Finite Group Databasebeta

← #10815519

Subgroups — #10815519

Order 768.

1086 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#103C211yesmaximal, characteristic
2Q8×#217C211yesmaximal, characteristic
3C2×#5228C211yesmaximal, characteristic
4#82162C211yesmaximal, characteristic
5C2×#4789C211yesmaximal, characteristic, complemented
6#71972C211yesmaximal, characteristic
7#71967C211yesmaximal, characteristic
8#71556C211yesmaximal, characteristic
9#70281C211yesmaximal, characteristic, complemented
10#70259C211yesmaximal, characteristic, complemented
11#70256C211yesmaximal, characteristic, complemented
12#70241C211yesmaximal, characteristic, complemented
13#70239C211yesmaximal, characteristic, complemented
14#70237C211yesmaximal, characteristic, complemented
15#70220C211yesmaximal, characteristic, complemented
16#22859—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C22×#216C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C2×#924C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×C3×#103—24no—
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31#5229—21no—
32#5229—21no—
33#5229C2211yescharacteristic
34#5229C2211yescharacteristic
35#5228C418yescomplemented
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38#5212—21no—
39#5212C2211yescharacteristic
40#5210—21no—
41C4×#217—21no—
42C4×#217—21no—
43C4×#217C2211yescharacteristic
44C2×#830C2211yescharacteristic
45C2×#816C2211yescharacteristic
46C2×#816C2211yescharacteristic
47C2×#815C2211yescharacteristic
48C2×#815C2211yescharacteristic
49#4978—21no—
50#4976—21no—
51Dic3×C4⋊C4C2211yescharacteristic
52#4973C2211yescharacteristic
53#4972C2211yescharacteristic
54C2×#808C2211yescharacteristic
55C2×#808C2211yescharacteristic
56#4789—24no—
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#451—31no—
64C2×#441—31no—
65C22×#103D311yescharacteristic
66#2360—31no—
67Q8×C22⋊C4—31no—
68#1971—31no—
69#1935—31no—
70#1927—31no—
71#1892—31no—
72#1878—31no—
73#1878—31no—
74#1865—31no—
75#1853—31no—
76#1763—31no—
77#1752—31no—
78C22×C3×Q8—21no—
79C22×C3×Q8—22no—
80C22×C3×Q8—22no—
81C22×C3×Q8—22no—
82C22×C3×Q8—41no—
83C22×C3×Q8—41no—
84C22×C3×Q8C2×C414yes—
85C22×C3×Q8C2311yescharacteristic
86C22×C3×Q8C2311yescharacteristic
87C23×C3×C4—21no—
88C23×C3×C4—21no—
89C23×C3×C4C2311yescharacteristic
90C22×Dic6C2311yescharacteristic
91C22×C3×C8—22no—
92C22×C3×C8—22no—
93C22×C3×C8—41no—
94C22×C3×C8C2311yescharacteristic
95C22×C3×C8C2311yescharacteristic
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—41no—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×#217D411yescharacteristic
104C2×#217D411yescharacteristic
105Q8×Dic3—22no—
106Q8×Dic3—22no—
107Q8×Dic3—41no—
108Q8×Dic3—41no—
109C2×#216—22no—
110C2×#216—22no—
111C2×#216—41no—
112C2×#216—41no—
113C2×#216C2×C414yescomplemented
114C2×#216C2×C414yescomplemented
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120#924—22no—
121#924C2×C414yes—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133#887—22no—
134#887—41no—
135C3×#103—22no—
136C3×#103—22no—
137C3×#103—22no—
138C3×#103D414yes—
139#835C2×C414yescomplemented
140#835C2×C414yescomplemented
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150C3⋊#95D411yescharacteristic
151C3⋊#95D411yescharacteristic
152#830C2×C414yes—
153#816—22no—
154#816—41no—
155#815—22no—
156#815—41no—
157#808—22no—
158#808—41no—
159C23×Q8C2×D311yescharacteristic
160C22×Q16—31no—
161C22×Q16—31no—
162C23×C8C2×D311yescharacteristic
163C2×C22⋊Q8—31no—
164C2×C4×Q8—31no—
165C2×C4×Q8—31no—
166C2×C4×Q8—31no—
167C22×C4⋊C4C2×D311yescharacteristic
168#451—38no—
169#441—64no—
170C2×#106—31no—
171C2×#106—31no—
172C2×#103—24no—
173C2×#103—31no—
174C2×#103—31no—
175C2×#103—31no—
176C2×#103—31no—
177C2×#103C2×D311yescharacteristic
178C2×#103C2×D311yescharacteristic
179C2×#103C2×D311yescharacteristic
180C2×#103C2×D311yescharacteristic
181C2×C22⋊C8—31no—
182C2×#97—31no—
183C2×#97—31no—
184#391—31no—
185#391—31no—
186#391—31no—
187#391—61no—
188#391—61no—
189#389—31no—
190#389—61no—
191#387—61no—
192#385—61no—
193#380—31no—
194#380—61no—
195C4×C4⋊C4—31no—
