Finite Group Databasebeta

← #10831725

Subgroups — #10831725

Order 768.

1036 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5569C211yesmaximal, characteristic, complemented
2C2×#5164C211yesmaximal, characteristic
3C2×#4761C211yesmaximal, characteristic
4#76364C211yesmaximal, characteristic
5C3×#1880C211yesmaximal, characteristic
6#72048C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic, complemented
8#70238C211yesmaximal, characteristic, complemented
9#70222C211yesmaximal, characteristic, complemented
10#70164C211yesmaximal, characteristic
11#69639C211yesmaximal, characteristic
12#68829C211yesmaximal, characteristic, complemented
13#68806C211yesmaximal, characteristic, complemented
14#68634C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#24340—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22#5569—24no—
23#5569—24no—
24C2×#890—21no—
25C2×#890C2211yescharacteristic
26C2×#888—21no—
27C2×#888C2211yescharacteristic
28C2×#870—21no—
29C2×#870C2211yescharacteristic
30C2×C4×Dic6—21no—
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37#5221C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40#5164—24no—
41C2×#818C2211yescharacteristic
42C2×#817C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45#4984C2211yescharacteristic
46#4977C2211yescharacteristic
47C2×#808C2211yescharacteristic
48#4950—21no—
49#4949—21no—
50#4947—21no—
51#4946—21no—
52#4928C2211yescharacteristic
53#4761—24no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557—31no—
65C2×#476—31no—
66C2×#440—31no—
67#2580—31no—
68#2059—31no—
69#2059—31no—
70#2025—31no—
71#2025—31no—
72#1993—31no—
73#1985—31no—
74#1880—31no—
75#1880D311yescharacteristic
76#1865—31no—
77#1865—31no—
78#1805—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6—21no—
83C22×Dic6D411yescharacteristic
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16Q814yescomplemented
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101Q8×Dic3—22no—
102Q8×Dic3—22no—
103Q8×Dic3—22no—
104Q8×Dic3D414yes—
105D4×Dic3—22no—
106D4×Dic3—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3D411yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4D411yescharacteristic
122C2×Dic3⋊C4D411yescharacteristic
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3C2311yescharacteristic
126C2×C4×Dic3C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129#890—28no—
130#890—22no—
131#890—22no—
132#888—28no—
133#888—22no—
134#888D414yes—
135#870—28no—
136#870—22no—
137#870—22no—
138C4×Dic6—28no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C3×#103—22no—
142C3×#102—22no—
143#835—22no—
144#835—22no—
145#832—22no—
146#832—22no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95—21no—
151#818—22no—
152#817—22no—
153#815—22no—
154Dic3⋊C8—22no—
155#808—22no—
156C22×SD16C2×D311yescharacteristic
157#557—64no—
158#557—64no—
159C2×C4⋊Q8—31no—
160C2×C4⋊Q8—61no—
161C2×#125—31no—
162C2×#125—31no—
163C2×#125—61no—
164C2×#125—61no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—61no—
169C2×C4×Q8C2×D311yescharacteristic
170C2×C4×D4—31no—
171#476—64no—
172#440—64no—
173C2×#107—31no—
174C2×#106—31no—
175C2×#106—31no—
176C2×C4⋊C8—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103C2×D311yescharacteristic
180C2×#102—31no—
181C2×#102—31no—
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#340—31no—
198#340C2×D311yescharacteristic
199#336—31no—
200#336—31no—
201#336—31no—
202#336C2×D311yescharacteristic
203C24×C3—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8C2×D412yes—
208C2×C3×Q8C2×Q812yes—
209C2×C3×Q8#4112yes—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4C2×Q812yes—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
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
232C22×Dic3C2×D411yescharacteristic
233C2×Dic6—22no—
234C2×Dic6—24no—
235C2×Dic6—41no—
236C2×Dic6—41no—
237C2×Dic6—42no—
238C2×Dic6—42no—
239C3×SD16—28no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8C2×Q812yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C2×D412yes—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—81no—
253C3×C42—24no—
254C3×C42—21no—
255C3×C42—21no—
256#217—42no—
257#217—42no—
258#217—81no—
259C4⋊Dic3—24no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—22no—
264C4⋊Dic3—24no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—81no—
270C4⋊Dic3C2×D412yes—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—22no—
276Dic3⋊C4—22no—
277Dic3⋊C4—24no—
278Dic3⋊C4—81no—
279C4×Dic3—24no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—24no—
283C4×Dic3C2×D412yes—
284C4×Dic3C2×D412yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C22×Q8—31no—
292C22×Q8—61no—
293C22×Q8C22×D311yescharacteristic
294C22×D4C22×D311yescharacteristic
295C23×C4—61no—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×SD16Dic614yescomplemented
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—34no—
304C4⋊Q8—62no—
305C4⋊Q8—68no—
306#125—62no—
