Finite Group Databasebeta

← #10858273

Subgroups — #10858273

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2351C211yesmaximal, characteristic, complemented
2C2×#5021C211yesmaximal, characteristic, complemented
3C2×C4×#215C211yesmaximal, characteristic
4#73675C211yesmaximal, characteristic
5#71633C211yesmaximal, characteristic
6#71633C211yesmaximal, characteristic
7#71629C211yesmaximal, characteristic, complemented
8#71628C211yesmaximal, characteristic, complemented
9#69630C211yesmaximal, characteristic
10#69625C211yesmaximal, characteristic
11#69619C211yesmaximal, characteristic, complemented
12#69605C211yesmaximal, characteristic, complemented
13#69558C211yesmaximal, characteristic
14#69541C211yesmaximal, characteristic, complemented
15#69482C211yesmaximal, characteristic, complemented
16#26377—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#896—21no—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C3×#389—21no—
25C3×#389—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#382—21no—
29C3×#382C2211yescharacteristic
30C3×C4×C4⋊C4—21no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C4×#215—24no—
36#5021—24no—
37#4994—21no—
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4939C2211yescharacteristic
47#4939C2211yescharacteristic
48#4928C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4673—21no—
53#4668—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#477—31no—
63C2×C4×SD16—31no—
64#2452—31no—
65#2351D311yescharacteristic, complemented
66#2092—31no—
67#2085—31no—
68#2067—31no—
69#2058—31no—
70#1942—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1865—31no—
75#1863—31no—
76#1862—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6—21no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—41no—
86C3×C4×Q8—41no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#215—22no—
104C2×#215—22no—
105C2×#215Q814yescomplemented
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#896—44no—
122C4×D12—22no—
123C4×D12—41no—
124#869—22no—
125#869—41no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132#835—22no—
133#835—22no—
134#835D414yes—
135C3⋊#95—21no—
136C3⋊#95—21no—
137#809—22no—
138#809—22no—
139#809—41no—
140#808—22no—
141#808—41no—
142#807—22no—
143#807—22no—
144#807—22no—
145C4⋊(C3⋊C8)—22no—
146C4×C3⋊C8—22no—
147C22×SD16—31no—
148C2×C4⋊Q8—31no—
149C2×C22⋊Q8—61no—
150C2×C4×Q8C2×D311yescharacteristic
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154#477—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8—31no—
169#396—61no—
170#393—61no—
171#390—31no—
172#390—31no—
173#389—21no—
174#389—21no—
175#389—31no—
176#389—31no—
177#389C2×D311yescharacteristic
178#389C2×D311yescharacteristic
179#385—61no—
180#382—21no—
181#382C2×D311yescharacteristic
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2×D311yescharacteristic
184#340—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#336—31no—
189#336—31no—
190#336—31no—
191#336—31no—
192C23×D3—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8C2×D412yes—
197C2×C3×Q8C2×Q812yes—
198C2×C3×Q8#4112yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×Q812yes—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3—81no—
222C2×Dic6—42no—
223C2×Dic6—42no—
224C2×Dic6—81no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
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—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4#4112yes—
238C3×C42—24no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—42no—
246C3×C42—42no—
247#215—28no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—81no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3C2×D412yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D412yes—
273C2×C3⋊C8C2×Q812yes—
274C22×Q8—61no—
275C22×Q8C22×D311yescharacteristic
276C22×D4—31no—
277C23×C4—61no—
278C2×SD16—34no—
279C2×SD16—62no—
280C2×SD16—62no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8—31no—
285C4⋊Q8—62no—
286C4⋊Q8—121no—
287C22⋊Q8—124no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—41no—
293C4×Q8—41no—
294C4×D4—62no—
295C4×D4—121no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4#6711yescharacteristic
308C2×C4⋊C4#6711yescharacteristic
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—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42C22×D311yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#107—62no—
325#107—62no—
326#106—62no—
327#106—121no—
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—21no—
338#95—21no—
339#95—21no—
340#95—61no—
341#95—61no—
