Finite Group Databasebeta

← #10844741

Subgroups — #10844741

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
1C2×C3×C8×Q8C211yesmaximal, characteristic, complemented
2#82484C211yesmaximal, characteristic
3#82480C211yesmaximal, characteristic, complemented
4C2×#4845C211yesmaximal, characteristic
5#70488C211yesmaximal, characteristic, complemented
6#70487C211yesmaximal, characteristic, complemented
7#70484C211yesmaximal, characteristic
8#70482C211yesmaximal, characteristic
9#70455C211yesmaximal, characteristic, complemented
10#69775C211yesmaximal, characteristic, complemented
11#69774C211yesmaximal, characteristic
12#69773C211yesmaximal, characteristic
13#69768C211yesmaximal, characteristic
14#69767C211yesmaximal, characteristic, complemented
15C4×#816C211yesmaximal, characteristic, complemented
16#25297—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×Dic12C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×C8×Q8—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29#5231—21no—
30#5230—21no—
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4992—21no—
42#4990—21no—
43#4990—21no—
44#4985—21no—
45#4978—21no—
46#4978—21no—
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4936C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4929C2211yescharacteristic
52#4929C2211yescharacteristic
53#4929C2211yescharacteristic
54#4929C2211yescharacteristic
55#4928C2211yescharacteristic
56#4928C2211yescharacteristic
57#4845—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#464—31no—
63C2×C8×Q8D311yescharacteristic, complemented
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×#103—31no—
77C22×C3×Q8D411yescharacteristic
78C22×D12C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×Dic12—22no—
110C2×Dic12—22no—
111C2×Dic12—41no—
112C2×#204—22no—
113C2×#204—22no—
114C2×#204—41no—
115#875—22no—
116#875—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C3×C4⋊C8—22no—
120C3×C4⋊C8—22no—
121C3×C4⋊C8D414yes—
122C3×C4×C8—22no—
123C3×C4×C8—22no—
124C3×C4×C8—22no—
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—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#821—22no—
139#821—22no—
140#821—41no—
141#818—41no—
142#817—41no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—41no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153#464—64no—
154C8×Q8—24no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8C2×D311yescharacteristic
158C2×C4⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8C2×D311yescharacteristic
169C2×C4×C8C2×D311yescharacteristic
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⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C23×D3—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D412yes—
239C2×C3×C8C2×D412yes—
240C2×C3×C8#4112yes—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266Dic3⋊C4—42no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C4×Dic3—42no—
270C4×Dic3—81no—
271Dic12—44no—
272#204—44no—
273C22×Q8—31no—
274C22×Q8—31no—
275C22×Q8D1211yescharacteristic
276C22×D4—31no—
277C2×Q16—62no—
278C2×Q16—62no—
279C2×Q16—121no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—121no—
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
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—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4D1211yescharacteristic
302C2×C4⋊C4D1211yescharacteristic
303C2×C4⋊C4D1211yescharacteristic
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—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#106—121no—
317C4⋊C8—22no—
318C4⋊C8—22no—
319C4⋊C8#6714yes—
320#103—62no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—121no—
325#102—62no—
326#102—62no—
327#102—121no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—22no—
331#95—61no—
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—28no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4#12112yes—
368C2×C3×C4#12112yes—
369C2×C3×C4#13512yes—
370C2×C3×C4#13512yes—
371C2×C3×C4C22×D411yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C42−1+411yescharacteristic
377C2×Dic3—22no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398D12—41no—
399D12—41no—
400D12—41no—
401D12—41no—
402Dic6—22no—
403Dic6—22no—
404Dic6—41no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—22no—
412C3×C8—22no—
413C3×C8—41no—
414C24—61no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—32no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4C2×D1211yescharacteristic
444C22×C4C2×D1211yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—124no—
451SD16—124no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8#23912yes—
462C2×C8C2×#6712yes—
463C2×C8C2×#6712yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—122no—
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—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—122no—
491C42—241no—
492C42#23512yes—
493C42C2×#6712yes—
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—22no—
511C3×C4—22no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4#52911yescharacteristic
515C3×C4#54211yescharacteristic
516C3×C4#54611yescharacteristic
517C3×C4#54811yescharacteristic
518Dic3—42no—
519Dic3—81no—
520Dic3—81no—
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C22×D1211yescharacteristic
533Q8—28no—
534Q8—62no—
535Q8—62no—
536Q8—121no—
537Q8—121no—
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—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—62no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4#90412yes—
582C2×C4#90412yes—
583C2×C4#93012yes—
584C2×C4#93012yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×#23911yescharacteristic
587C2×C4#100711yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4#100911yescharacteristic
590C2×C4C22×#6711yescharacteristic
591C8—22no—
592C8—22no—
593C8—22no—
594C8—41no—
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—
610C22#557611yescharacteristic
611C22#557711yescharacteristic
612C22C2×#90411yescharacteristic
613C22C2×#90411yescharacteristic
614C22#575111yescharacteristic
615C22#575311yescharacteristic
616C22C2×#93011yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—122no—
624C4—241no—
625C4—241no—
626C4—241no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#558011yescharacteristic
631C4#558211yescharacteristic
632C4#581711yescharacteristic
633C4#582911yescharacteristic
634C3#2529711yescharacteristic, complemented, Sylow
635C2—481no—
636C2#7991211yescharacteristic
637C2#7991311yescharacteristic
638C2#7993411yescharacteristic
639C2#7993511yescharacteristic
640C2#8249511yescharacteristic
641C2#8374211yescharacteristic
642C2#8380811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C8×Q8C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×C8×Q8D3
hypocenterC3#25297
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C8×Q8C2semidirect
3. #82480C2semidirect
5. #70488C2semidirect
6. #70487C2semidirect
9. #70455C2semidirect
10. #69775C2semidirect
14. #69767C2semidirect
15. C4×#816C2semidirect
63. C2×C8×Q8D3semidirect
634. C3#25297semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×Q82
#824842
#824802
C2×#48452
#704882
#704872
#704842
#704822
#704552
#697752
#697742
#697732
#697682
#697672
C4×#8162
#252973