Finite Group Databasebeta

← #10824490

Subgroups — #10824490

Order 768.

844 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×#4710C211yesmaximal, characteristic
2C2×C4×#204C211yesmaximal, characteristic
3#73415C211yesmaximal, characteristic
4#73388C211yesmaximal, characteristic, complemented
5C3×#1768C211yesmaximal, characteristic, complemented
6#70489C211yesmaximal, characteristic, complemented
7#70488C211yesmaximal, characteristic, complemented
8#70482C211yesmaximal, characteristic
9#70482C211yesmaximal, characteristic
10#69813C211yesmaximal, characteristic, complemented
11C4×#816C211yesmaximal, characteristic, complemented
12#68687C211yesmaximal, characteristic
13#68641C211yesmaximal, characteristic, complemented
14#68637C211yesmaximal, characteristic
15#68631C211yesmaximal, characteristic, complemented
16#23689—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#376C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#4939C2211yescharacteristic
39#4939C2211yescharacteristic
40#4936C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4929C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4710—24no—
47C4×#204—24no—
48#4669—21no—
49#4669—21no—
50#4667—21no—
51#4664—21no—
52#4653—21no—
53#4653—21no—
54#4653—21no—
55#4653—21no—
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#440—31no—
63C2×C4×SD16—31no—
64#2327—31no—
65#2319—31no—
66#2085—31no—
67#2073—31no—
68#2068—31no—
69#2066—31no—
70#2055—31no—
71#2055—31no—
72#2043—31no—
73#2013—31no—
74#1768D311yescharacteristic, complemented
75#1763—31no—
76C4×#103—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×#204—22no—
108C2×#204—22no—
109C2×#204—22no—
110C2×#204—22no—
111C2×#204—22no—
112C2×#204—22no—
113C4×D12—22no—
114C4×D12D414yes—
115#869—22no—
116#869—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C3×C4⋊C8—22no—
122C3×C4⋊C8—22no—
123C3×#97—22no—
124C3×C4×C8—22no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
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#818—22no—
141#817—22no—
142#816—22no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154#440—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97C2×D311yescharacteristic
169C2×C4×C8C2×D311yescharacteristic
170#391—31no—
171#391—31no—
172#390—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#387—61no—
179#387—61no—
180#387—61no—
181#387—61no—
182#382—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193C23×D3—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12C2×D412yes—
214C2×C4×D3—24no—
215C2×C4×D3—81no—
216C2×C4×D3—81no—
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#4112yes—
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×C8—21no—
239C2×C3×C8#4112yes—
240C2×C3×C8#4112yes—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42C2×D412yes—
246C3×C42#4112yes—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—81no—
249(C2×D3)⋊C4—81no—
250(C2×D3)⋊C4—81no—
251(C2×D3)⋊C4—81no—
252C4⋊Dic3—24no—
253C4⋊Dic3—24no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2×D412yes—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266Dic3⋊C4—81no—
267Dic3⋊C4—81no—
268C4×Dic3—42no—
269C4×Dic3—42no—
270C4×Dic3—81no—
271#204—28no—
272#204—28no—
273C22×Q8—31no—
274C22×Q8—31no—
275C22×D4—31no—
276C23×C4—61no—
277C2×SD16—62no—
278C2×SD16—62no—
279C2×SD16—62no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8C22×D311yescharacteristic
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C4⋊Q8—62no—
288C4⋊Q8—62no—
289C4×Q8—62no—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×D4—34no—
294C4×D4—62no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317#107—62no—
318#106—62no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321#103—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#102—62no—
328#102—62no—
329#97—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#95D1211yescharacteristic
341#95D1211yescharacteristic
342#95D1211yescharacteristic
343#95D1211yescharacteristic
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C2×C3×C4—21no—
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—21no—
368C2×C3×C4#12112yes—
369C2×C3×C4#12112yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
