Finite Group Databasebeta

← #10835943

Subgroups — #10835943

Order 768.

870 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5162C211yesmaximal, characteristic, complemented
2C2×#4690C211yesmaximal, characteristic, complemented
3#76641C211yesmaximal, characteristic, complemented
4#76430C211yesmaximal, characteristic
5C3×#1968C211yesmaximal, characteristic, complemented
6#72054C211yesmaximal, characteristic
7#70582C211yesmaximal, characteristic, complemented
8#70564C211yesmaximal, characteristic, complemented
9#70275C211yesmaximal, characteristic, complemented
10#70229C211yesmaximal, characteristic, complemented
11#70205C211yesmaximal, characteristic
12#70154C211yesmaximal, characteristic, complemented
13#69767C211yesmaximal, characteristic, complemented
14#69675C211yesmaximal, characteristic, complemented
15#69649C211yesmaximal, characteristic, complemented
16#24666—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#938C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C2×#897C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5162—24no—
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4987C2211yescharacteristic
42#4977C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4958—21no—
47#4957—21no—
48#4946—21no—
49#4944—21no—
50#4929C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×#803C2211yescharacteristic
53#4690—24no—
54#4675—21no—
55#4675—21no—
56#4657—21no—
57#4657—21no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#461—31no—
63C2×#433—31no—
64#2621—31no—
65#2324—31no—
66#2036—31no—
67#2035—31no—
68#2000—31no—
69#2000—31no—
70#1981—31no—
71#1973—31no—
72#1968D311yescharacteristic, complemented
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1804—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C23×Dic3—21no—
81C22×C4×D3C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—41no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#946—41no—
96#946—41no—
97#944—22no—
98#944—22no—
99#938—22no—
100#938—22no—
101D4×Dic3—22no—
102D4×Dic3—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×#214—41no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4D411yescharacteristic
110C2×(C2×D3)⋊C4D411yescharacteristic
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4D411yescharacteristic
120C2×Dic3⋊C4D411yescharacteristic
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#897—41no—
126#897—41no—
127#872—22no—
128#872—22no—
129C3×#106—41no—
130C3×#103—41no—
131C3×#102—41no—
132C3×C4×C8—22no—
133C3×#95—21no—
134C3×#95—21no—
135#835—22no—
136#835—22no—
137#832—41no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145(C2×D3)⋊C8—22no—
146(C2×D3)⋊C8—22no—
147#816—22no—
148#816—22no—
149Dic3⋊C8—22no—
150Dic3⋊C8D414yescomplemented
151#810—41no—
152#807—41no—
153#803—22no—
154C22×SD16—31no—
155C22×SD16C2×D311yescharacteristic
156C2×C4⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8—31no—
159C2×#121—31no—
160C2×C4×D4—31no—
161C2×#117—31no—
162#461—64no—
163#433—64no—
164C2×#107—31no—
165C2×#106C2×D311yescharacteristic
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103C2×D311yescharacteristic
174C2×#102—31no—
175C2×#102C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178C2×#97—31no—
179C2×C4×C8C2×D311yescharacteristic
180#398—61no—
181#398—61no—
182#397—61no—
183#397—61no—
184#391—31no—
185#391C2×D311yescharacteristic
186#390—61no—
187#390C2×D311yescharacteristic
188#389—31no—
189#388—61no—
190#386—61no—
191#385—61no—
192#382—31no—
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3—21no—
200C23×D3C2×D411yescharacteristic
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C2×#67—24no—
220C2×#67—81no—
221C2×#67—81no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—42no—
227C22×Dic3—81no—
228C22×Dic3—81no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C3×SD16—44no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8C2×D412yes—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C22⋊C4—81no—
256C3×C22⋊C4—81no—
257C3×C42—21no—
258C3×C42—21no—
259#217—24no—
260#217—42no—
261#214—44no—
262(C2×D3)⋊C4—22no—
263(C2×D3)⋊C4—22no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—81no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4⋊Dic3#4112yes—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279Dic3⋊C4—81no—
280C4×Dic3—81no—
281C4×Dic3C2×D412yes—
282C4×Dic3#4112yes—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8C2×D412yes—
289C22×Q8—31no—
290C22×Q8C22×D311yescharacteristic
291C22×D4—61no—
292C22×D4C22×D311yescharacteristic
293C23×C4—31no—
294C23×C4—61no—
295C2×SD16—22no—
296C2×SD16—22no—
297C2×SD16—41no—
298C2×SD16—62no—
299C2×SD16—62no—
300C2×SD16—121no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C4⋊Q8—62no—
306C4⋊Q8—62no—
307C22⋊Q8—121no—
308C22⋊Q8—121no—
309C22⋊Q8—121no—
310C22⋊Q8—121no—
311#121—62no—
312#121—62no—
313C4×D4—62no—
314C4×D4—62no—
315#117—62no—
316#117—62no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C42—31no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#107—121no—
342#106—41no—
343C4⋊C8—34no—
344C4⋊C8—62no—
345#103—41no—
346#103—62no—
347#103—62no—
348#103—62no—
349#103—62no—