196C4×C22⋊C4—31no—
197C4×C22⋊C4—61no—
198C4×C22⋊C4—61no—
199#340—31no—
200#340—31no—
201#340—31no—
202#340—31no—
203#336—31no—
204#336—31no—
205C24×C3C2×D411yescharacteristic
206C2×C3×Q8—24no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—28no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—42no—
217C2×C3×Q8—42no—
218C2×C3×Q8—44no—
219C2×C3×Q8—44no—
220C2×C3×Q8—44no—
221C2×C3×Q8C22×C412yes—
222C2×C3×Q8C22×C412yes—
223C2×C3×Q8C2×D412yes—
224C2×C3×Q8#4112yes—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—24no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—42no—
235C22×C3×C4—42no—
236C22×C3×C4—42no—
237C22×C3×C4—42no—
238C22×C3×C4—81no—
239C22×C3×C4C22×C412yes—
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4#4112yes—
242C22×C3×C4C2411yescharacteristic
243C22×Dic3—21no—
244C22×Dic3—21no—
245C22×Dic3—21no—
246C22×Dic3—21no—
247C22×Dic3C2×D411yescharacteristic
248C22×Dic3C2×D411yescharacteristic
249C22×Dic3C2×D411yescharacteristic
250C22×Dic3C2×D411yescharacteristic
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6—21no—
256C2×Dic6C22×C412yes—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—44no—
263C2×C3×C8C2×D412yes—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—42no—
268C3×C4⋊C4C2×D412yes—
269#217—24no—
270#217—22no—
271#217—22no—
272#217—42no—
273#216—28no—
274#216—44no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—42no—
281C4⋊Dic3—42no—
282C4⋊Dic3C22×C412yes—
283Dic3⋊C4—24no—
284Dic3⋊C4—42no—
285Dic3⋊C4—81no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—21no—
290C4×Dic3—42no—
291C4×Dic3—42no—
292C4×Dic3—42no—
293C4×Dic3—81no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8C22×C412yes—
299C2×C3⋊C8C22×C412yes—
300C22×Q8—21no—
301C22×Q8—22no—
302C22×Q8—22no—
303C22×Q8—22no—
304C22×Q8—31no—
305C22×Q8—41no—
306C22×Q8—41no—
307C22×Q8C4×D314yes—
308C22×Q8C22×D311yescharacteristic
309C22×Q8C22×D311yescharacteristic
310C23×C4—21no—
311C23×C4—21no—
312C23×C4C22×D311yescharacteristic
313C2×Q16—34no—
314C2×Q16—34no—
315C2×Q16—62no—
316C2×Q16—62no—
317C2×Q16—121no—
318C2×Q16—121no—
319C22×C8—22no—
320C22×C8—22no—
321C22×C8—31no—
322C22×C8—31no—
323C22×C8—41no—
324C22×C8C22×D311yescharacteristic
325C22×C8C22×D311yescharacteristic
326C22⋊Q8—34no—
327C22⋊Q8—62no—
328C4×Q8—62no—
329C4×Q8—62no—
330C4×Q8—62no—
331C4×Q8—121no—
332C4×Q8—121no—
333C4×Q8—121no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C22⋊C4—31no—
348C2×C22⋊C4—31no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—61no—
354C2×C42—61no—
355C2×C42—61no—
356C2×C42—61no—
357#106—62no—
358#106—121no—
359#103—22no—
360#103—22no—
361#103—22no—
362#103—34no—
363#103—34no—
364#103—62no—
365#103—121no—
366#103#6714yes—
367C22⋊C8—34no—
368#97—62no—
369#97—121no—
370#95—31no—
371#95—31no—
372#95—61no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377#95—61no—
378#95—61no—
379#95—61no—
380#95—61no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—21no—
384C23×C3—21no—
385C23×C3C4×D412yes—
386C23×C3#13712yes—
387C23×C3#13712yes—
388C23×C3C22×D411yescharacteristic
389C3×Q8—22no—
390C3×Q8—22no—
391C3×Q8—22no—
392C3×Q8—22no—
393C3×Q8—22no—
394C3×Q8—48no—
395C3×Q8—41no—
396C3×Q8—41no—
397C3×Q8C4×D414yes—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—48no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—84no—
418C2×C3×C4C4×D412yes—
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—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449Dic6—22no—
450Dic6—22no—
451Dic6—41no—
452Dic6—41no—
453C3×C8—22no—
454C3×C8—41no—
455C3⋊C8—22no—
456C3⋊C8—22no—
457C24D3×D411yescharacteristic
458C2×Q8—24no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—28no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—32no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8—61no—
477C2×Q8—61no—
478C2×Q8—61no—
479C2×Q8C2×C4×D312yes—
480C2×Q8C2×C4×D312yes—
481C2×Q8#23512yes—
482C2×Q8C2×#6712yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—31no—
493C22×C4—31no—
494C22×C4—31no—
495C22×C4—31no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—61no—
501C22×C4—61no—
