Finite Group Databasebeta

← #10817469

Subgroups — #10817469

Order 768.

1034 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4872C211yesmaximal, characteristic
2C2×#4852C211yesmaximal, characteristic, complemented
3C2×#4844C211yesmaximal, characteristic
4#74228C211yesmaximal, characteristic, complemented
5#73654C211yesmaximal, characteristic, complemented
6C3×#1949C211yesmaximal, characteristic, complemented
7#70642C211yesmaximal, characteristic, complemented
8#70584C211yesmaximal, characteristic, complemented
9#70465C211yesmaximal, characteristic, complemented
10#69551C211yesmaximal, characteristic, complemented
11C4×#809C211yesmaximal, characteristic, complemented
12#68897C211yesmaximal, characteristic
13#68896C211yesmaximal, characteristic, complemented
14#68852C211yesmaximal, characteristic, complemented
15#68850C211yesmaximal, characteristic, complemented
16#22999—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#887C2211yescharacteristic
23C2×#875C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C3×#382C2211yescharacteristic
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34#4988C2211yescharacteristic
35#4973C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44#4940—21no—
45C4×(C2×D3)⋊C4—21no—
46C4×C4⋊Dic3C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4872—24no—
49#4852—24no—
50#4844C418yescomplemented
51#4670—21no—
52#4668—21no—
53#4668—21no—
54#4668—21no—
55#4667—21no—
56D3×#95—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#463—31no—
64C2×C4×SD16—31no—
65C2×C4×D8—31no—
66#2367—31no—
67#2349—31no—
68#1958—31no—
69#1949D311yescharacteristic, complemented
70#1934—31no—
71#1933—31no—
72#1879—31no—
73#1877—31no—
74#1864—31no—
75#1863—31no—
76C8×C4⋊C4—31no—
77C4×#102—31no—
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3D411yescharacteristic
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C2×D24—22no—
113C2×D24—22no—
114C2×D24C2×C414yes—
115C2×#204—22no—
116C2×#204—22no—
117C2×#204C2×C414yescomplemented
118#893—22no—
119#893—22no—
120#893—22no—
121#893—41no—
122#893—41no—
123#893—41no—
124#887—22no—
125#887—41no—
126#875—22no—
127#875C2×C414yescomplemented
128C4×D12—22no—
129C4×D12C2×C414yescomplemented
130C3×#107—22no—
131C3×#106—22no—
132C3×C4⋊C8C2×C414yescomplemented
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#821C2×C414yescomplemented
143#816C2×C414yescomplemented
144C8×Dic3—22no—
145C8×Dic3—22no—
146#810—22no—
147#810—41no—
148#809—22no—
149#809—22no—
150#809—41no—
151#809D414yes—
152C4×C3⋊C8—22no—
153C22×SD16—31no—
154C22×D8—31no—
155C2×#124—31no—
156C2×C4×Q8—31no—
157C2×C4×D4—31no—
158C2×C4×D4—31no—
159C2×C4×D4—31no—
160C2×C4×D4—31no—
161#463—38no—
162C4×SD16—64no—
163C4×D8—64no—
164C2×#107C2×D311yescharacteristic
165C2×#106C2×D311yescharacteristic
166C2×C4⋊C8C2×D311yescharacteristic
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#102—31no—
175C2×C4×C8—31no—
176C2×C4×C8—31no—
177C2×C4×C8—31no—
178#391—31no—
179#390—31no—
180#389—31no—
181#389—31no—
182#388—61no—
183#388—61no—
184#385—61no—
185#385—61no—
186#385—61no—
187#382C2×D311yescharacteristic
188#380—61no—
189C4×C4⋊C4—31no—
190C4×C4⋊C4C2×D311yescharacteristic
191C4×C22⋊C4—61no—
192C2×#95—61no—
193#340—31no—
194#340—31no—
195#340C2×D311yescharacteristic
196#340C2×D311yescharacteristic
197#336—31no—
198#336—31no—
199C23×D3—21no—
200C23×D3—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C22×C412yes—
