Finite Group Databasebeta

← #10858293

Subgroups — #10858293

Order 768.

848 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4847C211yesmaximal, characteristic
2C2×C4×#204C211yesmaximal, characteristic
3#73770C211yesmaximal, characteristic
4#73676C211yesmaximal, characteristic, complemented
5C3×#1916C211yesmaximal, characteristic, complemented
6#70488C211yesmaximal, characteristic, complemented
7#69769C211yesmaximal, characteristic, complemented
8#68693C211yesmaximal, characteristic, complemented
9#68683C211yesmaximal, characteristic, complemented
10#68672C211yesmaximal, characteristic
11#68672C211yesmaximal, characteristic
12#68671C211yesmaximal, characteristic
13#68670C211yesmaximal, characteristic
14#68641C211yesmaximal, characteristic, complemented
15#68634C211yesmaximal, characteristic, complemented
16#26378—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37#4989—21no—
38#4989—21no—
39#4983—21no—
40#4976—21no—
41#4936C2211yescharacteristic
42#4929C2211yescharacteristic
43#4929C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4847—24no—
49C4×#204—24no—
50#4667—21no—
51D3×#95—21no—
52#4658—21no—
53#4652—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#478—31no—
63C2×C4×SD16—31no—
64#2453—31no—
65#2349—31no—
66#2097—31no—
67#2085—31no—
68#2066—31no—
69#2059—31no—
70#1941—31no—
71#1916D311yescharacteristic, complemented
72#1877—31no—
73#1877—31no—
74#1864—31no—
75#1863—31no—
76#1862—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3C2311yescharacteristic
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×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×#204—22no—
109C2×#204—22no—
110C2×#204Q814yes—
111C4×D12—22no—
112C4×D12—22no—
113C4×D12—22no—
114C4×D12—22no—
115C4×D12—41no—
116C4×D12—41no—
117#870—22no—
118#870—41no—
119C4×Dic6—22no—
120C4×Dic6—41no—
121C3×C4⋊C8—22no—
122C3×C4×C8—22no—
123C3×#95D411yescharacteristic
124C3×#95D411yescharacteristic
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—22no—
137#821D414yes—
138#818—22no—
139#818—41no—
140#817—22no—
141#817—22no—
142#817—22no—
143#816—22no—
144#816—22no—
145#816—41no—
146C22×SD16—31no—
147C2×#125—31no—
148C2×C4×Q8—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4—31no—
152#478—64no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#106—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C4×C8C2×D311yescharacteristic
167#400—61no—
168#398—61no—
169#391—31no—
170#391—31no—
171#389—31no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#387—61no—
177#385—61no—
178#385—61no—
179#382—31no—
180#380—61no—
181C4×C4⋊C4C2×D311yescharacteristic
182C2×#95—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#336C2×D311yescharacteristic
188#336C2×D311yescharacteristic
189#336C2×D311yescharacteristic
190#336C2×D311yescharacteristic
191C23×D3—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12C2×D412yes—
210C2×D12C2×Q812yes—
211C2×D12#4112yes—
212C2×C4×D3—24no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×C4×D3—81no—
218C2×C4×D3—81no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6C2×Q812yes—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
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×C8C2×D412yes—
238C2×C3×C8C2×Q812yes—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—41no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D412yes—
261C4⋊Dic3#4112yes—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267Dic3⋊C4—81no—
268Dic3⋊C4—81no—
269C4×Dic3—42no—
270#204—28no—
271C22×Q8—31no—
272C22×D4—31no—
273C23×C4—61no—
274C23×C4—61no—
275C23×C4—61no—
276C2×SD16—34no—
277C2×SD16—62no—
278C2×SD16—62no—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283#125—62no—
284#125—121no—
285C4×Q8—62no—
286C4×Q8—121no—
287C4×D4—62no—
288C4×D4—62no—
289C4×D4—62no—
290C4×D4—62no—
291C4×D4—121no—
292C4×D4—121no—
293C2×C4⋊C4—31no—
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⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4D1211yescharacteristic
307C2×C4⋊C4D1211yescharacteristic
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313C2×C42C22×D311yescharacteristic
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#107—121no—
317#106—62no—
318#106—62no—
319#106—62no—
320C4⋊C8—22no—
321#103—62no—
322#103—62no—
323#103—121no—
324#102—34no—
325#102—62no—
326#102—62no—
327C4×C8—22no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95D1211yescharacteristic
339#95D1211yescharacteristic
340C23×C3C22×D411yescharacteristic
