Finite Group Databasebeta

← #10845213

Subgroups — #10845213

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5163C211yesmaximal, characteristic
2#82343C211yesmaximal, characteristic
3C2×#4764C211yesmaximal, characteristic
4#76542C211yesmaximal, characteristic, complemented
5C3×#1877C211yesmaximal, characteristic, complemented
6#72051C211yesmaximal, characteristic, complemented
7#71605C211yesmaximal, characteristic, complemented
8#70276C211yesmaximal, characteristic, complemented
9#70220C211yesmaximal, characteristic, complemented
10#70165C211yesmaximal, characteristic, complemented
11#69680C211yesmaximal, characteristic, complemented
12#68892C211yesmaximal, characteristic, complemented
13#68801C211yesmaximal, characteristic, complemented
14#68637C211yesmaximal, characteristic, complemented
15#68618C211yesmaximal, characteristic, complemented
16#25332—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#940C2211yescharacteristic
22C2×C4×#67—21no—
23C2×#893C2211yescharacteristic
24C2×#890C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5213—21no—
34#5163—24no—
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4957—21no—
44#4956—21no—
45#4939C2211yescharacteristic
46#4936C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49#4764—24no—
50#4657—21no—
51#4657—21no—
52#4655—21no—
53#4655—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490—31no—
63C2×#460—31no—
64#2565—31no—
65#2366—31no—
66#2073—31no—
67#2071—31no—
68#2055—31no—
69#2055—31no—
70#1989—31no—
71#1940—31no—
72#1877—31no—
73#1877D311yescharacteristic, complemented
74#1804—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×#67—21no—
81C22×#67—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—41no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×D4—22no—
90C3×C4×D4—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99C2×#215—22no—
100C2×#215—22no—
101C2×#215—41no—
102#940—22no—
103#940D414yes—
104C4×#67—44no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#893—22no—
125#893—41no—
126#890—22no—
127#890—22no—
128C3×#103—22no—
129C3×#102—22no—
130C3×#95—21no—
131C3×#95—21no—
132#835—22no—
133#832—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#821—22no—
143#821—22no—
144#815—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—41no—
147#809—22no—
148C4⋊(C3⋊C8)—41no—
149C22×SD16—31no—
150C22×SD16C2×D311yescharacteristic
151C2×#127—31no—
152C2×#125—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155C2×C4×D4—61no—
156C2×C4×D4C2×D311yescharacteristic
157#490—64no—
158#460—64no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102C2×D311yescharacteristic
170C2×#97—31no—
171#398—61no—
172#398—61no—
173#391—31no—
174#391C2×D311yescharacteristic
175#390—31no—
176#390—31no—
177#390—61no—
178#387—61no—
179#387—61no—
180#386—61no—
181#386—61no—
182#384—31no—
183#382—31no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#336—31no—
193#336C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3—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×D4C2×D412yes—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4—81no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C2×#67—24no—
218C2×#67—24no—
219C2×#67—42no—
220C2×#67—42no—
221C2×#67—81no—
222C2×#67—81no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12C2×D412yes—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—81no—
239C3×SD16—44no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4#4112yes—
250C3×C22⋊C4—42no—
251C3×C42—21no—
252C3×C42—21no—
253#217—42no—
254#217—81no—
255#215—44no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
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—41no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3C2×D412yes—
277C4×Dic3#4112yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—61no—
287C22×D4—61no—
288C22×D4C22×D311yescharacteristic
289C23×C4—21no—
290C23×C4—61no—
291C2×SD16—22no—
292C2×SD16—22no—
293C2×SD16—41no—
294C2×SD16—62no—
295C2×SD16—62no—
296C2×SD16—121no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#127—34no—
302#127—62no—
303#125—62no—
304#125—62no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×D4—22no—
308C4×D4—41no—
309C4×D4—62no—
310C4×D4—121no—
311C4×D4—124no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42C22×D311yescharacteristic
335C4⋊C8—62no—
336C4⋊C8—121no—
337C4⋊C8—121no—
338#103—22no—
339#103—62no—
340#102—22no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346#97—62no—
