Finite Group Databasebeta

← #10835913

Subgroups — #10835913

Order 768.

814 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5182C211yesmaximal, characteristic, complemented
2C2×#4690C211yesmaximal, characteristic
3#76641C211yesmaximal, characteristic
4#76611C211yesmaximal, characteristic
5C3×#1969C211yesmaximal, characteristic, complemented
6#72071C211yesmaximal, characteristic
7#70635C211yesmaximal, characteristic
8#70616C211yesmaximal, characteristic, complemented
9#70274C211yesmaximal, characteristic
10#70273C211yesmaximal, characteristic
11#70229C211yesmaximal, characteristic, complemented
12#70222C211yesmaximal, characteristic, complemented
13#69767C211yesmaximal, characteristic, complemented
14#69678C211yesmaximal, characteristic, complemented
15#69675C211yesmaximal, characteristic, complemented
16#24664—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×#946C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×#944C2211yescharacteristic
22C22×#216C2211yescharacteristic
23C2×#897C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#391C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5182—24no—
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4987C2211yescharacteristic
42#4977C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4929C2211yescharacteristic
47#4928C2211yescharacteristic
48C2×#803C2211yescharacteristic
49#4690—24no—
50#4675—21no—
51#4675—21no—
52#4670—21no—
53#4669—21no—
54#4657—21no—
55#4657—21no—
56#4652—21no—
57#4647—21no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#462—31no—
63C2×#433—31no—
64#2621—31no—
65#2323—31no—
66#2036—31no—
67#2035—31no—
68#2001—31no—
69#2001—31no—
70#1981—31no—
71#1971—31no—
72#1969D311yescharacteristic, complemented
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1805—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—41no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#946—22no—
90#946—22no—
91#946—41no—
92#946—41no—
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95#944—22no—
96#944—22no—
97C2×#216—22no—
98C2×#216—22no—
99C2×#216—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
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
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#897—41no—
124#897—41no—
125#872—22no—
126#872—22no—
127C3×#107—41no—
128C3×#103—41no—
129C3×#103—41no—
130C3×C4×C8—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135#835—22no—
136#835—22no—
137#835—41no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145(C2×D3)⋊C8—22no—
146(C2×D3)⋊C8—22no—
147#816—22no—
148#816—22no—
149Dic3⋊C8—22no—
150Dic3⋊C8D414yes—
151#810—41no—
152#808—41no—
153#803—22no—
154C22×Q16—31no—
155C22×Q16C2×D311yescharacteristic
156C2×C4⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8—31no—
159C2×C22⋊Q8—31no—
160C2×C4×Q8—31no—
161C2×#117—31no—
162#462—64no—
163#433—64no—
164C2×#107C2×D311yescharacteristic
165C2×#106—31no—
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103—31no—
172C2×#103—31no—
173C2×#103—31no—
174C2×#103C2×D311yescharacteristic
175C2×#103C2×D311yescharacteristic
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178C2×#97—31no—
179C2×C4×C8C2×D311yescharacteristic
180#398—61no—
181#398—61no—
182#397—61no—
183#397—61no—
184#391—31no—
185#391C2×D311yescharacteristic
186#391C2×D311yescharacteristic
187#389—31no—
188#389—61no—
189#388—61no—
190#388—61no—
191#387—61no—
192#382—31no—
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C23×D3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8#4112yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
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
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C3×Q16—44no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C42—21no—
250C3×C42—21no—
251#216—44no—
252(C2×D3)⋊C4—24no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—22no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—81no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264C4⋊Dic3—42no—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—22no—
268Dic3⋊C4—22no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272Dic3⋊C4—81no—
273C4×Dic3—42no—
274C4×Dic3—81no—
275C4×Dic3C2×D412yes—
276C4×Dic3#4112yes—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C22×Q8—31no—
284C22×Q8C22×D311yescharacteristic
285C22×Q8C22×D311yescharacteristic
286C23×C4—31no—
287C2×Q16—22no—
288C2×Q16—22no—
289C2×Q16—41no—
290C2×Q16—62no—
291C2×Q16—62no—
292C2×Q16—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299C22⋊Q8—62no—
300C22⋊Q8—62no—
301C22⋊Q8—121no—
302C22⋊Q8—121no—
303C22⋊Q8—121no—
304C22⋊Q8—121no—
305C4×Q8—62no—
