Finite Group Databasebeta

← #10824662

Subgroups — #10824662

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#946C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic, complemented
3C2×#4685C211yesmaximal, characteristic, complemented
4C4×#895C211yesmaximal, characteristic
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70280C211yesmaximal, characteristic, complemented
7#70275C211yesmaximal, characteristic, complemented
8#70272C211yesmaximal, characteristic
9#70262C211yesmaximal, characteristic, complemented
10C4×#821C211yesmaximal, characteristic, complemented
11C4×#807C211yesmaximal, characteristic, complemented
12#68901C211yesmaximal, characteristic, complemented
13#68891C211yesmaximal, characteristic, complemented
14#68822C211yesmaximal, characteristic
15#68809C211yesmaximal, characteristic, complemented
16#23701—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C2×#895C2211yescharacteristic
21C2×C4×D12—21no—
22C2×C4×D12C2211yescharacteristic
23C2×C42×D3C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C4×#215—24no—
37#4989—21no—
38#4988C2211yescharacteristic
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4984C2211yescharacteristic
42#4978—21no—
43#4972C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C4×(C2×D3)⋊C4—21no—
49C4×(C2×D3)⋊C4—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51C4×Dic3⋊C4—21no—
52C2×C4×C3⋊C8C2211yescharacteristic
53#4685—24no—
54#4670—21no—
55#4665—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×C4×SD16—31no—
63C2×#431—31no—
64C4×C22⋊Q8—31no—
65C4×#121—31no—
66#2088—31no—
67#2077—31no—
68#2068—31no—
69#2067—31no—
70#2055—31no—
71#2036—31no—
72#2033—31no—
73#2000—31no—
74C4×#107—31no—
75C4×#103D311yescharacteristic, complemented
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3D411yescharacteristic
82C22×C4×D3D411yescharacteristic
83C22×C4×D3C2311yescharacteristic
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×C4⋊C4C2311yescharacteristic
92C2×C3×C42—21no—
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95#946—22no—
96#946—22no—
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
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⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#895—22no—
120#895—22no—
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—44no—
124C42×D3—22no—
125C42×D3—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×C4×C8—22no—
129C3×#95—21no—
130#835—22no—
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(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144#809—22no—
145#808—22no—
146#807—22no—
147#807—22no—
148C4×C3⋊C8D414yescomplemented
149C22×SD16—31no—
150C2×C22⋊Q8—31no—
151C2×#121—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4—31no—
154C2×C4×D4—61no—
155C22×C42—31no—
156C4×SD16—64no—
157#431—64no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×C22⋊C8—31no—
170C2×C22⋊C8—31no—
171C2×C4×C8—31no—
172C2×C4×C8C2×D311yescharacteristic
173#391—31no—
174#390—31no—
175#389—31no—
176#389—31no—
177#388—61no—
178#387—61no—
179#386—61no—
180#385—61no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4—61no—
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C22⋊C4—61no—
187C4×C22⋊C4—61no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3C2×D411yescharacteristic
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—24no—
220C2×D12—21no—
221C2×D12—42no—
222C2×D12—42no—
223C2×D12#4112yes—
224C2×C4×D3—22no—
225C2×C4×D3—22no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4#4112yes—
252C3×C42—42no—
253C3×C42—42no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256#215—28no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—42no—
270C4⋊Dic3#4112yes—
271Dic3⋊C4—24no—
272Dic3⋊C4—42no—
273C4×Dic3—22no—
274C4×Dic3—22no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—42no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C2×C3⋊C8C2×D412yes—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—61no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298C22⋊Q8—62no—
299C22⋊Q8—62no—
300#121—62no—
301#121—62no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—124no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—62no—
330C2×C42—62no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—62no—
334#107—62no—
335#106—62no—
336C4⋊C8—62no—
337C4⋊C8—62no—
338#103—22no—
339#103—22no—
340#103—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344C22⋊C8—62no—
345C22⋊C8—62no—
346C4×C8—22no—
347C4×C8—34no—
