Finite Group Databasebeta

← #10728687

Subgroups — #10728687

Order 768.

1718 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#887C211yesmaximal, characteristic
2C2×C3×#398C211yesmaximal, characteristic
3C2×#4983C211yesmaximal, characteristic
4C2×#4979C212yesmaximal
5C2×#4972C211yesmaximal, characteristic
6#71565C211yesmaximal, characteristic
7#71564C212yesmaximal
8#69410C211yesmaximal, characteristic
9#69409C212yesmaximal
10#69406C212yesmaximal
11#69402C211yesmaximal, characteristic
12#14880—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×#887—24no—
26C2×#887—24no—
27C2×#887—24no—
28C3×#398—24no—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4983—24no—
41#4979—28no—
42#4972—24no—
43C22×C4×Q8—31no—
44C2×#398—31no—
45C2×#398D311yescharacteristic
46C2×#395—32no—
47C2×#389—31no—
48#1441—32no—
49#1441—31no—
50#1441—31no—
51#1436—32no—
52#1436—32no—
53#1434—31no—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C22×Dic6—22no—
66C22×Dic6—22no—
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6D414yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4D414yes—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—24no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—24no—
94C2×Dic3⋊C4D414yes—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3D414yes—
104#887—232no—
105C3×#95—22no—
106C3×#95—24no—
107C3⋊#95—24no—
108C3⋊#95—22no—
109C3⋊#95—24no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95Q818yescomplemented
116C23×Q8—31no—
117C2×C4×Q8—64no—
118C2×C4×Q8—64no—
119C2×C4×Q8—64no—
120C22×C4⋊C4—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—32no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4C2×D312yes—
126C22×C4⋊C4C2×D311yescharacteristic
127C22×C4⋊C4C2×D311yescharacteristic
128C22×C42—31no—
129C22×C42—31no—
130C22×C42—31no—
131#398—24no—
132#398—64no—
133#395—68no—
134#389—64no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—31no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95C2×D311yescharacteristic
145C2×#95C2×D312yes—
146C24×C3C2411yescharacteristic
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×Q814yes—
177C22×C3×C4#4112yes—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×D412yes—
209C22×Dic3C2×D412yes—
210C22×Dic3C2×D412yes—
211C22×Dic3C2×Q814yes—
212C22×Dic3C2×Q814yes—
213C22×Dic3#4114yes—
214C22×Dic3#4112yes—
215C22×Dic3#4112yes—
216C2×Dic6—28no—
217C2×Dic6—28no—
218C2×Dic6—28no—
219C2×Dic6—28no—
220C2×Dic6—28no—
221C2×Dic6—28no—
222C2×Dic6—28no—
223C3×C4⋊C4—28no—
224C3×C4⋊C4—44no—
225C3×C4⋊C4—48no—
226C4⋊Dic3—28no—
227Dic3⋊C4—28no—
228Dic3⋊C4—28no—
229Dic3⋊C4—48no—
230Dic3⋊C4—48no—
231C4×Dic3—28no—
232C4×Dic3—28no—
233C4×Dic3—28no—
234C22×Q8—34no—
235C22×Q8—62no—
236C22×Q8—62no—
237C22×Q8—62no—
238C22×Q8—62no—
239C22×Q8—62no—
240C22×Q8—62no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—31no—
246C23×C4—32no—
247C23×C4C22×D312yes—
248C23×C4C22×D312yes—
249C23×C4C22×D311yescharacteristic
250C23×C4C22×D311yescharacteristic
251C23×C4C22×D311yescharacteristic
252C4×Q8—632no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—64no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4#6714yes—
276C2×C42—34no—
277C2×C42—62no—
278C2×C42—62no—
279C2×C42—62no—
280C2×C42—62no—
281C2×C42—62no—
282C2×C42—62no—
283C2×C42—62no—
284C2×C42—62no—
285#95—22no—
286#95—24no—
287#95—38no—
288#95—64no—
289#95—64no—
290#95—64no—
291#95—64no—
292#95—64no—
293#95—64no—
294#95—64no—
295#95—62no—
296C23×C3C22×D411yescharacteristic
297C23×C3C22×D411yescharacteristic
298C23×C3C22×D411yescharacteristic
299C23×C3C22×D411yescharacteristic
300C23×C3C22×Q812yes—
301C23×C3C2×#4111yescharacteristic
302C23×C3C2×#4111yescharacteristic
303C23×C3C2×#4111yescharacteristic
304C23×C3C2×#4112yes—
305C23×C32−1+412yes—
306C23×C32−1+412yes—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4#12014yes—
337C2×C3×C4#12114yes—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—22no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3#12114yes—
374C2×Dic3#12114yes—
375C2×Dic3C4⋊Q814yes—
