Finite Group Databasebeta

← #10845470

Subgroups — #10845470

Order 768.

860 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#435C211yesmaximal, characteristic
2#82382C211yesmaximal, characteristic
3#82375C211yesmaximal, characteristic
4C2×#4850C211yesmaximal, characteristic, complemented
5#71989C211yesmaximal, characteristic, complemented
6#71988C211yesmaximal, characteristic, complemented
7#71972C211yesmaximal, characteristic, complemented
8#71971C211yesmaximal, characteristic, complemented
9#70455C211yesmaximal, characteristic, complemented
10#69939C211yesmaximal, characteristic, complemented
11#69938C211yesmaximal, characteristic, complemented
12#69937C211yesmaximal, characteristic, complemented
13#69936C211yesmaximal, characteristic, complemented
14#69828C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#25352—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#936—21no—
21C2×#924C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#435—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5222—21no—
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35#5212C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#817C2211yescharacteristic
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45#4983—21no—
46#4973—21no—
47#4953—21no—
48#4953—21no—
49#4944—21no—
50#4944—21no—
51#4936C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4850—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#477—31no—
63C2×#435D311yescharacteristic
64#2547—31no—
65#2536—31no—
66#2076—31no—
67#2075—31no—
68#2036—31no—
69#2035—31no—
70#1993—31no—
71#1949—31no—
72#1894—31no—
73#1892—31no—
74#1809—31no—
75#1808—31no—
76#1763—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6—21no—
81C22×Dic6C2311yescharacteristic
82C2×C3×M16—22no—
83C2×C3×M16—22no—
84C2×C3×M16—41no—
85C22×C3×C8—22no—
86C22×C3×C8—22no—
87C22×C3×C8—41no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106#936—44no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115#924—22no—
116#924—22no—
117#924—41no—
118#924—41no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124#869—22no—
125#869—22no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×C4⋊C8D414yes—
129C3×C22⋊C8—22no—
130C3×C22⋊C8—22no—
131C3×#97—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141#818—22no—
142#818—22no—
143#818—41no—
144#817—41no—
145#816—22no—
146#816—22no—
147#816—22no—
148#816—22no—
149C22×M16C2×D311yescharacteristic
150C23×C8C2×D311yescharacteristic
151C2×C4⋊Q8—31no—
152C2×#124—61no—
153C2×C22⋊Q8—31no—
154C2×C22⋊Q8—31no—
155C2×C4×Q8—31no—
156C2×C4×D4C2×D311yescharacteristic
157#477—64no—
158#435—24no—
159C2×#107—31no—
160C2×#107—31no—
161C2×#107—31no—
162C2×#106—31no—
163C2×C4⋊C8C2×D311yescharacteristic
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×C22⋊C8C2×D311yescharacteristic
169C2×C22⋊C8C2×D311yescharacteristic
170C2×#97C2×D311yescharacteristic
171#398—61no—
172#397—61no—
173#397—61no—
174#394—61no—
175#391—61no—
176#388—61no—
177#388—61no—
178#382—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184#380—31no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3C2×D411yescharacteristic
195C24×C3C2×D411yescharacteristic
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—41no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—42no—
230C2×Dic6—42no—
231C2×Dic6—42no—
232C2×Dic6#4112yes—
233C3×M16—44no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—44no—
244C2×C3×C8C2×D412yes—
245C2×C3×C8C2×D412yes—
246C2×C3×C8#4112yes—
247C3×C4⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252#217—42no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—42no—
258#217—42no—
259#217—42no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278C4×Dic3—42no—
279C4×Dic3—42no—
280C22×Q8—31no—
281C22×Q8—61no—
282C22×D4D1211yescharacteristic
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D311yescharacteristic
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—41no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—41no—
291C22×C8C22×D311yescharacteristic
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C4⋊Q8—62no—
296C4⋊Q8—62no—
297#124—124no—
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—121no—
301C22⋊Q8—121no—
302C4×Q8—62no—
303C4×Q8—62no—
304C4×D4—22no—
305C4×D4—22no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4D1211yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4D1211yescharacteristic
327C2×C22⋊C4D1211yescharacteristic
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#107—62no—
332#107—62no—