348#95—21no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—22no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3#13512yes—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12112yes—
390C2×C3×C4#12112yes—
391C2×C3×C4AGL(1,ℤ/8)12yes—
392C2×C3×C4#13712yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3#13512yes—
417D12—22no—
418D12—22no—
419D12—48no—
420D12—41no—
421D12—41no—
422C4×D3—28no—
423C4×D3—41no—
424C4×D3—41no—
425C4×D3—41no—
426C4×D3—41no—
427C4×D3—44no—
428C4×D3—84no—
429C3×C8—41no—
430C3×C8—41no—
431C3⋊C8—22no—
432C3⋊C8—22no—
433C24—31no—
434C24—61no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8#23512yes—
441C2×D4—32no—
442C2×D4—64no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—122no—
449C2×D4—122no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—68no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—32no—
486C2×C8—32no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—42no—
498C4⋊C4—64no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4#23512yes—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C42—42no—
515C42—42no—
516C42—62no—
517C42—62no—
518C42—61no—
519C42—61no—
520C42—122no—
521C42#23512yes—
522C42D3×D412yes—
523C22×C3C2×#12111yescharacteristic
524C22×C3#52511yescharacteristic
525C22×C3#52511yescharacteristic
526C22×C3C2×#13511yescharacteristic
527C22×C3C2×#13511yescharacteristic
528C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
529C22×C3C2×#13711yescharacteristic
530C2×D3—22no—
531C2×D3—22no—
532C2×D3—22no—
533C2×D3—41no—
534C2×D3—41no—
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—42no—
546C3×C4—42no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4#46814yes—
550C3×C4#52511yescharacteristic
551C3×C4#52911yescharacteristic
552C3×C4#52911yescharacteristic
553C3×C4#53211yescharacteristic
554Dic3—22no—
555Dic3—42no—
556Dic3—41no—
557Dic3—41no—
558Dic3—81no—
559Dic3—81no—
560C23—32no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23—62no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C2×D3×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579D4—62no—
580D4—62no—
581D4—128no—
582D4—121no—
583D4—121no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—32no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—68no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—62no—
608C2×C4—61no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—124no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—244no—
623C2×C4#88412yes—
624C2×C4#88412yes—
625C2×C4#91412yes—
626C2×C4#91812yes—
627C2×C4C2×#23511yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—41no—
634C8—41no—
635C8—62no—
636C8—62no—
637C2×C3#239211yescharacteristic
638C2×C3#305011yescharacteristic
639C2×C3C2×#46811yescharacteristic
640C2×C3#310511yescharacteristic
641C2×C3#310611yescharacteristic
642C2×C3#310711yescharacteristic
643C2×C3#310811yescharacteristic
644D3—42no—
645D3—82no—
646C22—62no—
647C22—62no—
648C22—62no—
649C22—121no—
650C22—121no—
651C22—121no—
652C22—121no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×#88411yescharacteristic
661C22#554711yescharacteristic
662C22#554711yescharacteristic
663C22C2×#91411yescharacteristic
664C22C2×#91811yescharacteristic
665C22D3×#13511yescharacteristic
666C22D3×#13511yescharacteristic
667C4—22no—
668C4—42no—
669C4—42no—
670C4—62no—
671C4—81no—
672C4—81no—
673C4—122no—
674C4—121no—
675C4—121no—
676C4—241no—
677C4—241no—
678C4#481214yes—
679C4#547911yescharacteristic
680C4#552811yescharacteristic
681C4#554711yescharacteristic
682C4#554811yescharacteristic
683C3#2370111yescharacteristic, complemented, Sylow
684C2—122no—
685C2—242no—
686C2#7358111yescharacteristic
687C2C2×#481211yescharacteristic
688C2#7883811yescharacteristic
689C2#7965411yescharacteristic
690C2#7969511yescharacteristic
691C2#7972311yescharacteristic
692C2#7974711yescharacteristic
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenterC4×#103D3
hypocenterC3#23701
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. C4×#946C2semidirect
2. C2×C4×#215C2semidirect
3. C2×#4685C2semidirect
5. C3×C4×#103C2semidirect
6. #70280C2semidirect
7. #70275C2semidirect
9. #70262C2semidirect
10. C4×#821C2semidirect
11. C4×#807C2semidirect
12. #68901C2semidirect
13. #68891C2semidirect
15. #68809C2semidirect
75. C4×#103D3semidirect
148. C4×C3⋊C8D4semidirect
683. C3#23701semidirect

Maximal subgroups

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

subgroupindex
C4×#9462
C2×C4×#2152
C2×#46852
C4×#8952
C3×C4×#1032
#702802
#702752
#702722
#702622
C4×#8212
C4×#8072
#689012
#688912
#688222
#688092
#237013