—
638C2×C4—62no—
639C2×C4—64no—
640C2×C4—62no—
641C2×C4—68no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—62no—
645C2×C4—64no—
646C2×C4—64no—
647C2×C4—62no—
648C2×C4—62no—
649C2×C4—62no—
650C2×C4—64no—
651C2×C4—62no—
652C2×C4—84no—
653C2×C4—244no—
654C2×C4—244no—
655C2×C4C2×D3×D412yes—
656C2×C4C2×D3×D412yes—
657C2×C4C2×D3×D412yes—
658C2×C4C2×#23712yes—
659C2×C4D3×#4112yes—
660C2×C3—44no—
661C2×C3C2×#52511yescharacteristic
662C2×C3#348611yescharacteristic
663C2×C3#350011yescharacteristic
664C2×C3#350712yes—
665C2×C3#351512yes—
666D3—42no—
667C22—28no—
668C22—24no—
669C22—42no—
670C22—42no—
671C22—44no—
672C22—62no—
673C22—64no—
674C22—121no—
675C22—121no—
676C22—122no—
677C22C22×D3×D411yescharacteristic
678C22C22×#23711yescharacteristic
679C22C2×D3×#4111yescharacteristic
680C22#596412yes—
681C22D3×2−1+412yes—
682C4—22no—
683C4—22no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—64no—
689C4—64no—
690C4—62no—
691C4—64no—
692C4—121no—
693C4—122no—
694C4—121no—
695C4—124no—
696C4#560614yes—
697C3#3571211yescharacteristic, complemented, Sylow
698C2—44no—
699C2—122no—
700C2C2×#560611yescharacteristic
701C2#8607611yescharacteristic
702C2#8635511yescharacteristic
703C2#8668212yes—
704C2#8700712yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2547C2
prime radicalC22×#237C22
hypercenter#2547D3
hypocenterC3#35712
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#5721C2semidirect
3. C3×#2547C2semidirect
4. #83261C2semidirect
6. #82955C2semidirect
7. #82388C2semidirect
8. #77543C2semidirect
9. #77537C2semidirect
10. #77533C2semidirect
11. #77460C2semidirect
12. #76615C2semidirect
13. #76504C2semidirect
14. #73908C2semidirect
15. #73905C2semidirect
16. #73668C2semidirect
17. #73643C2semidirect
93. #2547D3semidirect
697. C3#35712semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C2×#55462
C3×#25472
#832612
#829592
#829552
#823882
#775432
#775372
#775332
#774602
#766152
#765042
#739082
#739052
#736682
#736432
#357123