Finite Group Databasebeta

← #10831799

Subgroups — #10831799

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic, complemented
2C2×#5164C211yesmaximal, characteristic
3C2×#4766C211yesmaximal, characteristic
4#76346C211yesmaximal, characteristic
5C3×#1880C211yesmaximal, characteristic
6#72048C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic, complemented
8#70236C211yesmaximal, characteristic, complemented
9#70221C211yesmaximal, characteristic, complemented
10#70164C211yesmaximal, characteristic
11#69638C211yesmaximal, characteristic
12#68829C211yesmaximal, characteristic, complemented
13#68805C211yesmaximal, characteristic, complemented
14#68632C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#24346—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22#5570—24no—
23#5570—24no—
24C2×#890—21no—
25C2×#890—21no—
26C2×#890C2211yescharacteristic
27C2×#890C2211yescharacteristic
28C2×#878—21no—
29C2×#869—21no—
30C2×#869C2211yescharacteristic
31C2×C4×Dic6—21no—
32C2×C4×Dic6C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C3×#390C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38#5221C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41#5164—24no—
42C2×#818C2211yescharacteristic
43C2×#818C2211yescharacteristic
44C2×#815C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46#4984C2211yescharacteristic
47#4975C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4948—21no—
50#4948—21no—
51#4942—21no—
52#4928C2211yescharacteristic
53#4766—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×#478—31no—
66C2×#440—31no—
67#2538—31no—
68#2059—31no—
69#2057—31no—
70#2025—31no—
71#2023—31no—
72#1992—31no—
73#1985—31no—
74#1880—31no—
75#1880D311yescharacteristic
76#1865—31no—
77#1863—31no—
78#1803—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⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4D411yescharacteristic
124C2×Dic3⋊C4D411yescharacteristic
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131#890—28no—
132#890—28no—
133#890—22no—
134#890—22no—
135#890—22no—
136#890D414yes—
137#878—44no—
138#869—28no—
139#869—22no—
140#869—22no—
141C4×Dic6—28no—
142C4×Dic6—22no—
143C4×Dic6—22no—
144C3×#103—22no—
145C3×#102—22no—
146#835—22no—
147#835—22no—
148#832—22no—
149#832—22no—
150C3⋊#95—21no—
151C3⋊#95—21no—
152#818—22no—
153#818—22no—
154#815—22no—
155Dic3⋊C8—22no—
156#808—22no—
157C22×SD16C2×D311yescharacteristic
158#557—64no—
159#557—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—61no—
162C2×#125—31no—
163C2×#125—31no—
164C2×#125—61no—
165C2×#125—61no—
166C2×C22⋊Q8—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—61no—
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4—31no—
173#478—64no—
174#440—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103C2×D311yescharacteristic
182C2×#102—31no—
183C2×#102—31no—
184C2×#102C2×D311yescharacteristic
185C2×#97—31no—
186#396—61no—
187#396—61no—
188#390—31no—
189#390C2×D311yescharacteristic
190#389—31no—
191#389—31no—
192#384—31no—
193#380—61no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199#340C2×D311yescharacteristic
200#336—31no—
201#336—31no—
202#336—31no—
203#336C2×D311yescharacteristic
204C24×C3—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8C2×D412yes—
209C2×C3×Q8C2×Q812yes—
210C2×C3×Q8#4112yes—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4C2×Q812yes—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—42no—
228C22×Dic3—42no—
229C22×Dic3—42no—
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C2×Dic6—24no—
235C2×Dic6—22no—
236C2×Dic6—22no—
237C2×Dic6—41no—
238C2×Dic6—42no—
239C2×Dic6—42no—
240C3×SD16—28no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8C2×Q812yes—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2×D412yes—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—81no—
254C3×C42—24no—
255C3×C42—21no—
256C3×C42—21no—
257#217—42no—
258#217—42no—
259#217—81no—
260C4⋊Dic3—24no—
261C4⋊Dic3—24no—
262C4⋊Dic3—24no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—22no—
266C4⋊Dic3—24no—
267C4⋊Dic3—24no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4⋊Dic3C2×D412yes—
274C4⋊Dic3C2×D412yes—
275Dic3⋊C4—24no—
276Dic3⋊C4—22no—
277Dic3⋊C4—22no—
278Dic3⋊C4—24no—
279Dic3⋊C4—42no—
280Dic3⋊C4—42no—
281Dic3⋊C4—81no—
282C4×Dic3—24no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—24no—
286C4×Dic3C2×D412yes—
287C4×Dic3C2×D412yes—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C22×Q8—31no—
295C22×Q8—61no—
296C22×Q8C22×D311yescharacteristic
297C22×D4C22×D311yescharacteristic
298C23×C4—61no—
299C2×SD16—22no—
300C2×SD16—22no—
301C2×SD16Dic614yescomplemented
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—68no—
309#125—34no—
310#125—62no—
311#125—62no—
312#125—68no—
313#125—68no—
314#125—62no—
315C22⋊Q8—124no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—34no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—68no—
