Finite Group Databasebeta

← #10858354

Subgroups — #10858354

Order 768.

932 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×Dic3×D8C211yesmaximal, characteristic
2D4×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#4766C211yesmaximal, characteristic, complemented
4#76325C211yesmaximal, characteristic, complemented
5C3×#1879C211yesmaximal, characteristic
6#72037C211yesmaximal, characteristic
7#71607C211yesmaximal, characteristic, complemented
8#70193C211yesmaximal, characteristic, complemented
9#70169C211yesmaximal, characteristic, complemented
10#70168C211yesmaximal, characteristic, complemented
11#69656C211yesmaximal, characteristic, complemented
12#68858C211yesmaximal, characteristic, complemented
13#68829C211yesmaximal, characteristic, complemented
14#68639C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#26382—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×C4⋊Dic3—21no—
23C2×#890C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5212—21no—
33Dic3×D8—24no—
34C2×#818C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4950—21no—
45#4950—21no—
46#4950—21no—
47#4949—21no—
48#4948—21no—
49#4942—21no—
50#4935C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52#4928C2211yescharacteristic
53#4766—24no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×C4×D8—31no—
64#2454—31no—
65D4×C4⋊C4—31no—
66#2094—31no—
67#2083—31no—
68#2065—31no—
69#2059—31no—
70#1941—31no—
71#1916—31no—
72#1879—31no—
73#1879D311yescharacteristic
74#1864—31no—
75#1864—31no—
76#1862—31no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C23×Dic3—21no—
83C2×C3×D8—22no—
84C2×C3×D8—22no—
85C2×C3×D8Q814yes—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101D4×Dic3—22no—
102D4×Dic3—22no—
103D4×Dic3—22no—
104D4×Dic3—41no—
105D4×Dic3—41no—
106D4×Dic3—41no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—44no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126#890—22no—
127#890—41no—
128C3×#102—22no—
129C3×#102—22no—
130#832—22no—
131#832—22no—
132#832—41no—
133#832D414yes—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#818—22no—
141#818—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#808—22no—
145#808—22no—
146#808—41no—
147C22×D8C2×D311yescharacteristic
148C2×#125—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152C2×C4×D4C2×D311yescharacteristic
153C22×C4⋊C4—61no—
154#478—64no—
155C4×D8—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×C4×C8—31no—
169#399—61no—
170#396—61no—
171#390—31no—
172#390C2×D311yescharacteristic
173#389—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#385—61no—
178#385—61no—
179#385—61no—
180#384—31no—
181#380—61no—
182#380—61no—
183C4×C4⋊C4—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#336—31no—
189#336—31no—
190#336—31no—
191#336C2×D311yescharacteristic
192C24×C3—21no—
193C24×C3—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4C2×D412yes—
203C2×C3×D4C2×Q812yes—
204C2×C3×D4C2×Q812yes—
205C2×C3×D4#4112yes—
206C22×C3×C4—24no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—42no—
210C22×C3×C4—42no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—42no—
222C22×Dic3—42no—
223C22×Dic3—42no—
224C22×Dic3—81no—
225C22×Dic3—81no—
226C22×Dic3—81no—
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C3×D8—28no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×Q812yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4#4112yes—
242C3×C22⋊C4—24no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—81no—
245C3×C42—21no—
246C3×C42—21no—
247#217—42no—
248#217—42no—
249#217—42no—
250#217—81no—
251#217—81no—
252C4⋊Dic3—22no—
253C4⋊Dic3—21no—
254C4⋊Dic3—22no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—42no—
262C4⋊Dic3—42no—
263C4⋊Dic3—42no—
264C4⋊Dic3—42no—
265C4⋊Dic3—81no—
266C4⋊Dic3C2×D412yes—
267Dic3⋊C4—42no—
268Dic3⋊C4—81no—
269Dic3⋊C4—81no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8C2×D412yes—
280C22×D4C22×D311yescharacteristic
281C22×D4C22×D311yescharacteristic
282C23×C4—21no—
283C23×C4—61no—
284C23×C4—61no—
285C23×C4—61no—
286C2×D8—22no—
287C2×D8—22no—
288C2×D8Dic614yes—
289C22×C8—31no—
290C22×C8—31no—
291C22×C8C22×D311yescharacteristic
292C22×C8C22×D311yescharacteristic
293#125—62no—
294#125—121no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—62no—
300C4×D4—121no—
301C4×D4—121no—
302C4×D4—121no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—124no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42C22×D311yescharacteristic
331#107—62no—
332#107—62no—
333#106—62no—
334#106—62no—
