Finite Group Databasebeta

← #10830051

Subgroups — #10830051

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×#5054C211yesmaximal, characteristic
3#82480C211yesmaximal, characteristic, complemented
4C2×C4×#216C211yesmaximal, characteristic, complemented
5#71793C211yesmaximal, characteristic, complemented
6#71781C211yesmaximal, characteristic, complemented
7#71729C211yesmaximal, characteristic
8#71725C211yesmaximal, characteristic
9#71722C211yesmaximal, characteristic
10#71634C211yesmaximal, characteristic, complemented
11#71632C211yesmaximal, characteristic, complemented
12#71628C211yesmaximal, characteristic, complemented
13#71627C211yesmaximal, characteristic, complemented
14#71612C211yesmaximal, characteristic
15#71606C211yesmaximal, characteristic
16#24195—31nomaximal, Sylow
17C3×#556—24no—
18C3×#556—24no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126—21no—
21C2×C3×#126—21no—
22C2×C3×#124C2211yescharacteristic
23C2×C3×#124C2211yescharacteristic
24C2×C3×C22⋊Q8—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117—21no—
30C22×#216C2211yescharacteristic
31C22×#216C2211yescharacteristic
32C22×#214C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34#5230—21no—
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#835C2211yescharacteristic
41C2×#832C2211yescharacteristic
42C2×#832C2211yescharacteristic
43C2×#832C2211yescharacteristic
44#5054—24no—
45C4×#216—24no—
46#4985—21no—
47#4978—21no—
48#4978—21no—
49C2×#810C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4936C2211yescharacteristic
52#4929C2211yescharacteristic
53#4929C2211yescharacteristic
54#4928C2211yescharacteristic
55#4928C2211yescharacteristic
56C2×C4⋊(C3⋊C8)C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556D311yescharacteristic
65C2×#464—31no—
66C2×C4×Q16—31no—
67#2581—31no—
68#2043—31no—
69#2042—31no—
70#2016—31no—
71#2015—31no—
72#1984—31no—
73#1958—31no—
74#1878—31no—
75#1877—31no—
76#1863—31no—
77#1862—31no—
78#1805—31no—
79C22×C3×Q8D411yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4C2311yescharacteristic
83C23×C3×C4—21no—
84C22×Dic6C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—22no—
87C3×#126—28no—
88C3×#126—28no—
89C3×#124—22no—
90C3×#124—22no—
91C3×#124—22no—
92C3×#124D414yes—
93C3×C22⋊Q8—28no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8—22no—
99C3×C4×Q8D414yes—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×#117—28no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4D411yescharacteristic
108C2×C3×C4⋊C4D411yescharacteristic
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×#216—22no—
122C2×#216—22no—
123C2×#216—22no—
124C2×#216—22no—
125C2×#216—22no—
126C2×#216—41no—
127C2×#214—22no—
128C2×#214—22no—
129C2×#214—41no—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2311yescharacteristic
133C2×Dic3⋊C4—21no—
134C2×Dic3⋊C4—21no—
135C2×Dic3⋊C4—21no—
136C2×C4×Dic3—21no—
137C22×C3⋊C8C2311yescharacteristic
138C22×C3⋊C8C2311yescharacteristic
139C22×C3⋊C8C2311yescharacteristic
140C22×C3⋊C8C2311yescharacteristic
141C4×Dic6—22no—
142C4×Dic6—22no—
143#835—22no—
144#835—22no—
145#835—22no—
146#835—41no—
147#835—41no—
148#835—41no—
149#832—22no—
150#832—22no—
151#832—41no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156C3⋊#95—21no—
157C3⋊#95—21no—
158C3⋊#95—21no—
159C3⋊#95—21no—
160#810—22no—
161#807—22no—
162C4⋊(C3⋊C8)—22no—
163C4×C3⋊C8—22no—
164C22×Q16—31no—
165C22×Q16—31no—
166C22×SD16—31no—
167#556—24no—
168#556—24no—
169C2×C4⋊Q8C2×D311yescharacteristic
170C2×#126—21no—
171C2×#126—21no—
172C2×#124C2×D311yescharacteristic
173C2×#124C2×D311yescharacteristic
174C2×C22⋊Q8—21no—
175C2×C4×Q8—31no—
176C2×C4×Q8C2×D311yescharacteristic
177C2×C4×Q8C2×D311yescharacteristic
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×D4C2×D311yescharacteristic
180C2×#117—21no—
181#464—64no—
182C4×Q16—64no—
183C2×#107—31no—
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103—31no—
191C2×#103—31no—
192C2×#102—31no—
193C2×#102—31no—
194C2×#102—31no—
195C2×C4×C8—31no—
