Finite Group Databasebeta

← #10844928

Subgroups — #10844928

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
1#83179C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic
3C2×#4701C211yesmaximal, characteristic, complemented
4#74191C211yesmaximal, characteristic, complemented
5C3×#1961C211yesmaximal, characteristic
6#70217C211yesmaximal, characteristic, complemented
7#70217C211yesmaximal, characteristic, complemented
8#70194C211yesmaximal, characteristic, complemented
9#70193C211yesmaximal, characteristic, complemented
10#70192C211yesmaximal, characteristic, complemented
11#70189C211yesmaximal, characteristic, complemented
12#70187C211yesmaximal, characteristic, complemented
13#70185C211yesmaximal, characteristic, complemented
14#69813C211yesmaximal, characteristic, complemented
15#69540C211yesmaximal, characteristic, complemented
16#25310—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×#936—21no—
19C22×#214C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×#382C2211yescharacteristic
29#5222—21no—
30#5222—21no—
31#5221—21no—
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#815C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43C2×C8×Dic3C2211yescharacteristic
44#5075—24no—
45#4984C2211yescharacteristic
46#4984C2211yescharacteristic
47#4973C2211yescharacteristic
48#4973C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53#4701—24no—
54#4659—21no—
55#4656—21no—
56#4653—21no—
57#4653—21no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#460—31no—
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#1961D311yescharacteristic
73#1949—31no—
74#1936—31no—
75#1934—31no—
76#1763—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C22×Dic6D411yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#127—22no—
86C3×#127—22no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98#936—44no—
99C2×#214—22no—
100C2×#214—22no—
101C2×#214—22no—
102C2×#214—22no—
103C2×#214—41no—
104C2×#214—41no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×Dic3⋊C4D411yescharacteristic
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C4×Dic6—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#102—41no—
127C3×#102—41no—
128C3×#97—22no—
129C3×#95—21no—
130C3×#95—21no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#816—22no—
141#816—22no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146#808—41no—
147#807—41no—
148C4⋊(C3⋊C8)D414yes—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127C2×D311yescharacteristic
152C2×#124—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155#460—64no—
156#446—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×#97—31no—
171C2×#97C2×D311yescharacteristic
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#382C2×D311yescharacteristic
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—21no—
195C24×C3—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×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—42no—
209C2×C3×D4—42no—
210C2×C3×D4—42no—
211C2×C3×D4—42no—
212C2×C3×D4—81no—
213C2×C3×D4—81no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—22no—
232C2×Dic6—21no—
233C2×Dic6—22no—
234C2×Dic6—41no—
235C2×Dic6#4112yes—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8#4112yes—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C42C2×D412yes—
247C3×C42#4112yes—
248#217—42no—
249#217—42no—
250#217—42no—
251#217—42no—
252#217—81no—
253#217—81no—
254#217—81no—
255#217—81no—
256#214—44no—
257#214—44no—
258C4⋊Dic3—22no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—42no—
266C4⋊Dic3—42no—
267Dic3⋊C4—22no—
268Dic3⋊C4—22no—
269Dic3⋊C4—42no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8C2×D412yes—
282C22×Q8—31no—
283C22×Q8—31no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4C22×D311yescharacteristic
287C22×D4C22×D311yescharacteristic
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—121no—
293C2×SD16—121no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#127—22no—
299#127—22no—
300#124—124no—
301C4×Q8—62no—
302C4×Q8—62no—
303C4×Q8—62no—
304C4×Q8—62no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
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—31no—
325C2×C42—31no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—121no—
330#106—121no—
331C4⋊C8—34no—
332C4⋊C8—62no—
333C4⋊C8—62no—
334#103—62no—
335#103—62no—
336#102—22no—
337#102—22no—
338#102—41no—
339#102—41no—
340#102—62no—
341#102—62no—
342#97—22no—
343#97—62no—
344C4×C8—62no—
345#95—21no—
346#95—21no—
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×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—84no—
378C3×D4—84no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12112yes—
390C2×C3×C4#12112yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396C2×C3×C42−1+411yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
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—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3AGL(1,ℤ/8)12yes—
421Dic6—22no—
422Dic6—22no—
423Dic6—44no—
424Dic6—41no—
425Dic6—41no—
426C3×C8—22no—
427C3×C8—41no—
428C3⋊C8—22no—
429C3⋊C8—22no—
430C24—21no—
431C24—21no—
432C2×Q8—32no—
433C2×Q8—62no—
434C2×Q8—62no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—121no—
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—124no—
473SD16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—32no—
480C2×C8—32no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8#23712yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
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—61no—
511C42—61no—
512C42—61no—
513C42—61no—
514C42—122no—
515C42—122no—
516C42#23512yes—
517C42D3×D412yes—
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—81no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54811yescharacteristic
547Dic3—22no—
548Dic3—42no—
549Dic3—41no—
550Dic3—41no—
551Dic3—41no—
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—62no—
571Q8—62no—
572Q8—124no—
573Q8—121no—
574Q8—121no—
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—32no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—62no—
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—61no—
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#88412yes—
620C2×C4#88412yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4C2×#23711yescharacteristic
624C2×C4#100411yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4#101011yescharacteristic
627C8—22no—
628C8—41no—
629C8—62no—
630C8—62no—
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—
654C22C2×#88411yescharacteristic
655C22#567711yescharacteristic
656C22#571811yescharacteristic
657C22D3×AGL(1,ℤ/8)11yescharacteristic
658C22#577311yescharacteristic
659C22#577511yescharacteristic
660C22#577511yescharacteristic
661C4—22no—
662C4—22no—
663C4—62no—
664C4—81no—
665C4—81no—
666C4—121no—
667C4—121no—
668C4—121no—
669C4—122no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#555711yescharacteristic
674C4#556111yescharacteristic
675C4#559811yescharacteristic
676C4#561211yescharacteristic
677C3#2531011yescharacteristic, complemented, Sylow
678C2—161no—
679C2—161no—
680C2#7644311yescharacteristic
681C2#7893011yescharacteristic
682C2#7893111yescharacteristic
683C2#8170911yescharacteristic
684C2#8171111yescharacteristic
685C2#8183911yescharacteristic
686C2#8184111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1961C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#1961D3
hypocenterC3#25310
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
3. C2×#4701C2semidirect
4. #74191C2semidirect
6. #70217C2semidirect
7. #70217C2semidirect
8. #70194C2semidirect
9. #70193C2semidirect
10. #70192C2semidirect
11. #70189C2semidirect
12. #70187C2semidirect
13. #70185C2semidirect
14. #69813C2semidirect
15. #69540C2semidirect
677. C3#25310semidirect

Maximal subgroups

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

subgroupindex
#831792
C2×#50752
C2×#47012
#741912
C3×#19612
#702172
#701942
#701932
#701922
#701892
#701872
#701852
#698132
#695402
#253103