350#103—121no—
351#102—41no—
352#102—121no—
353C22⋊C8—62no—
354C22⋊C8—62no—
355#97—62no—
356C4×C8—22no—
357#95—21no—
358#95—21no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3—22no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3C22×D411yescharacteristic
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—21no—
380C22×D3—21no—
381C3×Q8—41no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×Q8—41no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—41no—
388C3×D4—41no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4H(5,𝔽2)11yescharacteristic
409C2×C3×C4H(5,𝔽2)11yescharacteristic
410#67—84no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—22no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—84no—
431C2×Dic3#12112yes—
432C2×Dic3#12112yes—
433C4×D3—41no—
434C4×D3—41no—
435C4×D3—41no—
436C4×D3—41no—
437Dic6—41no—
438Dic6—41no—
439Dic6—41no—
440Dic6—41no—
441C3×C8—22no—
442C3×C8—41no—
443C3⋊C8—22no—
444C3⋊C8—41no—
445C24—21no—
446C24—31no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—64no—
465C2×D4—241no—
466C2×D4—241no—
467C2×D4#23512yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—122no—
485C22×C4—241no—
486C22×C4—241no—
487C22×C4D3×D411yescharacteristic
488C22×C4D3×D411yescharacteristic
489C22×C4C23×D311yescharacteristic
490SD16—44no—
491SD16—124no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—32no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8D3×D412yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—32no—
509C4⋊C4—62no—
510C4⋊C4—62no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—122no—
517C4⋊C4—122no—
518C4⋊C4—122no—
519C4⋊C4—122no—
520C4⋊C4—241no—
521C22⋊C4—62no—
522C22⋊C4—62no—
523C22⋊C4—64no—
524C22⋊C4—81no—
525C22⋊C4—81no—
526C22⋊C4—122no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—241no—
530C42—21no—
531C42—21no—
532C42—32no—
533C42—32no—
534C42—241no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3—81no—
542C22×C3C2×#12111yescharacteristic
543C22×C3#52511yescharacteristic
544C22×C3#53511yescharacteristic
545C22×C3#57711yescharacteristic
546C22×C3#57711yescharacteristic
547C22×C3#57711yescharacteristic
548C22×C3#57811yescharacteristic
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4—81no—
561C3×C4#54611yescharacteristic
562C3×C4#55011yescharacteristic
563C3×C4#55311yescharacteristic
564C3×C4#55511yescharacteristic
565Dic3—22no—
566Dic3—22no—
567Dic3—42no—
568Dic3—41no—
569Dic3—81no—
570Dic3—81no—
571Dic3—81no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23C2×D3×D411yescharacteristic
587Q8—41no—
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8—121no—
592Q8—121no—
593Q8—121no—
594Q8—121no—
595D4—22no—
596D4—22no—
597D4—41no—
598D4—41no—
599D4—244no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—32no—
607C2×C4—32no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—62no—
624C2×C4—61no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—244no—
641C2×C4C2×#23511yescharacteristic
642C2×C4C2×D3×D411yescharacteristic
643C2×C4#100411yescharacteristic
644C2×C4#100411yescharacteristic
645C2×C4D3×#4111yescharacteristic
646C2×C4D3×#4111yescharacteristic
647C8—22no—
648C8—41no—
649C8—62no—
650C8—121no—
651C2×C3—82no—
652C2×C3#241411yescharacteristic
653C2×C3#305711yescharacteristic
654C2×C3#306311yescharacteristic
655C2×C3#312011yescharacteristic
656C2×C3#312311yescharacteristic
657C2×C3#322111yescharacteristic
658C2×C3#322511yescharacteristic
659D3—81no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—121no—
668C22—121no—
669C22—121no—
670C22—121no—
671C22—121no—
672C22—121no—
673C22—121no—
674C22#554711yescharacteristic
675C22D3×#12111yescharacteristic
676C22#571511yescharacteristic
677C22#576211yescharacteristic
678C22#576211yescharacteristic
679C22#577011yescharacteristic
680C22#577111yescharacteristic
681C4—41no—
682C4—41no—
683C4—62no—
684C4—62no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—121no—
689C4—122no—
690C4—241no—
691C4—241no—
692C4—241no—
693C4#549011yescharacteristic
694C4#554011yescharacteristic
695C4#559911yescharacteristic
696C4#567011yescharacteristic
697C3#2466611yescharacteristic, complemented, Sylow
698C2—82no—
699C2—241no—
700C2#7631111yescharacteristic
701C2#7965711yescharacteristic
702C2#7965911yescharacteristic
703C2#8065911yescharacteristic
704C2#8066211yescharacteristic
705C2#8176811yescharacteristic
706C2#8182811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1968C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1968D3
hypocenterC3#24666
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
1. C2×#5162C2semidirect
2. C2×#4690C2semidirect
3. #76641C2semidirect
5. C3×#1968C2semidirect
7. #70582C2semidirect
8. #70564C2semidirect
9. #70275C2semidirect
10. #70229C2semidirect
12. #70154C2semidirect
13. #69767C2semidirect
14. #69675C2semidirect
15. #69649C2semidirect
72. #1968D3semidirect
150. Dic3⋊C8D4semidirect
697. C3#24666semidirect

Maximal subgroups

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

subgroupindex
C2×#51622
C2×#46902
#766412
#764302
C3×#19682
#720542
#705822
#705642
#702752
#702292
#702052
#701542
#697672
#696752
#696492
#246663