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
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#13512yes—
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×Q811yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C42−1+411yescharacteristic
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—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—
393D12—22no—
394D12—22no—
395D12—22no—
396D12C22⋊Q814yes—
397C4×D3—84no—
398C4×D3—84no—
399C4×D3—84no—
400Dic6—22no—
401Dic6—22no—
402Dic6—41no—
403Dic6—41no—
404C3×C8—22no—
405C3×C8—22no—
406C3×C8—41no—
407C3×C8—41no—
408C24—61no—
409C2×Q8—32no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×D4—32no—
416C2×D4—32no—
417C2×D4—32no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—64no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—122no—
433C22×C4—241no—
434C22×C4—241no—
435C22×C4—241no—
436C22×C4—241no—
437C22×C4C2×D1211yescharacteristic
438C22×C4C2×D1211yescharacteristic
439C22×C4C2×D1211yescharacteristic
440C22×C4C2×D1211yescharacteristic
441C22×C4C2×D1211yescharacteristic
442C22×C4C2×D1211yescharacteristic
443C22×C4C23×D311yescharacteristic
444SD16—68no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8D3×D412yes—
456C2×C8D3×Q812yes—
457C4⋊C4—22no—
458C4⋊C4—32no—
459C4⋊C4—32no—
460C4⋊C4—41no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—122no—
472C4⋊C4—122no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—241no—
477C4⋊C4—241no—
478C22⋊C4—64no—
479C22⋊C4—122no—
480C22⋊C4—122no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—122no—
488C22×C3#52511yescharacteristic
489C22×C3D4211yescharacteristic
490C22×C3D4×Q811yescharacteristic
491C22×C3C2×#13511yescharacteristic
492C22×C3#57511yescharacteristic
493C22×C3#57711yescharacteristic
494C22×C3#57711yescharacteristic
495C2×D3—24no—
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C3×C4—22no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4C2×C22⋊Q811yescharacteristic
509C3×C4#53911yescharacteristic
510C3×C4#54711yescharacteristic
511C3×C4#55411yescharacteristic
512Dic3—42no—
513Dic3—42no—
514Dic3—42no—
515Dic3—81no—
516Dic3—81no—
517Dic3—81no—
518Dic3—81no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23C22×D1211yescharacteristic
527Q8—62no—
528Q8—62no—
529Q8—121no—
530Q8—121no—
531D4—34no—
532D4—62no—
533D4—62no—
534D4—62no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
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—244no—
574C2×C4—244no—
575C2×C4—244no—
576C2×C4#90412yes—
577C2×C4C2×D3×D411yescharacteristic
578C2×C4C2×D3×Q811yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C2×C4#100911yescharacteristic
582C2×C4#101011yescharacteristic
583C8—22no—
584C8—22no—
585C8—41no—
586C8—41no—
587C2×C3#244911yescharacteristic
588C2×C3#310211yescharacteristic
589C2×C3#312311yescharacteristic
590C2×C3#326711yescharacteristic
591C2×C3#328211yescharacteristic
592C2×C3#338111yescharacteristic
593C2×C3#338811yescharacteristic
594D3—44no—
595C22—64no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22#552411yescharacteristic
603C22D4×D1211yescharacteristic
604C22Q8×D1211yescharacteristic
605C22C2×#90411yescharacteristic
606C22#574811yescharacteristic
607C22#575111yescharacteristic
608C22#575111yescharacteristic
609C4—22no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—122no—
616C4—122no—
617C4—122no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4—241no—
622C4D3×C22⋊Q811yescharacteristic
623C4#565511yescharacteristic
624C4#569011yescharacteristic
625C4#570711yescharacteristic
626C3#2637811yescharacteristic, complemented, Sylow
627C2—124no—
628C2#7532111yescharacteristic
629C2#7929911yescharacteristic
630C2#7938711yescharacteristic
631C2#7943511yescharacteristic
632C2#7949611yescharacteristic
633C2#7993111yescharacteristic
634C2#7997811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1916C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1916D3
hypocenterC3#26378
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
4. #73676C2semidirect
5. C3×#1916C2semidirect
6. #70488C2semidirect
7. #69769C2semidirect
8. #68693C2semidirect
9. #68683C2semidirect
14. #68641C2semidirect
15. #68634C2semidirect
71. #1916D3semidirect
626. C3#26378semidirect

Maximal subgroups

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

subgroupindex
C2×#48472
C2×C4×#2042
#737702
#736762
C3×#19162
#704882
#697692
#686932
#686832
#686722
#686712
#686702
#686412
#686342
#263783