322C4×Q8—62no—
323C4×Q8—62no—
324C4×Q8—62no—
325C4×D4—62no—
326C4×D4—62no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4—61no—
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×C42—21no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—61no—
357C2×C42—61no—
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—62no—
362C4⋊C8—62no—
363#103—22no—
364#103—62no—
365#103—62no—
366#102—22no—
367#102—62no—
368#102—62no—
369#97—62no—
370#95—61no—
371#95—61no—
372C23×C3—22no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3C22×D411yescharacteristic
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8C22⋊Q814yescomplemented
384C3×D4—22no—
385C3×D4—22no—
386C3×D4—41no—
387C3×D4—41no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4AGL(1,ℤ/8)12yes—
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×Q811yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
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—41no—
431C2×Dic3—84no—
432C2×Dic3#12012yes—
433C2×Dic3#12012yes—
434Dic6—44no—
435Dic6—48no—
436C3×C8—22no—
437C3×C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—62no—
445C2×Q8—64no—
446C2×Q8—62no—
447C2×Q8—121no—
448C2×Q8—122no—
449C2×Q8—122no—
450C2×Q8C2×Dic612yes—
451C2×Q8D3×D412yes—
452C2×Q8#23712yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4C2×Dic612yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—28no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8C2×Dic612yes—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—32no—
494C4⋊C4—32no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—62no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—122no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—241no—
515C4⋊C4D3×D412yes—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—241no—
521C42—24no—
522C42—21no—
523C42—21no—
524C42—32no—
525C42—32no—
526C42—64no—
527C42—64no—
528C42—61no—
529C42—61no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3C2×#12011yescharacteristic
538C22×C3#54811yescharacteristic
539C22×C3D4×Q811yescharacteristic
540C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
541C22×C3#57511yescharacteristic
542C22×C3#57711yescharacteristic
543C22×C3#57811yescharacteristic
544C3×C4—24no—
545C3×C4—22no—
546C3×C4—41no—
547C3×C4—42no—
548C3×C4—81no—
549C3×C4C2×C22⋊Q811yescharacteristic
550C3×C4#53611yescharacteristic
551C3×C4#54811yescharacteristic
552C3×C4D4×Q811yescharacteristic
553Dic3—22no—
554Dic3—22no—
555Dic3—41no—
556Dic3—42no—
557Dic3—42no—
558Dic3—41no—
559Dic3—81no—
560Dic3—81no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C2×D3×D411yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—124no—
573Q8—128no—
574Q8#87814yescomplemented
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—32no—
589C2×C4—32no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—61no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—244no—
618C2×C4#91412yes—
619C2×C4C22×Dic611yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×D3×D411yescharacteristic
622C2×C4C2×#23711yescharacteristic
623C2×C4#101011yescharacteristic
624C2×C4#101011yescharacteristic
625C8—22no—
626C8—41no—
627C8—121no—
628C8—121no—
629C2×C3—82no—
630C2×C3#240911yescharacteristic
631C2×C3#300611yescharacteristic
632C2×C3#301311yescharacteristic
633C2×C3#336811yescharacteristic
634C2×C3#337711yescharacteristic
635C2×C3#338611yescharacteristic
636C2×C3#338811yescharacteristic
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22D4×Dic611yescharacteristic
645C22D3×#12011yescharacteristic
646C22#559811yescharacteristic
647C22C2×#91411yescharacteristic
648C22#577111yescharacteristic
649C22#577511yescharacteristic
650C22#577711yescharacteristic
651C4—22no—
652C4—24no—
653C4—41no—
654C4—42no—
655C4—62no—
656C4—62no—
657C4—81no—
658C4—121no—
659C4—122no—
660C4—122no—
661C4—121no—
662C4—241no—
663C4—241no—
664C4C2×#87811yescharacteristic
665C4D4×Dic611yescharacteristic
666C4#559811yescharacteristic
667C4#564811yescharacteristic
668C3#2434611yescharacteristic, complemented, Sylow
669C2—82no—
670C2#7468911yescharacteristic
671C2#7963511yescharacteristic
672C2#7976111yescharacteristic
673C2#8004011yescharacteristic
674C2#8008811yescharacteristic
675C2#8037111yescharacteristic
676C2#8058711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1880D3
hypocenterC3#24346
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×#5570C2semidirect
7. #71629C2semidirect
8. #70236C2semidirect
9. #70221C2semidirect
12. #68829C2semidirect
13. #68805C2semidirect
14. #68632C2semidirect
15. #68620C2semidirect
86. C2×C3×SD16Q8semidirect
301. C2×SD16Dic6semidirect
383. C3×Q8C22⋊Q8semidirect
574. Q8#878semidirect
668. C3#24346semidirect

Maximal subgroups

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

subgroupindex
C2×#55702
C2×#51642
C2×#47662
#763462
C3×#18802
#720482
#716292
#702362
#702212
#701642
#696382
#688292
#688052
#686322
#686202
#243463