Finite Group Databasebeta

← #10819879

Subgroups — #10819879

Order 768.

986 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4715C211yesmaximal, characteristic
2C2×#4710C211yesmaximal, characteristic
3C2×C4×Dic12C211yesmaximal, characteristic, complemented
4#73655C211yesmaximal, characteristic
5#73651C211yesmaximal, characteristic, complemented
6C3×#1771C211yesmaximal, characteristic, complemented
7#70458C211yesmaximal, characteristic
8#70456C211yesmaximal, characteristic, complemented
9#69824C211yesmaximal, characteristic, complemented
10#69814C211yesmaximal, characteristic, complemented
11#69774C211yesmaximal, characteristic
12#68686C211yesmaximal, characteristic
13#68685C211yesmaximal, characteristic, complemented
14#68685C211yesmaximal, characteristic, complemented
15#68672C211yesmaximal, characteristic
16#23194—31nomaximal, Sylow
17C22×Dic12C2211yescharacteristic
18C22×Dic12C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#875C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#4990—21no—
43#4989—21no—
44Dic3×C4⋊C4—21no—
45#4972—21no—
46#4939C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49#4929C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52#4715C418yescomplemented
53#4710—24no—
54C4×Dic12—24no—
55#4669—21no—
56#4666—21no—
57#4654—21no—
58#4654—21no—
59C3×#336C2211yescharacteristic
60C3×#336C2211yescharacteristic
61#4551C2211yescharacteristic
62#4551C2211yescharacteristic
63C2×#497—31no—
64C2×#440—31no—
65C2×C4×Q16—31no—
66#2367—31no—
67#2365—31no—
68#1983—31no—
69#1981—31no—
70#1971—31no—
71#1968—31no—
72#1880—31no—
73#1878—31no—
74#1878—31no—
75#1877—31no—
76#1776—31no—
77#1771D311yescharacteristic, complemented
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×Dic12—22no—
112C2×Dic12—22no—
113C2×Dic12—41no—
114C2×Dic12—41no—
115C2×Dic12C2×C414yescomplemented
116C2×Dic12C2×C414yescomplemented
117C2×#204—22no—
118C2×#204—22no—
119C2×#204—41no—
120C2×#204—41no—
121C2×#204C2×C414yes—
122C2×#204C2×C414yes—
123#875—22no—
124#875C2×C414yes—
125C4×D12—22no—
126C4×D12—41no—
127#869—22no—
128#869C2×C414yescomplemented
129C4×Dic6—22no—
130C4×Dic6—22no—
131C4×Dic6—22no—
132C4×Dic6—41no—
133C4×Dic6—41no—
134C4×Dic6—41no—
135C3×C4⋊C8—22no—
136C3×#97—22no—
137C3×#97C2×C414yescomplemented
138C3×C4×C8—22no—
139C3×#95D411yescharacteristic
140C3×#95D411yescharacteristic
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#821—22no—
150#821—41no—
151#818D414yes—
152#817—22no—
153#816—22no—
154#816—22no—
155#816—22no—
156#816—41no—
157#816—41no—
158#816—41no—
159C22×Q16—31no—
160C22×Q16—31no—
161C22×SD16—31no—
162C22×SD16—31no—
163C2×C4⋊Q8—31no—
164C2×#124—31no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×D4—31no—
169#497—38no—
170#440—64no—
171C4×Q16—64no—
172C2×#107—31no—
173C2×#106—31no—
174C2×C4⋊C8C2×D311yescharacteristic
175C2×#103—31no—
176C2×#103—31no—
177C2×#103—31no—
178C2×#103—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#102—31no—
182C2×#102—31no—
183C2×#97C2×D311yescharacteristic
184C2×#97C2×D311yescharacteristic
185C2×C4×C8C2×D311yescharacteristic
186#391—31no—
187#391—31no—
188#391—31no—
189#390—31no—
190#389—61no—
191#388—61no—
192#388—61no—
193#387—61no—
194#387—61no—
195#385—61no—
196C4×C4⋊C4—31no—
197C4×C4⋊C4—31no—
198C4×C4⋊C4—61no—
199C4×C4⋊C4C2×D311yescharacteristic
200C4×C22⋊C4—61no—
201#340—31no—
202#340—31no—
203#336C2×D311yescharacteristic
204#336C2×D311yescharacteristic
205C23×D3—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C22×C412yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6C22×C412yes—