196#400—61no—
197#395—61no—
198#391—31no—
199#391—31no—
200#389—31no—
201#389—31no—
202#387—61no—
203#387—61no—
204#382—31no—
205#340—31no—
206#340—31no—
207#336—31no—
208#336—31no—
209#336—31no—
210#336—31no—
211C24×C3—21no—
212C2×C3×Q8—22no—
213C2×C3×Q8—22no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8—41no—
225C2×C3×Q8C2×D412yes—
226C2×C3×Q8C2×D412yes—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4C2×D412yes—
233C22×C3×C4—24no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—21no—
237C22×C3×C4—42no—
238C22×C3×C4—42no—
239C22×C3×C4C2×D411yescharacteristic
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4C2×D411yescharacteristic
242C22×C3×C4C2×D411yescharacteristic
243C22×C3×C4C2×D411yescharacteristic
244C22×C3×C4C2×D411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C22×Dic3—21no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6#4112yes—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4—22no—
261C3×C4⋊C4—22no—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4C2×D412yes—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—24no—
269C3×C22⋊C4—24no—
270C3×C42—24no—
271C3×C42—24no—
272C3×C42—21no—
273C3×C42—21no—
274C3×C42—21no—
275C3×C42—21no—
276C3×C42C2×D412yes—
277C3×C42C2×D412yes—
278#216—28no—
279#216—44no—
280#214—44no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3—21no—
286C4⋊Dic3—21no—
287Dic3⋊C4—42no—
288Dic3⋊C4—42no—
289Dic3⋊C4—42no—
290C4×Dic3—42no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8#4112yes—
303C22×Q8—31no—
304C22×Q8#6711yescharacteristic
305C22×Q8C22×D311yescharacteristic
306C22×Q8C22×D311yescharacteristic
307C22×D4C22×D311yescharacteristic
308C23×C4—21no—
309C2×Q16—62no—
310C2×Q16—62no—
311C2×Q16—62no—
312C2×Q16—62no—
313C2×Q16—62no—
314C2×Q16—121no—
315C2×SD16—62no—
316C2×SD16—62no—
317C2×SD16—121no—
318C22×C8—31no—
319C22×C8—31no—
320C22×C8—31no—
321C22×C8—31no—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324#126—28no—
325#126—28no—
326#124—22no—
327#124—22no—
328#124—22no—
329#124#6714yes—
330C22⋊Q8—28no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8—62no—
337C4×Q8—62no—
338C4×Q8#6714yes—
339C4×D4—22no—
340C4×D4—22no—
341#117—28no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4#6711yescharacteristic
353C2×C4⋊C4#6711yescharacteristic
354C2×C4⋊C4#6711yescharacteristic
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—61no—
364C2×C42C22×D311yescharacteristic
365C2×C42C22×D311yescharacteristic
366C2×C42C22×D311yescharacteristic
367#107—62no—
368C4⋊C8—62no—
369#103—62no—
370#103—62no—
371#103—62no—
372#103—62no—
373#103—121no—
374#103—121no—
375#103—121no—
376#102—62no—
377#102—62no—
378#102—121no—
379C4×C8—62no—
380#95—61no—
381#95—61no—
382#95—61no—
383#95—61no—
384#95—61no—
385#95—61no—
386#95—61no—
387#95—61no—
388C23×C3—22no—
389C23×C3—41no—
390C23×C3—41no—
391C23×C3—41no—
392C23×C3—41no—
393C23×C3—41no—
394C23×C3—41no—
395C23×C3C22×D411yescharacteristic
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—22no—
399C3×Q8—22no—
400C3×Q8—44no—
401C3×Q8—41no—
402C3×Q8—41no—
403C3×Q8—41no—
404C3×Q8—41no—
405C3×D4—22no—
406C3×D4—22no—
407C3×D4—41no—
408C3×D4—41no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—21no—
424C2×C3×C4—21no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—48no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4—41no—
431C2×C3×C4—41no—
432C2×C3×C4—41no—
433C2×C3×C4—41no—
434C2×C3×C4#12012yes—
435C2×C3×C4#12012yes—
436C2×C3×C4#13512yes—
437C2×C3×C4C22×D411yescharacteristic
438C2×C3×C4C22×D411yescharacteristic
439C2×C3×C4C2×#4111yescharacteristic
440C2×C3×C4C2×#4111yescharacteristic
441C2×C3×C4H(5,𝔽2)11yescharacteristic
442C2×C3×C42−1+411yescharacteristic
443C2×Dic3—22no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C2×Dic3—41no—
453C2×Dic3—41no—
454C2×Dic3—41no—
455Dic6—22no—
456Dic6—22no—
457Dic6—41no—
458Dic6—41no—
459C3⋊C8—22no—
460C3⋊C8—41no—
461C3⋊C8—41no—
462C3⋊C8—41no—
463C24—21no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—32no—
