Finite Group Databasebeta

← #10799731

Subgroups — #10799731

Order 768.

1142 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5729C211yesmaximal, characteristic
2C2×#4839C211yesmaximal, characteristic
3#73750C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic, complemented
5#71779C211yesmaximal, characteristic, complemented
6#70461C211yesmaximal, characteristic, complemented
7#69491C211yesmaximal, characteristic, complemented
8#68867C211yesmaximal, characteristic
9#68867C211yesmaximal, characteristic
10#68866C211yesmaximal, characteristic
11D3×#340C211yesmaximal, characteristic
12#68802C211yesmaximal, characteristic, complemented
13#68801C211yesmaximal, characteristic, complemented
14#68800C211yesmaximal, characteristic, complemented
15#68799C211yesmaximal, characteristic, complemented
16#21315—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#946—21no—
19C2×#944—21no—
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22#5729—24no—
23#5729C418yescomplemented
24C2×#896—21no—
25C2×#892C2211yescharacteristic
26C2×#892C2211yescharacteristic
27C2×D3×C4⋊C4C2211yescharacteristic
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×#890C2211yescharacteristic
30C2×#889—21no—
31C2×#888C2211yescharacteristic
32C2×#872C2211yescharacteristic
33C2×#870C2211yescharacteristic
34C2×C3×C4⋊C8C2211yescharacteristic
35C3×C4×C4⋊C4C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40C4×(C2×D3)⋊C4—21no—
41C4×C4⋊Dic3C2211yescharacteristic
42C4×Dic3⋊C4—21no—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4839—24no—
45#4666—21no—
46#4649—21no—
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#544—31no—
64C2×#424—31no—
65#2274—31no—
66#1989—31no—
67#1989—31no—
68#1989—31no—
69#1989—31no—
70#1988—31no—
71#1988D311yescharacteristic, complemented
72#1987—31no—
73#1987—31no—
74#1736—31no—
75#1736—31no—
76#1735—31no—
77C2×#340—31no—
78C22×C3×Q8—21no—
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C4×D3C2311yescharacteristic
82C22×Dic6—21no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8C2×C414yescomplemented
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94#946—28no—
95#944—28no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)—22no—
124C2×D3×C8—22no—
125C2×D3×C8—22no—
126C2×D3×C8—22no—
127#896—28no—
128#892—22no—
129#892—22no—
130#892C2×C414yes—
131#892Q814yescomplemented
132D3×C4⋊C4—22no—
133D3×C4⋊C4—22no—
134D3×C4⋊C4C2×C414yes—
135D3×C4⋊C4Q814yescomplemented
136#890—22no—
137#890C2×C414yescomplemented
138#889—28no—
139#888—22no—
140#888C2×C414yescomplemented
141#872C2×C414yes—
142#872D414yes—
143#870—22no—
144#870C2×C414yescomplemented
145C3×C4⋊C8—22no—
146C3×#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150Dic3⋊C8—22no—
151Dic3⋊C8—22no—
152C4⋊(C3⋊C8)—22no—
153C22×M16—31no—
154C23×C8—31no—
155#544—38no—
156#544—64no—
157C2×C4⋊Q8—31no—
158C2×C4⋊Q8—61no—
159C2×C4⋊Q8C2×D311yescharacteristic
160C2×#125—31no—
161C2×#125—31no—
162C2×#125—61no—
163C2×C22⋊Q8—61no—
164C2×C22⋊Q8—61no—
165C2×#117—31no—
166C2×#117—31no—
167C2×#117—31no—
168C22×C4⋊C4—31no—
169C22×C4⋊C4—31no—
170#424—64no—
171C2×C4⋊C8—31no—
172C2×C4⋊C8—31no—
173C2×C4⋊C8—31no—
174C2×C4⋊C8C2×D311yescharacteristic
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178C4×C4⋊C4—61no—
179C4×C4⋊C4—61no—
180C4×C4⋊C4C2×D311yescharacteristic
181C4×C22⋊C4—61no—
182C4×C22⋊C4—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340—31no—
195#340C2×D311yescharacteristic
196#340C2×D311yescharacteristic
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C23×D3C2×D411yescharacteristic