347#95—21no—
348#95—21no—
349#95—31no—
350#95—31no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3C22×D411yescharacteristic
364C22×D3—22no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—41no—
378C3×D4—41no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—84no—
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
401#67—48no—
402#67—84no—
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#12112yes—
422C2×Dic3#12112yes—
423C2×Dic3AGL(1,ℤ/8)12yes—
424D12—22no—
425D12—22no—
426D12—41no—
427D12—41no—
428C4×D3—84no—
429C3×C8—41no—
430C3×C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—21no—
434C24—61no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8#23512yes—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—32no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—241no—
457C2×D4—241no—
458C2×D4D3×D412yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—81no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—44no—
479SD16—124no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—62no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—121no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4#23712yes—
509C22⋊C4—42no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C42—21no—
516C42—21no—
517C42—32no—
518C42—32no—
519C42—61no—
520C42—61no—
521C42—122no—
522C42—122no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3C2×#12111yescharacteristic
531C22×C3#54611yescharacteristic
532C22×C3#54811yescharacteristic
533C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57711yescharacteristic
536C22×C3#57711yescharacteristic
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—42no—
545C3×C4—42no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4—81no—
549C3×C4#52911yescharacteristic
550C3×C4#54211yescharacteristic
551C3×C4#54611yescharacteristic
552C3×C4#54711yescharacteristic
553Dic3—22no—
554Dic3—22no—
555Dic3—22no—
556Dic3—41no—
557Dic3—81no—
558Dic3—81no—
559Dic3—81no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—62no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C2×D3×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579D4—22no—
580D4—22no—
581D4—41no—
582D4—41no—
583D4—62no—
584D4—62no—
585D4—128no—
586D4—121no—
587D4—121no—
588D4—244no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—32no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—84no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—244no—
627C2×C4C2×#23511yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4#100411yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4#101011yescharacteristic
633C8—41no—
634C8—41no—
635C8—121no—
636C8—121no—
637C2×C3—82no—
638C2×C3#242711yescharacteristic
639C2×C3#303611yescharacteristic
640C2×C3#304211yescharacteristic
641C2×C3#331011yescharacteristic
642C2×C3#331711yescharacteristic
643C2×C3#336811yescharacteristic
644C2×C3#337111yescharacteristic
645D3—82no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22#555211yescharacteristic
661C22#559111yescharacteristic
662C22D3×#12111yescharacteristic
663C22#576211yescharacteristic
664C22#576811yescharacteristic
665C22D3×AGL(1,ℤ/8)11yescharacteristic
666C22#577811yescharacteristic
667C4—41no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—62no—
672C4—62no—
673C4—62no—
674C4—81no—
675C4—121no—
676C4—241no—
677C4—241no—
678C4—241no—
679C4#549011yescharacteristic
680C4#553811yescharacteristic
681C4#561111yescharacteristic
682C4#566211yescharacteristic
683C3#2533211yescharacteristic, complemented, Sylow
684C2—82no—
685C2—242no—
686C2#7469711yescharacteristic
687C2#7962711yescharacteristic
688C2#7975211yescharacteristic
689C2#8016111yescharacteristic
690C2#8025611yescharacteristic
691C2#8043011yescharacteristic
692C2#8053411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#1877D3
hypocenterC3#25332
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
4. #76542C2semidirect
5. C3×#1877C2semidirect
6. #72051C2semidirect
7. #71605C2semidirect
8. #70276C2semidirect
9. #70220C2semidirect
10. #70165C2semidirect
11. #69680C2semidirect
12. #68892C2semidirect
13. #68801C2semidirect
14. #68637C2semidirect
15. #68618C2semidirect
73. #1877D3semidirect
683. C3#25332semidirect

Maximal subgroups

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

subgroupindex
C2×#51632
#823432
C2×#47642
#765422
C3×#18772
#720512
#716052
#702762
#702202
#701652
#696802
#688922
#688012
#686372
#686182
#253323