Finite Group Databasebeta

← #10819884

Subgroups — #10819884

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
1C3×#2367C211yesmaximal, characteristic
2C2×#5060C211yesmaximal, characteristic
3C2×C4×#216C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic
5#73651C211yesmaximal, characteristic, complemented
6#71721C211yesmaximal, characteristic
7#71632C211yesmaximal, characteristic, complemented
8#71612C211yesmaximal, characteristic
9#69690C211yesmaximal, characteristic, complemented
10#69678C211yesmaximal, characteristic, complemented
11#69660C211yesmaximal, characteristic
12#69646C211yesmaximal, characteristic
13#69486C211yesmaximal, characteristic, complemented
14#69481C211yesmaximal, characteristic, complemented
15#69478C211yesmaximal, characteristic, complemented
16#23194—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#388—21no—
30C3×#388—21no—
31C3×#385—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C4⋊C4C2211yescharacteristic
34C3×C4×C22⋊C4—21no—
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39#5060C418yescomplemented
40C4×#216—24no—
41#5018—24no—
42Dic3×C4⋊C4—21no—
43#4972—21no—
44C2×#810C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4929C2211yescharacteristic
52#4929C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×#803C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57#4654—21no—
58#4654—21no—
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63C2×#497—31no—
64C2×#440—31no—
65C2×C4×Q16—31no—
66#2367D311yescharacteristic
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#1771—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C22×Dic6C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C3×#124—22no—
84C3×#124C2×C414yes—
85C3×C4×Q8—22no—
86C3×C4×Q8—41no—
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#216—22no—
105C2×#216—22no—
106C2×#216—41no—
107C2×#216—41no—
108C2×#216C2×C414yescomplemented
109C2×#216C2×C414yescomplemented
110C2×#214—22no—
111C2×#214—22no—
112C2×#214—41no—
113C2×#214—41no—
114C2×#214C2×C414yes—
115C2×#214C2×C414yes—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×C4×Dic3—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129#869—22no—
130#869C2×C414yescomplemented
131C4×Dic6—22no—
132C4×Dic6—22no—
133C4×Dic6—41no—
134C4×Dic6—41no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95D411yescharacteristic
139C3×#95D411yescharacteristic
140#835—22no—
141#835—41no—
142#832—22no—
143#832—41no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#810—22no—
150#810—22no—
151#810—41no—
152#810—41no—
153#808—22no—
154#807D414yes—
155C4⋊(C3⋊C8)—22no—
156#803—22no—
157#803C2×C414yescomplemented
158C4×C3⋊C8—22no—
159C22×Q16—31no—
160C22×Q16—31no—
161C22×SD16—31no—
162C22×SD16—31no—
163C2×C4⋊Q8—31no—
164C2×#124C2×D311yescharacteristic
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C2×C4×Q8C2×D311yescharacteristic
168C2×C4×D4C2×D311yescharacteristic
169#497—38no—
170#440—64no—
171C4×Q16—64no—
172C2×#107—31no—
173C2×#106—31no—
174C2×C4⋊C8—31no—
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×#97—31no—
184C2×#97—31no—
185C2×C4×C8—31no—
186#391—31no—
187#391—31no—
188#391C2×D311yescharacteristic
189#390C2×D311yescharacteristic
190#389—61no—
191#388—21no—
192#388—21no—
193#387—61no—
194#387—61no—
195#385—21no—
196C4×C4⋊C4—31no—
197C4×C4⋊C4—61no—
198C4×C4⋊C4C2×D311yescharacteristic
199C4×C4⋊C4C2×D311yescharacteristic
200C4×C22⋊C4—21no—
201#340—31no—
202#340—31no—
203#336—31no—
204#336—31no—
205C24×C3—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8C22×C412yes—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C22×C412yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—42no—
222C22×C3×C4—42no—
223C22×C3×C4—81no—
224C22×C3×C4C2×D411yescharacteristic
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—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6C22×C412yes—
246C2×Dic6C22×C412yes—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—41no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4C2×D412yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—42no—
258C3×C22⋊C4—81no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—42no—
264C3×C42—81no—
265C3×C42C22×C412yes—
266C3×C42#4112yes—
267#216—28no—
268#216—44no—
269#214—28no—
270#214—44no—
271C4⋊Dic3—24no—
272C4⋊Dic3—24no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3C2×D412yes—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279C4×Dic3—42no—
280C4×Dic3—42no—
281C4×Dic3—42no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8C22×C412yes—
291C2×C3⋊C8C22×C412yes—
292C2×C3⋊C8C2×D412yes—
293C2×C3⋊C8#4112yes—
294C22×Q8—31no—
295C22×Q8—31no—
296C22×Q8C22×D311yescharacteristic
297C22×D4C22×D311yescharacteristic
298C23×C4—21no—
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×C8—31no—
312C22×C8—31no—
313C22×C8—31no—
314C22×C8—31no—
315C4⋊Q8—34no—
316C4⋊Q8—62no—
317#124—22no—
318#124C4×D314yes—
319C4×Q8—22no—
320C4×Q8—41no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—121no—
324C4×Q8—121no—
325C4×D4—22no—
326C4×D4—41no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4#6711yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42—21no—
344C2×C42—21no—
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—62no—
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—34no—