335#106—121no—
336C4⋊C8—62no—
337#102—22no—
338#102—22no—
339#102—34no—
340#102—62no—
341#102—62no—
342#102—121no—
343C4×C8—62no—
344#95—31no—
345#95—31no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3C22×D411yescharacteristic
365C3×D4—22no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4C22⋊Q814yes—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—48no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#13512yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×Q811yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—84no—
416C2×Dic3—84no—
417C2×Dic3—84no—
418C2×Dic3AGL(1,ℤ/8)12yes—
419C3×C8—22no—
420C3×C8—41no—
421C3⋊C8—22no—
422C3⋊C8—41no—
423C24—21no—
424C24—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C2×Dic612yes—
434C2×D4C2×Dic612yes—
435C2×D4D3×D412yes—
436C2×D4#23712yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—241no—
457C22×C4—241no—
458C22×C4—241no—
459C22×C4D3×D411yescharacteristic
460C22×C4D3×D411yescharacteristic
461C22×C4C23×D311yescharacteristic
462D8—28no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—32no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8C2×Dic612yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—32no—
479C4⋊C4—62no—
480C4⋊C4—62no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C4⋊C4—122no—
493C4⋊C4—241no—
494C4⋊C4—241no—
495C4⋊C4—241no—
496C4⋊C4#23712yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C42—21no—
506C42—21no—
507C42—61no—
508C42—61no—
509C42—61no—
510C42—61no—
511C22×C3—24no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3#52511yescharacteristic
526C22×C3D4211yescharacteristic
527C22×C3D4×Q811yescharacteristic
528C22×C3C2×#13511yescharacteristic
529C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
530C22×C3#57611yescharacteristic
531C22×C3#57711yescharacteristic
532C3×C4—22no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—81no—
536C3×C4C2×C22⋊Q811yescharacteristic
537C3×C4#53911yescharacteristic
538C3×C4#54711yescharacteristic
539C3×C4#55411yescharacteristic
540Dic3—22no—
541Dic3—41no—
542Dic3—42no—
543Dic3—41no—
544Dic3—41no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D3×D411yescharacteristic
563D4—22no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4#87814yes—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—32no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—62no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—244no—
608C2×C4—244no—
609C2×C4—244no—
610C2×C4#91212yes—
611C2×C4C22×Dic611yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4C2×#23711yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4#100911yescharacteristic
616C2×C4#101011yescharacteristic
617C8—22no—
618C8—41no—
619C8—62no—
620C8—121no—
621C2×C3—44no—
622C2×C3—82no—
623C2×C3#244911yescharacteristic
624C2×C3#310511yescharacteristic
625C2×C3#311911yescharacteristic
626C2×C3#326411yescharacteristic
627C2×C3#328411yescharacteristic
628C2×C3#338011yescharacteristic
629C2×C3#338411yescharacteristic
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22D4×Dic611yescharacteristic
645C22#558611yescharacteristic
646C22#560911yescharacteristic
647C22C2×#91211yescharacteristic
648C22#576911yescharacteristic
649C22D3×AGL(1,ℤ/8)11yescharacteristic
650C22#577511yescharacteristic
651C4—22no—
652C4—42no—
653C4—41no—
654C4—62no—
655C4—81no—
656C4—121no—
657C4—122no—
658C4—121no—
659C4—121no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4C2×#87811yescharacteristic
664C4#554311yescharacteristic
665C4#561111yescharacteristic
666C4#566511yescharacteristic
667C3#2638211yescharacteristic, complemented, Sylow
668C2—44no—
669C2—82no—
670C2#7471011yescharacteristic
671C2#7957411yescharacteristic
672C2#7985511yescharacteristic
673C2#7998911yescharacteristic
674C2#8007211yescharacteristic
675C2#8035111yescharacteristic
676C2#8055211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1879C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenter#1879D3
hypocenterC3#26382
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊Dic3C2semidirect
3. C2×#4766C2semidirect
4. #76325C2semidirect
7. #71607C2semidirect
8. #70193C2semidirect
9. #70169C2semidirect
10. #70168C2semidirect
11. #69656C2semidirect
12. #68858C2semidirect
13. #68829C2semidirect
14. #68639C2semidirect
15. #68610C2semidirect
667. C3#26382semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D82
D4×C4⋊Dic32
C2×#47662
#763252
C3×#18792
#720372
#716072
#701932
#701692
#701682
#696562
#688582
#688292
#686392
#686102
#263823