Finite Group Databasebeta

← #10831710

Subgroups — #10831710

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5690C211yesmaximal, characteristic, complemented
2C2×#4759C211yesmaximal, characteristic, complemented
3C2×#4785C211yesmaximal, characteristic
4#76364C211yesmaximal, characteristic, complemented
5C3×#2025C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#70208C211yesmaximal, characteristic
8#70164C211yesmaximal, characteristic, complemented
9#69657C211yesmaximal, characteristic
10#68894C211yesmaximal, characteristic, complemented
11#68893C211yesmaximal, characteristic, complemented
12#68829C211yesmaximal, characteristic, complemented
13#68806C211yesmaximal, characteristic, complemented
14#68670C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#24339—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C22×#213C2211yescharacteristic
20#5690—24no—
21#5690—24no—
22C2×#897—21no—
23C2×#896—21no—
24C2×#896—21no—
25C2×#893C2211yescharacteristic
26C2×#893C2211yescharacteristic
27C2×D3×C4⋊C4—21no—
28C2×#890C2211yescharacteristic
29C2×#888C2211yescharacteristic
30C2×C4×D12C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C3×#390C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#815C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#4984C2211yescharacteristic
43#4977C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4950—21no—
48#4949—21no—
49#4947—21no—
50#4946—21no—
51#4928C2211yescharacteristic
52#4785—24no—
53#4759—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#554—31no—
65C2×#474—31no—
66C2×#442—31no—
67#2580—31no—
68#2059—31no—
69#2059—31no—
70#2025—31no—
71#2025D311yescharacteristic, complemented
72#1993—31no—
73#1985—31no—
74#1879—31no—
75#1879—31no—
76#1862—31no—
77#1862—31no—
78#1804—31no—
79C22×C3×D4C2311yescharacteristic
80C23×Dic3—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C4×D3—21no—
85C22×Dic6—21no—
86C22×Dic6D411yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#125—22no—
90C3×#125—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102D4×Dic3—22no—
103D4×Dic3—22no—
104C2×#213—22no—
105C2×#213—22no—
106C2×#213Q814yescomplemented
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4D411yescharacteristic
120C2×Dic3⋊C4D411yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#897—28no—
126#896—28no—
127#896—28no—
128#893—22no—
129#893—22no—
130#893—22no—
131#893D414yes—
132D3×C4⋊C4—28no—
133#890—22no—
134#890—22no—
135#888—22no—
136#888D414yes—
137C4×D12—22no—
138C4×D12—22no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C3×#102—22no—
142C3×#102—22no—
143#832—22no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148#821—22no—
149#821—22no—
150#818—22no—
151#815—22no—
152Dic3⋊C8—22no—
153#809—22no—
154#808—22no—
155#807—22no—
156C22×D8—31no—
157#554—64no—
158#554—64no—
159C2×C4⋊Q8—31no—
160C2×#125—31no—
161C2×#125C2×D311yescharacteristic
162C2×C22⋊Q8—61no—
163C2×C22⋊Q8—61no—
164C2×C22⋊Q8—61no—
165C2×C4×Q8—31no—
166C2×C4×D4—31no—
167C2×C4×D4—31no—
168C2×C4×D4—31no—
169C2×C4×D4—31no—
170C22×C4⋊C4—61no—
171#474—64no—
172#442—64no—
173C2×#107—31no—
174C2×#107—31no—
175C2×#106—31no—
176C2×C4⋊C8—31no—
177C2×#102—31no—
178C2×#102—31no—
179C2×#102—31no—
180C2×#102—31no—
181C2×#102C2×D311yescharacteristic
182C2×#102C2×D311yescharacteristic
183C2×#97—31no—
184#399—61no—
185#393—61no—
186#390—31no—
187#390C2×D311yescharacteristic
188#389—31no—
189#389—31no—
190#385—61no—
191#385—61no—
192#382—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199#336—31no—
200#336—31no—
201#336—31no—
202#336C2×D311yescharacteristic
203C24×C3—21no—
204C23×D3—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4C2×Q812yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—42no—
221C22×Dic3—81no—
222C22×Dic3—81no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12C2×D412yes—
