Finite Group Databasebeta

← #10835933

Subgroups — #10835933

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2620C211yesmaximal, characteristic
2C2×#5054C211yesmaximal, characteristic, complemented
3#82235C211yesmaximal, characteristic, complemented
4C2×#5000C211yesmaximal, characteristic, complemented
5#71721C211yesmaximal, characteristic, complemented
6#71700C211yesmaximal, characteristic, complemented
7#71695C211yesmaximal, characteristic, complemented
8#71685C211yesmaximal, characteristic
9#71679C211yesmaximal, characteristic
10#69677C211yesmaximal, characteristic, complemented
11#69675C211yesmaximal, characteristic, complemented
12#69649C211yesmaximal, characteristic, complemented
13#69644C211yesmaximal, characteristic, complemented
14#69588C211yesmaximal, characteristic, complemented
15#69567C211yesmaximal, characteristic, complemented
16#24665—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×#121C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#396—21no—
26C3×#396—21no—
27C3×#394—21no—
28C3×#394—21no—
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#384C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5212C2211yescharacteristic
39C2×#830C2211yescharacteristic
40C2×#830C2211yescharacteristic
41#5054—24no—
42#5000—24no—
43#4978—21no—
44#4976—21no—
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4945—21no—
50#4944—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52#4929C2211yescharacteristic
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×#464—31no—
63C2×#433—31no—
64#2620D311yescharacteristic
65#2323—31no—
66#2036—31no—
67#2033—31no—
68#2001—31no—
69#2000—31no—
70#1983—31no—
71#1973—31no—
72#1969—31no—
73#1947—31no—
74#1935—31no—
75#1934—31no—
76#1803—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C3×#124—22no—
83C3×#124—22no—
84C3×C22⋊Q8—41no—
85C3×C22⋊Q8—41no—
86C3×#121—41no—
87C3×#121—41no—
88C3×#117—22no—
89C3×#117—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4D411yescharacteristic
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×#217—21no—
108C2×#217—21no—
109C2×#216—22no—
110C2×#216—22no—
111C2×#216—41no—
112C2×#214—22no—
113C2×#214—22no—
114C2×#214—41no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118#924—22no—
119#924—22no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129C4×Dic6—22no—
130C4×Dic6—22no—
131C3×#95—21no—
132C3×#95—21no—
133#835—22no—
134#835—22no—
135#835—41no—
136#832—22no—
137#832—22no—
138#832—41no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#830—22no—
145#830—22no—
146#810—41no—
147#810—41no—
148#807—41no—
149#807—41no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)D414yescomplemented
152#803—22no—
153C4×C3⋊C8—22no—
154C22×Q16—31no—
155C22×SD16—31no—
156C2×#124C2×D311yescharacteristic
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8C2×D311yescharacteristic
159C2×#121C2×D311yescharacteristic
160C2×C4×Q8—31no—
161C2×#117C2×D311yescharacteristic
162#464—64no—
163#433—64no—
164C2×#107—31no—
165C2×#107—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×#102—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178C2×#97—31no—
179C2×C4×C8—31no—
180#396—21no—
181#396—21no—
182#394—21no—
183#394—21no—
184#391—31no—
185#391C2×D311yescharacteristic
186#390C2×D311yescharacteristic
187#389—31no—
188#389—61no—
189#388—61no—
190#388—61no—
191#387—61no—
192#384C2×D311yescharacteristic
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199C24×C3—21no—
200C24×C3C2×D411yescharacteristic
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×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—42no—
214C2×C3×D4—81no—
215C2×C3×D4—81no—
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×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6#4112yes—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—81no—
249C3×C4⋊C4—81no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—81no—
257C3×C22⋊C4—81no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42C2×D412yes—
261C3×C42#4112yes—
262#217—24no—
263#217—81no—
264#216—44no—
265#214—44no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3#4112yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—42no—
274Dic3⋊C4—81no—
275C4×Dic3—42no—
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×Q8—31no—
290C22×Q8C22×D311yescharacteristic
291C22×D4—21no—
292C22×D4C22×D311yescharacteristic
293C23×C4C22×D311yescharacteristic
294C2×Q16—62no—
295C2×Q16—62no—
296C2×Q16—121no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×SD16—121no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8—31no—
304#124—22no—
305#124—22no—
306C22⋊Q8—41no—
307C22⋊Q8—41no—
308C22⋊Q8—62no—
309C22⋊Q8—62no—
310#121—41no—
311#121—41no—
312C4×Q8—62no—
313C4×Q8—62no—
314#117—22no—
315#117—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
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—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4#6711yescharacteristic
338C2×C22⋊C4#6711yescharacteristic
339C2×C42—61no—
340C2×C42—61no—
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343#107—121no—