223C2×D12C22×C412yes—
224C2×D12C2×D412yes—
225C2×D12#4112yes—
226C2×C4×D3—24no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—81no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6C22×C412yes—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C22×C412yes—
249C2×C3×C8C22×C412yes—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4C2×D412yes—
258C3×C42—81no—
259C3×C42C22×C412yes—
260C3×C42#4112yes—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—21no—
270C4⋊Dic3—22no—
271C4⋊Dic3—22no—
272C4⋊Dic3C22×C412yes—
273Dic3⋊C4—42no—
274Dic3⋊C4—81no—
275Dic3⋊C4—81no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—42no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8C2×D412yes—
287D24—28no—
288#204—28no—
289C22×Q8—31no—
290C22×D4—31no—
291C22×D4—31no—
292C23×C4—61no—
293C23×C4—61no—
294C23×C4—61no—
295C23×C4—61no—
296C2×SD16—34no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×D8—34no—
300C2×D8—62no—
301C2×D8—62no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306#124—34no—
307#124—62no—
308C4×Q8—62no—
309C4×Q8—121no—
310C4×D4—34no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—121no—
316C4×D4—121no—
317C4×D4—121no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C42—21no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#107—22no—
342#106—22no—
343C4⋊C8C4×D314yescomplemented
344#103—34no—
345#103—62no—
346#103—121no—
347#102—34no—
348#102—34no—
349#102—62no—
350#102—62no—
351#102—121no—
352C4×C8—62no—
353C4×C8—62no—
354C4×C8—62no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—31no—
359#95—31no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3C22×D411yescharacteristic
365C22×D3—22no—
366C22×D3—22no—
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4C4×D412yes—
396C2×C3×C4C4×D412yes—
397C2×C3×C4C23×C411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4H(5,𝔽2)11yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3#13512yes—
420C2×Dic3#13512yes—
421D12—22no—
422D12—22no—
423D12—22no—
424D12—22no—
425D12—22no—
426D12—41no—
427D12—41no—
428D12C4×D414yes—
429C4×D3—84no—
430C4×D3—84no—
431C4×D3—84no—
432C4×D3—84no—
433Dic6—22no—
434Dic6—22no—
435Dic6—41no—
436Dic6—41no—
437C3×C8—22no—
438C3×C8—22no—
439C3⋊C8—22no—
440C3⋊C8—41no—
441C24—61no—
442C24—61no—
443C2×Q8—32no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8—61no—
449C2×D4—32no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—32no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—31no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—64no—
472C22×C4—122no—
473C22×C4—122no—
474C22×C4—122no—
475C22×C4—122no—
476C22×C4—122no—
477C22×C4—122no—
478C22×C4—241no—
479C22×C4—241no—
480C22×C4—241no—
481C22×C4—241no—
482C22×C4—241no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486SD16—68no—
487D8—68no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—32no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8C2×C4×D312yes—
499C2×C8C2×C4×D312yes—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—32no—
508C4⋊C4—62no—
509C4⋊C4—62no—
510C4⋊C4—61no—
511C4⋊C4—122no—
512C4⋊C4—241no—
513C4⋊C4—241no—
514C4⋊C4D3×D412yes—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—241no—
523C42—61no—
524C42—61no—
525C42—61no—
526C42—61no—
527C42—81no—
528C42—122no—
529C42C2×C4×D312yes—
530C42#23912yes—
531C22×C3C2×C4×D411yescharacteristic
532C22×C3D4211yescharacteristic
533C22×C3#54811yescharacteristic
