Finite Group Databasebeta

← #10859124

Subgroups — #10859124

Order 768.

748 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
1C3×#2441C211yesmaximal, characteristic
2#73465C211yesmaximal, characteristic
3#71744C211yesmaximal, characteristic, complemented
4#71738C211yesmaximal, characteristic, complemented
5#71585C211yesmaximal, characteristic
6#71581C211yesmaximal, characteristic
7#69634C211yesmaximal, characteristic, complemented
8#69632C211yesmaximal, characteristic, complemented
9#69624C211yesmaximal, characteristic, complemented
10#69602C211yesmaximal, characteristic, complemented
11#69588C211yesmaximal, characteristic, complemented
12#69572C211yesmaximal, characteristic, complemented
13#69516C211yesmaximal, characteristic
14#69505C211yesmaximal, characteristic
15#69450C211yesmaximal, characteristic
16#26434—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#399—21no—
24C3×#396—21no—
25C3×#389—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#387—21no—
29C3×#384C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×C4×C22⋊C4C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4946—21no—
40Dic3×C22⋊C4—21no—
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4655—21no—
47#4647—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2441D311yescharacteristic
61#2360—31no—
62#2100—31no—
63#2088—31no—
64#2066—31no—
65#2026—31no—
66#1989—31no—
67#1920—31no—
68#1894—31no—
69#1873—31no—
70#1865—31no—
71#1853—31no—
72#1851—31no—
73#1803—31no—
74#1736—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6C2311yescharacteristic
78C3×#125—22no—
79C3×#125—41no—
80C3×#117—22no—
81C3×#117—41no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—41no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106#924—22no—
107#924Q814yes—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×#208—22no—
114C2×#208—41no—
115C2×#208—41no—
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C4×Dic6—22no—
121C4×Dic6—41no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#830—22no—
132#810—22no—
133#810—41no—
134#810—41no—
135#810—41no—
136#808—22no—
137#808—41no—
138C4⋊(C3⋊C8)—41no—
139C4⋊(C3⋊C8)—41no—
140C22×M16—31no—
141C2×#125C2×D311yescharacteristic
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117C2×D311yescharacteristic
145C22×C4⋊C4—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×C22⋊C8—31no—
155#399—21no—
156#396—21no—
157#391—31no—
158#389—21no—
159#389—31no—
160#389—61no—
161#389C2×D311yescharacteristic
162#389C2×D311yescharacteristic
163#387—21no—
164#387—61no—
165#385—61no—
166#384C2×D311yescharacteristic
167#380C2×D311yescharacteristic
168C4×C4⋊C4—31no—
169C4×C22⋊C4—61no—
170C4×C22⋊C4C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×Q812yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—81no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—81no—
225C3×C4⋊C4—81no—
226C3×C4⋊C4#4112yes—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—41no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—81no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—42no—
236#217—24no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—42no—
242C4⋊Dic3C2×Q812yes—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—42no—
246Dic3⋊C4—81no—
247C4×Dic3—42no—
248C4×Dic3—81no—
249#208—44no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C22×Q8—31no—
263C23×C4—61no—
264C23×C4C22×D311yescharacteristic
265C2×M16—62no—
266C2×M16—121no—
267C2×M16—121no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272#125—22no—
273#125—41no—
274C22⋊Q8—34no—
275C22⋊Q8—62no—
276C4×Q8—62no—
277C4×Q8—121no—
278#117—22no—
279#117—41no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—121no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4#6711yescharacteristic
303C2×C22⋊C4#6711yescharacteristic
304C2×C42—21no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42C22×D311yescharacteristic
308C2×C42C22×D311yescharacteristic
309#106—62no—
310#106—121no—
311C4⋊C8—121no—
312C4⋊C8—121no—
313#103—62no—
314#103—121no—
315#103—121no—
316#103—121no—
317C22⋊C8—62no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95#6711yescharacteristic
326#95#6711yescharacteristic
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3C22⋊Q812yes—
334C23×C3C22×D411yescharacteristic
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4C22⋊Q812yes—
365C2×C3×C4#13712yes—
366C2×C3×C4C22×Q811yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—84no—
385Dic6—22no—
386Dic6—22no—
387Dic6—41no—
388Dic6—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C24C2×#6711yescharacteristic
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—241no—
415C22×C4D3×Q812yes—
416C22×C4#23912yes—
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C23×D311yescharacteristic
423M16—124no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—32no—
444C4⋊C4—32no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—64no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—122no—
457C4⋊C4—122no—
458C4⋊C4—241no—
459C4⋊C4#23712yes—
460C22⋊C4—22no—
461C22⋊C4—41no—
462C22⋊C4—42no—
463C22⋊C4—64no—
464C22⋊C4—81no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—42no—
470C42—122no—
471C42—241no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3C2×C22⋊Q811yescharacteristic
480C22×C3#54611yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3C2×#13711yescharacteristic
483C22×C3#57511yescharacteristic
484C22×C3#57711yescharacteristic
485C22×C3#57711yescharacteristic
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—41no—
489C3×C4—42no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4—81no—
495C3×C4#52911yescharacteristic
496C3×C4#53811yescharacteristic
497C3×C4#55411yescharacteristic
498C3×C4#55411yescharacteristic
499Dic3—42no—
500Dic3—42no—
501Dic3—42no—
502Dic3—81no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23#94612yes—
510C23C22×#6711yescharacteristic
511Q8—62no—
512Q8—62no—
513Q8—121no—
514Q8—121no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—244no—
557C2×C4#94612yes—
558C2×C4#95112yes—
559C2×C4C2×#23711yescharacteristic
560C2×C4#100411yescharacteristic
561C2×C4C2×D3×Q811yescharacteristic
562C2×C4C2×#23911yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C2×C4#100911yescharacteristic
565C8—121no—
566C8—121no—
567C8—121no—
568C8—121no—
569C2×C3—42no—
570C2×C3#245011yescharacteristic
571C2×C3#307811yescharacteristic
572C2×C3#308411yescharacteristic
573C2×C3#331711yescharacteristic
574C2×C3#332011yescharacteristic
575C2×C3#332311yescharacteristic
576C2×C3#333111yescharacteristic
577C22—22no—
578C22—22no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22#580611yescharacteristic
585C22C2×#94611yescharacteristic
586C22#582211yescharacteristic
587C22C2×#95111yescharacteristic
588C22#582911yescharacteristic
589C22#583611yescharacteristic
590C22#583711yescharacteristic
591C4—22no—
592C4—22no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—122no—
601C4—122no—
602C4—122no—
603C4—241no—
604C4#562411yescharacteristic
605C4#563511yescharacteristic
606C4#563611yescharacteristic
607C4#564411yescharacteristic
608C3#2643411yescharacteristic, complemented, Sylow
609C2—42no—
610C2#8276511yescharacteristic
611C2#8336711yescharacteristic
612C2#8347611yescharacteristic
613C2#8351511yescharacteristic
614C2#8357311yescharacteristic
615C2#8360111yescharacteristic
616C2#8362011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2441C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2441D3
hypocenterC3#26434
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
3. #71744C2semidirect
4. #71738C2semidirect
7. #69634C2semidirect
8. #69632C2semidirect
9. #69624C2semidirect
10. #69602C2semidirect
11. #69588C2semidirect
12. #69572C2semidirect
608. C3#26434semidirect

Maximal subgroups

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

subgroupindex
C3×#24412
#734652
#717442
#717382
#715852
#715812
#696342
#696322
#696242
#696022
#695882
#695722
#695162
#695052
#694502
#264343