306C4×Q8—62no—
307#117—62no—
308#117—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C22⋊C4—31no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C42—31no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—41no—
334#106—121no—
335C4⋊C8—34no—
336C4⋊C8—62no—
337#103—41no—
338#103—41no—
339#103—62no—
340#103—62no—
341#103—62no—
342#103—62no—
343#103—121no—
344#103—121no—
345C22⋊C8—62no—
346C22⋊C8—62no—
347#97—62no—
348C4×C8—22no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3C22×D411yescharacteristic
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8—41no—
375C3×Q8—41no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
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—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4H(5,𝔽2)11yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—41no—
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#12112yes—
420C2×Dic3#12112yes—
421C4×D3—41no—
422C4×D3—41no—
423C4×D3—41no—
424C4×D3—41no—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—41no—
431C3⋊C8—22no—
432C3⋊C8—41no—
433C24—31no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8#23512yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472Q16—44no—
473Q16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—32no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8D3×D412yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—32no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—64no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—122no—
500C4⋊C4—122no—
501C4⋊C4—122no—
502C4⋊C4—122no—
503C4⋊C4—122no—
504C4⋊C4—241no—
505C22⋊C4—62no—
506C22⋊C4—62no—
507C22⋊C4—64no—
508C22⋊C4—122no—
509C22⋊C4—122no—
510C22⋊C4—241no—
511C42—21no—
512C42—21no—
513C42—32no—
514C42—32no—
515C42—122no—
516C42—241no—
517C22×C3C2×#12111yescharacteristic
518C22×C3#52511yescharacteristic
519C22×C3#53511yescharacteristic
520C22×C3#57711yescharacteristic
521C22×C3#57811yescharacteristic
522C22×C3#57811yescharacteristic
523C22×C3#57811yescharacteristic
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—42no—
532C3×C4—41no—
533C3×C4—41no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4#54611yescharacteristic
538C3×C4#55011yescharacteristic
539C3×C4#55311yescharacteristic
540C3×C4#55511yescharacteristic
541Dic3—22no—
542Dic3—22no—
543Dic3—42no—
544Dic3—41no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23C2×D3×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—121no—
565Q8—121no—
566Q8—121no—
567Q8—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—62no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4C2×#23511yescharacteristic
612C2×C4C2×D3×D411yescharacteristic
613C2×C4#100411yescharacteristic
614C2×C4#100411yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C8—22no—
618C8—41no—
619C8—62no—
620C8—121no—
621C2×C3#241411yescharacteristic
622C2×C3#305911yescharacteristic
623C2×C3#306311yescharacteristic
624C2×C3#312011yescharacteristic
625C2×C3#312711yescharacteristic
626C2×C3#322311yescharacteristic
627C2×C3#322511yescharacteristic
628D3—81no—
629C22—121no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22#554711yescharacteristic
637C22D3×#12111yescharacteristic
638C22#571511yescharacteristic
639C22#576611yescharacteristic
640C22#576611yescharacteristic
641C22#577011yescharacteristic
642C22#577111yescharacteristic
643C4—42no—
644C4—41no—
645C4—41no—
646C4—62no—
647C4—62no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—121no—
652C4—122no—
653C4—241no—
654C4—241no—
655C4—241no—
656C4#549011yescharacteristic
657C4#554011yescharacteristic
658C4#559911yescharacteristic
659C4#567011yescharacteristic
660C3#2466411yescharacteristic, complemented, Sylow
661C2—241no—
662C2#7631111yescharacteristic
663C2#7972411yescharacteristic
664C2#7974811yescharacteristic
665C2#8076011yescharacteristic
666C2#8076211yescharacteristic
667C2#8180411yescharacteristic
668C2#8182811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1969C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1969D3
hypocenterC3#24664
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1969

Complements

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

NH ≅ G/Ntype
1. C2×#5182C2semidirect
5. C3×#1969C2semidirect
8. #70616C2semidirect
11. #70229C2semidirect
12. #70222C2semidirect
13. #69767C2semidirect
14. #69678C2semidirect
15. #69675C2semidirect
72. #1969D3semidirect
660. C3#24664semidirect

Maximal subgroups

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

subgroupindex
C2×#51822
C2×#46902
#766412
#766112
C3×#19692
#720712
#706352
#706162
#702742
#702732
#702292
#702222
#697672
#696782
#696752
#246643