342#95#6711yescharacteristic
343#95#6711yescharacteristic
344C23×C3C22×D411yescharacteristic
345C22×D3—22no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8C22⋊Q814yescomplemented
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4#13512yes—
385C2×C3×C4AGL(1,ℤ/8)12yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×Q811yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4H(5,𝔽2)11yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C2×Dic3—22no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401D12—22no—
402D12—22no—
403D12—41no—
404D12—41no—
405C4×D3—84no—
406Dic6—84no—
407C3⋊C8—22no—
408C3⋊C8—22no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C24—61no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—122no—
416C2×Q8—122no—
417C2×Q8—241no—
418C2×Q8#23512yes—
419C2×Q8D3×D412yes—
420C2×Q8D3×Q812yes—
421C2×D4—32no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—122no—
435C22×C4—122no—
436C22×C4—241no—
437C22×C4C2×#6711yescharacteristic
438C22×C4C2×#6711yescharacteristic
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6711yescharacteristic
442C22×C4C2×#6711yescharacteristic
443C22×C4C23×D311yescharacteristic
444SD16—68no—
445C2×C8—32no—
446C2×C8—32no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—32no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4#23512yes—
479C22⋊C4—122no—
480C22⋊C4—122no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—241no—
484C42—24no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C22×C3#52511yescharacteristic
494C22×C3D4211yescharacteristic
495C22×C3D4×Q811yescharacteristic
496C22×C3C2×#13511yescharacteristic
497C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
498C22×C3#57711yescharacteristic
499C22×C3#57811yescharacteristic
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C3×C4—24no—
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4C2×C22⋊Q811yescharacteristic
518C3×C4#53911yescharacteristic
519C3×C4#54711yescharacteristic
520C3×C4#55411yescharacteristic
521Dic3—42no—
522Dic3—81no—
523Dic3—81no—
524C23—62no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23—121no—
529C23—121no—
530C23—121no—
531C23C22×#6711yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—22no—
535Q8—244no—
536Q8#89614yescomplemented
537D4—62no—
538D4—62no—
539D4—121no—
540D4—121no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
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—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—62no—
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—244no—
579C2×C4#94912yes—
580C2×C4#95012yes—
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4#100411yescharacteristic
585C2×C4C2×D3×Q811yescharacteristic
586C2×C4#100711yescharacteristic
587C8—62no—
588C8—62no—
589C8—121no—
590C8—121no—
591C2×C3#244911yescharacteristic
592C2×C3#310611yescharacteristic
593C2×C3#312111yescharacteristic
594C2×C3#326511yescharacteristic
595C2×C3#328511yescharacteristic
596C2×C3#338111yescharacteristic
597C2×C3#338611yescharacteristic
598D3—82no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22#579111yescharacteristic
607C22D4×#6711yescharacteristic
608C22Q8×#6711yescharacteristic
609C22C2×#94911yescharacteristic
610C22C2×#95011yescharacteristic
611C22#583711yescharacteristic
612C22#583811yescharacteristic
613C4—22no—
614C4—22no—
615C4—24no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—122no—
624C4—241no—
625C4—241no—
626C4#550711yescharacteristic
627C4C2×#89611yescharacteristic
628C4#551111yescharacteristic
629C4#551511yescharacteristic
630C3#2637711yescharacteristic, complemented, Sylow
631C2—242no—
632C2#8217311yescharacteristic
633C2#8252911yescharacteristic
634C2#8255211yescharacteristic
635C2#8354211yescharacteristic
636C2#8354911yescharacteristic
637C2#8369311yescharacteristic
638C2#8369411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2351C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2351D3
hypocenterC3#26377
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C3×#2351C2semidirect
2. C2×#5021C2semidirect
7. #71629C2semidirect
8. #71628C2semidirect
11. #69619C2semidirect
12. #69605C2semidirect
14. #69541C2semidirect
15. #69482C2semidirect
65. #2351D3semidirect
105. C2×#215Q8semidirect
355. C3×Q8C22⋊Q8semidirect
536. Q8#896semidirect
630. C3#26377semidirect

Maximal subgroups

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

subgroupindex
C3×#23512
C2×#50212
C2×C4×#2152
#736752
#716332
#716292
#716282
#696302
#696252
#696192
#696052
#695582
#695412
#694822
#263773