245C2×Dic6C22×C412yes—
246C2×Dic6C22×C412yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8C22×C412yes—
256C2×C3×C8C22×C412yes—
257C2×C3×C8C2×D412yes—
258C2×C3×C8#4112yes—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—41no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42C22×C412yes—
266C3×C42#4112yes—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—42no—
270(C2×D3)⋊C4—42no—
271(C2×D3)⋊C4—81no—
272C4⋊Dic3—24no—
273C4⋊Dic3—24no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3C2×D412yes—
281C4⋊Dic3C2×D412yes—
282Dic3⋊C4—42no—
283Dic3⋊C4—42no—
284Dic3⋊C4—42no—
285C4×Dic3—42no—
286C4×Dic3—42no—
287C4×Dic3—42no—
288C4×Dic3—42no—
289C4×Dic3—81no—
290Dic12—28no—
291Dic12—44no—
292#204—28no—
293#204—44no—
294C22×Q8—31no—
295C22×Q8—31no—
296C22×Q8—31no—
297C22×D4—31no—
298C23×C4—61no—
299C2×Q16—34no—
300C2×Q16—34no—
301C2×Q16—62no—
302C2×Q16—62no—
303C2×Q16—121no—
304C2×Q16—121no—
305C2×SD16—34no—
306C2×SD16—34no—
307C2×SD16—62no—
308C2×SD16—62no—
309C2×SD16—121no—
310C2×SD16—121no—
311C22×C8C22×D311yescharacteristic
312C22×C8C22×D311yescharacteristic
313C22×C8C22×D311yescharacteristic
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—34no—
316C4⋊Q8—62no—
317#124—34no—
318#124—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×Q8—121no—
323C4×Q8—121no—
324C4×Q8—121no—
325C4×D4—62no—
326C4×D4—121no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4D1211yescharacteristic
337C2×C4⋊C4D1211yescharacteristic
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C42—61no—
344C2×C42—61no—
345C2×C42—61no—
346C2×C42—61no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D311yescharacteristic
351#107—34no—
352#106—62no—
353C4⋊C8—22no—
354#103—62no—
355#103—62no—
356#103—62no—
357#103—121no—
358#103—121no—
359#103—121no—
360#102—62no—
361#102—121no—
362#97—22no—
363#97C4×D314yescomplemented
364C4×C8—22no—
365#95—61no—
366#95—61no—
367#95—61no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95D1211yescharacteristic
374#95D1211yescharacteristic
375C23×C3C22×D411yescharacteristic
376C22×D3—22no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
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×C4C4×D412yes—
399C2×C3×C4C4×D412yes—
400C2×C3×C4#13712yes—
401C2×C3×C4C23×C411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
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—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428D12—22no—
429D12—22no—
430D12—41no—
431D12—41no—
432C4×D3—84no—
433Dic6—22no—
434Dic6—22no—
435Dic6—22no—
436Dic6—22no—
437Dic6—22no—
438Dic6—22no—
439Dic6—41no—
440Dic6—41no—
441Dic6—41no—
442Dic6—41no—
443Dic6—41no—
444Dic6—41no—
445C3×C8—22no—
446C3×C8—41no—
447C3×C8—41no—
448C3×C8C4×D414yes—
449C24—61no—
450C2×Q8—32no—
451C2×Q8—32no—
452C2×Q8—32no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×D4—32no—
469C2×D4—61no—
470C2×D4—61no—
471C2×D4—61no—
472C2×D4—61no—
473C2×D4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—122no—
482C22×C4—122no—
483C22×C4—241no—
484C22×C4C2×D1211yescharacteristic
485C22×C4C2×D1211yescharacteristic
486C22×C4C2×D1211yescharacteristic
487C22×C4C2×D1211yescharacteristic
488C22×C4C2×D1211yescharacteristic
489C22×C4C2×D1211yescharacteristic
490C22×C4C23×D311yescharacteristic
491Q16—68no—
492Q16—124no—
493SD16—68no—
494SD16—124no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8C2×C4×D312yes—
504C2×C8C2×C4×D312yes—
505C2×C8D3×D412yes—
506C2×C8#23912yes—
507C4⋊C4—22no—
508C4⋊C4—32no—
509C4⋊C4—32no—
510C4⋊C4—41no—
511C4⋊C4—64no—
