Finite Group Databasebeta

← #10835939

Subgroups — #10835939

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
2#82786C211yesmaximal, characteristic, complemented
3#82633C211yesmaximal, characteristic
4C2×#4720C211yesmaximal, characteristic, complemented
5C3×#2000C211yesmaximal, characteristic
6#72049C211yesmaximal, characteristic, complemented
7#71695C211yesmaximal, characteristic, complemented
8#71685C211yesmaximal, characteristic
9#70255C211yesmaximal, characteristic, complemented
10#70229C211yesmaximal, characteristic, complemented
11#70192C211yesmaximal, characteristic, complemented
12#70154C211yesmaximal, characteristic, complemented
13#69936C211yesmaximal, characteristic, complemented
14#68851C211yesmaximal, characteristic, complemented
15#68797C211yesmaximal, characteristic, complemented
16#24666—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5221—21no—
32#5221C2211yescharacteristic
33#5220—21no—
34C2×#832C2211yescharacteristic
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5162—24no—
38C2×#817C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43C2×Dic3⋊C8C2211yescharacteristic
44C2×C8×Dic3C2211yescharacteristic
45#4984C2211yescharacteristic
46#4982—21no—
47#4982—21no—
48#4977C2211yescharacteristic
49Dic3×C4⋊C4C2211yescharacteristic
50#4973C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4952—21no—
54#4952—21no—
55#4951—21no—
56#4950—21no—
57#4720—24no—
58C3×#340C2211yescharacteristic
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#2000D311yescharacteristic
70#1981—31no—
71#1973—31no—
72#1968—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1804—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—41no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C22⋊Q8—41no—
89C3×C22⋊Q8—41no—
90C3×#121—22no—
91C3×#121—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217D411yescharacteristic
103C2×#217D411yescharacteristic
104#944—22no—
105#944—22no—
106D4×Dic3—22no—
107D4×Dic3—22no—
108C2×#214—22no—
109C2×#214—22no—
110C2×#214—41no—
111#926—22no—
112#926—22no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119#924—41no—
120#924—41no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3C2311yescharacteristic
126C2×C4×Dic3C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129C3×#103—22no—
130C3×#102—41no—
131C3×C22⋊C8—22no—
132#835—22no—
133#835—41no—
134#832—41no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#830—22no—
145#817—41no—
146#816—22no—
147#816—41no—
148#815—22no—
149Dic3⋊C8—22no—
150Dic3⋊C8D414yescomplemented
151C8×Dic3—22no—
152#810—22no—
153#807—41no—
154C22×SD16—31no—
155C22×SD16C2×D311yescharacteristic
156C2×C4⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8C2×D311yescharacteristic
159C2×#121C2×D311yescharacteristic
160C2×C4×D4—31no—
161C2×#117—31no—
162#461—64no—
163#433—64no—
164C2×#107—31no—
165C2×#106—31no—
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⋊C8C2×D311yescharacteristic
178C2×#97—31no—
179C2×C4×C8—31no—
180#398—61no—
181#398—61no—
182#397—61no—
183#397—61no—
184#391—31no—
185#391—31no—
186#390—31no—
187#390—61no—
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#340C2×D311yescharacteristic
199C24×C3—21no—
200C24×C3C2×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—24no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—81no—
214C2×C3×D4—81no—
215C2×C3×D4#4112yes—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—42no—
232C22×Dic3—81no—
233C22×Dic3—81no—
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
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—42no—
255C3×C4⋊C4#4112yes—
256C3×C22⋊C4—24no—
257C3×C22⋊C4—42no—
258#217—22no—
259#217—22no—
260#217—42no—
261#217—42no—
262#217—81no—
263#217—81no—
264#217—81no—
265#214—44no—
266C4⋊Dic3—22no—
267C4⋊Dic3—22no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—81no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278C4×Dic3—21no—
279C4×Dic3—21no—
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—21no—
292C22×D4C22×D311yescharacteristic
293C23×C4—61no—
294C23×C4C22×D311yescharacteristic
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—41no—
308C22⋊Q8—41no—
309C22⋊Q8—121no—
310C22⋊Q8—121no—
311#121—22no—
312#121—22no—
313C4×D4—62no—
314C4×D4—62no—
315#117—62no—
316#117—62no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
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—31no—
340C2×C42—61no—
341#107—121no—
342#106—121no—
343C4⋊C8—34no—
344C4⋊C8—62no—
345#103—22no—
346#103—62no—
347#103—62no—
348#103—62no—
349#103—121no—
350#103—121no—
351#102—41no—
352#102—121no—
353C22⋊C8—22no—
354C22⋊C8—62no—
355#97—62no—
356C4×C8—62no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3—21no—
