Finite Group Databasebeta

← #10831798

Subgroups — #10831798

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5567C211yesmaximal, characteristic, complemented
2C3×#2538C211yesmaximal, characteristic
3C2×#5009C211yesmaximal, characteristic
4C2×#5018C211yesmaximal, characteristic
5#71635C211yesmaximal, characteristic, complemented
6#69660C211yesmaximal, characteristic
7#69660C211yesmaximal, characteristic
8#69639C211yesmaximal, characteristic
9#69638C211yesmaximal, characteristic
10#69632C211yesmaximal, characteristic, complemented
11#69631C211yesmaximal, characteristic, complemented
12#69606C211yesmaximal, characteristic, complemented
13#69604C211yesmaximal, characteristic, complemented
14#69588C211yesmaximal, characteristic, complemented
15#69583C211yesmaximal, characteristic, complemented
16#24346—31nomaximal, Sylow
17C2×C3×C22⋊Q8—21no—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#944—21no—
21C22×#214C2211yescharacteristic
22#5567—24no—
23#5567—24no—
24C2×#889—21no—
25C2×#889—21no—
26C2×#887—21no—
27C2×#870C2211yescharacteristic
28C2×#870C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C3×#396—21no—
34C3×#396—21no—
35C3×#390C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C3×#389C2211yescharacteristic
38C3×#389C2211yescharacteristic
39C3×#384C2211yescharacteristic
40C3×#380—21no—
41C2×#832C2211yescharacteristic
42C2×#832C2211yescharacteristic
43C2×#832C2211yescharacteristic
44#5018—24no—
45#5009—24no—
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51C2×#807C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557—31no—
65C2×#478—31no—
66C2×#440—31no—
67#2538D311yescharacteristic
68#2059—31no—
69#2057—31no—
70#2025—31no—
71#2023—31no—
72#1992—31no—
73#1985—31no—
74#1880—31no—
75#1880—31no—
76#1865—31no—
77#1863—31no—
78#1803—31no—
79C22×C3×Q8D411yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C22×Dic6—21no—
83C22×Dic6C2311yescharacteristic
84C3×C22⋊Q8—44no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C2×C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107#944—28no—
108C2×#214—22no—
109C2×#214—22no—
110C2×#214Q814yescomplemented
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
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#889—28no—
130#889—28no—
131#887—28no—
132#870—22no—
133#870—22no—
134#870—22no—
135#870D414yes—
136#869—22no—
137#869—22no—
138C4×Dic6—22no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C4×Dic6—22no—
142C4×Dic6—22no—
143C4×Dic6D414yes—
144C3×#95—21no—
145C3×#95—21no—
146#832—22no—
147#832—22no—
148#832—22no—
149#810—22no—
150#810—22no—
151#810—22no—
152#808—22no—
153#807—22no—
154#807—22no—
155C4⋊(C3⋊C8)—22no—
156#803—22no—
157C22×SD16—31no—
158#557—64no—
159#557—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—61no—
162C2×#125—31no—
163C2×#125—31no—
164C2×#125—61no—
165C2×#125—61no—
166C2×C22⋊Q8—21no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—61no—
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4C2×D311yescharacteristic
173#478—64no—
174#440—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×C4⋊C8—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#97—31no—
186#396—21no—
187#396—21no—
188#390C2×D311yescharacteristic
189#390C2×D311yescharacteristic
190#389C2×D311yescharacteristic
191#389C2×D311yescharacteristic
192#384C2×D311yescharacteristic
193#380—21no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
199#340—31no—
200#336—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204C24×C3—21no—
205C2×C3×Q8—22no—
206C2×C3×Q8—22no—
207C2×C3×Q8—41no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4C2×Q812yes—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—42no—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—24no—
235C2×Dic6—42no—
236C2×Dic6—42no—
237C2×Dic6C2×D412yes—
238C2×Dic6C2×Q812yes—
239C2×Dic6#4112yes—
240C3×C4⋊C4—22no—
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—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—81no—
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×C42—21no—
261C3×C42—21no—
262C3×C42C2×D412yes—
263C3×C42C2×D412yes—
264#214—28no—
265C4⋊Dic3—24no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3C2×D412yes—
270C4⋊Dic3C2×D412yes—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279C4×Dic3—24no—
280C4×Dic3—24no—
281C4×Dic3—24no—
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⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8C2×Q812yes—
294C22×Q8—31no—
295C22×Q8—61no—
296C22×Q8#6711yescharacteristic
297C22×D4C22×D311yescharacteristic
298C23×C4—21no—
299C2×SD16—34no—
300C2×SD16—62no—
301C2×SD16—62no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—31no—
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—68no—
309#125—34no—
310#125—62no—
311#125—62no—
312#125—68no—
313#125—68no—