200C2×C3×Q8—24no—
201C2×C3×Q8—81no—
202C2×C3×Q8—81no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C2×C4×D3—24no—
218C2×C4×D3—24no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×C4×D3—81no—
222C2×C4×D3—81no—
223C2×C4×D3C22×C412yes—
224C2×C4×D3C2×D412yes—
225C2×C4×D3C2×Q812yes—
226C2×C4×D3C2×Q812yes—
227C2×C4×D3#4112yes—
228C2×C4×D3#4112yes—
229C2×Dic6—24no—
230C2×Dic6—81no—
231C2×Dic6—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C22×C412yes—
244C3×C4⋊C4C22×C412yes—
245C3×C4⋊C4C2×Q812yes—
246C3×C4⋊C4C2×Q812yes—
247C3×C42—81no—
248C3×C42C22×C412yes—
249C3×C42C2×D412yes—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4C22⋊C414yes—
253(C2×D3)⋊C4C22⋊C414yes—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—24no—
259C4⋊Dic3C22×C412yes—
260C4⋊Dic3C22×C412yes—
261C4⋊Dic3C2×Q812yes—
262C4⋊Dic3C2×Q812yes—
263Dic3⋊C4—24no—
264Dic3⋊C4—24no—
265Dic3⋊C4—24no—
266Dic3⋊C4—24no—
267Dic3⋊C4—24no—
268Dic3⋊C4—24no—
269Dic3⋊C4C22⋊C414yes—
270Dic3⋊C4C22⋊C414yes—
271C4×Dic3—24no—
272C4×Dic3—24no—
273C4×Dic3—81no—
274C4×Dic3—81no—
275C4×Dic3—81no—
276C4×Dic3C22×C412yes—
277C4×Dic3C2×D412yes—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284ΓL(1,𝔽25)—44no—
285D3×C8—44no—
286C22×Q8—21no—
287C22×Q8—61no—
288C23×C4—31no—
289C23×C4—61no—
290C23×C4—61no—
291C2×M16—62no—
292C2×M16—62no—
293C2×M16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8—62no—
297C22×C8—62no—
298C22×C8—62no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301C4⋊Q8—22no—
302C4⋊Q8—34no—
303C4⋊Q8—62no—
304C4⋊Q8—68no—
305C4⋊Q8C4×D314yescomplemented
306#125—34no—
307#125—34no—
308#125—62no—
309#125—68no—
310#125—62no—
311C22⋊Q8—68no—
312C22⋊Q8—68no—
313#117—34no—
314#117—34no—
315#117—34no—
316#117—34no—
317#117—62no—
318#117—62no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—62no—
329C2×C4⋊C4—62no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C22⋊C4—31no—
342C2×C22⋊C4—31no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—21no—
346C2×C42—31no—
347C2×C42—61no—
348C2×C42—61no—
349C2×C42—61no—
350C2×C42—61no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353C4⋊C8—22no—
354C4⋊C8—62no—
355C4⋊C8—62no—
356C4⋊C8—62no—
357#95—21no—
358#95—61no—
359#95—61no—
360#95—61no—
361C23×C3C22×D411yescharacteristic
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3C2×C22⋊C412yes—
367C22×D3C22⋊Q812yes—
368C22×D3C22⋊Q812yes—
369C3×Q8—84no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4C2×C22⋊C412yes—
387C2×C3×C4C2×C22⋊C412yes—
388C2×C3×C4C23×C411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×Q811yescharacteristic
391C2×C3×C4C22×Q811yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3C2×C22⋊C412yes—
411C2×Dic3C2×C22⋊C412yes—
412C2×Dic3C2×C22⋊C412yes—
413C2×Dic3C22⋊Q812yes—
414C2×Dic3C22⋊Q812yes—
415C4×D3—84no—
416C4×D3—84no—
417C4×D3C4×Q814yes—
418C4×D3C4×Q814yes—
419C4×D3#12414yes—
420C4×D3C4⋊Q814yes—
421Dic6—84no—
422C3×C8—41no—
423C3×C8—41no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—31no—
427C2×Q8—24no—
428C2×Q8—64no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×Q8—241no—
432C2×Q8—241no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—64no—
447C22×C4—64no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—241no—
453C22×C4—241no—
454C22×C4—241no—
455C22×C4—241no—
456C22×C4D3×D411yescharacteristic
457C22×C4D3×D411yescharacteristic
458C22×C4C23×D311yescharacteristic
459M16—124no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—124no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—34no—
479C4⋊C4—34no—
480C4⋊C4—32no—
481C4⋊C4—32no—
