Finite Group Databasebeta

← #10795112

Subgroups — #10795112

Order 768.

1350 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5522C211yesmaximal, characteristic
2C2×#4705C211yesmaximal, characteristic, complemented
3C2×#4685C211yesmaximal, characteristic, complemented
4C3×#1750C211yesmaximal, characteristic, complemented
5#71582C211yesmaximal, characteristic, complemented
6#69956C212yesmaximal
7#69908C212yesmaximal, complemented
8#69830C211yesmaximal, characteristic, complemented
9#69753C211yesmaximal, characteristic, complemented
10#68665C212yesmaximal, complemented
11#68662C212yesmaximal, complemented
12#20721—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16#5522—24no—
17#5522—24no—
18C2×#893C411yescharacteristic
19C2×#893C411yescharacteristic
20C2×#887C411yescharacteristic
21C2×#887C411yescharacteristic
22C2×#67⋊C4C411yescharacteristic
23C2×#67⋊C4C411yescharacteristic
24C2×#67⋊C4C411yescharacteristic
25C2×#67⋊C4C411yescharacteristic
26C2×#872C2211yescharacteristic
27C2×#872C2211yescharacteristic
28C2×C42×D3C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C2×C3×C22⋊C8C2212yes—
31C2×C3×#97C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33C2×#830C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38C2×Dic3⋊C8C2212yes—
39C2×Dic3⋊C8C2212yes—
40C2×#803C2211yescharacteristic
41C2×C4×C3⋊C8C2211yescharacteristic
42#4705—24no—
43#4685—24no—
44C3×#336C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4524C2212yes—
48C2×C4×#41—31no—
49C2×#432—31no—
50C2×#431—31no—
51#1818—32no—
52#1797—32no—
53#1797—32no—
54#1780—32no—
55#1750—31no—
56#1750—31no—
57#1750—31no—
58#1750D311yescharacteristic, complemented
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C411yescharacteristic
61C22×#67C2×C411yescharacteristic
62C23×Dic3C2×C411yescharacteristic
63C23×Dic3C2×C411yescharacteristic
64C2×#235—24no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—24no—
68C22×D12C2×C411yescharacteristic
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3C2×C411yescharacteristic
71C22×C4×D3C2×C411yescharacteristic
72C22×C4×D3C2×C411yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C22×C4×D3C2311yescharacteristic
75C22×Dic6C2×C411yescharacteristic
76C22×C3×C8C2312yes—
77C22×C3×C8C2312yes—
78C3×#117—22no—
79C3×#117—22no—
80C2×C3×C4⋊C4C2×C411yescharacteristic
81C2×C3×C4⋊C4C2×C411yescharacteristic
82C2×C3×C22⋊C4C2×C411yescharacteristic
83C2×C3×C22⋊C4C2×C411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4C2×C411yescharacteristic
87C2×(C2×D3)⋊C4C2×C411yescharacteristic
88C2×(C2×D3)⋊C4C2×C411yescharacteristic
89C2×(C2×D3)⋊C4C2×C411yescharacteristic
90C2×Dic3⋊C4C2×C411yescharacteristic
91C2×Dic3⋊C4C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C411yescharacteristic
93C2×Dic3⋊C4C2×C411yescharacteristic
94C2×C4×Dic3—22no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3C2×C411yescharacteristic
97C2×C4×Dic3C2×C411yescharacteristic
98C2×C4×Dic3C2×C411yescharacteristic
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2312yes—
106#893—24no—
107#893—24no—
108#887—24no—
109#887—24no—
110#67⋊C4—24no—
111#67⋊C4—24no—
112#67⋊C4—24no—
113#67⋊C4—24no—
114#872—22no—
115#872—22no—
116#872—22no—
117#872—22no—
118C42×D3—22no—
119C42×D3—22no—
120C42×D3D414yes—
121C42×D3D414yes—
122C3×C22⋊C8—24no—
123C3×#97—22no—
124C3×C4×C8—22no—
125#830—24no—
126(C2×D3)⋊C8—24no—
127(C2×D3)⋊C8—24no—
128(C2×D3)⋊C8—24no—
129(C2×D3)⋊C8—24no—
130Dic3⋊C8—24no—
131Dic3⋊C8—24no—
132#803—22no—
133C4×C3⋊C8—22no—
134C22×#41—31no—
135C4×#41—64no—
136C4×#41—64no—
137C2×C4×Q8—31no—
138C2×C4×Q8—31no—
139C2×C4×D4—31no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×#117—31no—
146C2×#117—31no—
147C2×#117C2×D311yescharacteristic
148C22×C42—31no—
149C22×C42—31no—
150C22×C42—31no—
151#432—64no—
152#431—64no—
153C2×C4⋊C8—32no—
154C2×C4⋊C8—32no—
155C2×C22⋊C8—32no—
156C2×C22⋊C8—32no—
157C2×C22⋊C8—32no—
