Finite Group Databasebeta

← #10855136

Subgroups — #10855136

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82704C211yesmaximal, characteristic
2#82161C211yesmaximal, characteristic, complemented
3C2×#4964C211yesmaximal, characteristic, complemented
4C2×#4779C211yesmaximal, characteristic, complemented
5C3×#1874C211yesmaximal, characteristic, complemented
6#71682C211yesmaximal, characteristic, complemented
7#70206C211yesmaximal, characteristic, complemented
8#70181C211yesmaximal, characteristic, complemented
9#70174C211yesmaximal, characteristic, complemented
10#69953C211yesmaximal, characteristic, complemented
11#69929C211yesmaximal, characteristic, complemented
12#68877C211yesmaximal, characteristic, complemented
13#68836C211yesmaximal, characteristic, complemented
14#68833C211yesmaximal, characteristic, complemented
15#68787C211yesmaximal, characteristic, complemented
16#26156—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#938C2211yescharacteristic
20C2×C4×#67C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C2×#892C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5223—21no—
28#5222—21no—
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5213—21no—
33#5212C2211yescharacteristic
34C4×#217—21no—
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#817C2211yescharacteristic
39#4990—21no—
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42#4978—21no—
43#4977C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45#4972C2211yescharacteristic
46#4964—24no—
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4957—21no—
52#4945—21no—
53#4779—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#482—31no—
63C2×#428—31no—
64#2414—31no—
65#2352—31no—
66#2092—31no—
67#2085—31no—
68#2077—31no—
69#2033—31no—
70#1926—31no—
71#1918—31no—
72#1895—31no—
73#1874D311yescharacteristic, complemented
74#1870—31no—
75#1854—31no—
76#1854—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×C3×C4C2311yescharacteristic
81C22×#67—21no—
82C22×#67D411yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—22no—
87C3×#121—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—41no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217D411yescharacteristic
101#938—22no—
102#938—41no—
103C4×#67—22no—
104C4×#67—22no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4D411yescharacteristic
109#926—22no—
110#926—41no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2311yescharacteristic
121C2×#208—22no—
122C2×#208—22no—
123C2×#208—22no—
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126#892—22no—
127#892—41no—
128C3×#102—22no—
129C3×#102—41no—
130C3×C22⋊C8—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142#818—41no—
143#817—22no—
144#808—22no—
145#808—22no—
146#807—22no—
147#807D414yescomplemented
148C22×M16—31no—
149C2×#121—31no—
150C2×#121C2×D311yescharacteristic
151C2×C4×D4—31no—
152C2×#117—31no—
153C2×#117—31no—
154C22×C4⋊C4C2×D311yescharacteristic
155#482—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102C2×D311yescharacteristic
166C2×#102C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170#394—61no—
171#392—61no—
172#390—31no—
173#390—61no—
174#389—31no—
175#388—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#382—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—42no—
203C2×C3×D4—81no—
204C2×C3×D4—81no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—81no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4#4112yes—
217C22×C3×C4#4112yes—
218C22×C3×C4C2411yescharacteristic
219C2×#67—22no—
220C2×#67—41no—
221C2×#67—41no—
222C2×#67—42no—
223C2×#67—42no—
224C2×#67—81no—
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4C2×D412yes—
251C3×C22⋊C4—42no—
252#217—22no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—42no—
257#217—81no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3C2×D412yes—
270Dic3⋊C4—22no—
271Dic3⋊C4—81no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C4×Dic3—81no—
277#208—28no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C2×C3⋊C8#4112yes—
284C22×D4—21no—
285C22×D4—31no—
286C22×D4—61no—
287C22×D4C22×D311yescharacteristic
288C23×C4—21no—
289C23×C4—31no—
290C23×C4C22×D311yescharacteristic
291C2×M16—62no—
292C2×M16—62no—
293C2×M16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#121—22no—
299#121—22no—
300#121—62no—
301#121—121no—
302C4×D4—62no—
303C4×D4—62no—
304#117—62no—
305#117—62no—
306#117—121no—
307#117—121no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—41no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42—61no—
335#107—34no—
336#107—62no—
