Finite Group Databasebeta

← #10844922

Subgroups — #10844922

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic
2#83179C211yesmaximal, characteristic
3#82481C211yesmaximal, characteristic, complemented
4C2×#4828C211yesmaximal, characteristic, complemented
5#72051C211yesmaximal, characteristic, complemented
6#72051C211yesmaximal, characteristic, complemented
7#72050C211yesmaximal, characteristic, complemented
8#72049C211yesmaximal, characteristic, complemented
9#71765C211yesmaximal, characteristic
10#71631C211yesmaximal, characteristic, complemented
11#70455C211yesmaximal, characteristic, complemented
12#70188C211yesmaximal, characteristic, complemented
13#70186C211yesmaximal, characteristic, complemented
14#70180C211yesmaximal, characteristic, complemented
15#70179C211yesmaximal, characteristic, complemented
16#25310—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#936—21no—
22C2×C4×Dic6C2211yescharacteristic
23C3×#460—24no—
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27#5229C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5222—21no—
32#5222—21no—
33#5221—21no—
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36#5221C2211yescharacteristic
37#5221C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41C2×#832C2211yescharacteristic
42C2×#818C2211yescharacteristic
43C2×#817C2211yescharacteristic
44C2×#815C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46C2×Dic3⋊C8C2211yescharacteristic
47C2×C8×Dic3C2211yescharacteristic
48#4986—21no—
49#4979—21no—
50#4972C2211yescharacteristic
51#4972C2211yescharacteristic
52#4936C2211yescharacteristic
53C4×C4⋊Dic3C2211yescharacteristic
54#4933—21no—
55#4933—21no—
56C2×#803C2211yescharacteristic
57#4828—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#460D311yescharacteristic
63C2×#446—31no—
64#2581—31no—
65#2548—31no—
66#2067—31no—
67#2065—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1961—31no—
73#1949—31no—
74#1936—31no—
75#1934—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C22×Dic6D411yescharacteristic
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C2×C3×SD16—41no—
88C2×C3×SD16—41no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#127—22no—
92C3×#127—22no—
93C3×C4×Q8—22no—
94C3×C4×Q8—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107#936—44no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×C4⋊C8D414yes—
125C3×#102—22no—
126C3×#102—22no—
127#835—22no—
128#835—22no—
129#832—22no—
130#832—22no—
131#832—41no—
132#832—41no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#818—41no—
143#817—41no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147C8×Dic3—22no—
148#803—22no—
149C22×SD16C2×D311yescharacteristic
150C22×SD16C2×D311yescharacteristic
151C2×#127C2×D311yescharacteristic
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8C2×D311yescharacteristic
155#460—24no—
156#446—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8—31no—
173#400—61no—
174#395—61no—
175#394—61no—
176#394—61no—
177#391—31no—
178#391—31no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#390—61no—
184#389—31no—
185#389—31no—
186#387—61no—
187#387—61no—
188#382—31no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—42no—
215C2×C3×D4—42no—
216C2×C3×D4—42no—
217C2×C3×D4—42no—
218C2×C3×D4—81no—
219C2×C3×D4—81no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6—41no—
236C3×SD16—44no—
237C3×SD16—44no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C2×D412yes—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C42—42no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—42no—
254#217—81no—
255#217—81no—
256#217—81no—
257#217—81no—
258C4⋊Dic3—22no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267Dic3⋊C4—22no—
268Dic3⋊C4—22no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8#4112yes—
282C22×Q8—31no—
283C22×Q8C22×D311yescharacteristic
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C22×D311yescharacteristic
287C22×D4C22×D311yescharacteristic
288C2×SD16—22no—
289C2×SD16—22no—
290C2×SD16—22no—
291C2×SD16—22no—
292C2×SD16—41no—
293C2×SD16—41no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#127—22no—
299#127—22no—
300#124—124no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—62no—
304C4×Q8—62no—
305C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
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×C42—21no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—121no—
