Finite Group Databasebeta

← #10751712

Subgroups — #10751712

Order 768.

1598 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#944C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic
3C2×#4979C211yesmaximal, characteristic
4C2×#4978C211yesmaximal, characteristic
5#72134C211yesmaximal, characteristic
6#71568C212yesmaximal
7#71568C212yesmaximal
8#69424C211yesmaximal, characteristic
9#69414C212yesmaximal
10#69409C212yesmaximal
11#69402C211yesmaximal, characteristic
12#16603—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C2×#944—24no—
18C2×#944—24no—
19C2×#944—24no—
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C3×#395—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4979—24no—
41#4978—24no—
42C22×C4⋊Q8—31no—
43C2×#395—31no—
44C2×#395D311yescharacteristic
45C2×#387—31no—
46#1446—31no—
47#1446—31no—
48#1444—32no—
49#1444—32no—
50#1442—32no—
51#1441—32no—
52#1434—31no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—42no—
56C22×C3×Q8—42no—
57C22×C3×Q8D414yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C22×Dic6—22no—
70C22×Dic6—22no—
71C22×Dic6—42no—
72C22×Dic6—42no—
73C22×Dic6D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4—42no—
80#944—232no—
81C2×Dic3⋊C4—24no—
82C2×Dic3⋊C4—24no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—24no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—42no—
91C2×Dic3⋊C4—42no—
92C2×Dic3⋊C4Q814yescomplemented
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3D414yes—
96C3×#95—24no—
97C3×#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C3⋊#95Q818yescomplemented
110C23×Q8—31no—
111C23×Q8C2×D311yescharacteristic
112C2×C4⋊Q8—64no—
113C2×C4⋊Q8—64no—
114C2×C4⋊Q8—64no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—32no—
117C22×C4⋊C4—32no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4C2×D311yescharacteristic
120C22×C4⋊C4C2×D312yes—
121C22×C42—31no—
122#395—24no—
123#395—64no—
124#387—64no—
125C2×#95—32no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—31no—
130C2×#95—31no—
131C2×#95—31no—
132C2×#95—31no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95C2×D311yescharacteristic
138C2×#95C2×D312yes—
139C24×C3C2411yescharacteristic
140C2×C3×Q8—28no—
141C2×C3×Q8—48no—
142C2×C3×Q8—48no—
143C2×C3×Q8—44no—
144C2×C3×Q8—44no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×Q814yes—
175C22×C3×C4C2×Q812yes—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D414yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×Q814yes—
208C22×Dic3C2×Q812yes—
209C22×Dic3C2×Q812yes—
210C22×Dic3C2×Q814yes—
211C22×Dic3#4112yes—
212C2×Dic6—28no—
213C2×Dic6—48no—
214C2×Dic6—48no—
215C2×Dic6—44no—
216C2×Dic6—44no—
217C3×C4⋊C4—48no—
218C3×C4⋊C4—44no—
219Dic3⋊C4—216no—
220Dic3⋊C4—216no—
221Dic3⋊C4—28no—
222Dic3⋊C4—48no—
223C4×Dic3—28no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—34no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8—62no—
230C22×Q8—62no—
231C22×Q8—122no—
232C22×Q8—122no—
233C22×Q8#6714yes—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4—31no—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D311yescharacteristic
242C23×C4C22×D312yes—
243C23×C4C22×D312yes—
244C23×C4C22×D311yescharacteristic
245C4⋊Q8—632no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—122no—
263C2×C4⋊C4—122no—
264C2×C42—34no—
265C2×C42—62no—
266C2×C42—62no—
267#95—24no—
268#95—22no—
269#95—38no—
270#95—64no—
271#95—62no—
272#95—62no—
273#95—62no—
274#95—64no—
275#95—62no—
276#95—62no—
277#95—122no—
278#95—122no—
279#95—122no—
280#95—122no—
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3C22×D411yescharacteristic
284C23×C3C22×Q811yescharacteristic
285C23×C3C22×Q812yes—
286C23×C3C2×#4111yescharacteristic
287C23×C3H(5,𝔽2)12yes—
288C23×C3H(5,𝔽2)12yes—
289C23×C32−1+412yes—
290C23×C32−1+411yescharacteristic
291C23×C32−1+411yescharacteristic
292C3×Q8—416no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4#12014yes—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3C22⋊Q814yes—
