Finite Group Databasebeta

← #10845274

Subgroups — #10845274

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#489C211yesmaximal, characteristic
2#83243C211yesmaximal, characteristic, complemented
3#82955C211yesmaximal, characteristic
4C2×#5054C211yesmaximal, characteristic
5#71780C211yesmaximal, characteristic, complemented
6#70285C211yesmaximal, characteristic, complemented
7#70239C211yesmaximal, characteristic, complemented
8#70233C211yesmaximal, characteristic, complemented
9#70221C211yesmaximal, characteristic, complemented
10#70217C211yesmaximal, characteristic
11#70171C211yesmaximal, characteristic
12#70165C211yesmaximal, characteristic
13#70153C211yesmaximal, characteristic
14#69836C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#25336—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#489—24no—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5230—21no—
31#5230—21no—
32#5229C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#5054—24no—
41#4975C2211yescharacteristic
42#4975C2211yescharacteristic
43#4973C2211yescharacteristic
44#4973C2211yescharacteristic
45#4972—21no—
46#4972—21no—
47#4951—21no—
48#4951—21no—
49#4950—21no—
50Dic3×C22⋊C4—21no—
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×#489D311yescharacteristic
63C2×#464—31no—
64#2567—31no—
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2056—31no—
69#2055—31no—
70#1988—31no—
71#1940—31no—
72#1878—31no—
73#1877—31no—
74#1803—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×Dic3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8—22no—
86C3×#124—22no—
87C3×#124D414yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C42C2311yescharacteristic
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97Q8×Dic3—22no—
98Q8×Dic3—41no—
99C2×#216—22no—
100C2×#216—22no—
101C2×#216—41no—
102D4×Dic3—22no—
103D4×Dic3—41no—
104C2×#214—22no—
105C2×#214—22no—
106C2×#214—41no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3D411yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C4×Dic6—22no—
126C4×Dic6—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#102—22no—
130C3×#102—22no—
131C3×#97—22no—
132#835—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143#816—22no—
144#816—22no—
145Dic3⋊C8—41no—
146Dic3⋊C8—41no—
147C4⋊(C3⋊C8)—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#489—24no—
156#464—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169#395—61no—
170#395—61no—
171#391—31no—
172#391—31no—
173#391—31no—
174#390—31no—
175#389—61no—
176#389—61no—
177#388—61no—
178#388—61no—
179#385—61no—
180#384—31no—
181#384—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—42no—
200C2×C3×Q8—81no—
201C2×C3×Q8—81no—
202C2×C3×Q8C2×D412yes—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4C2×D412yes—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—81no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C42C2×D412yes—
246C3×C42#4112yes—
247#217—42no—
248#217—42no—
249#217—81no—
250#216—44no—
251#214—44no—
252C4⋊Dic3—21no—
253C4⋊Dic3—22no—
254C4⋊Dic3—22no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—42no—
260C4⋊Dic3#4112yes—
261Dic3⋊C4—42no—
262Dic3⋊C4—81no—
263Dic3⋊C4—81no—
264Dic3⋊C4—81no—
265Dic3⋊C4—81no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8—21no—
280C22×Q8—31no—
281C22×Q8C22×D311yescharacteristic
282C22×D4C22×D311yescharacteristic
283C23×C4—61no—
284C2×Q16—62no—
285C2×Q16—62no—
286C2×Q16—121no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296#124—22no—
297#124#6714yes—
298C4×Q8—62no—
299C4×Q8—62no—
300C4×Q8—62no—
301C4×Q8—121no—
302C4×D4—62no—
303C4×D4—121no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—61no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329C4⋊C8—62no—
330C4⋊C8—121no—
331C4⋊C8—121no—
332#103—22no—
333#103—22no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—22no—
338#102—22no—
339#102—62no—
340#97—22no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—22no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—84no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—41no—
366C3×D4—41no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—22no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4#12112yes—
381C2×C3×C4#12112yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
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—84no—
411C2×Dic3#13712yes—
412C2×Dic3#13712yes—
413Dic6—22no—
414Dic6—22no—
415Dic6—41no—
416Dic6—41no—
417C3×C8—41no—
418C3×C8—41no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—32no—
428C2×Q8—42no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—81no—
435C2×Q8—81no—
436C2×Q8C2×#6712yes—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×#6712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—241no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—124no—
461SD16—124no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—32no—
479C4⋊C4—42no—
480C4⋊C4—62no—
481C4⋊C4—62no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—241no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—61no—
499C42—61no—
500C42—61no—
501C42—61no—
502C42—122no—
503C42—122no—
504C42—241no—
505C42#23712yes—
506C42C2×#6712yes—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3C2×#12111yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3#54811yescharacteristic
517C22×C3C2×#13711yescharacteristic
518C22×C3C2×#13711yescharacteristic
519C22×C3#57611yescharacteristic
520C22×C3#57711yescharacteristic
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—42no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4#52911yescharacteristic
527C3×C4#54211yescharacteristic
528C3×C4#54611yescharacteristic
529C3×C4#54711yescharacteristic
530Dic3—22no—
531Dic3—22no—
532Dic3—42no—
533Dic3—41no—
534Dic3—42no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×D3×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—62no—
551Q8—62no—
552Q8—84no—
553Q8—121no—
554Q8—121no—
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—244no—
597C2×C4#93912yes—
598C2×C4#93912yes—
599C2×C4C2×#23711yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4#100911yescharacteristic
603C2×C4#101011yescharacteristic
604C2×C4C22×#6711yescharacteristic
605C8—41no—
606C8—41no—
607C8—121no—
608C8—121no—
609C2×C3—82no—
610C2×C3#242711yescharacteristic
611C2×C3#303911yescharacteristic
612C2×C3#304511yescharacteristic
613C2×C3#331411yescharacteristic
614C2×C3#331611yescharacteristic
615C2×C3#336911yescharacteristic
616C2×C3#337511yescharacteristic
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22#567611yescharacteristic
625C22#567711yescharacteristic
626C22#577411yescharacteristic
627C22#577511yescharacteristic
628C22D3×#13711yescharacteristic
629C22D3×#13711yescharacteristic
630C22C2×#93911yescharacteristic
631C4—22no—
632C4—22no—
633C4—42no—
634C4—62no—
635C4—62no—
636C4—81no—
637C4—81no—
638C4—121no—
639C4—122no—
640C4—121no—
641C4—122no—
642C4—241no—
643C4—241no—
644C4#568011yescharacteristic
645C4#568211yescharacteristic
646C4#582411yescharacteristic
647C4#582911yescharacteristic
648C3#2533611yescharacteristic, complemented, Sylow
649C2—82no—
650C2#8167111yescharacteristic
651C2#8167511yescharacteristic
652C2#8170011yescharacteristic
653C2#8170111yescharacteristic
654C2#8299611yescharacteristic
655C2#8393611yescharacteristic
656C2#8396711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#489D3
hypocenterC3#25336
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. #83243C2semidirect
5. #71780C2semidirect
6. #70285C2semidirect
7. #70239C2semidirect
8. #70233C2semidirect
9. #70221C2semidirect
14. #69836C2semidirect
15. #69813C2semidirect
648. C3#25336semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
#832432
#829552
C2×#50542
#717802
#702852
#702392
#702332
#702212
#702172
#701712
#701652
#701532
#698362
#698132
#253363