Finite Group Databasebeta

← #10845214

Subgroups — #10845214

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×C3×#490C211yesmaximal, characteristic
2#83173C211yesmaximal, characteristic
3C2×#5075C211yesmaximal, characteristic
4#83039C211yesmaximal, characteristic, complemented
5#71738C211yesmaximal, characteristic, complemented
6#70217C211yesmaximal, characteristic, complemented
7#70217C211yesmaximal, characteristic, complemented
8#70171C211yesmaximal, characteristic, complemented
9#70171C211yesmaximal, characteristic, complemented
10#70166C211yesmaximal, characteristic, complemented
11#70165C211yesmaximal, characteristic, complemented
12#70154C211yesmaximal, characteristic, complemented
13#70153C211yesmaximal, characteristic, complemented
14#69836C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#25332—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×D4×Dic3—21no—
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#490—24no—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31#5221—21no—
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34#5220—21no—
35#5220—21no—
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#5075—24no—
43#4982—21no—
44#4982—21no—
45#4978—21no—
46#4978—21no—
47#4977C2211yescharacteristic
48#4975C2211yescharacteristic
49#4973C2211yescharacteristic
50#4973C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490D311yescharacteristic
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#1877—31no—
74#1804—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C23×Dic3—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#127—22no—
87C3×#127D414yes—
88C3×#125—22no—
89C3×#125—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101D4×Dic3—22no—
102D4×Dic3—22no—
103D4×Dic3—41no—
104D4×Dic3—41no—
105D4×Dic3—44no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214—22no—
109C2×#214—22no—
110C2×#214—41no—
111C2×#214—41no—
112C2×C4⋊Dic3D411yescharacteristic
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#97—22no—
133#832—22no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95D411yescharacteristic
143C3⋊#95D411yescharacteristic
144#816—22no—
145#816—22no—
146Dic3⋊C8—41no—
147Dic3⋊C8—41no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127C2×D311yescharacteristic
152C2×#125C2×D311yescharacteristic
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—61no—
157#490—24no—
158#460—64no—
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×#97C2×D311yescharacteristic
171#398—61no—
172#398—61no—
173#391—31no—
174#391—31no—
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#340—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×D4—24no—
197C2×C3×D4—24no—
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—81no—
211C2×C3×D4—81no—
212C2×C3×D4C2×D412yes—
213C2×C3×D4C2×D412yes—
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×Dic3—42no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C22×Dic3—42no—
227C22×Dic3—81no—
228C22×Dic3—81no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6#4112yes—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—81no—
258#217—81no—
259#214—44no—
260#214—44no—
261C4⋊Dic3—21no—
262C4⋊Dic3—22no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—41no—
268C4⋊Dic3#4112yes—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—42no—
277C4×Dic3—42no—
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×Q8—31no—
285C22×D4—21no—
286C22×D4—21no—
287C22×D4C22×D311yescharacteristic
288C22×D4C22×D311yescharacteristic
289C23×C4—61no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×SD16—121no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#127—22no—
302#127#6714yes—
303#125—22no—
304#125—22no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—121no—
310C4×D4—121no—
311C4×D4—124no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—61no—
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—62no—
339#103—62no—
340#102—22no—
341#102—22no—
342#102—22no—
343#102—22no—
344#102—62no—
345#102—62no—
346#97—22no—
347#95—31no—
348#95—31no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—22no—
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×C3—41no—
369C23×C3—41no—
370C23×C3C22×D411yescharacteristic
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—48no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—84no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4#12112yes—
392C2×C3×C4#12112yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—22no—
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—84no—
423C2×Dic3—84no—
424C2×Dic3AGL(1,ℤ/8)12yes—
425Dic6—22no—
426Dic6—22no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—41no—
430C3×C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—21no—
434C24—21no—
435C2×Q8—32no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—24no—
442C2×D4—24no—
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—81no—
456C2×D4—81no—
457C2×D4C2×#6712yes—
458C2×D4C2×#6712yes—
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×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4—241no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
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—21no—
496C4⋊C4—32no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—62no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—121no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C42—61no—
516C42—61no—
517C42—61no—
518C42—61no—
519C42—122no—
520C42—122no—
521C42#23712yes—
522C42C2×#6712yes—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3C2×#12111yescharacteristic
538C22×C3#54611yescharacteristic
539C22×C3#54811yescharacteristic
540C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
541C22×C3#57511yescharacteristic
542C22×C3#57711yescharacteristic
543C22×C3#57711yescharacteristic
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—81no—
547C3×C4—81no—
548C3×C4#52911yescharacteristic
549C3×C4#54211yescharacteristic
550C3×C4#54611yescharacteristic
551C3×C4#54711yescharacteristic
552Dic3—22no—
553Dic3—42no—
554Dic3—41no—
555Dic3—42no—
556Dic3—41no—
557Dic3—41no—
558Dic3—81no—
559Dic3—81no—
560C23—22no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23C2×D3×D411yescharacteristic
575Q8—62no—
576Q8—62no—
577Q8—121no—
578Q8—121no—
579D4—22no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4—48no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—32no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—244no—
624C2×C4—244no—
625C2×C4#93912yes—
626C2×C4#93912yes—
627C2×C4C2×#23711yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4#100911yescharacteristic
631C2×C4#101011yescharacteristic
632C2×C4C22×#6711yescharacteristic
633C8—41no—
634C8—41no—
635C8—121no—
636C8—121no—
637C2×C3—82no—
638C2×C3—82no—
639C2×C3#242711yescharacteristic
640C2×C3#303611yescharacteristic
641C2×C3#304211yescharacteristic
642C2×C3#331011yescharacteristic
643C2×C3#331711yescharacteristic
644C2×C3#336811yescharacteristic
645C2×C3#337111yescharacteristic
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22#567611yescharacteristic
661C22#567711yescharacteristic
662C22D3×AGL(1,ℤ/8)11yescharacteristic
663C22#577311yescharacteristic
664C22#577511yescharacteristic
665C22#577511yescharacteristic
666C22C2×#93911yescharacteristic
667C4—22no—
668C4—22no—
669C4—62no—
670C4—81no—
671C4—81no—
672C4—122no—
673C4—121no—
674C4—122no—
675C4—121no—
676C4—121no—
677C4—241no—
678C4—241no—
679C4#568011yescharacteristic
680C4#568211yescharacteristic
681C4#582411yescharacteristic
682C4#582911yescharacteristic
683C3#2533211yescharacteristic, complemented, Sylow
684C2—82no—
685C2—82no—
686C2#8170911yescharacteristic
687C2#8171111yescharacteristic
688C2#8172511yescharacteristic
689C2#8172611yescharacteristic
690C2#8299611yescharacteristic
691C2#8393111yescharacteristic
692C2#8393511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#490C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#490D3
hypocenterC3#25332
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
4. #83039C2semidirect
5. #71738C2semidirect
6. #70217C2semidirect
7. #70217C2semidirect
8. #70171C2semidirect
9. #70171C2semidirect
10. #70166C2semidirect
11. #70165C2semidirect
12. #70154C2semidirect
13. #70153C2semidirect
14. #69836C2semidirect
15. #69813C2semidirect
683. C3#25332semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4902
#831732
C2×#50752
#830392
#717382
#702172
#701712
#701662
#701652
#701542
#701532
#698362
#698132
#253323