376Dic6—232no—
377C24C23×D311yescharacteristic
378C2×Q8—68no—
379C2×Q8—68no—
380C2×Q8—68no—
381C2×Q8—68no—
382C2×Q8—68no—
383C2×Q8—68no—
384C2×Q8—68no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—34no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—32no—
417C22×C4—34no—
418C22×C4—34no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—62no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4D3×D412yes—
447C22×C4D3×D412yes—
448C22×C4D3×D412yes—
449C22×C4#23712yes—
450C22×C4D3×Q814yes—
451C22×C4C2×#6712yes—
452C22×C4C2×#6712yes—
453C22×C4C2×#6712yes—
454C4⋊C4—28no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—68no—
458C4⋊C4—68no—
459C4⋊C4—68no—
460C4⋊C4—128no—
461C4⋊C4—128no—
462C42—68no—
463C42—68no—
464C42—68no—
465C22×C3C2×#12011yescharacteristic
466C22×C3C2×#12111yescharacteristic
467C22×C3C2×#12111yescharacteristic
468C22×C3C2×#12111yescharacteristic
469C22×C3#52511yescharacteristic
470C22×C3#52511yescharacteristic
471C22×C3#52511yescharacteristic
472C22×C3#52511yescharacteristic
473C22×C3C2×C4⋊Q811yescharacteristic
474C22×C3#53312yes—
475C22×C3#53312yes—
476C22×C3#53611yescharacteristic
477C22×C3#53611yescharacteristic
478C22×C3#54112yes—
479C22×C3#54112yes—
480C22×C3#54712yes—
481C22×C3#54712yes—
482C22×C3#54812yes—
483C22×C3D4×Q812yes—
484C22×C3D4×Q812yes—
485C22×C3D4×Q812yes—
486C22×C3#55212yes—
487C22×C3#55712yes—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—42no—
491C3×C4—44no—
492C3×C4—82no—
493Dic3—24no—
494Dic3—24no—
495Dic3—24no—
496Dic3—42no—
497Dic3—44no—
498Dic3—44no—
499C23C2×D3×D411yescharacteristic
500C23C2×D3×D411yescharacteristic
501C23C2×D3×D411yescharacteristic
502C23C2×#23711yescharacteristic
503C23C2×D3×Q812yes—
504C23#100712yes—
505C23D3×#4112yes—
506C23D3×#4111yescharacteristic
507C23D3×#4111yescharacteristic
508C23#101012yes—
509C23C22×#6711yescharacteristic
510Q8—632no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—34no—
527C2×C4—34no—
528C2×C4—34no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—62no—
544C2×C4—64no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—62no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—121no—
570C2×C4—122no—
571C2×C4—121no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4#93714yes—
579C2×C4#93814yes—
580C2×C3#239911yescharacteristic
581C2×C3#240611yescharacteristic
582C2×C3#242912yes—
583C2×C3#242912yes—
584C2×C3#243411yescharacteristic
585C2×C3#246212yes—
586C2×C3#246212yes—
587C2×C3#253611yescharacteristic
588C2×C3#254611yescharacteristic
589C2×C3#263411yescharacteristic
590C2×C3#284711yescharacteristic
591C22D3×#12111yescharacteristic
592C22D3×#12111yescharacteristic
593C22#560611yescharacteristic
594C22#560611yescharacteristic
595C22#562212yes—
596C22#562512yes—
597C22#563212yes—
598C22#563312yes—
599C22#564811yescharacteristic
600C22#565512yes—
601C22#565611yescharacteristic
602C22#568512yes—
603C22#568812yes—
604C22#569312yes—
605C22#569612yes—
606C22D3×C4⋊Q811yescharacteristic
607C22#572411yescharacteristic
608C22#572712yes—
609C22C2×#93711yescharacteristic
610C22C2×#93811yescharacteristic
611C22Q8×#6712yes—
612C22#582412yes—
613C22#582811yescharacteristic
614C4—24no—
615C4—24no—
616C4—44no—
617C4—42no—
618C4—64no—
619C4—64no—
620C4—64no—
621C4—82no—
622C4—122no—
623C4—124no—
624C4—124no—
625C3#1488011yescharacteristic, complemented, Sylow
626C2#7774011yescharacteristic
627C2#7774412yes—
628C2#7775112yes—
629C2#7778411yescharacteristic
630C2#7778511yescharacteristic
631C2#7782311yescharacteristic
632C2#8272311yescharacteristic
633C2#8275712yes—
634C2#8284111yescharacteristic
635C2#8303112yes—
636C2#8323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#398C2
hypercenterC2×#398D3
hypocenterC3#14880
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
115. C3⋊#95Q8semidirect
625. C3#14880semidirect

Maximal subgroups

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

subgroupindex
C22×#8872
C2×C3×#3982
C2×#49832
C2×#49792
C2×#49722
#715652
#715642
#694102
#694092
#694062
#694022
#148803