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—
397D12—22no—
398D12—22no—
399D12—41no—
400D12—41no—
401C4×D3—84no—
402Dic6—22no—
403Dic6—22no—
404Dic6—22no—
405Dic6—22no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—22no—
412C3×C8—41no—
413C3×C8—41no—
414C24—61no—
415C2×Q8—32no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
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×D4—32no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—64no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—241no—
442C22×C4—241no—
443C22×C4C2×D1211yescharacteristic
444C22×C4C2×D1211yescharacteristic
445C22×C4C2×D1211yescharacteristic
446C22×C4C2×D1211yescharacteristic
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—68no—
451SD16—68no—
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—21no—
462C2×C8#23512yes—
463C2×C8#23512yes—
464C4⋊C4—32no—
465C4⋊C4—64no—
466C4⋊C4—64no—
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—122no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4—241no—
480C22⋊C4—64no—
481C22⋊C4—241no—
482C22⋊C4—241no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—122no—
490C42—122no—
491C42—241no—
492C42D3×D412yes—
493C42#23912yes—
494C22×C3C2×#12111yescharacteristic
495C22×C3#52511yescharacteristic
496C22×C3#53211yescharacteristic
497C22×C3#57711yescharacteristic
498C22×C3#57711yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4#52511yescharacteristic
515C3×C4#52511yescharacteristic
516C3×C4#52911yescharacteristic
517C3×C4#52911yescharacteristic
518Dic3—42no—
519Dic3—42no—
520Dic3—42no—
521Dic3—81no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—62no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23C22×D1211yescharacteristic
533Q8—62no—
534Q8—62no—
535Q8—62no—
536Q8—62no—
537Q8—121no—
538Q8—121no—
539Q8—121no—
540Q8—121no—
541D4—62no—
542D4—62no—
543D4—121no—
544D4—121no—
545C2×C4—21no—
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—21no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
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—121no—
582C2×C4—244no—
583C2×C4#89512yes—
584C2×C4#89512yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×#23511yescharacteristic
587C2×C4C2×D3×D411yescharacteristic
588C2×C4C2×#23911yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C8—22no—
592C8—22no—
593C8—41no—
594C8—41no—
595C2×C3#239211yescharacteristic
596C2×C3#304211yescharacteristic
597C2×C3#304611yescharacteristic
598C2×C3#312111yescharacteristic
599C2×C3#312611yescharacteristic
600C2×C3#317411yescharacteristic
601C2×C3#317811yescharacteristic
602D3—82no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—241no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22#547711yescharacteristic
611C22C2×#89511yescharacteristic
612C22#552411yescharacteristic
613C22#575111yescharacteristic
614C22#575111yescharacteristic
615C22#575111yescharacteristic
616C22#575311yescharacteristic
617C4—22no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—122no—
624C4—122no—
625C4—122no—
626C4—241no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#554711yescharacteristic
631C4#554711yescharacteristic
632C4#558011yescharacteristic
633C4#558011yescharacteristic
634C3#2368911yescharacteristic, complemented, Sylow
635C2—242no—
636C2#7377611yescharacteristic
637C2#7856711yescharacteristic
638C2#7857011yescharacteristic
639C2#7908711yescharacteristic
640C2#7908911yescharacteristic
641C2#7939411yescharacteristic
642C2#7939511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1768C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1768D3
hypocenterC3#23689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
4. #73388C2semidirect
5. C3×#1768C2semidirect
6. #70489C2semidirect
7. #70488C2semidirect
10. #69813C2semidirect
11. C4×#816C2semidirect
13. #68641C2semidirect
15. #68631C2semidirect
74. #1768D3semidirect
634. C3#23689semidirect

Maximal subgroups

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

subgroupindex
C2×#47102
C2×C4×#2042
#734152
#733882
C3×#17682
#704892
#704882
#704822
#698132
C4×#8162
#686872
#686412
#686372
#686312
#236893