Finite Group Databasebeta

← #10858296

Subgroups — #10858296

Order 768.

828 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
1#83030C211yesmaximal, characteristic
2C2×#5063C211yesmaximal, characteristic, complemented
3C2×#4852C211yesmaximal, characteristic
4#75981C211yesmaximal, characteristic, complemented
5C3×#2059C211yesmaximal, characteristic, complemented
6#70642C211yesmaximal, characteristic, complemented
7#70641C211yesmaximal, characteristic
8#70567C211yesmaximal, characteristic, complemented
9#69630C211yesmaximal, characteristic
10#68909C211yesmaximal, characteristic
11#68906C211yesmaximal, characteristic, complemented
12#68895C211yesmaximal, characteristic
13#68859C211yesmaximal, characteristic, complemented
14#68686C211yesmaximal, characteristic
15#68611C211yesmaximal, characteristic, complemented
16#26379—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#887C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#5063—24no—
30#4994—21no—
31#4993—21no—
32#4989—21no—
33#4988C2211yescharacteristic
34#4987—21no—
35#4984C2211yescharacteristic
36#4984C2211yescharacteristic
37#4976—21no—
38#4975—21no—
39#4975C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48#4939C2211yescharacteristic
49C4×Dic3⋊C4C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4852—24no—
52#4649—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#336C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×C4×SD16—31no—
64#2454—31no—
65Q8×C4⋊C4—31no—
66#2093—31no—
67#2084—31no—
68#2067—31no—
69#2059D311yescharacteristic, complemented
70#1942—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1864—31no—
75#1863—31no—
76#1863—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#125—22no—
83C3×#125—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
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×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×#204—22no—
117C2×#204—22no—
118C2×#204Q814yes—
119#893—22no—
120#893—41no—
121#887—22no—
122#887—22no—
123#887—22no—
124#887—41no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C3×#107—41no—
128C3×#106—22no—
129C3×#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#821—41no—
136#816—22no—
137C8×Dic3—22no—
138#810—22no—
139#810D414yes—
140#809—22no—
141#809—22no—
142#808—22no—
143#807—22no—
144#807—22no—
145C4⋊(C3⋊C8)—22no—
146C22×SD16—31no—
147C2×#125C2×D311yescharacteristic
148C2×C4×Q8—31no—
149C2×C4×Q8—31no—
150C2×C4×Q8—31no—
151C2×C4×D4—31no—
152#479—64no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C4×C8—31no—
167#399—61no—
168#396—61no—
169#390—31no—
170#390—31no—
171#389—31no—
172#389—31no—
173#389—31no—
174#389—61no—
175#389—61no—
176#389C2×D311yescharacteristic
177#385—61no—
178#384—31no—
179#384—61no—
180#380—61no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183#340—31no—
184#340—31no—
185#340C2×D311yescharacteristic
186#340C2×D311yescharacteristic
187#336—31no—
188#336—31no—
189#336—31no—
190#336C2×D311yescharacteristic
191C23×D3—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C2×Q812yes—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—81no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6C2×D412yes—
219C2×Dic6C2×Q812yes—
220C2×Dic6#4112yes—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8C2×Q812yes—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4—81no—
236C3×C4⋊C4C2×D412yes—
237C3×C4⋊C4#4112yes—
238C3×C42—21no—
239C3×C42—21no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—81no—
243(C2×D3)⋊C4—81no—
244(C2×D3)⋊C4—81no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—81no—
250C4⋊Dic3—81no—
251Dic3⋊C4—24no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259C4×Dic3—24no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272#204—28no—
273C22×Q8—31no—
274C22×D4—31no—
275C23×C4—61no—
276C2×SD16—34no—
277C2×SD16—62no—
278C2×SD16—62no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283#125—22no—
284#125—41no—
285C4×Q8—62no—
286C4×Q8—62no—
287C4×Q8—62no—
288C4×Q8—62no—
289C4×Q8—121no—
290C4×Q8—121no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
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×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—41no—
324#107—62no—
325#107—62no—
326#106—22no—
327#106—62no—
328C4⋊C8—62no—
329#103—34no—
