Finite Group Databasebeta

← #10813406

Subgroups — #10813406

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2352C211yesmaximal, characteristic
2#82335C211yesmaximal, characteristic
3C2×#5012C211yesmaximal, characteristic
4C2×#4995C211yesmaximal, characteristic, complemented
5C2×#4964C211yesmaximal, characteristic, complemented
6#71576C211yesmaximal, characteristic, complemented
7#71555C211yesmaximal, characteristic, complemented
8#69549C211yesmaximal, characteristic, complemented
9#69540C211yesmaximal, characteristic, complemented
10#69513C211yesmaximal, characteristic, complemented
11#69513C211yesmaximal, characteristic, complemented
12#69506C211yesmaximal, characteristic, complemented
13#69504C211yesmaximal, characteristic, complemented
14C4×#807C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#22664—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#388—21no—
25C3×#387—21no—
26C3×#386—21no—
27C3×#382C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C22⋊C4—21no—
32#5220—21no—
33#5212C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5012C418yescomplemented
38#4995—24no—
39#4980—21no—
40#4964—24no—
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4951—21no—
52Dic3×C22⋊C4—21no—
53#4936C2211yescharacteristic
54C4×C4⋊Dic3C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#436—31no—
64C2×#428—31no—
65C2×#428—31no—
66#2352—31no—
67#2352D311yescharacteristic
68#1949—31no—
69#1949—31no—
70#1854—31no—
71#1854—31no—
72#1852—31no—
73#1852—31no—
74#1846—31no—
75#1809—31no—
76#1809—31no—
77C4×#107—31no—
78C22×C3×D4D411yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C3×C4×D4C2×C414yes—
84C3×C4×D4Q814yes—
85C3×#117—22no—
86C3×#117—41no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—41no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106#926—22no—
107#926—41no—
108C2×C4⋊Dic3—22no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—41no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×#208—22no—
118C2×#208—22no—
119C2×#208—22no—
120C2×#208—22no—
121C2×#208C2×C414yescomplemented
122C2×#208C2×C414yescomplemented
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#830C2×C414yescomplemented
138#830C2×C414yescomplemented
139#808—22no—
140#808—22no—
141#808—22no—
142#808—22no—
143#807—22no—
144#807—22no—
145#807—22no—
146#807—41no—
147#807—41no—
148#807D414yescomplemented
149C4⋊(C3⋊C8)C2×C414yescomplemented
150C4×C3⋊C8C2×C414yescomplemented
151C22×M16—31no—
152C22×M16—31no—
153C2×C4×D4C2×D311yescharacteristic
154C2×#117—31no—
155C2×#117C2×D311yescharacteristic
156C22×C4⋊C4—31no—
157C22×C4⋊C4C2×D311yescharacteristic
158#436—38no—
159#428—64no—
160#428—64no—
161C2×#107—31no—
162C2×#107—31no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#107—31no—
166C2×#107—31no—
167C2×#106—31no—
168C2×#106—31no—
169C2×#106—31no—
170C2×#106—31no—
171C2×C4⋊C8—31no—
172C2×C22⋊C8—31no—
173C2×C22⋊C8—31no—
174C2×C4×C8—31no—
175#388—21no—
176#388—61no—
177#387—21no—
178#387—61no—
179#386—21no—
180#386—61no—
181#382—31no—
182#382C2×D311yescharacteristic
183#380—31no—
184#380—31no—
185#380C2×D311yescharacteristic
186#380C2×D311yescharacteristic
187C4×C4⋊C4—31no—
188C4×C4⋊C4C2×D311yescharacteristic
189C4×C22⋊C4—21no—
190C4×C22⋊C4—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197C24×C3C2×D411yescharacteristic
198C24×C3C2×D411yescharacteristic
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4C4⋊C414yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—42no—
215C22×C3×C4—42no—
216C22×C3×C4—81no—
217C22×C3×C4C22×C412yes—
218C22×C3×C4C22×C412yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×Q812yes—
224C22×C3×C4C2×Q812yes—
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—42no—
231C22×Dic3—42no—
232C22×Dic3—81no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4C4⋊C414yes—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—81no—
246C3×C22⋊C4C4⋊C414yes—
247C3×C22⋊C4C4⋊C414yes—
248C3×C42—42no—
249C3×C42—81no—
250C3×C42C22×C412yes—
251C3×C42C2×Q812yes—
252#217—42no—
253#217—42no—
254#217—81no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—42no—
263C4⋊Dic3C2×D412yes—
264C4×Dic3—42no—
265C4×Dic3—81no—
266#208—28no—
267#208—28no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8C22×C412yes—
275C2×C3⋊C8C22×C412yes—
276C2×C3⋊C8C22×C412yes—
277C2×C3⋊C8C22×C412yes—
278C2×C3⋊C8C2×D412yes—
279C2×C3⋊C8#4112yes—
280C22×D4#6711yescharacteristic
281C23×C4—21no—
282C23×C4—61no—
283C23×C4C22×D311yescharacteristic
284C23×C4C22×D311yescharacteristic
285C2×M16—34no—
286C2×M16—34no—
287C2×M16—62no—
288C2×M16—62no—
289C2×M16—62no—
290C2×M16—62no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8—31no—
294C22×C8—31no—
295C4×D4Dic614yes—
296C4×D4C4×D314yes—
297#117—22no—
298#117—41no—
299#117—62no—
300#117—121no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—41no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—121no—
311C2×C4⋊C4#6711yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4#6711yescharacteristic
323C2×C22⋊C4#6711yescharacteristic
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—34no—
330#107—62no—
331#107—62no—
332#107—62no—
333#107—121no—
334#107—121no—
335#106—62no—