158C2×C22⋊C8—32no—
159C2×C22⋊C8—32no—
160C2×C22⋊C8C2×D312yes—
161C2×#97—31no—
162C2×#97C2×D311yescharacteristic
163C2×C4×C8—31no—
164C2×C4×C8C2×D311yescharacteristic
165#336—32no—
166#336—32no—
167#336—32no—
168#336C2×D312yes—
169C24×C3C22×C411yescharacteristic
170C23×D3C22×C411yescharacteristic
171C23×D3C22×C411yescharacteristic
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C22×C411yescharacteristic
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C22×C411yescharacteristic
179C22×C3×C4C22×C411yescharacteristic
180C22×C3×C4#4114yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183C2×#67—22no—
184C2×#67—22no—
185C2×#67—22no—
186C2×#67—22no—
187C2×#67—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3—42no—
192C22×Dic3C22×C411yescharacteristic
193C22×Dic3C22×C411yescharacteristic
194C22×Dic3C22×C411yescharacteristic
195C22×Dic3C22×C411yescharacteristic
196C22×Dic3C22×C411yescharacteristic
197C22×Dic3C22×C411yescharacteristic
198#235—216no—
199#235—216no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—24no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—24no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3C22⋊C414yes—
215C2×C4×D3C22⋊C414yes—
216C2×C4×D3C2×D412yes—
217C2×C4×D3C2×D412yes—
218C2×C4×D3#4114yes—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42C2×D412yes—
235C3×C42C2×D412yes—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—22no—
240Dic3⋊C4—22no—
241Dic3⋊C4—22no—
242Dic3⋊C4—22no—
243Dic3⋊C4—22no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—22no—
249C4×Dic3—22no—
250C4×Dic3C22⋊C414yes—
251C4×Dic3C22⋊C414yes—
252C4×Dic3C2×D412yes—
253C4×Dic3C2×D412yes—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—22no—
256C2×C3⋊C8—22no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—22no—
260C2×#41—62no—
261C2×#41—62no—
262C2×#41—64no—
263C2×#41—64no—
264C22×Q8—31no—
265C22×D4—31no—
266C22×D4—31no—
267C22×D4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—31no—
271C23×C4—31no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—31no—
276C23×C4C22×D311yescharacteristic
277C22×C8—32no—
278C22×C8—32no—
279C22×C8C22×D312yes—
280C22×C8C22×D312yes—
281C4×Q8—64no—
282C4×Q8—64no—
283C4×D4—64no—
284C4×D4—64no—
285C4×D4—64no—
286C4×D4—64no—
287C4×D4—64no—
288C4×D4—64no—
289#117—22no—
290#117—22no—
291#117—62no—
292#117—62no—
293#117—62no—
294#117—62no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4C4×D311yescharacteristic
300C2×C4⋊C4C4×D311yescharacteristic
301C2×C22⋊C4—31no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4C4×D311yescharacteristic
306C2×C22⋊C4C4×D311yescharacteristic
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—34no—
311C2×C42—34no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—62no—
318C2×C42—62no—
319C2×C42—62no—
320C2×C42—62no—
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323C4⋊C8—64no—
324C4⋊C8—64no—
325C22⋊C8—24no—
326C22⋊C8—64no—
327C22⋊C8—64no—
328C22⋊C8—64no—
329C22⋊C8—64no—
330C22⋊C8—64no—
331#97—22no—
332#97—62no—
333C4×C8—22no—
334C4×C8—62no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3#11712yes—
340C23×C3C23×C411yescharacteristic
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—22no—
348C22×D3—22no—
349C22×D3C2×C22⋊C412yes—
350C22×D3#11712yes—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4C2×C22⋊C412yes—
362C2×C3×C4C2×C22⋊C412yes—
363C2×C3×C4#11712yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4112yes—
367C2×C3×C4C2×#4112yes—
368#67—28no—
369#67—28no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—44no—
380C2×Dic3—44no—
381C2×Dic3C2×C22⋊C412yes—
382C2×Dic3C2×C22⋊C412yes—
383C2×Dic3C2×C22⋊C412yes—
384C2×Dic3#11712yes—
385C2×Dic3#13114yes—
386D12—28no—
387C4×D3—24no—
388C4×D3—24no—
389C4×D3—24no—
390C4×D3—24no—
391C4×D3—28no—
392C4×D3—44no—
393C4×D3—44no—
394C4×D3—44no—
395Dic6—28no—
396C3×C8—42no—
397C3×C8—42no—
398C3⋊C8—42no—