330#106—121no—
331C4⋊C8—62no—
332C4⋊C8—62no—
333C4⋊C8#6714yes—
334#103—62no—
335#103—62no—
336#102—22no—
337#102—22no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—62no—
343#97—62no—
344C4×C8—62no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
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×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3C22×D411yescharacteristic
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—84no—
382C3×D4—84no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4#12112yes—
394C2×C3×C4#12112yes—
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C42−1+411yescharacteristic
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3AGL(1,ℤ/8)12yes—
425Dic6—44no—
426C3×C8—22no—
427C3×C8—22no—
428C3⋊C8—22no—
429C3⋊C8—41no—
430C24—21no—
431C24—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—62no—
438C2×Q8—62no—
439C2×Q8—121no—
440C2×Q8#23712yes—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—81no—
458C2×D4—81no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—44no—
473SD16—44no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—32no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8C2×#6712yes—
485C2×C8C2×#6712yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C42—42no—
511C42—61no—
512C42—61no—
513C42—61no—
514C42—61no—
515C42—122no—
516C42#23712yes—
517C42C2×#6712yes—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3C2×#12111yescharacteristic
533C22×C3#54811yescharacteristic
534C22×C3#55011yescharacteristic
535C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
536C22×C3#57511yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57711yescharacteristic
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—42no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54811yescharacteristic
547Dic3—22no—
548Dic3—41no—
549Dic3—41no—
550Dic3—41no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554Dic3—81no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23C2×D3×D411yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—124no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—84no—
584D4—84no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—32no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
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—61no—
605C2×C4—61no—
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—121no—
618C2×C4—121no—
619C2×C4#93812yes—
620C2×C4#93812yes—
621C2×C4C2×#23711yescharacteristic
622C2×C4C2×#23711yescharacteristic
623C2×C4#100411yescharacteristic
624C2×C4D3×#4111yescharacteristic
625C2×C4#101011yescharacteristic
626C2×C4C22×#6711yescharacteristic
627C8—22no—
628C8—22no—
629C8—62no—
630C8—121no—
631C2×C3—161no—
632C2×C3—161no—
633C2×C3#242611yescharacteristic
634C2×C3#305511yescharacteristic
635C2×C3#305711yescharacteristic
636C2×C3#336811yescharacteristic
637C2×C3#337111yescharacteristic
638C2×C3#339711yescharacteristic
639C2×C3#340111yescharacteristic
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22#559811yescharacteristic
655C22#559911yescharacteristic
656C22#576211yescharacteristic
657C22#576211yescharacteristic
658C22D3×AGL(1,ℤ/8)11yescharacteristic
659C22#577311yescharacteristic
660C22C2×#93811yescharacteristic
661C4—22no—
662C4—22no—
663C4—42no—
664C4—62no—
665C4—81no—
666C4—121no—
667C4—121no—
668C4—121no—
669C4—241no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#560011yescharacteristic
674C4#560311yescharacteristic
675C4#580611yescharacteristic
676C4#583111yescharacteristic
677C3#2531011yescharacteristic, complemented, Sylow
678C2—161no—
679C2—161no—
680C2#8037211yescharacteristic
681C2#8037311yescharacteristic
682C2#8037611yescharacteristic
683C2#8037711yescharacteristic
684C2#8264611yescharacteristic
685C2#8386311yescharacteristic
686C2#8395011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#460D3
hypocenterC3#25310
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
3. #82481C2semidirect
4. C2×#4828C2semidirect
5. #72051C2semidirect
6. #72051C2semidirect
7. #72050C2semidirect
8. #72049C2semidirect
10. #71631C2semidirect
11. #70455C2semidirect
12. #70188C2semidirect
13. #70186C2semidirect
14. #70180C2semidirect
15. #70179C2semidirect
677. C3#25310semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4602
#831792
#824812
C2×#48282
#720512
#720502
#720492
#717652
#716312
#704552
#701882
#701862
#701802
#701792
#253103