534C22×C3C2×#13511yescharacteristic
535C22×C3C2×#13511yescharacteristic
536C22×C3#57611yescharacteristic
537C22×C3#57711yescharacteristic
538C2×D3—24no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—42no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4—81no—
558C3×C4C2×C4×D411yescharacteristic
559C3×C4#52011yescharacteristic
560C3×C4#54711yescharacteristic
561C3×C4#55311yescharacteristic
562Dic3—22no—
563Dic3—22no—
564Dic3—42no—
565Dic3—42no—
566Dic3—41no—
567Dic3—41no—
568C23—62no—
569C23—62no—
570C23—62no—
571C23—62no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23C2×D3×D411yescharacteristic
583Q8—62no—
584Q8—62no—
585Q8—121no—
586Q8—121no—
587D4—34no—
588D4—62no—
589D4—62no—
590D4—62no—
591D4—62no—
592D4—62no—
593D4—121no—
594D4—121no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—244no—
630C2×C4—244no—
631C2×C4—244no—
632C2×C4—244no—
633C2×C4#89312yes—
634C2×C4#89312yes—
635C2×C4C22×C4×D311yescharacteristic
636C2×C4C2×D3×D411yescharacteristic
637C2×C4C2×#23911yescharacteristic
638C2×C4#100711yescharacteristic
639C2×C4D3×#4111yescharacteristic
640C2×C4#100911yescharacteristic
641C8—22no—
642C8—22no—
643C8—62no—
644C8—121no—
645C2×C3#236111yescharacteristic
646C2×C3C4×#13511yescharacteristic
647C2×C3#294611yescharacteristic
648C2×C3#328011yescharacteristic
649C2×C3#328211yescharacteristic
650C2×C3#336011yescharacteristic
651C2×C3#336111yescharacteristic
652D3—44no—
653D3—82no—
654C22—64no—
655C22—122no—
656C22—122no—
657C22—122no—
658C22—122no—
659C22—122no—
660C22—122no—
661C22—122no—
662C22—122no—
663C22—122no—
664C22—241no—
665C22—241no—
666C22—241no—
667C22—241no—
668C22C2×#89311yescharacteristic
669C22#560911yescharacteristic
670C22#563311yescharacteristic
671C22D3×#13511yescharacteristic
672C22D3×#13511yescharacteristic
673C22#577411yescharacteristic
674C22#577811yescharacteristic
675C4—22no—
676C4—22no—
677C4—42no—
678C4—62no—
679C4—62no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—122no—
684C4—121no—
685C4—122no—
686C4—121no—
687C4C4×D3×D411yescharacteristic
688C4#557411yescharacteristic
689C4#569511yescharacteristic
690C4#572611yescharacteristic
691C3#2299911yescharacteristic, complemented, Sylow
692C2—124no—
693C2—242no—
694C2#7620611yescharacteristic
695C2#7916511yescharacteristic
696C2#7920111yescharacteristic
697C2#8055711yescharacteristic
698C2#8063011yescharacteristic
699C2#8115011yescharacteristic
700C2#8121511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1949C2
socleC23×C3C22×D4
prime radicalC22×D24C22
cosocleC22×C3×C4C24
hypercenter#1949D3
hypocenterC3#22999
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. C2×#4852C2semidirect
4. #74228C2semidirect
5. #73654C2semidirect
6. C3×#1949C2semidirect
7. #70642C2semidirect
8. #70584C2semidirect
9. #70465C2semidirect
10. #69551C2semidirect
11. C4×#809C2semidirect
13. #68896C2semidirect
14. #68852C2semidirect
15. #68850C2semidirect
50. #4844C4semidirect
69. #1949D3semidirect
117. C2×#204C2×C4semidirect
127. #875C2×C4semidirect
129. C4×D12C2×C4semidirect
132. C3×C4⋊C8C2×C4semidirect
142. #821C2×C4semidirect
143. #816C2×C4semidirect
343. C4⋊C8C4×D3semidirect
691. C3#22999semidirect

Maximal subgroups

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

subgroupindex
C2×#48722
C2×#48522
C2×#48442
#742282
#736542
C3×#19492
#706422
#705842
#704652
#695512
C4×#8092
#688972
#688962
#688522
#688502
#229993