Finite Group Databasebeta

← #10728691

Subgroups — #10728691

Order 768.

1718 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic
3C2×#5230C211yesmaximal, characteristic
4C2×#4982C212yesmaximal
5C2×#4976C211yesmaximal, characteristic
6#71565C211yesmaximal, characteristic
7#71564C212yesmaximal
8#69411C212yesmaximal
9#69410C211yesmaximal, characteristic
10#69405C212yesmaximal
11#69403C211yesmaximal, characteristic
12#14880—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×Q8×Dic3—28no—
17C2×Q8×Dic3—24no—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×C4×Dic3C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C3×#395—24no—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29#5230—24no—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4982—28no—
41#4976—24no—
42C22×C4×Q8—31no—
43C2×#398—32no—
44C2×#395—31no—
45C2×#395D311yescharacteristic
46C2×#389—31no—
47#1441—31no—
48#1441—32no—
49#1441—31no—
50#1436—32no—
51#1436—32no—
52#1434—31no—
53C22×C3×Q8—24no—
54C22×C3×Q8—22no—
55C22×C3×Q8—24no—
56C22×C3×Q8—22no—
57C22×C3×Q8D414yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—22no—
75Q8×Dic3—232no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—24no—
78C2×C4⋊Dic3—24no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3D418yes—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—24no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—24no—
92C2×C4×Dic3—24no—
93C2×C4×Dic3—24no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3D414yes—
97C3×#95—22no—
98C3×#95—24no—
99C3⋊#95—22no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95Q818yescomplemented
109C23×Q8C2×D311yescharacteristic
110C2×C4×Q8—64no—
111C2×C4×Q8—68no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4C2×D312yes—
119C22×C4⋊C4C2×D311yescharacteristic
120C22×C42—31no—
121C22×C42—32no—
122#398—68no—
123#395—24no—
124#395—64no—
125#389—64no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—31no—
133C2×#95—31no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95C2×D311yescharacteristic
137C2×#95C2×D312yes—
138C24×C3C2411yescharacteristic
139C2×C3×Q8—28no—
140C2×C3×Q8—28no—
141C2×C3×Q8—216no—
142C2×C3×Q8—216no—
143C2×C3×Q8—28no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
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—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4C2×D414yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×Q814yes—
172C22×C3×C4#4114yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—22no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—21no—
204C22×Dic3C2×D414yes—
205C22×Dic3C2×D414yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×D412yes—
208C22×Dic3C2×Q814yes—
209C22×Dic3C2×Q814yes—
210C22×Dic3#4112yes—
211C3×C4⋊C4—44no—
212C3×C4⋊C4—48no—
213C4⋊Dic3—28no—
214C4⋊Dic3—216no—
215Dic3⋊C4—28no—
216Dic3⋊C4—44no—
217Dic3⋊C4—48no—
218Dic3⋊C4—44no—
219C4×Dic3—216no—
220C4×Dic3—28no—
221C22×Q8—22no—
222C22×Q8—24no—
223C22×Q8—22no—
224C22×Q8—24no—
225C22×Q8#6714yes—
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—32no—
230C23×C4—32no—
231C23×C4—31no—
232C23×C4C22×D311yescharacteristic
233C23×C4C22×D312yes—
234C23×C4C22×D311yescharacteristic
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D312yes—
237C4×Q8—632no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—38no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—64no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C42—34no—
261C2×C42—64no—
262C2×C42—64no—
263C2×C42—64no—
264C2×C42—64no—
265#95—22no—
266#95—24no—
267#95—38no—
268#95—64no—
269#95—62no—
270#95—62no—
271#95—64no—
272#95—64no—
273#95—64no—
274#95—64no—
275#95—62no—
276#95—64no—
277C23×C3C22×D411yescharacteristic
278C23×C3C22×D412yes—
279C23×C3C22×D411yescharacteristic
280C23×C3C22×Q812yes—
281C23×C3C2×#4112yes—
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C32−1+412yes—
286C23×C32−1+411yescharacteristic
287C23×C32−1+411yescharacteristic
288C3×Q8—232no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4#12118yes—
319C2×C3×C4#12114yes—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—22no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—42no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3#12014yes—