336#106—62no—
337#106—62no—
338#106—62no—
339C4⋊C8—34no—
340C22⋊C8—34no—
341C22⋊C8—34no—
342C4×C8—34no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3C2×C4⋊C412yes—
366C23×C3C2×C4⋊C412yes—
367C23×C3C22×D411yescharacteristic
368C3×D4—44no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
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×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—84no—
398C2×C3×C4C2×C4⋊C412yes—
399C2×C3×C4C2×C4⋊C412yes—
400C2×C3×C4C2×C4⋊C412yes—
401C2×C3×C4C2×C4⋊C412yes—
402C2×C3×C4C23×C411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×Q811yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408C2×Dic3—22no—
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—41no—
420C2×Dic3—84no—
421C3⋊C8—22no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C3⋊C8C4×D414yes—
425C24C2×#6711yescharacteristic
426C24C2×#6711yescharacteristic
427C2×D4—41no—
428C2×D4—41no—
429C2×D4Dic3⋊C414yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—81no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—241no—
452C22×C4C2×Dic612yes—
453C22×C4C2×Dic612yes—
454C22×C4C2×C4×D312yes—
455C22×C4C2×C4×D312yes—
456C22×C4C2×#6711yescharacteristic
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C23×D311yescharacteristic
461M16—68no—
462M16—68no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—32no—
466C2×C8—32no—
467C2×C8—32no—
468C2×C8—32no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—42no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—122no—
492C4⋊C4Dic3⋊C414yes—
493C4⋊C4D3×D412yes—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—241no—
500C22⋊C4Dic3⋊C414yes—
501C22⋊C4Dic3⋊C414yes—
502C42—42no—
503C42—81no—
504C42—122no—
505C42—241no—
506C42C2×Dic612yes—
507C42C2×C4×D312yes—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—41no—
522C22×C3C22×C4⋊C411yescharacteristic
523C22×C3D4211yescharacteristic
524C22×C3#54711yescharacteristic
525C22×C3#57511yescharacteristic
526C22×C3#57511yescharacteristic
527C22×C3#57611yescharacteristic
528C22×C3#57811yescharacteristic
529C3×C4—22no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—42no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4C2×C4×D411yescharacteristic
538C3×C4#52011yescharacteristic
539C3×C4D4×Q811yescharacteristic
540C3×C4#55411yescharacteristic
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544Dic3—81no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23C2×Dic3⋊C412yes—
558C23C2×Dic3⋊C412yes—
559C23C22×#6711yescharacteristic
560D4—44no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
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—41no—
589C2×C4—62no—
590C2×C4—84no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—244no—
603C2×C4C2×Dic3⋊C412yes—
604C2×C4C2×Dic3⋊C412yes—
605C2×C4C2×Dic3⋊C412yes—
606C2×C4C2×Dic3⋊C412yes—
607C2×C4C22×Dic611yescharacteristic
608C2×C4C22×C4×D311yescharacteristic
609C2×C4C2×D3×D411yescharacteristic
610C2×C4#100411yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4#101011yescharacteristic
613C8—34no—
614C8—62no—
615C8—62no—
616C8—62no—
617C2×C3—42no—
618C2×C3—42no—
619C2×C3D4×C4⋊C411yescharacteristic
620C2×C3#291111yescharacteristic
621C2×C3#291311yescharacteristic
622C2×C3#327311yescharacteristic
623C2×C3#327711yescharacteristic
624C2×C3#335211yescharacteristic
625C2×C3#335511yescharacteristic
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22C22×Dic3⋊C411yescharacteristic
641C22D4×#6711yescharacteristic
642C22#582211yescharacteristic
643C22#582211yescharacteristic
644C22#582411yescharacteristic
645C22#583511yescharacteristic
646C22#583811yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—122no—
656C4—241no—
657C4—241no—
658C4—241no—
659C4D4×Dic611yescharacteristic
660C4#553911yescharacteristic
661C4C4×D3×D411yescharacteristic
662C4#554611yescharacteristic
663C3#2266411yescharacteristic, complemented, Sylow
664C2—42no—
665C2—42no—
666C2#8220711yescharacteristic
667C2D4×Dic3⋊C411yescharacteristic
668C2#8256211yescharacteristic
669C2#8350411yescharacteristic
670C2#8354811yescharacteristic
671C2#8359711yescharacteristic
672C2#8361311yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2352C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
cosocleC22×C3×C4C24
hypercenter#2352D3
hypocenterC3#22664
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2352

Complements

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

NH ≅ G/Ntype
4. C2×#4995C2semidirect
5. C2×#4964C2semidirect
6. #71576C2semidirect
7. #71555C2semidirect
8. #69549C2semidirect
9. #69540C2semidirect
10. #69513C2semidirect
11. #69513C2semidirect
12. #69506C2semidirect
13. #69504C2semidirect
14. C4×#807C2semidirect
15. #69468C2semidirect
37. #5012C4semidirect
121. C2×#208C2×C4semidirect
122. C2×#208C2×C4semidirect
137. #830C2×C4semidirect
138. #830C2×C4semidirect
148. #807D4semidirect
149. C4⋊(C3⋊C8)C2×C4semidirect
150. C4×C3⋊C8C2×C4semidirect
663. C3#22664semidirect

Maximal subgroups

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

subgroupindex
C3×#23522
#823352
C2×#50122
C2×#49952
C2×#49642
#715762
#715552
#695492
#695402
#695132
#695062
#695042
C4×#8072
#694682
#226643