477C2×Q8—41no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8C2×#6712yes—
484C2×Q8C2×#6712yes—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4C2×#6712yes—
491C22×C4—24no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—61no—
498C22×C4—61no—
499C22×C4—61no—
500C22×C4—61no—
501C22×C4C2×#6711yescharacteristic
502C22×C4C2×#6711yescharacteristic
503C22×C4C2×#6711yescharacteristic
504C22×C4C2×#6711yescharacteristic
505C22×C4C2×#6711yescharacteristic
506C22×C4C2×#6711yescharacteristic
507C22×C4C23×D311yescharacteristic
508Q16—68no—
509Q16—124no—
510SD16—124no—
511C2×C8—32no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—61no—
535C4⋊C4—61no—
536C4⋊C4—61no—
537C4⋊C4—122no—
538C4⋊C4—122no—
539C4⋊C4—122no—
540C4⋊C4C2×#6712yes—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C42—24no—
547C42—24no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—122no—
553C42C2×#6712yes—
554C42C2×#6712yes—
555C22×C3—42no—
556C22×C3—42no—
557C22×C3—42no—
558C22×C3—81no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3—81no—
562C22×C3C2×#12011yescharacteristic
563C22×C3#54811yescharacteristic
564C22×C3#55011yescharacteristic
565C22×C3C2×#13511yescharacteristic
566C22×C3#57511yescharacteristic
567C22×C3#57811yescharacteristic
568C22×C3#57811yescharacteristic
569C3×C4—22no—
570C3×C4—22no—
571C3×C4—22no—
572C3×C4—42no—
573C3×C4—42no—
574C3×C4—41no—
575C3×C4—42no—
576C3×C4—41no—
577C3×C4—41no—
578C3×C4#52511yescharacteristic
579C3×C4#53711yescharacteristic
580C3×C4#54611yescharacteristic
581C3×C4#54711yescharacteristic
582Dic3—42no—
583Dic3—81no—
584Dic3—81no—
585Dic3—81no—
586C23—22no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23C22×#6711yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—22no—
597Q8—22no—
598Q8—44no—
599Q8—41no—
600Q8—41no—
601Q8—41no—
602Q8—41no—
603Q8—62no—
604Q8—62no—
605Q8—121no—
606Q8—121no—
607D4—22no—
608D4—22no—
609D4—41no—
610D4—41no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—48no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—62no—
637C2×C4—121no—
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#95012yes—
649C2×C4#95312yes—
650C2×C4#95312yes—
651C2×C4D3×#4111yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C2×C4#100911yescharacteristic
654C2×C4#101011yescharacteristic
655C2×C4C22×#6711yescharacteristic
656C2×C4C22×#6711yescharacteristic
657C8—62no—
658C8—121no—
659C8—121no—
660C8—121no—
661C2×C3—82no—
662C2×C3#240711yescharacteristic
663C2×C3#300211yescharacteristic
664C2×C3#301111yescharacteristic
665C2×C3#336411yescharacteristic
666C2×C3#337711yescharacteristic
667C2×C3#339311yescharacteristic
668C2×C3#340211yescharacteristic
669C22—42no—
670C22—42no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22C2×#95011yescharacteristic
677C22#582211yescharacteristic
678C22#583111yescharacteristic
679C22#583411yescharacteristic
680C22#583811yescharacteristic
681C22#583811yescharacteristic
682C22C2×#95311yescharacteristic
683C4—22no—
684C4—22no—
685C4—22no—
686C4—41no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—41no—
691C4—41no—
692C4—122no—
693C4—241no—
694C4—241no—
695C4—241no—
696C4#582411yescharacteristic
697C4#582811yescharacteristic
698C4#582911yescharacteristic
699C4#583311yescharacteristic
700C3#2419511yescharacteristic, complemented, Sylow
701C2—82no—
702C2#8367011yescharacteristic
703C2#8367611yescharacteristic
704C2#8367811yescharacteristic
705C2#8368111yescharacteristic
706C2#8408011yescharacteristic
707C2#8409711yescharacteristic
708C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#24195
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
3. #82480C2semidirect
4. C2×C4×#216C2semidirect
5. #71793C2semidirect
6. #71781C2semidirect
10. #71634C2semidirect
11. #71632C2semidirect
12. #71628C2semidirect
13. #71627C2semidirect
700. C3#24195semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50542
#824802
C2×C4×#2162
#717932
#717812
#717292
#717252
#717222
#716342
#716322
#716282
#716272
#716122
#716062
#241953