512C4⋊C4—64no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C4⋊C4—122no—
522C22⋊C4—64no—
523C22⋊C4—64no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—122no—
532C42—122no—
533C42—122no—
534C42—122no—
535C42—241no—
536C42C2×C4×D312yes—
537C42#23512yes—
538C22×C3C2×C4×D411yescharacteristic
539C22×C3D4211yescharacteristic
540C22×C3#55011yescharacteristic
541C22×C3C2×#13711yescharacteristic
542C22×C3#57511yescharacteristic
543C22×C3#57711yescharacteristic
544C22×C3#57811yescharacteristic
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—22no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4C2×C4×D411yescharacteristic
559C3×C4#51811yescharacteristic
560C3×C4#55011yescharacteristic
561C3×C4#55911yescharacteristic
562Dic3—42no—
563Dic3—42no—
564Dic3—42no—
565Dic3—42no—
566Dic3—42no—
567Dic3—81no—
568Dic3—81no—
569C23—62no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23C22×D1211yescharacteristic
577Q8—62no—
578Q8—62no—
579Q8—62no—
580Q8—62no—
581Q8—62no—
582Q8—62no—
583Q8—121no—
584Q8—121no—
585Q8—121no—
586Q8—121no—
587Q8—121no—
588Q8—121no—
589D4—62no—
590D4—62no—
591D4—121no—
592D4—121no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
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—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
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—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—244no—
630C2×C4C4×D1212yes—
631C2×C4C4×D1212yes—
632C2×C4#90912yes—
633C2×C4C22×C4×D311yescharacteristic
634C2×C4C2×#23511yescharacteristic
635C2×C4C2×D3×D411yescharacteristic
636C2×C4C2×#23911yescharacteristic
637C2×C4#100911yescharacteristic
638C2×C4#100911yescharacteristic
639C8—22no—
640C8—41no—
641C8—41no—
642C8#89314yes—
643C2×C3#236611yescharacteristic
644C2×C3C4×#13711yescharacteristic
645C2×C3#294711yescharacteristic
646C2×C3#326711yescharacteristic
647C2×C3#327011yescharacteristic
648C2×C3#340011yescharacteristic
649C2×C3#340211yescharacteristic
650D3—82no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22C2×C4×D1211yescharacteristic
659C22D4×D1211yescharacteristic
660C22#557711yescharacteristic
661C22C2×#90911yescharacteristic
662C22#574811yescharacteristic
663C22#575111yescharacteristic
664C22#575311yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—122no—
672C4—122no—
673C4—122no—
674C4—122no—
675C4—122no—
676C4—241no—
677C4—241no—
678C4C2×#89311yescharacteristic
679C4#552111yescharacteristic
680C4#555311yescharacteristic
681C4#558511yescharacteristic
682C3#2319411yescharacteristic, complemented, Sylow
683C2—242no—
684C2#7365311yescharacteristic
685C2C4×#90911yescharacteristic
686C2#7857111yescharacteristic
687C2#7949611yescharacteristic
688C2#7950711yescharacteristic
689C2#7995311yescharacteristic
690C2#7995511yescharacteristic
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1771C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1771D3
hypocenterC3#23194
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic12C2semidirect
5. #73651C2semidirect
6. C3×#1771C2semidirect
8. #70456C2semidirect
9. #69824C2semidirect
10. #69814C2semidirect
13. #68685C2semidirect
14. #68685C2semidirect
52. #4715C4semidirect
77. #1771D3semidirect
115. C2×Dic12C2×C4semidirect
116. C2×Dic12C2×C4semidirect
128. #869C2×C4semidirect
137. C3×#97C2×C4semidirect
363. #97C4×D3semidirect
682. C3#23194semidirect

Maximal subgroups

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

subgroupindex
C2×#47152
C2×#47102
C2×C4×Dic122
#736552
#736512
C3×#17712
#704582
#704562
#698242
#698142
#697742
#686862
#686852
#686722
#231943