359C2×Dic3C4⋊Q814yes—
360Dic6—416no—
361C24C23×D311yescharacteristic
362C2×Q8—28no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—68no—
368C2×Q8—128no—
369C2×Q8—128no—
370C2×Q8—124no—
371C2×Q8—124no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—34no—
400C22×C4—32no—
401C22×C4—32no—
402C22×C4—34no—
403C22×C4—34no—
404C22×C4—32no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4D3×D412yes—
434C22×C4D3×D412yes—
435C22×C4D3×Q814yes—
436C22×C4D3×Q812yes—
437C22×C4C2×#6714yes—
438C22×C4C2×#6712yes—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—68no—
442C4⋊C4—616no—
443C4⋊C4—616no—
444C4⋊C4—128no—
445C42—68no—
446C22×C3C2×#12011yescharacteristic
447C22×C3C2×C22⋊Q811yescharacteristic
448C22×C3C2×C4⋊Q811yescharacteristic
449C22×C3#53412yes—
450C22×C3#53412yes—
451C22×C3#53511yescharacteristic
452C22×C3#53511yescharacteristic
453C22×C3#53611yescharacteristic
454C22×C3#53611yescharacteristic
455C22×C3#54212yes—
456C22×C3#54412yes—
457C22×C3#54412yes—
458C22×C3#54412yes—
459C22×C3#54811yescharacteristic
460C22×C3#54811yescharacteristic
461C22×C3D4×Q811yescharacteristic
462C22×C3D4×Q812yes—
463C22×C3D4×Q811yescharacteristic
464C22×C3D4×Q812yes—
465C22×C3#55412yes—
466C22×C3Q8212yes—
467C22×C3#56312yes—
468C22×C3#56312yes—
469C3×C4—24no—
470C3×C4—44no—
471C3×C4—42no—
472C3×C4—44no—
473C3×C4—81no—
474Dic3—24no—
475Dic3—24no—
476Dic3—44no—
477Dic3—42no—
478Dic3—44no—
479Dic3—81no—
480C23C2×D3×D411yescharacteristic
481C23C2×D3×D411yescharacteristic
482C23#100412yes—
483C23#100412yes—
484C23C2×D3×Q811yescharacteristic
485C23C2×D3×Q812yes—
486C23#100711yescharacteristic
487C23#100712yes—
488C23D3×#4111yescharacteristic
489C23#101011yescharacteristic
490C23C22×#6711yescharacteristic
491Q8—416no—
492Q8—1216no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—34no—
507C2×C4—34no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—64no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—121no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—121no—
554C2×C4—122no—
555C2×C4—121no—
556C2×C4—122no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4#93714yes—
560C2×C3#240911yescharacteristic
561C2×C3#243411yescharacteristic
562C2×C3#256711yescharacteristic
563C2×C3#261311yescharacteristic
564C2×C3#262811yescharacteristic
565C2×C3#266011yescharacteristic
566C2×C3#272912yes—
567C2×C3#272912yes—
568C2×C3#273412yes—
569C2×C3#273412yes—
570C2×C3#282111yescharacteristic
571C22#558912yes—
572C22#559211yescharacteristic
573C22#559211yescharacteristic
574C22#562312yes—
575C22D3×C22⋊Q811yescharacteristic
576C22#562811yescharacteristic
577C22#563211yescharacteristic
578C22#563311yescharacteristic
579C22#563612yes—
580C22#564012yes—
581C22#564612yes—
582C22#572212yes—
583C22D3×C4⋊Q811yescharacteristic
584C22#572411yescharacteristic
585C22#572712yes—
586C22#572912yes—
587C22#572912yes—
588C22#573512yes—
589C22C2×#93711yescharacteristic
590C22#580412yes—
591C22Q8×#6712yes—
592C22Q8×#6711yescharacteristic
593C22#583111yescharacteristic
594C4—24no—
595C4—42no—
596C4—44no—
597C4—44no—
598C4—64no—
599C4—64no—
600C4—81no—
601C4—122no—
602C4—124no—
603C4—124no—
604C4—241no—
605C3#1660311yescharacteristic, complemented, Sylow
606C2#7718911yescharacteristic
607C2#7720211yescharacteristic
608C2#7720811yescharacteristic
609C2#7722312yes—
610C2#7722412yes—
611C2#7727611yescharacteristic
612C2#8260711yescharacteristic
613C2#8275111yescharacteristic
614C2#8280512yes—
615C2#8323211yescharacteristic
616C2#8325012yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#395C2
hypercenterC2×#395D3
hypocenterC3#16603
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
92. C2×Dic3⋊C4Q8semidirect
109. C3⋊#95Q8semidirect
605. C3#16603semidirect

Maximal subgroups

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

subgroupindex
C22×#9442
C2×C3×#3952
C2×#49792
C2×#49782
#721342
#715682
#694242
#694142
#694092
#694022
#166033