Finite Group Databasebeta

← #10750420

Subgroups — #10750420

Order 768.

1638 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic
2C2×C3×#398C211yesmaximal, characteristic
3C2×#4981C211yesmaximal, characteristic
4C2×#4979C212yesmaximal
5C2×#4978C211yesmaximal, characteristic
6#71569C211yesmaximal, characteristic
7#71568C212yesmaximal
8#69424C212yesmaximal
9#69423C211yesmaximal, characteristic
10#69414C212yesmaximal
11#69402C211yesmaximal, characteristic
12#16507—31nomaximal, Sylow
13C23×Dic6C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22C2×#888—24no—
23C2×#888—24no—
24C2×#888—24no—
25C3×#398—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4981—24no—
41#4979—28no—
42#4978—24no—
43C22×C4⋊Q8—31no—
44C2×#398—31no—
45C2×#398D311yescharacteristic
46C2×#395—32no—
47C2×#387—31no—
48#1446—32no—
49#1445—31no—
50#1445—31no—
51#1444—32no—
52#1442—32no—
53#1434—31no—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2312yes—
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2311yescharacteristic
65C22×Dic6—24no—
66C22×Dic6—24no—
67C22×Dic6—24no—
68C22×Dic6—24no—
69C22×Dic6—24no—
70C22×Dic6—42no—
71C22×Dic6—42no—
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—24no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—42no—
80C2×C3×C4⋊C4D414yes—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3D414yes—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—42no—
92C2×Dic3⋊C4D414yes—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—22no—
95C2×C4×Dic3D414yes—
96#888—232no—
97C3×#95—22no—
98C3×#95—24no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C3⋊#95Q818yescomplemented
111C23×Q8—32no—
112C2×C4⋊Q8—64no—
113C2×C4⋊Q8—64no—
114C2×C4⋊Q8—64no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—32no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4C2×D311yescharacteristic
120C22×C4⋊C4C2×D312yes—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C42—31no—
123#398—24no—
124#398—64no—
125#395—68no—
126#387—64no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—31no—
134C2×#95—31no—
135C2×#95—31no—
136C2×#95—31no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95C2×D311yescharacteristic
140C2×#95C2×D312yes—
141C24×C3C2411yescharacteristic
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×Q814yes—
173C22×Dic3—21no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D414yes—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×Q814yes—
207C22×Dic3C2×Q814yes—
208C22×Dic3#4112yes—
209C2×Dic6—216no—
210C2×Dic6—48no—
211C2×Dic6—48no—
212C2×Dic6—48no—
213C2×Dic6—48no—
214C2×Dic6—48no—
215C2×Dic6—48no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—48no—
218C3×C4⋊C4—44no—
219C4⋊Dic3—28no—
220Dic3⋊C4—216no—
221Dic3⋊C4—28no—
222Dic3⋊C4—48no—
223C4×Dic3—28no—
224C22×Q8—64no—
225C22×Q8—64no—
226C22×Q8—64no—
227C22×Q8—64no—
228C22×Q8—64no—
229C22×Q8—122no—
230C22×Q8—122no—
231C23×C4—31no—
232C23×C4—31no—
233C23×C4—32no—
234C23×C4—31no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D311yescharacteristic
239C23×C4C22×D312yes—
240C23×C4C22×D312yes—
241C23×C4C22×D311yescharacteristic
242C4⋊Q8—632no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—34no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—64no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—122no—
263C2×C4⋊C4#6714yes—
264C2×C42—34no—
265C2×C42—62no—
266C2×C42—62no—
267#95—22no—
268#95—24no—
269#95—38no—
270#95—62no—
271#95—62no—
272#95—62no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—64no—
277#95—62no—
278#95—62no—
279#95—122no—
280#95—122no—
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3C22×D411yescharacteristic
284C23×C3C22×D411yescharacteristic
285C23×C3C22×Q812yes—
286C23×C3C2×#4111yescharacteristic
287C23×C3H(5,𝔽2)11yescharacteristic
288C23×C3H(5,𝔽2)11yescharacteristic
289C23×C32−1+412yes—
290C23×C32−1+412yes—
291C23×C32−1+412yes—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4#12014yes—
323C2×Dic3—22no—
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—24no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—41no—
