Finite Group Databasebeta

← #10835948

Subgroups — #10835948

Order 768.

870 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2621C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic, complemented
3#82236C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic, complemented
5#71778C211yesmaximal, characteristic, complemented
6#71699C211yesmaximal, characteristic
7#71699C211yesmaximal, characteristic
8#71695C211yesmaximal, characteristic, complemented
9#71694C211yesmaximal, characteristic, complemented
10#69679C211yesmaximal, characteristic, complemented
11#69676C211yesmaximal, characteristic, complemented
12#69675C211yesmaximal, characteristic, complemented
13#69674C211yesmaximal, characteristic, complemented
14#69556C211yesmaximal, characteristic, complemented
15#69540C211yesmaximal, characteristic, complemented
16#24666—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C22×#215C2211yescharacteristic
23C2×#930C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#398—21no—
26C3×#398—21no—
27C3×#397—21no—
28C3×#397—21no—
29C3×#391C2211yescharacteristic
30C3×#391C2211yescharacteristic
31C3×#382C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38#5212C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#830C2211yescharacteristic
41#5083—24no—
42#5000—24no—
43#4990—21no—
44#4989—21no—
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4957—21no—
50#4956—21no—
51#4939C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4928C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#461—31no—
63C2×#433—31no—
64#2621D311yescharacteristic, complemented
65#2324—31no—
66#2036—31no—
67#2035—31no—
68#2000—31no—
69#2000—31no—
70#1981—31no—
71#1973—31no—
72#1968—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1804—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—22no—
85C3×C22⋊Q8—41no—
86C3×C22⋊Q8—41no—
87C3×C22⋊Q8—41no—
88C3×C22⋊Q8—41no—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4D411yescharacteristic
102C2×C3×C22⋊C4D411yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×#217—21no—
106C2×#215—22no—
107C2×#215—22no—
108C2×#215—22no—
109C2×#215—22no—
110C2×#215—41no—
111C2×#215—41no—
112#930—22no—
113#930—22no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—21no—
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3—21no—
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C4×D12—22no—
128C4×D12—22no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135#835—22no—
136#835—22no—
137#835—22no—
138#835—22no—
139#835—41no—
140#835—41no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#830—22no—
145#830—22no—
146#809—41no—
147#809—41no—
148#808—41no—
149#807—41no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)D414yescomplemented
152#803—22no—
153C4×C3⋊C8—22no—
154C22×SD16—31no—
155C22×SD16—31no—
156C2×C4⋊Q8C2×D311yescharacteristic
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×C22⋊Q8C2×D311yescharacteristic
159C2×#121—31no—
160C2×C4×D4—31no—
161C2×#117C2×D311yescharacteristic
162#461—64no—
163#433—64no—
164C2×#107—31no—
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×#102—31no—
175C2×#102—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178C2×#97—31no—
179C2×C4×C8—31no—
180#398—21no—
181#398—21no—
182#397—21no—
183#397—21no—
184#391C2×D311yescharacteristic
185#391C2×D311yescharacteristic
186#390—31no—
187#390—61no—
188#389—31no—
189#388—61no—
190#386—61no—
191#385—61no—
192#382C2×D311yescharacteristic
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3C2×D411yescharacteristic
200C23×D3—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—21no—
212C2×C3×Q8—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C2×#67—24no—
230C2×#67—81no—
231C2×#67—81no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12#4112yes—
241C2×C4×D3—42no—
242C2×C4×D3—81no—
243C2×C4×D3—81no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4—42no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—42no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42C2×D412yes—
261C3×C42#4112yes—
262#217—81no—
263#215—44no—
264#215—44no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—81no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3#4112yes—
275Dic3⋊C4—81no—
276C4×Dic3—81no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D412yes—
288C2×C3⋊C8C2×D412yes—
289C22×Q8C22×D311yescharacteristic
290C22×Q8C22×D311yescharacteristic
291C22×D4—31no—
292C22×D4—61no—
293C23×C4—61no—
294C23×C4C22×D311yescharacteristic
295C2×SD16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×SD16—121no—
300C2×SD16—121no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8—31no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C22⋊Q8—41no—
308C22⋊Q8—41no—
309C22⋊Q8—41no—
310C22⋊Q8—41no—
311#121—62no—
312#121—62no—
313C4×D4—62no—
314C4×D4—62no—
315#117—22no—
316#117—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4#6711yescharacteristic
326C2×C4⋊C4#6711yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4#6711yescharacteristic
337C2×C22⋊C4#6711yescharacteristic
338C2×C42—61no—
339C2×C42C22×D311yescharacteristic
340C2×C42C22×D311yescharacteristic
341#107—121no—
