Finite Group Databasebeta

← #10751754

Subgroups — #10751754

Order 768.

1426 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×#889C211yesmaximal, characteristic
2C2×C3×#398C211yesmaximal, characteristic
3C2×#4983C211yesmaximal, characteristic
4C2×#4972C211yesmaximal, characteristic
5#71568C212yesmaximal
6#71568C212yesmaximal
7#71565C211yesmaximal, characteristic
8#69420C212yesmaximal
9#69414C212yesmaximal
10#69410C211yesmaximal, characteristic
11#69402C211yesmaximal, characteristic
12#16607—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2211yescharacteristic
19C22×Dic3⋊C4C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×#889—24no—
26C2×#889—24no—
27C2×#889—24no—
28C3×#398—24no—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
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#4972—24no—
42C22×#125—31no—
43C2×#398—31no—
44C2×#398D311yescharacteristic
45C2×#389—31no—
46#1444—32no—
47#1444—32no—
48#1444—32no—
49#1442—32no—
50#1441—31no—
51#1441—31no—
52#1434—31no—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2311yescharacteristic
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C23×Dic3C2312yes—
63C23×Dic3C2311yescharacteristic
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4D414yes—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—24no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—42no—
92C2×Dic3⋊C4—42no—
93C2×Dic3⋊C4—42no—
94C2×Dic3⋊C4—42no—
95C2×Dic3⋊C4D414yes—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3D414yes—
99#889—232no—
100C3×#95—22no—
101C3×#95—24no—
102C3⋊#95—22no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C3⋊#95Q818yes—
111C2×#125—64no—
112C2×#125—64no—
113C2×#125—64no—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—31no—
116C22×C4⋊C4—31no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4—31no—
119C22×C4⋊C4—31no—
120C22×C4⋊C4—32no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4C2×D311yescharacteristic
123C22×C4⋊C4C2×D311yescharacteristic
124C22×C4⋊C4C2×D312yes—
125C22×C42—31no—
126#398—24no—
127#398—64no—
128#389—64no—
129C2×#95—32no—
130C2×#95—31no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95C2×D311yescharacteristic
139C2×#95C2×D312yes—
140C24×C3C2411yescharacteristic
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
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×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×Q814yes—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×D412yes—
205C22×Dic3C2×D412yes—
206C22×Dic3C2×D412yes—
207C22×Dic3C2×Q814yes—
208C22×Dic3C2×Q814yes—
209C22×Dic3#4114yes—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—48no—
212C3×C4⋊C4—44no—
213C4⋊Dic3—28no—
214C4⋊Dic3—28no—
215Dic3⋊C4—28no—
216Dic3⋊C4—28no—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219Dic3⋊C4—48no—
220Dic3⋊C4—48no—
221C4×Dic3—28no—
222C23×C4—31no—
223C23×C4—31no—
224C23×C4—31no—
225C23×C4—31no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4C22×D312yes—
229C23×C4C22×D312yes—
230C23×C4C22×D311yescharacteristic
231C23×C4C22×D311yescharacteristic
232C23×C4C22×D311yescharacteristic
233#125—632no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—34no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4—64no—
261C2×C4⋊C4—122no—
262C2×C4⋊C4—122no—
263C2×C4⋊C4—122no—
264C2×C4⋊C4—122no—
265C2×C4⋊C4#6714yes—
266C2×C42—34no—
267C2×C42—62no—
268C2×C42—62no—
269#95—22no—
270#95—24no—
271#95—38no—
272#95—64no—
273#95—64no—
274#95—64no—
275#95—64no—
276#95—62no—
277#95—122no—
278#95—122no—
279#95—122no—
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3C22×Q812yes—
284C23×C3C2×#4112yes—
285C23×C3H(5,𝔽2)11yescharacteristic
286C23×C3H(5,𝔽2)11yescharacteristic
287C23×C3H(5,𝔽2)11yescharacteristic
288C23×C32−1+412yes—
289C23×C32−1+412yes—
290C23×C32−1+411yescharacteristic