333#107—121no—
334#106—121no—
335C4⋊C8#6714yes—
336#103—62no—
337#103—62no—
338#103—62no—
339#103—62no—
340C22⋊C8—22no—
341C22⋊C8—22no—
342#97—22no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3AGL(1,ℤ/8)12yes—
366C23×C3C22×D411yescharacteristic
367C3×D4—44no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4#12112yes—
387C2×C3×C4#12112yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C2×Dic3—22no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
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—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418Dic6—22no—
419Dic6—22no—
420Dic6—41no—
421Dic6—41no—
422Dic6—84no—
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—41no—
427C24C2×D1211yescharacteristic
428C24C2×D1211yescharacteristic
429C2×Q8—32no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—122no—
436C2×Q8—122no—
437C2×Q8—122no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—41no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4C2×D1211yescharacteristic
464C22×C4C2×D1211yescharacteristic
465C22×C4C23×D311yescharacteristic
466M16—44no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—44no—
477C2×C8#23712yes—
478C2×C8C2×#6712yes—
479C2×C8C2×#6712yes—
480C4⋊C4—22no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—122no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C42—122no—
510C42—122no—
511C42#23512yes—
512C42C2×#6712yes—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3C2×#12111yescharacteristic
528C22×C3#54611yescharacteristic
529C22×C3#54711yescharacteristic
530C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
531C22×C3#57511yescharacteristic
532C22×C3#57811yescharacteristic
533C22×C3#57811yescharacteristic
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4#52911yescharacteristic
539C3×C4#54211yescharacteristic
540C3×C4#54611yescharacteristic
541C3×C4#54811yescharacteristic
542Dic3—42no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548Dic3—81no—
549Dic3—81no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23#90812yes—
564C23C22×D1211yescharacteristic
565Q8—62no—
566Q8—62no—
567Q8—121no—
568Q8—121no—
569Q8—244no—
570D4—44no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—62no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
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—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4#93012yes—
614C2×C4#93012yes—
615C2×C4C2×#23511yescharacteristic
616C2×C4C2×#23711yescharacteristic
617C2×C4#100411yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C2×C4#101011yescharacteristic
620C2×C4C22×#6711yescharacteristic
621C8—22no—
622C8—22no—
623C8—22no—
624C8—41no—
625C2×C3—42no—
626C2×C3—81no—
627C2×C3#242711yescharacteristic
628C2×C3#305511yescharacteristic
629C2×C3#305911yescharacteristic
630C2×C3#330811yescharacteristic
631C2×C3#331311yescharacteristic
632C2×C3#334311yescharacteristic
633C2×C3#334911yescharacteristic
634C22—22no—
635C22—22no—
636C22—22no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22#555411yescharacteristic
649C22#555511yescharacteristic
650C22C2×#90811yescharacteristic
651C22#574811yescharacteristic
652C22#575311yescharacteristic
653C22#575311yescharacteristic
654C22C2×#93011yescharacteristic
655C4—22no—
656C4—22no—
657C4—41no—
658C4—41no—
659C4—122no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4—241no—
666C4—241no—
667C4#555711yescharacteristic
668C4#556111yescharacteristic
669C4#580611yescharacteristic
670C4#583111yescharacteristic
671C3#2535211yescharacteristic, complemented, Sylow
672C2—42no—
673C2—81no—
674C2#7946911yescharacteristic
675C2#7947511yescharacteristic
676C2#7950911yescharacteristic
677C2#7951211yescharacteristic
678C2#8240811yescharacteristic
679C2#8377211yescharacteristic
680C2#8380711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#435C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenterC2×#435D3
hypocenterC3#25352
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#435

Complements

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

NH ≅ G/Ntype
4. C2×#4850C2semidirect
5. #71989C2semidirect
6. #71988C2semidirect
7. #71972C2semidirect
8. #71971C2semidirect
9. #70455C2semidirect
10. #69939C2semidirect
11. #69938C2semidirect
12. #69937C2semidirect
13. #69936C2semidirect
14. #69828C2semidirect
15. #69813C2semidirect
671. C3#25352semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4352
#823822
#823752
C2×#48502
#719892
#719882
#719722
#719712
#704552
#699392
#699382
#699372
#699362
#698282
#698132
#253523