Finite Group Databasebeta

← #10855215

Subgroups — #10855215

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5185C211yesmaximal, characteristic
2Q8×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#4788C211yesmaximal, characteristic
4#76506C211yesmaximal, characteristic
5C3×#1878C211yesmaximal, characteristic, complemented
6#72065C211yesmaximal, characteristic, complemented
7#71633C211yesmaximal, characteristic
8#70253C211yesmaximal, characteristic, complemented
9#70245C211yesmaximal, characteristic
10#70237C211yesmaximal, characteristic, complemented
11#69685C211yesmaximal, characteristic, complemented
12#68895C211yesmaximal, characteristic
13#68841C211yesmaximal, characteristic, complemented
14#68635C211yesmaximal, characteristic
15#68610C211yesmaximal, characteristic, complemented
16#26161—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C22×#215C2211yescharacteristic
23C2×#893C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#391C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5185—24no—
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4984—21no—
37#4984—21no—
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4972C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4939C2211yescharacteristic
45#4935—21no—
46#4935C2211yescharacteristic
47C4×C4⋊Dic3—21no—
48C4×C4⋊Dic3C2211yescharacteristic
49#4788—24no—
50#4675—21no—
51#4674—21no—
52#4668—21no—
53#4664—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#464—31no—
63C2×C4×SD16—31no—
64#2447—31no—
65Q8×C4⋊C4—31no—
66#2096—31no—
67#2088—31no—
68#2071—31no—
69#2042—31no—
70#1933—31no—
71#1916—31no—
72#1880—31no—
73#1878—31no—
74#1878D311yescharacteristic, complemented
75#1865—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C2×C3×Q16—22no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93#947—22no—
94#947—41no—
95Q8×Dic3—22no—
96Q8×Dic3—22no—
97Q8×Dic3—41no—
98Q8×Dic3—41no—
99C2×#215—22no—
100C2×#215—22no—
101C2×#215D414yes—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#893—22no—
126#893—41no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#95—21no—
130C3×#95—21no—
131#835—22no—
132#835—22no—
133#835—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#821—22no—
142#821—22no—
143Dic3⋊C8—22no—
144C8×Dic3—22no—
145#809—22no—
146#808—22no—
147#808—41no—
148C22×Q16C2×D311yescharacteristic
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155#464—64no—
156C4×SD16—64no—
157C2×#106—31no—
158C2×#106—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—
169#397—61no—
170#397—61no—
171#391—31no—
172#391C2×D311yescharacteristic
173#390—31no—
174#389—31no—
175#389—31no—
176#389—31no—
177#389—61no—
178#389—61no—
179#385—61no—
180#384—31no—
181#384—61no—
182#381—61no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191#336—31no—
192#336C2×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—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8C2×D412yes—
204C2×C3×Q8#4112yes—
205C2×C3×Q8#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C2×D412yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C3×Q16—44no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4#4112yes—
241C3×C42—24no—
242C3×C42—21no—
243C3×C42—21no—
244#215—28no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—81no—
250C4⋊Dic3—22no—
251C4⋊Dic3—22no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3—42no—
260C4⋊Dic3—42no—
261C4⋊Dic3—81no—
262C4⋊Dic3—81no—
263C4⋊Dic3—81no—
264Dic3⋊C4—81no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8C2×D412yes—
277C22×Q8C22×D311yescharacteristic
278C22×Q8C22×D311yescharacteristic
279C22×D4—31no—
280C23×C4—61no—
281C2×Q16—22no—
282C2×Q16—22no—
283C2×Q16—22no—
284C2×SD16—34no—
285C2×SD16—62no—
286C2×SD16—62no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#124—62no—
292#124—121no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8—121no—
298C4×Q8—121no—
299C4×D4—62no—
300C4×D4—121no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—21no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#106—62no—
328#106—121no—
329C4⋊C8—62no—
330#103—22no—
331#103—22no—
332#103—62no—
333#103—62no—
334#103—121no—
335#102—62no—
336#102—62no—
337#102—62no—