399C3⋊C8—42no—
400C24—31no—
401C24—31no—
402C24C2×C4×D311yescharacteristic
403#41—616no—
404#41—616no—
405C2×Q8—62no—
406C2×Q8—62no—
407C2×Q8—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—64no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—64no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—64no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4C2×C4×D311yescharacteristic
447C22×C4C2×C4×D311yescharacteristic
448C22×C4C2×C4×D311yescharacteristic
449C22×C4C2×C4×D311yescharacteristic
450C22×C4C2×C4×D311yescharacteristic
451C22×C4#23514yes—
452C22×C4C23×D311yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—62no—
464C2×C8—62no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—62no—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C42—21no—
478C42—21no—
479C42—34no—
480C42—34no—
481C42—32no—
482C42—32no—
483C42—62no—
484C42—62no—
485C42—62no—
486C42—62no—
487C42—61no—
488C42—61no—
489C42D3×D412yes—
490C42D3×D412yes—
491C22×C3—24no—
492C22×C3—22no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3C22×C22⋊C411yescharacteristic
497C22×C3C2×#11711yescharacteristic
498C22×C3C2×#11711yescharacteristic
499C22×C3C2×#13112yes—
500C22×C3#56812yes—
501C2×D3—22no—
502C2×D3—22no—
503C2×D3—24no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—41no—
507C2×D3—41no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—42no—
511C3×C4—22no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4#52512yes—
517C3×C4#52512yes—
518Dic3—22no—
519Dic3—22no—
520Dic3—22no—
521Dic3—41no—
522Dic3—41no—
523Dic3#40814yes—
524C23—21no—
525C23—22no—
526C23—22no—
527C23—21no—
528C23—32no—
529C23—32no—
530C23—61no—
531C23—61no—
532C23—61no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—61no—
538C23#87212yes—
539C23C22×C4×D311yescharacteristic
540Q8—68no—
541D4—68no—
542D4—68no—
543D4—68no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—32no—
555C2×C4—34no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—68no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—124no—
574C2×C4—124no—
575C2×C4—124no—
576C2×C4—124no—
577C2×C4—124no—
578C2×C4#87212yes—
579C2×C4D3×C22⋊C412yes—
580C2×C4D3×C22⋊C412yes—
581C2×C4C2×#23512yes—
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4D3×#4112yes—
585C8—42no—
586C8—42no—
587C8—122no—
588C8—122no—
589C2×C3—42no—
590C2×C3#228011yescharacteristic
591C2×C3C2×#40811yescharacteristic
592C2×C3#287911yescharacteristic
593C2×C3#292512yes—
594C2×C3#292512yes—
595D3—42no—
596D3—42no—
597C22—22no—
598C22—24no—
599C22—42no—
600C22—41no—
601C22—41no—
602C22—64no—
603C22—62no—
604C22—62no—
605C22—62no—
606C22—62no—
607C22—122no—
608C22—121no—
609C22—121no—
610C22—122no—
611C22—122no—
612C22C2×#87211yescharacteristic
613C22C2×D3×C22⋊C411yescharacteristic
614C22D3×#11711yescharacteristic
615C22D3×#13112yes—
616C22#575012yes—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—34no—
621C4—41no—
622C4—41no—
623C4—62no—
624C4—62no—
625C4—62no—
626C4—121no—
627C4—121no—
628C4#554712yes—
629C4#554712yes—
630C3#2072111yescharacteristic, complemented, Sylow
631C2—42no—
632C2—122no—
633C2—122no—
634C2#7356111yescharacteristic
635C2#7855212yes—
636C2D3×#40811yescharacteristic
637C2#7866211yescharacteristic
638C2#7937512yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1750C2
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#1750D3
hypocenterC3#20721
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#4705C2semidirect
3. C2×#4685C2semidirect
4. C3×#1750C2semidirect
5. #71582C2semidirect
7. #69908C2semidirect
8. #69830C2semidirect
9. #69753C2semidirect
10. #68665C2semidirect
11. #68662C2semidirect
58. #1750D3semidirect
630. C3#20721semidirect

Maximal subgroups

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

subgroupindex
C2×#55222
C2×#47052
C2×#46852
C3×#17502
#715822
#699562
#699082
#698302
#697532
#686652
#686622
#207213