367C23×C3—22no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3C22×D411yescharacteristic
381C3×Q8—41no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×Q8—41no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—84no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
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×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×C3×C4H(5,𝔽2)11yescharacteristic
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
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—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—84no—
435C2×Dic3#12112yes—
436C2×Dic3#12112yes—
437Dic6—41no—
438Dic6—41no—
439Dic6—41no—
440Dic6—41no—
441C3×C8—22no—
442C3×C8—41no—
443C3⋊C8—22no—
444C3⋊C8—41no—
445C24—21no—
446C24D3×D411yescharacteristic
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—24no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—81no—
466C2×D4—81no—
467C2×D4#23712yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—122no—
486C22×C4—241no—
487C22×C4—241no—
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—42no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—61no—
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—241no—
520C4⋊C4#23712yes—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—62no—
524C22⋊C4—62no—
525C22⋊C4—122no—
526C22⋊C4—122no—
527C22⋊C4—241no—
528C22⋊C4—241no—
529C22⋊C4—241no—
530C42—32no—
531C42—32no—
532C42—61no—
533C42—61no—
534C42—241no—
535C22×C3—41no—
536C22×C3—42no—
537C22×C3—41no—
538C22×C3—42no—
539C22×C3—41no—
540C22×C3—42no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3C2×#12111yescharacteristic
550C22×C3#52511yescharacteristic
551C22×C3#53511yescharacteristic
552C22×C3#57711yescharacteristic
553C22×C3#57711yescharacteristic
554C22×C3#57711yescharacteristic
555C22×C3#57811yescharacteristic
556C3×C4—42no—
557C3×C4—41no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4#54611yescharacteristic
561C3×C4#55011yescharacteristic
562C3×C4#55311yescharacteristic
563C3×C4#55511yescharacteristic
564Dic3—22no—
565Dic3—22no—
566Dic3—41no—
567Dic3—41no—
568Dic3—81no—
569Dic3—81no—
570Dic3—81no—
571Dic3—81no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
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—84no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—32no—
605C2×C4—32no—
606C2×C4—41no—
607C2×C4—41no—
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—41no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
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×D3×D411yescharacteristic
642C2×C4C2×#23711yescharacteristic
643C2×C4C2×#23711yescharacteristic
644C2×C4#100411yescharacteristic
645C2×C4#100411yescharacteristic
646C2×C4D3×#4111yescharacteristic
647C8—22no—
648C8—41no—
649C8—62no—
650C8—121no—
651C2×C3—81no—
652C2×C3—82no—
653C2×C3#241411yescharacteristic
654C2×C3#305711yescharacteristic
655C2×C3#306311yescharacteristic
656C2×C3#312011yescharacteristic
657C2×C3#312311yescharacteristic
658C2×C3#322111yescharacteristic
659C2×C3#322511yescharacteristic
660C22—41no—
661C22—42no—
662C22—41no—
663C22—41no—
664C22—42no—
665C22—41no—
666C22—42no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22#558611yescharacteristic
675C22#559211yescharacteristic
676C22D3×#12111yescharacteristic
677C22#576211yescharacteristic
678C22#577011yescharacteristic
679C22#577511yescharacteristic
680C22#577911yescharacteristic
681C4—42no—
682C4—41no—
683C4—62no—
684C4—62no—
685C4—81no—
686C4—81no—
687C4—121no—
688C4—121no—
689C4—241no—
690C4—241no—
691C4—241no—
692C4—241no—
693C4#559111yescharacteristic
694C4#560111yescharacteristic
695C4#561611yescharacteristic
696C4#571811yescharacteristic
697C3#2466611yescharacteristic, complemented, Sylow
698C2—81no—
699C2—82no—
700C2#7667011yescharacteristic
701C2#8005911yescharacteristic
702C2#8012711yescharacteristic
703C2#8013711yescharacteristic
704C2#8021611yescharacteristic
705C2#8065911yescharacteristic
706C2#8081011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2000C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2000D3
hypocenterC3#24666
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2000

Complements

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

NH ≅ G/Ntype
1. C2×#5162C2semidirect
2. #82786C2semidirect
4. C2×#4720C2semidirect
6. #72049C2semidirect
7. #71695C2semidirect
9. #70255C2semidirect
10. #70229C2semidirect
11. #70192C2semidirect
12. #70154C2semidirect
13. #69936C2semidirect
14. #68851C2semidirect
15. #68797C2semidirect
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
#827862
#826332
C2×#47202
C3×#20002
#720492
#716952
#716852
#702552
#702292
#701922
#701542
#699362
#688512
#687972
#246663