482C4⋊C4—32no—
483C4⋊C4—32no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4C2×C4×D312yes—
496C4⋊C4C2×C4×D312yes—
497C4⋊C4D3×Q812yes—
498C4⋊C4D3×Q812yes—
499C22⋊C4—34no—
500C22⋊C4—34no—
501C22⋊C4—64no—
502C22⋊C4—64no—
503C42—32no—
504C42—32no—
505C42—64no—
506C42—64no—
507C42—81no—
508C42—241no—
509C42—241no—
510C42—241no—
511C42C2×C4×D312yes—
512C42D3×D412yes—
513C22×C3C22×C22⋊C411yescharacteristic
514C22×C3C2×C22⋊Q811yescharacteristic
515C22×C3C2×C22⋊Q811yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57811yescharacteristic
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3#39114yes—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—42no—
532C3×C4—42no—
533C3×C4C2×C4×Q811yescharacteristic
534C3×C4C2×C4×Q811yescharacteristic
535C3×C4C2×#12411yescharacteristic
536C3×C4C2×C4⋊Q811yescharacteristic
537Dic3—22no—
538Dic3—22no—
539Dic3—42no—
540Dic3—42no—
541Dic3—42no—
542Dic3—42no—
543Dic3#39114yes—
544C23—32no—
545C23—32no—
546C23—32no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23C2×D3×D411yescharacteristic
552Q8—84no—
553Q8—244no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—32no—
563C2×C4—32no—
564C2×C4—32no—
565C2×C4—34no—
566C2×C4—34no—
567C2×C4—34no—
568C2×C4—34no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—244no—
596C2×C4—244no—
597C2×C4D3×C22⋊C412yes—
598C2×C4D3×C22⋊C412yes—
599C2×C4C22×C4×D311yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C2×C4C2×D3×Q811yescharacteristic
602C2×C4C2×D3×Q811yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C8—41no—
606C8—41no—
607C8—121no—
608C8—121no—
609C2×C3C2×#39111yescharacteristic
610C2×C3#289711yescharacteristic
611C2×C3#289811yescharacteristic
612C2×C3#308111yescharacteristic
613C2×C3#308211yescharacteristic
614C2×C3#308311yescharacteristic
615C2×C3#308411yescharacteristic
616D3—24no—
617C22—34no—
618C22—62no—
619C22—62no—
620C22—62no—
621C22—62no—
622C22—62no—
623C22—62no—
624C22C2×D3×C22⋊C411yescharacteristic
625C22D3×C22⋊Q811yescharacteristic
626C22D3×C22⋊Q811yescharacteristic
627C22#575711yescharacteristic
628C22#576211yescharacteristic
629C22#577811yescharacteristic
630C22#577911yescharacteristic
631C4—22no—
632C4—22no—
633C4—34no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—62no—
639C4—62no—
640C4—122no—
641C4—122no—
642C4—122no—
643C4—122no—
644C4C4×D3×Q811yescharacteristic
645C4C4×D3×Q811yescharacteristic
646C4D3×#12411yescharacteristic
647C4D3×C4⋊Q811yescharacteristic
648C3#2131511yescharacteristic, complemented, Sylow
649C2—64no—
650C2D3×#39111yescharacteristic
651C2#7874211yescharacteristic
652C2#7895711yescharacteristic
653C2#8096711yescharacteristic
654C2#8096811yescharacteristic
655C2#8101311yescharacteristic
656C2#8101411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1988C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1988D3
hypocenterC3#21315
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
4. C3×#1988C2semidirect
5. #71779C2semidirect
6. #70461C2semidirect
7. #69491C2semidirect
12. #68802C2semidirect
13. #68801C2semidirect
14. #68800C2semidirect
15. #68799C2semidirect
23. #5729C4semidirect
71. #1988D3semidirect
86. C3×C4⋊Q8C2×C4semidirect
131. #892Q8semidirect
135. D3×C4⋊C4Q8semidirect
137. #890C2×C4semidirect
140. #888C2×C4semidirect
144. #870C2×C4semidirect
305. C4⋊Q8C4×D3semidirect
648. C3#21315semidirect

Maximal subgroups

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

subgroupindex
C2×#57292
C2×#48392
#737502
C3×#19882
#717792
#704612
#694912
#688672
#688662
D3×#3402
#688022
#688012
#688002
#687992
#213153