355C2×Dic3C4⋊Q814yes—
356C24C23×D311yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—28no—
361C2×Q8—216no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—32no—
386C22×C4—34no—
387C22×C4—32no—
388C22×C4—34no—
389C22×C4—34no—
390C22×C4—34no—
391C22×C4—32no—
392C22×C4—62no—
393C22×C4—61no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—61no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4—61no—
421C22×C4D3×D412yes—
422C22×C4D3×D414yes—
423C22×C4#23712yes—
424C22×C4#23714yes—
425C22×C4D3×Q814yes—
426C22×C4#23912yes—
427C22×C4C2×#6714yes—
428C22×C4C2×#6712yes—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C4⋊C4—616no—
432C4⋊C4—68no—
433C4⋊C4—68no—
434C4⋊C4—124no—
435C4⋊C4—124no—
436C4⋊C4—128no—
437C42—68no—
438C42—616no—
439C22×C3C2×#12011yescharacteristic
440C22×C3C2×#12112yes—
441C22×C3C2×#12111yescharacteristic
442C22×C3#52511yescharacteristic
443C22×C3#52512yes—
444C22×C3#52511yescharacteristic
445C22×C3C2×C4⋊Q811yescharacteristic
446C22×C3#53312yes—
447C22×C3#53312yes—
448C22×C3#53611yescharacteristic
449C22×C3#53611yescharacteristic
450C22×C3#54112yes—
451C22×C3#54112yes—
452C22×C3#54712yes—
453C22×C3#54712yes—
454C22×C3#54811yescharacteristic
455C22×C3#54811yescharacteristic
456C22×C3D4×Q812yes—
457C22×C3D4×Q812yes—
458C22×C3D4×Q812yes—
459C22×C3#55211yescharacteristic
460C22×C3#55211yescharacteristic
461C22×C3#55712yes—
462C3×C4—28no—
463C3×C4—24no—
464C3×C4—44no—
465C3×C4—42no—
466C3×C4—81no—
467Dic3—24no—
468Dic3—24no—
469Dic3—44no—
470Dic3—44no—
471Dic3—42no—
472Dic3—81no—
473C23C2×D3×D412yes—
474C23C2×D3×D411yescharacteristic
475C23C2×#23711yescharacteristic
476C23C2×#23712yes—
477C23C2×D3×Q812yes—
478C23C2×#23911yescharacteristic
479C23#100711yescharacteristic
480C23#100712yes—
481C23D3×#4111yescharacteristic
482C23#101011yescharacteristic
483C23C22×#6711yescharacteristic
484Q8—232no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—34no—
502C2×C4—34no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—64no—
524C2×C4—62no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—64no—
531C2×C4—62no—
532C2×C4—64no—
533C2×C4—121no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4—122no—
543C2×C4—121no—
544C2×C4—122no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—122no—
549C2×C4—121no—
550C2×C4#93914yes—
551C2×C4#93918yes—
552C2×C3#239911yescharacteristic
553C2×C3#240611yescharacteristic
554C2×C3#242912yes—
555C2×C3#242912yes—
556C2×C3#243411yescharacteristic
557C2×C3#246212yes—
558C2×C3#246212yes—
559C2×C3#253611yescharacteristic
560C2×C3#254611yescharacteristic
561C2×C3#263411yescharacteristic
562C2×C3#284711yescharacteristic
563C22#558611yescharacteristic
564C22#558612yes—
565C22D3×#12011yescharacteristic
566C22#562212yes—
567C22#562512yes—
568C22#562711yescharacteristic
569C22#562811yescharacteristic
570C22#563112yes—
571C22#563212yes—
572C22#563911yescharacteristic
573C22#564211yescharacteristic
574C22#572112yes—
575C22D3×C4⋊Q811yescharacteristic
576C22#572411yescharacteristic
577C22#572612yes—
578C22#572712yes—
579C22#572812yes—
580C22#573112yes—
581C22C2×#93911yescharacteristic
582C22C2×#93912yes—
583C22Q8×#6712yes—
584C22#581711yescharacteristic
585C22#583111yescharacteristic
586C4—28no—
587C4—24no—
588C4—44no—
589C4—42no—
590C4—64no—
591C4—64no—
592C4—81no—
593C4—124no—
594C4—122no—
595C4—124no—
596C4—241no—
597C3#1488011yescharacteristic, complemented, Sylow
598C2#7717111yescharacteristic
599C2#7717311yescharacteristic
600C2#7717712yes—
601C2#7717812yes—
602C2#7720411yescharacteristic
603C2#7724911yescharacteristic
604C2#8260111yescharacteristic
605C2#8275812yes—
606C2#8277711yescharacteristic
607C2#8323811yescharacteristic
608C2#8325412yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#395C2
hypercenterC2×#395D3
hypocenterC3#14880
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
108. C3⋊#95Q8semidirect
597. C3#14880semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×C3×#3952
C2×#52302
C2×#49822
C2×#49762
#715652
#715642
#694112
#694102
#694052
#694032
#148803