342#106—121no—
343C4⋊C8—34no—
344C4⋊C8—62no—
345#103—62no—
346#103—62no—
347#103—62no—
348#103—62no—
349#103—121no—
350#103—121no—
351#102—121no—
352#102—121no—
353C22⋊C8—62no—
354C22⋊C8—62no—
355#97—62no—
356C4×C8—62no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3C22×D411yescharacteristic
374C22×D3—22no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C3×Q8—41no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×Q8—41no—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—41no—
388C3×Q8—41no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4#12112yes—
419C2×C3×C4#12112yes—
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4H(5,𝔽2)11yescharacteristic
425C2×C3×C4H(5,𝔽2)11yescharacteristic
426#67—84no—
427C2×Dic3—22no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436D12—22no—
437D12—22no—
438D12—41no—
439D12—41no—
440C4×D3—84no—
441C3⋊C8—22no—
442C3⋊C8—22no—
443C3⋊C8—41no—
444C3⋊C8—41no—
445C24—61no—
446C24C2×#6711yescharacteristic
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×D4—32no—
460C2×D4—64no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—241no—
467C2×D4—241no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—122no—
482C22×C4—241no—
483C22×C4—241no—
484C22×C4C2×#6711yescharacteristic
485C22×C4C2×#6711yescharacteristic
486C22×C4C2×#6711yescharacteristic
487C22×C4C2×#6711yescharacteristic
488C22×C4C2×#6711yescharacteristic
489C22×C4C23×D311yescharacteristic
490SD16—124no—
491SD16—124no—
492C2×C8—32no—
493C2×C8—32no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—32no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—241no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—64no—
526C22⋊C4—122no—
527C22⋊C4—241no—
528C22⋊C4—241no—
529C22⋊C4—241no—
530C42—21no—
531C42—21no—
532C42—241no—
533C42D3×D412yes—
534C42#23712yes—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3C2×#12111yescharacteristic
543C22×C3#52511yescharacteristic
544C22×C3#53511yescharacteristic
545C22×C3#57711yescharacteristic
546C22×C3#57711yescharacteristic
547C22×C3#57711yescharacteristic
548C22×C3#57811yescharacteristic
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C2×D3—81no—
556C3×C4—22no—
557C3×C4—22no—
558C3×C4—41no—
559C3×C4—41no—
560C3×C4—41no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4—81no—
564C3×C4—81no—
565C3×C4#54611yescharacteristic
566C3×C4#55011yescharacteristic
567C3×C4#55311yescharacteristic
568C3×C4#55511yescharacteristic
569Dic3—42no—
570Dic3—81no—
571Dic3—81no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—62no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23—121no—
586C23C22×#6711yescharacteristic
587Q8—41no—
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8—41no—
592Q8—41no—
593Q8—41no—
594Q8—41no—
595D4—62no—
596D4—62no—
597D4—121no—
598D4—121no—
599D4—244no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—62no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—244no—
639C2×C4#93912yes—
640C2×C4#93912yes—
641C2×C4C2×D3×D411yescharacteristic
642C2×C4C2×#23711yescharacteristic
643C2×C4#100411yescharacteristic
644C2×C4#100411yescharacteristic
645C2×C4D3×#4111yescharacteristic
646C2×C4D3×#4111yescharacteristic
647C8—62no—
648C8—62no—
649C8—121no—
650C8—121no—
651C2×C3—81no—
652C2×C3#241411yescharacteristic
653C2×C3#305711yescharacteristic
654C2×C3#306311yescharacteristic
655C2×C3#312011yescharacteristic
656C2×C3#312311yescharacteristic
657C2×C3#322111yescharacteristic
658C2×C3#322511yescharacteristic
659D3—82no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—122no—
668C22—122no—
669C22—122no—
670C22—241no—
671C22—241no—
672C22—241no—
673C22—241no—
674C22C2×#93911yescharacteristic
675C22#580511yescharacteristic
676C22#582811yescharacteristic
677C22#583711yescharacteristic
678C22#583711yescharacteristic
679C22#583711yescharacteristic
680C22#583811yescharacteristic
681C4—22no—
682C4—22no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—122no—
691C4—241no—
692C4—241no—
693C4#559911yescharacteristic
694C4#561211yescharacteristic
695C4#561411yescharacteristic
696C4#561611yescharacteristic
697C3#2466611yescharacteristic, complemented, Sylow
698C2—81no—
699C2—242no—
700C2#8270411yescharacteristic
701C2#8339811yescharacteristic
702C2#8340011yescharacteristic
703C2#8340911yescharacteristic
704C2#8341011yescharacteristic
705C2#8344611yescharacteristic
706C2#8344711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2621C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2621D3
hypocenterC3#24666
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
1. C3×#2621C2semidirect
2. C2×#5083C2semidirect
4. C2×#5000C2semidirect
5. #71778C2semidirect
8. #71695C2semidirect
9. #71694C2semidirect
10. #69679C2semidirect
11. #69676C2semidirect
12. #69675C2semidirect
13. #69674C2semidirect
14. #69556C2semidirect
15. #69540C2semidirect
64. #2621D3semidirect
151. C4⋊(C3⋊C8)D4semidirect
697. C3#24666semidirect

Maximal subgroups

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

subgroupindex
C3×#26212
C2×#50832
#822362
C2×#50002
#717782
#716992
#716952
#716942
#696792
#696762
#696752
#696742
#695562
#695402
#246663