337#107—121no—
338#106—62no—
339#106—62no—
340#106—62no—
341#102—22no—
342#102—41no—
343#102—62no—
344#102—62no—
345C22⋊C8—22no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3#12312yes—
369C23×C3C22×D411yescharacteristic
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—84no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—84no—
398C2×C3×C4#12312yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4H(5,𝔽2)11yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405#67—44no—
406#67—84no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—41no—
432C3×C8—41no—
433C3×C8—41no—
434C3⋊C8—22no—
435C3⋊C8—22no—
436C24—21no—
437C24—31no—
438C24D3×D411yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—62no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—121no—
450C2×D4—121no—
451C2×D4—122no—
452C2×D4—122no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—81no—
477C22×C4#23512yes—
478C22×C4D3×D411yescharacteristic
479C22×C4#23712yes—
480C22×C4C23×D311yescharacteristic
481M16—68no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—32no—
489C2×C8—32no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—42no—
501C4⋊C4—62no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—241no—
510C4⋊C4D3×D412yes—
511C22⋊C4—42no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C42—61no—
523C42—61no—
524C42—122no—
525C42—122no—
526C42—241no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3C2×#12311yescharacteristic
542C22×C3D4211yescharacteristic
543C22×C3#54711yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57511yescharacteristic
546C22×C3#57611yescharacteristic
547C22×C3#57711yescharacteristic
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C3×C4—22no—
556C3×C4—42no—
557C3×C4—81no—
558C3×C4—81no—
559C3×C4#52511yescharacteristic
560C3×C4#53911yescharacteristic
561C3×C4#54811yescharacteristic
562C3×C4#55311yescharacteristic
563Dic3—41no—
564Dic3—41no—
565Dic3—41no—
566Dic3—42no—
567Dic3—81no—
568Dic3—81no—
569Dic3—81no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23#88312yes—
591C23C2×D3×D411yescharacteristic
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—84no—
597D4—124no—
598D4—244no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—62no—
624C2×C4—84no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4#88312yes—
641C2×C4C2×#23511yescharacteristic
642C2×C4C2×D3×D411yescharacteristic
643C2×C4C2×#23711yescharacteristic
644C2×C4#100411yescharacteristic
645C2×C4D3×#4111yescharacteristic
646C2×C4#101011yescharacteristic
647C8—41no—
648C8—41no—
649C8—62no—
650C8—62no—
651C2×C3—42no—
652C2×C3—161no—
653C2×C3#244311yescharacteristic
654C2×C3#309311yescharacteristic
655C2×C3#309611yescharacteristic
656C2×C3#327111yescharacteristic
657C2×C3#327311yescharacteristic
658C2×C3#334711yescharacteristic
659C2×C3#335111yescharacteristic
660D3—81no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—121no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22—121no—
681C22—121no—
682C22C2×#88311yescharacteristic
683C22#559011yescharacteristic
684C22#565511yescharacteristic
685C22#575711yescharacteristic
686C22#576211yescharacteristic
687C22#577311yescharacteristic
688C22#577711yescharacteristic
689C4—22no—
690C4—42no—
691C4—81no—
692C4—81no—
693C4—121no—
694C4—122no—
695C4—121no—
696C4—121no—
697C4—241no—
698C4—241no—
699C4—241no—
700C4#554711yescharacteristic
701C4#556411yescharacteristic
702C4#559811yescharacteristic
703C4#560111yescharacteristic
704C3#2615611yescharacteristic, complemented, Sylow
705C2—42no—
706C2—161no—
707C2—241no—
708C2#7470211yescharacteristic
709C2#7875611yescharacteristic
710C2#7889311yescharacteristic
711C2#8002311yescharacteristic
712C2#8006511yescharacteristic
713C2#8129611yescharacteristic
714C2#8133711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1874C2
socleC23×C3C22×D4
prime radical#82704C2
hypercenter#1874D3
hypocenterC3#26156
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. #82161C2semidirect
3. C2×#4964C2semidirect
4. C2×#4779C2semidirect
5. C3×#1874C2semidirect
6. #71682C2semidirect
7. #70206C2semidirect
8. #70181C2semidirect
9. #70174C2semidirect
10. #69953C2semidirect
11. #69929C2semidirect
12. #68877C2semidirect
13. #68836C2semidirect
14. #68833C2semidirect
15. #68787C2semidirect
73. #1874D3semidirect
147. #807D4semidirect
704. C3#26156semidirect

Maximal subgroups

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

subgroupindex
#827042
#821612
C2×#49642
C2×#47792
C3×#18742
#716822
#702062
#701812
#701742
#699532
#699292
#688772
#688362
#688332
#687872
#261563