363#97—62no—
364C4×C8—62no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372#95—61no—
373#95#6711yescharacteristic
374#95#6711yescharacteristic
375C23×C3—22no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3—41no—
382C23×C3C22×D411yescharacteristic
383C3×Q8—22no—
384C3×Q8—22no—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—41no—
390C3×D4—41no—
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—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
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—84no—
416C2×C3×C4C4×D412yes—
417C2×C3×C4C4×D412yes—
418C2×C3×C4#13712yes—
419C2×C3×C4C23×C411yescharacteristic
420C2×C3×C4C22×D411yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4H(5,𝔽2)11yescharacteristic
424C2×C3×C4H(5,𝔽2)11yescharacteristic
425C2×Dic3—22no—
426C2×Dic3—22no—
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—
436C2×Dic3—41no—
437Dic6—22no—
438Dic6—22no—
439Dic6—22no—
440Dic6—22no—
441Dic6—41no—
442Dic6—41no—
443Dic6—41no—
444Dic6—41no—
445C3⋊C8—22no—
446C3⋊C8—41no—
447C3⋊C8—41no—
448C3⋊C8C4×D414yes—
449C24—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—32no—
456C2×Q8—32no—
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×Q8C2×C4×D312yes—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4C2×C4×D312yes—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—81no—
484C22×C4C2×#6711yescharacteristic
485C22×C4C2×#6711yescharacteristic
486C22×C4C2×#6711yescharacteristic
487C22×C4C2×#6711yescharacteristic
488C22×C4C2×#6711yescharacteristic
489C22×C4C2×#6711yescharacteristic
490C22×C4C23×D311yescharacteristic
491Q16—68no—
492Q16—124no—
493SD16—68no—
494SD16—124no—
495C2×C8—32no—
496C2×C8—32no—
497C2×C8—32no—
498C2×C8—32no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—32no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—64no—
515C4⋊C4—64no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C4⋊C4D3×D412yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—81no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—42no—
532C42—81no—
533C42—122no—
534C42—122no—
535C42—122no—
536C42C2×C4×D312yes—
537C42#23512yes—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3C2×C4×D411yescharacteristic
546C22×C3D4211yescharacteristic
547C22×C3#55011yescharacteristic
548C22×C3C2×#13711yescharacteristic
549C22×C3#57511yescharacteristic
550C22×C3#57711yescharacteristic
551C22×C3#57811yescharacteristic
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—22no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—42no—
559C3×C4—41no—
560C3×C4—81no—
561C3×C4C2×C4×D411yescharacteristic
562C3×C4#51811yescharacteristic
563C3×C4#55011yescharacteristic
564C3×C4#55911yescharacteristic
565Dic3—42no—
566Dic3—42no—
567Dic3—42no—
568Dic3—81no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23C22×#6711yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—62no—
582Q8—62no—
583Q8—62no—
584Q8—62no—
585Q8—121no—
586Q8—121no—
587Q8—121no—
588Q8—121no—
589D4—22no—
590D4—22no—
591D4—41no—
592D4—41no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—84no—
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—121no—
630C2×C4C4×#6712yes—
631C2×C4C4×#6712yes—
632C2×C4#95112yes—
633C2×C4C22×C4×D311yescharacteristic
634C2×C4C2×#23511yescharacteristic
635C2×C4C2×D3×D411yescharacteristic
636C2×C4#100411yescharacteristic
637C2×C4D3×#4111yescharacteristic
638C2×C4#100911yescharacteristic
639C8—34no—
640C8—62no—
641C8—121no—
642C8—121no—
643C2×C3—82no—
644C2×C3#236611yescharacteristic
645C2×C3C4×#13711yescharacteristic
646C2×C3#294711yescharacteristic
647C2×C3#326711yescharacteristic
648C2×C3#327011yescharacteristic
649C2×C3#340011yescharacteristic
650C2×C3#340211yescharacteristic
651C22—42no—
652C22—42no—
653C22—42no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22C2×C4×#6711yescharacteristic
659C22D4×#6711yescharacteristic
660C22#582211yescharacteristic
661C22C2×#95111yescharacteristic
662C22#583411yescharacteristic
663C22#583611yescharacteristic
664C22#583811yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—81no—
674C4—122no—
675C4—122no—
676C4—122no—
677C4—241no—
678C4C4×D3×D411yescharacteristic
679C4#554511yescharacteristic
680C4#555311yescharacteristic
681C4#555911yescharacteristic
682C3#2319411yescharacteristic, complemented, Sylow
683C2—82no—
684C2#8234311yescharacteristic
685C2C4×#95111yescharacteristic
686C2#8257111yescharacteristic
687C2#8350311yescharacteristic
688C2#8354611yescharacteristic
689C2#8368111yescharacteristic
690C2#8372111yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2367C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#2367D3
hypocenterC3#23194
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
3. C2×C4×#216C2semidirect
5. #73651C2semidirect
7. #71632C2semidirect
9. #69690C2semidirect
10. #69678C2semidirect
13. #69486C2semidirect
14. #69481C2semidirect
15. #69478C2semidirect
39. #5060C4semidirect
108. C2×#216C2×C4semidirect
109. C2×#216C2×C4semidirect
130. #869C2×C4semidirect
157. #803C2×C4semidirect
682. C3#23194semidirect

Maximal subgroups

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

subgroupindex
C3×#23672
C2×#50602
C2×C4×#2162
C2×#50182
#736512
#717212
#716322
#716122
#696902
#696782
#696602
#696462
#694862
#694812
#694782
#231943