338C4×C8—62no—
339#95—21no—
340#95—21no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C3×Q8—22no—
357C3×Q8—22no—
358C3×Q8—22no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8#12314yes—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4#13512yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3#13712yes—
409D12—22no—
410D12—22no—
411D12—41no—
412D12—41no—
413C4×D3—84no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—22no—
417C3⋊C8—41no—
418C24—61no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8#23512yes—
429C2×Q8D3×D412yes—
430C2×Q8#23712yes—
431C2×D4—32no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—241no—
451C22×C4D3×D411yescharacteristic
452C22×C4D3×D411yescharacteristic
453C22×C4C23×D311yescharacteristic
454Q16—44no—
455SD16—68no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—32no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—62no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—241no—
484C4⋊C4—241no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C4⋊C4#23712yes—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C22⋊C4—241no—
493C42—24no—
494C42—21no—
495C42—21no—
496C42—61no—
497C42—61no—
498C42—61no—
499C42—61no—
500C42—122no—
501C42—122no—
502C22×C3#52511yescharacteristic
503C22×C3D4211yescharacteristic
504C22×C3#54811yescharacteristic
505C22×C3C2×#13511yescharacteristic
506C22×C3C2×#13711yescharacteristic
507C22×C3#57711yescharacteristic
508C22×C3#57711yescharacteristic
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C3×C4—22no—
517C3×C4—24no—
518C3×C4—41no—
519C3×C4—42no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4C2×#12311yescharacteristic
523C3×C4#53911yescharacteristic
524C3×C4#54711yescharacteristic
525C3×C4#55311yescharacteristic
526Dic3—22no—
527Dic3—41no—
528Dic3—41no—
529Dic3—41no—
530Dic3—81no—
531Dic3—81no—
532Dic3—81no—
533C23—62no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23—121no—
540C23C2×D3×D411yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8#88314yes—
549D4—62no—
550D4—62no—
551D4—121no—
552D4—121no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—32no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—244no—
592C2×C4#91912yes—
593C2×C4C2×#23511yescharacteristic
594C2×C4C2×D3×D411yescharacteristic
595C2×C4C2×#23711yescharacteristic
596C2×C4C2×#23711yescharacteristic
597C2×C4#100911yescharacteristic
598C2×C4#101011yescharacteristic
599C8—41no—
600C8—41no—
601C8—62no—
602C8—121no—
603C2×C3#244311yescharacteristic
604C2×C3#310811yescharacteristic
605C2×C3#312511yescharacteristic
606C2×C3#326611yescharacteristic
607C2×C3#328411yescharacteristic
608C2×C3#336511yescharacteristic
609C2×C3#336911yescharacteristic
610D3—82no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22#555211yescharacteristic
619C22#558611yescharacteristic
620C22#560911yescharacteristic
621C22C2×#91911yescharacteristic
622C22#577011yescharacteristic
623C22D3×#13711yescharacteristic
624C22#577811yescharacteristic
625C4—22no—
626C4—24no—
627C4—41no—
628C4—42no—
629C4—62no—
630C4—81no—
631C4—81no—
632C4—121no—
633C4—121no—
634C4—121no—
635C4—241no—
636C4—241no—
637C4—241no—
638C4C2×#88311yescharacteristic
639C4#556511yescharacteristic
640C4#561111yescharacteristic
641C4#566511yescharacteristic
642C3#2616111yescharacteristic, complemented, Sylow
643C2—242no—
644C2#7470411yescharacteristic
645C2#7970111yescharacteristic
646C2#7981511yescharacteristic
647C2#8017311yescharacteristic
648C2#8021911yescharacteristic
649C2#8050211yescharacteristic
650C2#8062711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1878C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1878D3
hypocenterC3#26161
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic3C2semidirect
5. C3×#1878C2semidirect
6. #72065C2semidirect
8. #70253C2semidirect
10. #70237C2semidirect
11. #69685C2semidirect
13. #68841C2semidirect
15. #68610C2semidirect
74. #1878D3semidirect
642. C3#26161semidirect

Maximal subgroups

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

subgroupindex
C2×#51852
Q8×C4⋊Dic32
C2×#47882
#765062
C3×#18782
#720652
#716332
#702532
#702452
#702372
#696852
#688952
#688412
#686352
#686102
#261613