344#107—121no—
345C4⋊C8—34no—
346C4⋊C8—62no—
347#103—62no—
348#103—62no—
349#103—121no—
350#103—121no—
351#103—121no—
352#102—62no—
353#102—62no—
354#102—121no—
355C22⋊C8—62no—
356C22⋊C8—62no—
357#97—62no—
358C4×C8—62no—
359#95—21no—
360#95—21no—
361#95—61no—
362#95—61no—
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×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3C22×D411yescharacteristic
381C3×Q8—41no—
382C3×Q8—41no—
383C3×Q8—41no—
384C3×Q8—41no—
385C3×D4—41no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—84no—
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—21no—
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#12112yes—
417C2×C3×C4#12112yes—
418C2×C3×C4C22×D411yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4H(5,𝔽2)11yescharacteristic
423C2×C3×C4H(5,𝔽2)11yescharacteristic
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436Dic6—22no—
437Dic6—22no—
438Dic6—41no—
439Dic6—41no—
440C3⋊C8—22no—
441C3⋊C8—22no—
442C3⋊C8—41no—
443C3⋊C8—41no—
444C24—21no—
445C24C2×#6711yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—32no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—81no—
466C2×D4—81no—
467C22×C4—21no—
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—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4C2×#6711yescharacteristic
481C22×C4C2×#6711yescharacteristic
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C23×D311yescharacteristic
486Q16—124no—
487SD16—124no—
488C2×C8—32no—
489C2×C8—32no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—32no—
507C4⋊C4—42no—
508C4⋊C4—64no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—122no—
517C4⋊C4—241no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—64no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—241no—
528C42—21no—
529C42—21no—
530C42—122no—
531C42—241no—
532C42D3×D412yes—
533C42#23712yes—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—81no—
546C22×C3—81no—
547C22×C3—81no—
548C22×C3C2×#12111yescharacteristic
549C22×C3#52511yescharacteristic
550C22×C3#53511yescharacteristic
551C22×C3#57611yescharacteristic
552C22×C3#57711yescharacteristic
553C22×C3#57811yescharacteristic
554C22×C3#57811yescharacteristic
555C3×C4—22no—
556C3×C4—22no—
557C3×C4—41no—
558C3×C4—41no—
559C3×C4—41no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4#54611yescharacteristic
564C3×C4#55011yescharacteristic
565C3×C4#55311yescharacteristic
566C3×C4#55511yescharacteristic
567Dic3—42no—
568Dic3—42no—
569Dic3—81no—
570Dic3—81no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23C22×#6711yescharacteristic
586Q8—41no—
587Q8—41no—
588Q8—41no—
589Q8—41no—
590Q8—62no—
591Q8—62no—
592Q8—121no—
593Q8—121no—
594D4—41no—
595D4—41no—
596D4—41no—
597D4—41no—
598D4—84no—
599C2×C4—21no—
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—41no—
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—62no—
626C2×C4—62no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4#93912yes—
638C2×C4#93912yes—
639C2×C4C2×D3×D411yescharacteristic
640C2×C4C2×#23711yescharacteristic
641C2×C4#100411yescharacteristic
642C2×C4#100411yescharacteristic
643C2×C4D3×#4111yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C8—62no—
646C8—62no—
647C8—121no—
648C8—121no—
649C2×C3—81no—
650C2×C3—161no—
651C2×C3#241411yescharacteristic
652C2×C3#306111yescharacteristic
653C2×C3#306311yescharacteristic
654C2×C3#312111yescharacteristic
655C2×C3#312411yescharacteristic
656C2×C3#322311yescharacteristic
657C2×C3#322411yescharacteristic
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22C2×#93911yescharacteristic
673C22#580511yescharacteristic
674C22#582811yescharacteristic
675C22#583511yescharacteristic
676C22#583611yescharacteristic
677C22#583811yescharacteristic
678C22#583811yescharacteristic
679C4—22no—
680C4—22no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—122no—
688C4—122no—
689C4—241no—
690C4—241no—
691C4#559911yescharacteristic
692C4#561211yescharacteristic
693C4#561411yescharacteristic
694C4#561611yescharacteristic
695C3#2466511yescharacteristic, complemented, Sylow
696C2—81no—
697C2—161no—
698C2#8270411yescharacteristic
699C2#8338911yescharacteristic
700C2#8341411yescharacteristic
701C2#8342111yescharacteristic
702C2#8342311yescharacteristic
703C2#8346111yescharacteristic
704C2#8346711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2620C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2620D3
hypocenterC3#24665
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
2. C2×#5054C2semidirect
3. #82235C2semidirect
4. C2×#5000C2semidirect
5. #71721C2semidirect
6. #71700C2semidirect
7. #71695C2semidirect
10. #69677C2semidirect
11. #69675C2semidirect
12. #69649C2semidirect
13. #69644C2semidirect
14. #69588C2semidirect
15. #69567C2semidirect
151. C4⋊(C3⋊C8)D4semidirect
695. C3#24665semidirect

Maximal subgroups

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

subgroupindex
C3×#26202
C2×#50542
#822352
C2×#50002
#717212
#717002
#716952
#716852
#716792
#696772
#696752
#696492
#696442
#695882
#695672
#246653