Finite Group Databasebeta

← #10858751

Subgroups — #10858751

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73492C211yesmaximal, characteristic, complemented
2#73469C211yesmaximal, characteristic
3C3×#1850C211yesmaximal, characteristic, complemented
4#71973C211yesmaximal, characteristic
5#71969C211yesmaximal, characteristic, complemented
6#70483C211yesmaximal, characteristic, complemented
7#70480C211yesmaximal, characteristic, complemented
8#70464C211yesmaximal, characteristic, complemented
9#70460C211yesmaximal, characteristic, complemented
10#69933C211yesmaximal, characteristic
11#69929C211yesmaximal, characteristic, complemented
12#68678C211yesmaximal, characteristic, complemented
13#68670C211yesmaximal, characteristic, complemented
14#68644C211yesmaximal, characteristic, complemented
15#68624C211yesmaximal, characteristic, complemented
16#26409—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#817C2211yescharacteristic
34#4957—21no—
35#4956—21no—
36#4948—21no—
37#4947—21no—
38#4939C2211yescharacteristic
39#4936C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4669—21no—
45#4666—21no—
46#4655—21no—
47#4646—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2439—31no—
61#2363—31no—
62#2101—31no—
63#2088—31no—
64#2065—31no—
65#2023—31no—
66#1988—31no—
67#1921—31no—
68#1893—31no—
69#1875—31no—
70#1862—31no—
71#1854—31no—
72#1850D311yescharacteristic, complemented
73#1804—31no—
74#1735—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C22×C3×C8—22no—
81C22×C3×C8—41no—
82C22×C3×C8—41no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94#930—22no—
95#930Q814yescomplemented
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100#926—22no—
101#926—22no—
102#926—41no—
103#926—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C4×D12—22no—
119C4×D12—41no—
120#869—22no—
121#869—41no—
122C3×C4⋊C8—41no—
123C3×C4⋊C8—41no—
124C3×C22⋊C8—22no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#821—22no—
135#821—41no—
136#821—41no—
137#821—41no—
138#818—22no—
139#817—41no—
140C23×C8C2×D311yescharacteristic
141C2×C4⋊Q8—31no—
142C2×#121—31no—
143C2×C4×D4—31no—
144C2×#117—31no—
145C2×#117—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×C4⋊C8C2×D311yescharacteristic
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×C22⋊C8C2×D311yescharacteristic
155#396—61no—
156#393—61no—
157#391—61no—
158#390—31no—
159#390—61no—
160#389—31no—
161#389—31no—
162#389—31no—
163#388—61no—
164#387—61no—
165#386—61no—
166#382—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C22⋊C4—61no—
170C4×C22⋊C4C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C24×C3C2×D411yescharacteristic
184C23×D3—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×Q812yes—
195C22×C3×C4#4112yes—
196C22×C3×C4C2411yescharacteristic
197C2×#67—24no—
198C2×#67—81no—
199C2×#67—81no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C2×Q812yes—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×Dic6—42no—
217C2×Dic6—42no—
218C2×Dic6—81no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—44no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—41no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238#217—42no—
239#217—42no—
240#217—81no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—81no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3C2×Q812yes—
258C4⋊Dic3#4112yes—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—81no—
262Dic3⋊C4—81no—
263C4×Dic3—42no—
264C4×Dic3—42no—
265C4×Dic3—81no—
266C22×Q8—61no—
267C22×D4—31no—
268C22×D4—61no—
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271C22×C8—22no—
272C22×C8—41no—
273C22×C8—41no—
274C22×C8C22×D311yescharacteristic
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4⋊Q8—62no—
279C4⋊Q8—121no—
280#121—34no—
281#121—62no—
282C4×D4—62no—
283C4×D4—121no—
284#117—62no—
285#117—62no—
286#117—121no—
287#117—121no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4D1211yescharacteristic
307C2×C22⋊C4D1211yescharacteristic
308C2×C42—61no—
309C2×C42—61no—
310C2×C42—61no—
311C2×C42C22×D311yescharacteristic
312C2×C42C22×D311yescharacteristic
313#107—62no—
314#106—121no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317#102—62no—
318#102—121no—
319#102—121no—
320#102—121no—
321C22⋊C8—22no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95D1211yescharacteristic