307#125—62no—
308#125—68no—
309#125—62no—
310#125—68no—
311#125—62no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—34no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—68no—
318C4×Q8—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×D4—62no—
322C4×D4—62no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C42—21no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—61no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353#107—62no—
354#106—62no—
355#106—62no—
356C4⋊C8—62no—
357#103—22no—
358#103—62no—
359#103—62no—
360#102—22no—
361#102—62no—
362#102—62no—
363#97—62no—
364#95—61no—
365#95—61no—
366#95—61no—
367#95—61no—
368C23×C3—22no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3C22×D411yescharacteristic
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8C22⋊Q814yes—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
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×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×Q811yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
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—84no—
428C2×Dic3#12012yes—
429C2×Dic3#12012yes—
430Dic6—44no—
431Dic6—48no—
432C3×C8—22no—
433C3×C8—41no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—64no—
441C2×Q8—62no—
442C2×Q8—121no—
443C2×Q8—121no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×Q8C2×Dic612yes—
447C2×Q8D3×D412yes—
448C2×Q8#23712yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4C2×Dic612yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—122no—
467C22×C4—241no—
468C22×C4—241no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8C2×Dic612yes—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—32no—
490C4⋊C4—32no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—62no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—64no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4D3×D412yes—
510C22⋊C4—42no—
511C22⋊C4—81no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C42—24no—
516C42—21no—
517C42—21no—
518C42—32no—
519C42—32no—
520C42—64no—
521C42—64no—
522C42—61no—
523C42—61no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3C2×#12011yescharacteristic
532C22×C3#54811yescharacteristic
533C22×C3D4×Q811yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57711yescharacteristic
538C3×C4—24no—
539C3×C4—41no—
540C3×C4—42no—
541C3×C4—41no—
542C3×C4—81no—
543C3×C4C2×C22⋊Q811yescharacteristic
544C3×C4#53611yescharacteristic
545C3×C4#54811yescharacteristic
546C3×C4D4×Q811yescharacteristic
547Dic3—22no—
548Dic3—22no—
549Dic3—41no—
550Dic3—42no—
551Dic3—41no—
552Dic3—42no—
553Dic3—81no—
554Dic3—81no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—22no—
566Q8—124no—
567Q8—128no—
568Q8#87814yes—
569D4—22no—
570D4—22no—
571D4—41no—
572D4—41no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—32no—
584C2×C4—32no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—61no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—244no—
613C2×C4C22×Dic611yescharacteristic
614C2×C4C2×D3×D411yescharacteristic
615C2×C4C2×D3×D411yescharacteristic
616C2×C4C2×#23711yescharacteristic
617C2×C4#101011yescharacteristic
618C2×C4#101011yescharacteristic
619C8—22no—
620C8—41no—
621C8—121no—
622C8—121no—
623C2×C3—82no—
624C2×C3#240911yescharacteristic
625C2×C3#300511yescharacteristic
626C2×C3#301011yescharacteristic
627C2×C3#337111yescharacteristic
628C2×C3#337111yescharacteristic
629C2×C3#338811yescharacteristic
630C2×C3#338811yescharacteristic
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22D4×Dic611yescharacteristic
639C22D3×#12011yescharacteristic
640C22#559811yescharacteristic
641C22#576211yescharacteristic
642C22#576811yescharacteristic
643C22#577311yescharacteristic
644C22#577811yescharacteristic
645C4—24no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—62no—
650C4—62no—
651C4—81no—
652C4—121no—
653C4—122no—
654C4—121no—
655C4—122no—
656C4—241no—
657C4—241no—
658C4C2×#87811yescharacteristic
659C4D4×Dic611yescharacteristic
660C4#559811yescharacteristic
661C4#564811yescharacteristic
662C3#2434011yescharacteristic, complemented, Sylow
663C2—82no—
664C2#7468911yescharacteristic
665C2#7963611yescharacteristic
666C2#7976211yescharacteristic
667C2#8004111yescharacteristic
668C2#8008311yescharacteristic
669C2#8037311yescharacteristic
670C2#8058311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1880D3
hypocenterC3#24340
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#5569C2semidirect
7. #71629C2semidirect
8. #70238C2semidirect
9. #70222C2semidirect
12. #68829C2semidirect
13. #68806C2semidirect
14. #68634C2semidirect
15. #68620C2semidirect
86. C2×C3×SD16Q8semidirect
298. C2×SD16Dic6semidirect
662. C3#24340semidirect

Maximal subgroups

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

subgroupindex
C2×#55692
C2×#51642
C2×#47612
#763642
C3×#18802
#720482
#716292
#702382
#702222
#701642
#696392
#688292
#688062
#686342
#686202
#243403