502C22×C4—61no—
503C22×C4—61no—
504C22×C4—81no—
505C22×C4C2×C4×D312yes—
506C22×C4D3×D411yescharacteristic
507C22×C4#23712yes—
508C22×C4C23×D311yescharacteristic
509Q16—68no—
510Q16—124no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—32no—
517C2×C8—32no—
518C2×C8—44no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C2×C8C2×#6712yes—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—32no—
528C4⋊C4—42no—
529C4⋊C4—64no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—61no—
535C4⋊C4—122no—
536C4⋊C4—122no—
537C4⋊C4—122no—
538C4⋊C4—241no—
539C4⋊C4C2×#6712yes—
540C22⋊C4—64no—
541C22⋊C4—62no—
542C22⋊C4—62no—
543C22⋊C4—122no—
544C42—61no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—122no—
549C42—122no—
550C42—122no—
551C42—241no—
552C22×C3—22no—
553C22×C3—22no—
554C22×C3—22no—
555C22×C3—22no—
556C22×C3—22no—
557C22×C3—22no—
558C22×C3#44114yescomplemented
559C22×C3C2×C4×D411yescharacteristic
560C22×C3D4211yescharacteristic
561C22×C3#54611yescharacteristic
562C22×C3C2×#13711yescharacteristic
563C22×C3C2×#13711yescharacteristic
564C22×C3#57711yescharacteristic
565C22×C3#57811yescharacteristic
566C3×C4—22no—
567C3×C4—24no—
568C3×C4—42no—
569C3×C4—42no—
570C3×C4—81no—
571C3×C4C2×C4×D411yescharacteristic
572C3×C4#51811yescharacteristic
573C3×C4#54611yescharacteristic
574C3×C4#55111yescharacteristic
575Dic3—42no—
576Dic3—42no—
577Dic3—41no—
578Dic3—41no—
579Dic3—41no—
580Dic3—41no—
581Dic3—81no—
582Dic3—81no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23#67⋊C412yes—
588C23#91512yes—
589C23#91812yes—
590C23C2×D3×D411yescharacteristic
591Q8—22no—
592Q8—22no—
593Q8—22no—
594Q8—22no—
595Q8—22no—
596Q8—48no—
597Q8—41no—
598Q8—41no—
599Q8—62no—
600Q8—62no—
601Q8—121no—
602Q8—121no—
603Q8C4×#6714yes—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—48no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—84no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4#67⋊C412yes—
649C2×C4C22×C4×D311yescharacteristic
650C2×C4C2×#23511yescharacteristic
651C2×C4C2×#23711yescharacteristic
652C2×C4#100411yescharacteristic
653C2×C4#100411yescharacteristic
654C2×C4C22×#6711yescharacteristic
655C8—22no—
656C8—41no—
657C8—62no—
658C8—62no—
659C2×C3—24no—
660C2×C3#235811yescharacteristic
661C2×C3C2×#44111yescharacteristic
662C2×C3#294711yescharacteristic
663C2×C3#326611yescharacteristic
664C2×C3#327011yescharacteristic
665C2×C3#331211yescharacteristic
666C2×C3#331611yescharacteristic
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22#478914yescomplemented
674C22C2×#67⋊C411yescharacteristic
675C22#549011yescharacteristic
676C22C2×#91511yescharacteristic
677C22#576211yescharacteristic
678C22C2×#91811yescharacteristic
679C22#576611yescharacteristic
680C22D4×#6711yescharacteristic
681C4—22no—
682C4—24no—
683C4—42no—
684C4—42no—
685C4—81no—
686C4—121no—
687C4—121no—
688C4—122no—
689C4—122no—
690C4—121no—
691C4—121no—
692C4—241no—
693C4—241no—
694C4#548611yescharacteristic
695C4#549511yescharacteristic
696C4C2×C4×#6711yescharacteristic
697C4#580611yescharacteristic
698C3#2285911yescharacteristic, complemented, Sylow
699C2—24no—
700C2C2×#478911yescharacteristic
701C2#7877411yescharacteristic
702C2#7881011yescharacteristic
703C2#7881111yescharacteristic
704C2#8209911yescharacteristic
705C2#8386611yescharacteristic
706C2#8388811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC22×C3×#103C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenterC22×#103D3
hypocenterC3#22859
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
5. C2×#4789C2semidirect
9. #70281C2semidirect
10. #70259C2semidirect
11. #70256C2semidirect
12. #70241C2semidirect
13. #70239C2semidirect
14. #70237C2semidirect
15. #70220C2semidirect
35. #5228C4semidirect
113. C2×#216C2×C4semidirect
114. C2×#216C2×C4semidirect
139. #835C2×C4semidirect
140. #835C2×C4semidirect
558. C22×C3#441semidirect
673. C22#4789semidirect
698. C3#22859semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1032
Q8×#2172
C2×#52282
#821622
C2×#47892
#719722
#719672
#715562
#702812
#702592
#702562
#702412
#702392
#702372
#702202
#228593