314#125—62no—
315C22⋊Q8—44no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—34no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—68no—
324C4×Q8—62no—
325C4×D4—22no—
326C4×D4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4#6711yescharacteristic
343C2×C4⋊C4#6711yescharacteristic
344C2×C4⋊C4#6711yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C42—61no—
354C2×C42—61no—
355C2×C42—61no—
356C2×C42C22×D311yescharacteristic
357C2×C42C22×D311yescharacteristic
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—62no—
362C4⋊C8—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#97—62no—
370#95—21no—
371#95—21no—
372C23×C3—22no—
373C23×C3—41no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3C22×D411yescharacteristic
380C3×Q8—44no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—41no—
384C3×D4—41no—
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×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
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—84no—
410C2×C3×C4#12012yes—
411C2×C3×C4#12012yes—
412C2×C3×C4AGL(1,ℤ/8)12yes—
413C2×C3×C4C22×D411yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×Q811yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C42−1+411yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431Dic6—22no—
432Dic6—22no—
433Dic6—22no—
434Dic6—48no—
435Dic6C22⋊Q814yescomplemented
436C3⋊C8—22no—
437C3⋊C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—32no—
444C2×Q8—32no—
445C2×Q8—32no—
446C2×Q8—41no—
447C2×Q8—64no—
448C2×Q8—61no—
449C2×Q8—61no—
450C2×Q8—61no—
451C2×Q8—122no—
452C2×Q8—122no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4D3×Q812yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477C2×C8—32no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—32no—
499C4⋊C4—32no—
500C4⋊C4—32no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—81no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—64no—
526C42—64no—
527C42—64no—
528C42D3×D412yes—
529C42D3×D412yes—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3C2×#12011yescharacteristic
538C22×C3#54811yescharacteristic
539C22×C3D4×Q811yescharacteristic
540C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
541C22×C3#57511yescharacteristic
542C22×C3#57711yescharacteristic
543C22×C3#57811yescharacteristic
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4—81no—
553C3×C4C2×C22⋊Q811yescharacteristic
554C3×C4#53611yescharacteristic
555C3×C4#54811yescharacteristic
556C3×C4D4×Q811yescharacteristic
557Dic3—24no—
558Dic3—42no—
559Dic3—42no—
560Dic3—42no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C22×#6711yescharacteristic
569Q8—34no—
570Q8—44no—
571Q8—62no—
572Q8—62no—
573Q8—62no—
574Q8—128no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—84no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4#93212yes—
617C2×C4#93712yes—
618C2×C4#93712yes—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×D3×Q811yescharacteristic
622C2×C4#100711yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4#101011yescharacteristic
625C8—62no—
626C8—121no—
627C8—121no—
628C8—121no—
629C2×C3—82no—
630C2×C3#240911yescharacteristic
631C2×C3#300611yescharacteristic
632C2×C3#301311yescharacteristic
633C2×C3#336811yescharacteristic
634C2×C3#337711yescharacteristic
635C2×C3#338611yescharacteristic
636C2×C3#338811yescharacteristic
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22C2×#93211yescharacteristic
645C22C2×#93711yescharacteristic
646C22Q8×#6711yescharacteristic
647C22#582211yescharacteristic
648C22#583111yescharacteristic
649C22#583611yescharacteristic
650C22#583811yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—64no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—122no—
662C4—122no—
663C4—122no—
664C4D3×C22⋊Q811yescharacteristic
665C4#562811yescharacteristic
666C4#563211yescharacteristic
667C4#563311yescharacteristic
668C3#2434611yescharacteristic, complemented, Sylow
669C2—82no—
670C2#8262411yescharacteristic
671C2#8275111yescharacteristic
672C2#8342711yescharacteristic
673C2#8355811yescharacteristic
674C2#8358711yescharacteristic
675C2#8363211yescharacteristic
676C2#8367811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2538C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2538D3
hypocenterC3#24346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#5567C2semidirect
5. #71635C2semidirect
10. #69632C2semidirect
11. #69631C2semidirect
12. #69606C2semidirect
13. #69604C2semidirect
14. #69588C2semidirect
15. #69583C2semidirect
110. C2×#214Q8semidirect
435. Dic6C22⋊Q8semidirect
668. C3#24346semidirect

Maximal subgroups

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

subgroupindex
C2×#55672
C3×#25382
C2×#50092
C2×#50182
#716352
#696602
#696392
#696382
#696322
#696312
#696062
#696042
#695882
#695832
#243463