231C2×D12C2×Q812yes—
232C2×D12#4112yes—
233C2×C4×D3—24no—
234C2×C4×D3—24no—
235C2×C4×D3—24no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—81no—
239C2×C4×D3—81no—
240C2×C4×D3—81no—
241C2×C4×D3—81no—
242C2×Dic6—22no—
243C2×Dic6—24no—
244C2×Dic6—41no—
245C2×Dic6—41no—
246C2×Dic6—81no—
247C2×Dic6—81no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4C2×D412yes—
261C3×C4⋊C4C2×D412yes—
262C3×C22⋊C4—42no—
263C3×C22⋊C4—81no—
264C3×C42—21no—
265C3×C42—21no—
266#217—42no—
267#217—42no—
268#217—81no—
269#213—28no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—24no—
272(C2×D3)⋊C4—24no—
273C4⋊Dic3—24no—
274C4⋊Dic3—21no—
275C4⋊Dic3—22no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—81no—
281C4⋊Dic3C2×D412yes—
282Dic3⋊C4—24no—
283Dic3⋊C4—22no—
284Dic3⋊C4—22no—
285Dic3⋊C4—24no—
286Dic3⋊C4—81no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3C2×D412yes—
290C4×Dic3C2×D412yes—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8C2×Q812yes—
297C22×Q8—31no—
298C22×Q8—61no—
299C22×D4—31no—
300C22×D4C22×D311yescharacteristic
301C23×C4—61no—
302C23×C4—61no—
303C23×C4—61no—
304C23×C4—61no—
305C2×D8—34no—
306C2×D8—62no—
307C2×D8—62no—
308C22×C8—31no—
309C22×C8—31no—
310C22×C8C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312C4⋊Q8—34no—
313C4⋊Q8—62no—
314#125—22no—
315#125—22no—
316#125—62no—
317#125—62no—
318C22⋊Q8—68no—
319C22⋊Q8—68no—
320C22⋊Q8—68no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×D4—34no—
324C4×D4—62no—
325C4×D4—62no—
326C4×D4—62no—
327C4×D4—62no—
328C4×D4—62no—
329C4×D4—62no—
330C4×D4—62no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—68no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—61no—
355C2×C22⋊C4—61no—
356C2×C42—31no—
357C2×C42—31no—
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—62no—
362C4⋊C8—62no—
363#102—22no—
364#102—22no—
365#102—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#97—62no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95—61no—
374C23×C3—22no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3C22×D411yescharacteristic
382C22×D3—22no—
383C22×D3—22no—
384C22×D3—22no—
385C22×D3—41no—
386C22×D3—41no—
387C22×D3—41no—
388C22×D3—41no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—41no—
392C3×D4—41no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C22×D411yescharacteristic
410C2×C3×C4C22×Q811yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C42−1+411yescharacteristic
413C2×C3×C42−1+411yescharacteristic
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
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—84no—
434C2×Dic3#12012yes—
435C2×Dic3#12012yes—
436D12—22no—
437D12—22no—
438D12—22no—
439D12C22⋊Q814yes—
440C4×D3—48no—
441C4×D3—84no—
442C4×D3—84no—
443Dic6—44no—
444Dic6—84no—
445C3×C8—41no—
446C3×C8—41no—
447C3⋊C8—22no—
448C3⋊C8—41no—
449C24—21no—
450C24—61no—
451C2×Q8—62no—
452C2×Q8—64no—
453C2×Q8—121no—
454C2×Q8—121no—
455C2×Q8—241no—
456C2×Q8—241no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—32no—
463C2×D4—32no—
464C2×D4—32no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—61no—
468C2×D4D3×Q812yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—64no—
477C22×C4—64no—
478C22×C4—64no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—122no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4—241no—
486C22×C4—241no—
487C22×C4—241no—
488C22×C4—241no—
489C22×C4—241no—
490C22×C4—241no—
491C22×C4D3×D411yescharacteristic
492C22×C4D3×D411yescharacteristic
493C22×C4C23×D311yescharacteristic
494D8—68no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—32no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—32no—
514C4⋊C4—62no—
515C4⋊C4—62no—
516C4⋊C4—62no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—61no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—241no—