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4#12014yes—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—24no—
325C2×Dic3—24no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—24no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3C4⋊Q814yes—
360C24C23×D311yescharacteristic
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—32no—
387C22×C4—34no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—61no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4D3×D412yes—
425C22×C4D3×D412yes—
426C22×C4D3×Q814yes—
427C22×C4C2×#6712yes—
428C22×C4C2×#6712yes—
429C22×C4C2×#6712yes—
430C4⋊C4—28no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—68no—
434C4⋊C4—68no—
435C4⋊C4—68no—
436C4⋊C4—68no—
437C4⋊C4—68no—
438C4⋊C4—68no—
439C4⋊C4—128no—
440C4⋊C4—128no—
441C42—68no—
442C22×C3C2×#12011yescharacteristic
443C22×C3#52511yescharacteristic
444C22×C3C2×C4⋊Q811yescharacteristic
445C22×C3#53511yescharacteristic
446C22×C3#53511yescharacteristic
447C22×C3#53511yescharacteristic
448C22×C3#53611yescharacteristic
449C22×C3#53611yescharacteristic
450C22×C3#53711yescharacteristic
451C22×C3#53711yescharacteristic
452C22×C3#53711yescharacteristic
453C22×C3#53912yes—
454C22×C3#53912yes—
455C22×C3#54412yes—
456C22×C3#54412yes—
457C22×C3#54712yes—
458C22×C3#54712yes—
459C22×C3D4×Q812yes—
460C22×C3D4×Q812yes—
461C22×C3#55412yes—
462C22×C3#55712yes—
463C22×C3#56312yes—
464C22×C3#56412yes—
465C3×C4—24no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4—82no—
470Dic3—24no—
471Dic3—42no—
472Dic3—42no—
473Dic3—44no—
474Dic3—42no—
475Dic3—44no—
476C23C2×D3×D411yescharacteristic
477C23C2×D3×D411yescharacteristic
478C23#100411yescharacteristic
479C23#100411yescharacteristic
480C23C2×D3×Q812yes—
481C23#100712yes—
482C23D3×#4112yes—
483C23#100911yescharacteristic
484C23#101012yes—
485C23#101011yescharacteristic
486C23C22×#6711yescharacteristic
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—34no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—62no—
519C2×C4—64no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—121no—
539C2×C4—122no—
540C2×C4—121no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4#93714yes—
556C2×C3#240611yescharacteristic
557C2×C3#243411yescharacteristic
558C2×C3#258411yescharacteristic
559C2×C3#261511yescharacteristic
560C2×C3#263311yescharacteristic
561C2×C3#266011yescharacteristic
562C2×C3#272012yes—
563C2×C3#272012yes—
564C2×C3#272612yes—
565C2×C3#272612yes—
566C2×C3#282211yescharacteristic
567C22#560711yescharacteristic
568C22#560711yescharacteristic
569C22#560811yescharacteristic
570C22#560811yescharacteristic
571C22#562312yes—
572C22#563612yes—
573C22#563812yes—
574C22#564612yes—
575C22#564811yescharacteristic
576C22#565512yes—
577C22#565611yescharacteristic
578C22#568512yes—
579C22C3⋊#56412yes—
580C22#568912yes—
581C22#569212yes—
582C22D3×C4⋊Q811yescharacteristic
583C22#572411yescharacteristic
584C22#572712yes—
585C22C2×#93711yescharacteristic
586C22#580511yescharacteristic
587C22Q8×#6712yes—
588C22#582412yes—
589C22#583311yescharacteristic
590C4—24no—
591C4—42no—
592C4—44no—
593C4—42no—
594C4—64no—
595C4—82no—
596C4—122no—
597C4—122no—
598C4—124no—
599C4—122no—
600C4—124no—
601C3#1660711yescharacteristic, complemented, Sylow
602C2#7779711yescharacteristic
603C2#7782211yescharacteristic
604C2#7783711yescharacteristic
605C2#7786112yes—
606C2#7786712yes—
607C2#7792611yescharacteristic
608C2#8270911yescharacteristic
609C2#8280312yes—
610C2#8284111yescharacteristic
611C2#8305412yes—
612C2#8323211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#398C2
hypercenterC2×#398D3
hypocenterC3#16607
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
601. C3#16607semidirect

Maximal subgroups

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

subgroupindex
C22×#8892
C2×C3×#3982
C2×#49832
C2×#49722
#715682
#715652
#694202
#694142
#694102
#694022
#166073