330#103—62no—
331#103—62no—
332#102—62no—
333#102—62no—
334#102—121no—
335C4×C8—62no—
336#95—21no—
337#95—31no—
338#95—31no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
369C2×C3×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C42−1+411yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3#13512yes—
394C2×Dic3#13712yes—
395D12—22no—
396D12—22no—
397D12—41no—
398D12—41no—
399C4×D3—84no—
400Dic6—22no—
401Dic6—22no—
402Dic6—22no—
403Dic6C22⋊Q814yes—
404C3×C8—22no—
405C3×C8—41no—
406C3⋊C8—22no—
407C3⋊C8—41no—
408C24—61no—
409C2×Q8—32no—
410C2×Q8—32no—
411C2×Q8—32no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×D4—32no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—122no—
433C22×C4—122no—
434C22×C4—241no—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4C23×D311yescharacteristic
438SD16—68no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—32no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8D3×Q812yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—64no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—81no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—241no—
471C4⋊C4—241no—
472C4⋊C4—241no—
473C4⋊C4—241no—
474C4⋊C4D3×D412yes—
475C4⋊C4#23912yes—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—241no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—21no—
482C42—21no—
483C42—64no—
484C42—61no—
485C42—61no—
486C42—61no—
487C42—61no—
488C42—122no—
489C42—122no—
490C22×C3#52511yescharacteristic
491C22×C3D4211yescharacteristic
492C22×C3D4×Q811yescharacteristic
493C22×C3C2×#13511yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57611yescharacteristic
496C22×C3#57711yescharacteristic
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4C2×C22⋊Q811yescharacteristic
511C3×C4#53911yescharacteristic
512C3×C4#54711yescharacteristic
513C3×C4#55411yescharacteristic
514Dic3—22no—
515Dic3—24no—
516Dic3—22no—
517Dic3—41no—
518Dic3—41no—
519Dic3—81no—
520Dic3—81no—
521C23—62no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23C2×D3×D411yescharacteristic
529Q8—34no—
530Q8—62no—
531Q8—62no—
532Q8—62no—
533D4—62no—
534D4—62no—
535D4—121no—
536D4—121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—32no—
546C2×C4—32no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—62no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—244no—
577C2×C4C2×D3×D411yescharacteristic
578C2×C4#100411yescharacteristic
579C2×C4C2×D3×Q811yescharacteristic
580C2×C4C2×#23911yescharacteristic
581C2×C4#100711yescharacteristic
582C2×C4D3×#4111yescharacteristic
583C8—22no—
584C8—41no—
585C8—62no—
586C8—121no—
587C2×C3#244911yescharacteristic
588C2×C3#310811yescharacteristic
589C2×C3#311911yescharacteristic
590C2×C3#326611yescharacteristic
591C2×C3#328711yescharacteristic
592C2×C3#338111yescharacteristic
593C2×C3#338511yescharacteristic
594D3—82no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22#559011yescharacteristic
603C22#562711yescharacteristic
604C22#572711yescharacteristic
605C22D3×#13511yescharacteristic
606C22#577011yescharacteristic
607C22#577411yescharacteristic
608C22D3×#13711yescharacteristic
609C4—41no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—62no—
614C4—62no—
615C4—64no—
616C4—81no—
617C4—81no—
618C4—121no—
619C4—121no—
620C4—241no—
621C4—241no—
622C4D3×C22⋊Q811yescharacteristic
623C4#563811yescharacteristic
624C4#572611yescharacteristic
625C4#573011yescharacteristic
626C3#2637911yescharacteristic, complemented, Sylow
627C2—242no—
628C2#7725211yescharacteristic
629C2#8013111yescharacteristic
630C2#8024111yescharacteristic
631C2#8114411yescharacteristic
632C2#8120511yescharacteristic
633C2#8191311yescharacteristic
634C2#8196511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2059D3
hypocenterC3#26379
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
2. C2×#5063C2semidirect
4. #75981C2semidirect
5. C3×#2059C2semidirect
6. #70642C2semidirect
8. #70567C2semidirect
11. #68906C2semidirect
13. #68859C2semidirect
15. #68611C2semidirect
69. #2059D3semidirect
626. C3#26379semidirect

Maximal subgroups

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

subgroupindex
#830302
C2×#50632
C2×#48522
#759812
C3×#20592
#706422
#706412
#705672
#696302
#689092
#689062
#688952
#688592
#686862
#686112
#263793