330#95D1211yescharacteristic
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3C22⋊Q812yes—
338C23×C3C22×D411yescharacteristic
339C22×D3—22no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4#13512yes—
365C2×C3×C4C22×Q811yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371#67—84no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393D12—22no—
394D12—22no—
395D12—41no—
396D12—41no—
397C4×D3—84no—
398Dic6—84no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8—41no—
403C24—61no—
404C24C2×D1211yescharacteristic
405C2×Q8—122no—
406C2×Q8—122no—
407C2×Q8—241no—
408C2×D4—32no—
409C2×D4—64no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—241no—
416C2×D4—241no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4C2×D1211yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4C2×D1211yescharacteristic
434C22×C4C2×D1211yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4#23712yes—
437C22×C4D3×Q812yes—
438C22×C4C23×D311yescharacteristic
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—44no—
452C4⋊C4—32no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—241no—
469C4⋊C4—241no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—64no—
473C22⋊C4—122no—
474C22⋊C4—122no—
475C22⋊C4—122no—
476C22⋊C4—122no—
477C22⋊C4—241no—
478C22⋊C4—241no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—122no—
484C42—122no—
485C42—241no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3C2×C22⋊Q811yescharacteristic
494C22×C3#54611yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3C2×#13511yescharacteristic
497C22×C3#57511yescharacteristic
498C22×C3#57611yescharacteristic
499C22×C3#57711yescharacteristic
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4#52911yescharacteristic
513C3×C4#53811yescharacteristic
514C3×C4#55411yescharacteristic
515C3×C4#55411yescharacteristic
516Dic3—42no—
517Dic3—42no—
518Dic3—42no—
519Dic3—81no—
520Dic3—81no—
521Dic3—81no—
522Dic3—81no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23#89712yes—
537C23C22×D1211yescharacteristic
538Q8—244no—
539D4—62no—
540D4—62no—
541D4—121no—
542D4—121no—
543D4—244no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—244no—
583C2×C4#89712yes—
584C2×C4#90412yes—
585C2×C4C2×#23711yescharacteristic
586C2×C4#100411yescharacteristic
587C2×C4#100411yescharacteristic
588C2×C4C2×D3×Q811yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—41no—
592C8—41no—
593C8—41no—
594C8—41no—
595C2×C3—42no—
596C2×C3#245011yescharacteristic
597C2×C3#307611yescharacteristic
598C2×C3#308111yescharacteristic
599C2×C3#329711yescharacteristic
600C2×C3#330411yescharacteristic
601C2×C3#331911yescharacteristic
602C2×C3#332111yescharacteristic
603D3—82no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22C2×#89711yescharacteristic
619C22#555411yescharacteristic
620C22#555411yescharacteristic
621C22C2×#90411yescharacteristic
622C22#574811yescharacteristic
623C22#575111yescharacteristic
624C22#575211yescharacteristic
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—122no—
631C4—122no—
632C4—122no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4—241no—
637C4#559311yescharacteristic
638C4#563611yescharacteristic
639C4#563611yescharacteristic
640C4#564911yescharacteristic
641C3#2640911yescharacteristic, complemented, Sylow
642C2—42no—
643C2—242no—
644C2#7442411yescharacteristic
645C2#7902511yescharacteristic
646C2#7909011yescharacteristic
647C2#7944111yescharacteristic
648C2#7945111yescharacteristic
649C2#7950611yescharacteristic
650C2#7951611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1850C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1850D3
hypocenterC3#26409
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. #73492C2semidirect
3. C3×#1850C2semidirect
5. #71969C2semidirect
6. #70483C2semidirect
7. #70480C2semidirect
8. #70464C2semidirect
9. #70460C2semidirect
11. #69929C2semidirect
12. #68678C2semidirect
13. #68670C2semidirect
14. #68644C2semidirect
15. #68624C2semidirect
72. #1850D3semidirect
95. #930Q8semidirect
641. C3#26409semidirect

Maximal subgroups

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

subgroupindex
#734922
#734692
C3×#18502
#719732
#719692
#704832
#704802
#704642
#704602
#699332
#699292
#686782
#686702
#686442
#686242
#264093