344C2×Dic3—42no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3#12114yes—
358C2×Dic3C4⋊Q814yes—
359Dic6—432no—
360C24C23×D311yescharacteristic
361C2×Q8—616no—
362C2×Q8—128no—
363C2×Q8—128no—
364C2×Q8—128no—
365C2×Q8—128no—
366C2×Q8—128no—
367C2×Q8—128no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—32no—
396C22×C4—34no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—62no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4D3×D412yes—
430C22×C4D3×D412yes—
431C22×C4D3×D412yes—
432C22×C4D3×Q814yes—
433C22×C4C2×#6714yes—
434C22×C4C2×#6712yes—
435C4⋊C4—28no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—68no—
439C4⋊C4—616no—
440C4⋊C4—68no—
441C4⋊C4—128no—
442C42—68no—
443C22×C3C2×#12011yescharacteristic
444C22×C3C2×#12111yescharacteristic
445C22×C3C2×C4⋊Q811yescharacteristic
446C22×C3#53511yescharacteristic
447C22×C3#53611yescharacteristic
448C22×C3#53611yescharacteristic
449C22×C3#53612yes—
450C22×C3#53611yescharacteristic
451C22×C3#53612yes—
452C22×C3#53612yes—
453C22×C3#54112yes—
454C22×C3#54412yes—
455C22×C3#54412yes—
456C22×C3D4211yescharacteristic
457C22×C3D4211yescharacteristic
458C22×C3#54611yescharacteristic
459C22×C3#54611yescharacteristic
460C22×C3D4×Q812yes—
461C22×C3D4×Q812yes—
462C22×C3D4×Q812yes—
463C22×C3#55712yes—
464C22×C3#56312yes—
465C22×C3#56312yes—
466C3×C4—24no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—44no—
470C3×C4—81no—
471Dic3—24no—
472Dic3—24no—
473Dic3—42no—
474Dic3—44no—
475Dic3—44no—
476Dic3—81no—
477C23C2×D3×D411yescharacteristic
478C23C2×D3×D411yescharacteristic
479C23C2×D3×D411yescharacteristic
480C23#100411yescharacteristic
481C23C2×D3×Q812yes—
482C23#100712yes—
483C23D3×#4111yescharacteristic
484C23#100911yescharacteristic
485C23#101012yes—
486C23#101012yes—
487C23C22×#6711yescharacteristic
488Q8—1232no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—34no—
503C2×C4—34no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—62no—
522C2×C4—64no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—64no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—64no—
535C2×C4—62no—
536C2×C4—64no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—122no—
540C2×C4—121no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—121no—
552C2×C4—122no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4#93714yes—
556C2×C3#240811yescharacteristic
557C2×C3#243411yescharacteristic
558C2×C3#254611yescharacteristic
559C2×C3#261411yescharacteristic
560C2×C3#262111yescharacteristic
561C2×C3#266011yescharacteristic
562C2×C3#267712yes—
563C2×C3#267712yes—
564C2×C3#271312yes—
565C2×C3#271312yes—
566C2×C3#282111yescharacteristic
567C22D3×#12111yescharacteristic
568C22#560711yescharacteristic
569C22#560911yescharacteristic
570C22#561211yescharacteristic
571C22#562312yes—
572C22#562812yes—
573C22#563212yes—
574C22#564612yes—
575C22#564811yescharacteristic
576C22#564812yes—
577C22#564811yescharacteristic
578C22#568512yes—
579C22#568912yes—
580C22#569612yes—
581C22#569912yes—
582C22D3×C4⋊Q811yescharacteristic
583C22#572411yescharacteristic
584C22#572712yes—
585C22C2×#93711yescharacteristic
586C22D4×#6711yescharacteristic
587C22Q8×#6712yes—
588C22#582512yes—
589C22#582911yescharacteristic
590C4—24no—
591C4—44no—
592C4—44no—
593C4—42no—
594C4—64no—
595C4—64no—
596C4—81no—
597C4—124no—
598C4—124no—
599C4—122no—
600C4—241no—
601C3#1650711yescharacteristic, complemented, Sylow
602C2#7778511yescharacteristic
603C2#7781811yescharacteristic
604C2#7783711yescharacteristic
605C2#7785612yes—
606C2#7792511yescharacteristic
607C2#7796812yes—
608C2#8264011yescharacteristic
609C2#8279112yes—
610C2#8286411yescharacteristic
611C2#8304812yes—
612C2#8323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#398C2
hypercenterC2×#398D3
hypocenterC3#16507
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
110. C3⋊#95Q8semidirect
601. C3#16507semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C2×C3×#3982
C2×#49812
C2×#49792
C2×#49782
#715692
#715682
#694242
#694232
#694142
#694022
#165073