Finite Group Databasebeta

← #10844747

Subgroups — #10844747

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82481C211yesmaximal, characteristic, complemented
2C2×#5028C211yesmaximal, characteristic
3C2×C8×Dic6C211yesmaximal, characteristic, complemented
4#73735C211yesmaximal, characteristic
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70285C211yesmaximal, characteristic, complemented
7#70280C211yesmaximal, characteristic
8#70258C211yesmaximal, characteristic, complemented
9#70257C211yesmaximal, characteristic
10#69773C211yesmaximal, characteristic
11#69468C211yesmaximal, characteristic, complemented
12#68912C211yesmaximal, characteristic, complemented
13#68901C211yesmaximal, characteristic
14#68852C211yesmaximal, characteristic
15#68850C211yesmaximal, characteristic, complemented
16#25297—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×#216C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26#5229C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#5028—24no—
37#4988C2211yescharacteristic
38#4988C2211yescharacteristic
39#4986—21no—
40#4979—21no—
41#4973C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4940—21no—
50#4940—21no—
51#4936C2211yescharacteristic
52#4933—21no—
53#4933—21no—
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C8×Dic6—24no—
56#4674—21no—
57#4673—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×C8×Q8—31no—
64#2592—31no—
65#2581—31no—
66#2067—31no—
67#2066—31no—
68#2056—31no—
69#2055—31no—
70#1969—31no—
71#1968—31no—
72#1958—31no—
73#1949—31no—
74#1934—31no—
75#1933—31no—
76C4×#103D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×Dic6D411yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#216—22no—
93C2×#216—22no—
94C2×#216—41no—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215—41no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4D411yescharacteristic
110C2×Dic3⋊C4D411yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#875—22no—
116#875—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×#103—22no—
120C3×#103—22no—
121C3×C4×C8—22no—
122C3×#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#821—22no—
136#821—41no—
137#816—22no—
138#816—41no—
139Dic3⋊C8—22no—
140Dic3⋊C8—22no—
141C8×Dic3—22no—
142C8×Dic3—22no—
143#810—22no—
144#809—22no—
145#808—41no—
146#807—41no—
147C4⋊(C3⋊C8)D414yes—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153#464—64no—
154C8×Q8—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8—31no—
169C2×C4×C8—31no—
170C2×C4×C8C2×D311yescharacteristic
171#400—61no—
172#397—61no—
173#395—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#391—31no—
178#391—31no—
179#390—31no—
180#390—31no—
181#389—31no—
182#389—31no—
183#388—61no—
184#388—61no—
185#387—61no—
186#387—61no—
187#382—31no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—22no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8#4112yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C42—42no—
243C3×C42—81no—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#216—44no—
247#215—44no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—81no—
253(C2×D3)⋊C4—81no—
254C4⋊Dic3—22no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272C2×C3⋊C8C2×D412yes—
273C22×Q8—31no—
274C22×Q8—31no—
275C22×Q8C22×D311yescharacteristic
276C22×D4—31no—
277C2×Q16—62no—
278C2×Q16—62no—
279C2×Q16—121no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—121no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287#124—62no—
288#124—62no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—62no—
292C4×Q8—62no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#106—121no—
317C4⋊C8—34no—
318C4⋊C8—62no—
319C4⋊C8—62no—
320#103—22no—
321#103—22no—
322#103—62no—
323#103—62no—
324#103—121no—
325#102—62no—
326#102—62no—
327#102—121no—
328C4×C8—22no—
329C4×C8—62no—
330C4×C8—62no—
331#95—21no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3C22×D411yescharacteristic
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4#12112yes—
373C2×C3×C4#12112yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C42−1+411yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3#13512yes—
400C2×Dic3#13512yes—
401D12—41no—
402D12—41no—
403D12—41no—
404D12—41no—
405Dic6—28no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—22no—
414C24—61no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—62no—
421C2×Q8—62no—
422C2×Q8—62no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8#23512yes—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—124no—
451SD16—124no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—32no—
458C2×C8—32no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8#23512yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—241no—
485C22⋊C4—241no—
486C42—42no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—61no—
491C42—81no—
492C42#23512yes—
493C42D3×D412yes—
494C22×C3C2×#12111yescharacteristic
495C22×C3#54811yescharacteristic
496C22×C3#55011yescharacteristic
497C22×C3C2×#13511yescharacteristic
498C22×C3C2×#13511yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52911yescharacteristic
515C3×C4#54211yescharacteristic
516C3×C4#54611yescharacteristic
517C3×C4#54811yescharacteristic
518Dic3—22no—
519Dic3—22no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537Q8—68no—
538Q8—121no—
539Q8—121no—
540Q8—121no—
541Q8—121no—
542D4—121no—
543D4—121no—
544D4—121no—
545D4—121no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—32no—
552C2×C4—32no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4#88412yes—
584C2×C4#88412yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×#23511yescharacteristic
587C2×C4C2×#23511yescharacteristic
588C2×C4C2×D3×D411yescharacteristic
589C2×C4#100911yescharacteristic
590C2×C4#101011yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—62no—
595C2×C3#242611yescharacteristic
596C2×C3#305011yescharacteristic
597C2×C3#306311yescharacteristic
598C2×C3#336411yescharacteristic
599C2×C3#336511yescharacteristic
600C2×C3#339311yescharacteristic
601C2×C3#339411yescharacteristic
602D3—161no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22C2×#88411yescharacteristic
611C22#555211yescharacteristic
612C22#555311yescharacteristic
613C22#576211yescharacteristic
614C22#576611yescharacteristic
615C22D3×#13511yescharacteristic
616C22D3×#13511yescharacteristic
617C4—22no—
618C4—22no—
619C4—42no—
620C4—62no—
621C4—62no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—121no—
626C4—121no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#547911yescharacteristic
631C4#553411yescharacteristic
632C4#555011yescharacteristic
633C4#555211yescharacteristic
634C3#2529711yescharacteristic, complemented, Sylow
635C2—481no—
636C2#7348511yescharacteristic
637C2#7880011yescharacteristic
638C2#7883811yescharacteristic
639C2#7962411yescharacteristic
640C2#7963911yescharacteristic
641C2#7970011yescharacteristic
642C2#7971011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC4×#103D3
hypocenterC3#25297
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. #82481C2semidirect
3. C2×C8×Dic6C2semidirect
5. C3×C4×#103C2semidirect
6. #70285C2semidirect
8. #70258C2semidirect
11. #69468C2semidirect
12. #68912C2semidirect
15. #68850C2semidirect
76. C4×#103D3semidirect
634. C3#25297semidirect

Maximal subgroups

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

subgroupindex
#824812
C2×#50282
C2×C8×Dic62
#737352
C3×C4×#1032
#702852
#702802
#702582
#702572
#697732
#694682
#689122
#689012
#688522
#688502
#252973