526C4⋊C4—241no—
527C4⋊C4D3×D412yes—
528C4⋊C4D3×D412yes—
529C22⋊C4—42no—
530C22⋊C4—64no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—81no—
534C22⋊C4—122no—
535C22⋊C4—122no—
536C22⋊C4—241no—
537C42—21no—
538C42—21no—
539C42—32no—
540C42—32no—
541C42—61no—
542C42—61no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3—81no—
549C22×C3—81no—
550C22×C3C2×#12011yescharacteristic
551C22×C3#54811yescharacteristic
552C22×C3D4×Q811yescharacteristic
553C22×C3#57511yescharacteristic
554C22×C3#57511yescharacteristic
555C22×C3#57611yescharacteristic
556C22×C3#57611yescharacteristic
557C2×D3—24no—
558C2×D3—42no—
559C2×D3—42no—
560C2×D3—42no—
561C2×D3—42no—
562C2×D3—42no—
563C2×D3—42no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—41no—
568C3×C4—81no—
569C3×C4C2×C22⋊Q811yescharacteristic
570C3×C4#53611yescharacteristic
571C3×C4#54811yescharacteristic
572C3×C4D4×Q811yescharacteristic
573Dic3—22no—
574Dic3—22no—
575Dic3—41no—
576Dic3—42no—
577Dic3—41no—
578Dic3—81no—
579Dic3—81no—
580C23—22no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—62no—
588C23—62no—
589C23—62no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23C2×D3×D411yescharacteristic
595Q8—124no—
596Q8—244no—
597D4—22no—
598D4—22no—
599D4—34no—
600D4—41no—
601D4—41no—
602D4—62no—
603D4—62no—
604D4—62no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—32no—
614C2×C4—32no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—62no—
633C2×C4—128no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—244no—
643C2×C4—244no—
644C2×C4—244no—
645C2×C4C2×D3×D411yescharacteristic
646C2×C4C2×D3×D411yescharacteristic
647C2×C4C2×D3×Q811yescharacteristic
648C2×C4D3×#4111yescharacteristic
649C2×C4#101011yescharacteristic
650C2×C4#101011yescharacteristic
651C8—41no—
652C8—41no—
653C8—62no—
654C8—121no—
655C2×C3—82no—
656C2×C3#240911yescharacteristic
657C2×C3#300511yescharacteristic
658C2×C3#300811yescharacteristic
659C2×C3#337011yescharacteristic
660C2×C3#337011yescharacteristic
661C2×C3#338911yescharacteristic
662C2×C3#338911yescharacteristic
663D3—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—64no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—122no—
673C22—122no—
674C22—122no—
675C22—122no—
676C22—122no—
677C22—122no—
678C22D3×#12011yescharacteristic
679C22#563211yescharacteristic
680C22#567711yescharacteristic
681C22#576811yescharacteristic
682C22#576911yescharacteristic
683C22#577311yescharacteristic
684C22#577411yescharacteristic
685C4—41no—
686C4—42no—
687C4—41no—
688C4—42no—
689C4—62no—
690C4—62no—
691C4—81no—
692C4—121no—
693C4—121no—
694C4—122no—
695C4—241no—
696C4—241no—
697C4D3×C22⋊Q811yescharacteristic
698C4#563211yescharacteristic
699C4#567511yescharacteristic
700C4#567711yescharacteristic
701C3#2433911yescharacteristic, complemented, Sylow
702C2—82no—
703C2—124no—
704C2#7678011yescharacteristic
705C2#8008311yescharacteristic
706C2#8008511yescharacteristic
707C2#8116211yescharacteristic
708C2#8116311yescharacteristic
709C2#8166011yescharacteristic
710C2#8171011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2025C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2025D3
hypocenterC3#24339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. C2×#5690C2semidirect
2. C2×#4759C2semidirect
4. #76364C2semidirect
5. C3×#2025C2semidirect
6. #71739C2semidirect
8. #70164C2semidirect
10. #68894C2semidirect
11. #68893C2semidirect
12. #68829C2semidirect
13. #68806C2semidirect
14. #68670C2semidirect
15. #68620C2semidirect
71. #2025D3semidirect
106. C2×#213Q8semidirect
701. C3#24339semidirect

Maximal subgroups

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

subgroupindex
C2×#56902
C2×#47592
C2×#47852
#763642
C3×#20252
#717392
#702082
#701642
#696572
#688942
#688932
#688292
#688062
#686702
#686202
#243393