Finite Group Databasebeta

← #10795125

Subgroups — #10795125

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#237C211yesmaximal, characteristic
2#73437C212yesmaximal, complemented
3C3×C4×#99C211yesmaximal, characteristic, complemented
4C4×#834C211yesmaximal, characteristic, complemented
5#70028C212yesmaximal, complemented
6#70027C212yesmaximal
7C4×#806C212yesmaximal, complemented
8#68959C212yesmaximal, complemented
9#68953C212yesmaximal, complemented
10#20723—31nomaximal, Sylow
11C22×#237C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C2×D4×Dic3C2211yescharacteristic
14C2×C4×#67C411yescharacteristic, complemented
15C2×C4×#67C2211yescharacteristic
16C22×C4×Dic3C411yescharacteristic, complemented
17C22×C4×Dic3C2211yescharacteristic
18C4×#237—22no—
19C4×#237—22no—
20C4×#237—22no—
21C4×#237—22no—
22C2×#892C411yescharacteristic, complemented
23C2×#887C411yescharacteristic, complemented
24C2×#67⋊C4C411yescharacteristic, complemented
25C2×#67⋊C4C2211yescharacteristic
26C2×#877C411yescharacteristic, complemented
27C2×#877C2211yescharacteristic
28C2×C42×D3C411yescharacteristic, complemented
29C2×C4×Dic6C411yescharacteristic, complemented
30C2×C3×#99C2212yes—
31C3×C4×C22⋊C4C2212yes—
32C2×#834C2212yes—
33C4×#217C2212yes—
34C2×#806C2214yes—
35#4943C2214yes—
36#4942C2214yes—
37#4932—22no—
38C4×Dic3⋊C4—22no—
39#4651—22no—
40#4650—22no—
41C3×#342C2212yes—
42#4527C2214yes—
43#4526C2212yes—
44C2×C4×#41—31no—
45#2283—32no—
46#1820—32no—
47#1799—32no—
48#1799—32no—
49#1782—32no—
50C4×#99—32no—
51C4×#99—31no—
52C4×#99D311yescharacteristic, complemented
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4C2×C411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×#67C2×C411yescharacteristic
57C22×#67C2311yescharacteristic
58C23×Dic3C2×C411yescharacteristic
59C23×Dic3C2×C411yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C2×#237—24no—
63C2×#237—24no—
64C2×#237—22no—
65C2×#237—22no—
66C22×C4×D3C2×C411yescharacteristic
67C22×C4×D3C2×C411yescharacteristic
68C22×C4×D3C2×C411yescharacteristic
69C22×Dic6C2×C411yescharacteristic
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C2×C3×C4⋊C4C2×C411yescharacteristic
73C2×C3×C22⋊C4C2×C411yescharacteristic
74C2×C3×C22⋊C4C2314yes—
75C2×C3×C22⋊C4C2311yescharacteristic
76C2×C3×C42—22no—
77C2×C3×C42C2×C411yescharacteristic
78C2×#217C2×C411yescharacteristic
79C2×#217C2314yes—
80C2×#217C2311yescharacteristic
81D4×Dic3—22no—
82D4×Dic3—22no—
83C4×#67—22no—
84C4×#67—22no—
85C4×#67—22no—
86C4×#67—22no—
87C2×(C2×D3)⋊C4C2×C411yescharacteristic
88C2×(C2×D3)⋊C4C2×C411yescharacteristic
89C2×C4⋊Dic3C2×C411yescharacteristic
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C411yescharacteristic
93C2×Dic3⋊C4C2×C411yescharacteristic
94C2×Dic3⋊C4C2×C411yescharacteristic
95C2×C4×Dic3—21no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2×C411yescharacteristic
104C2×C4×Dic3C2×C411yescharacteristic
105C2×C4×Dic3C2×C411yescharacteristic
106C2×C4×Dic3C2×C411yescharacteristic
107C2×C4×Dic3C2×C411yescharacteristic
108C2×C4×Dic3C2×C411yescharacteristic
109C2×C4×Dic3D412yes—
110C2×C4×Dic3D412yes—
111C2×C4×Dic3D412yes—
112C2×C4×Dic3D412yes—
113#892—22no—
114#892—22no—
115#887—22no—
116#887—22no—
117#67⋊C4—22no—
118#67⋊C4—22no—
119#67⋊C4—22no—
120#67⋊C4—22no—
121#877—22no—
122#877—22no—
123#877—22no—
124#877—22no—
125C42×D3—21no—
126C42×D3—21no—
127C42×D3—21no—
128C42×D3—21no—
129C4×Dic6—22no—
130C4×Dic6—22no—
131C3×#99—24no—
132C3×#95—22no—
133#834—24no—
134C3⋊#95—22no—
135C3⋊#95—24no—
136C3⋊#95—24no—
137C3⋊#95—24no—
138#806—28no—
139C22×#41—31no—
140C4×#41—62no—
141C4×#41—62no—
142C4×#41—62no—
143C4×#41—62no—
144C2×C4×Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×C4×D4—31no—
149C2×C4×D4—31no—
150C2×C4×D4—31no—
151C2×C4×D4C2×D311yescharacteristic
152C2×#117—31no—
153C2×#117—31no—
154C2×#117—31no—
155C22×C42—31no—
156C22×C42—31no—
157C22×C42—31no—
158C2×#99—32no—
159C2×#99—34no—
160C2×#99C2×D312yes—
161#383—62no—
162#382—62no—
163#381—34no—
164#380—34no—
165C4×C4⋊C4—62no—
166C4×C22⋊C4—32no—
167C4×C22⋊C4C2×D312yes—
168#376—62no—
169#342—32no—
170#342—34no—
171#342C2×D312yes—
172C24×C3C22×C411yescharacteristic
173C24×C3C2411yescharacteristic
174C23×D3C22×C411yescharacteristic
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4#4114yes—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C411yescharacteristic
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C2×#67—22no—
191C2×#67—22no—
192C2×#67—22no—
193C2×#67—22no—
194C2×#67—22no—
195C2×#67#4114yes—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—24no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3C22×C411yescharacteristic
208C22×Dic3C22×C411yescharacteristic
209C22×Dic3C22×C411yescharacteristic
210C22×Dic3C22×C411yescharacteristic
211C22×Dic3C22×C411yescharacteristic
212C22×Dic3C22×C411yescharacteristic
213C22×Dic3C22×C411yescharacteristic
214C22×Dic3C2×D412yes—
215C22×Dic3C2×D412yes—
216C22×Dic3C2×D412yes—
217C22×Dic3C2×D412yes—
218#237—216no—
219#237—216no—
220C2×C4×D3—22no—
221C2×C4×D3—22no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—22no—
233C3×C4⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—24no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—44no—
241#217—22no—
242#217—22no—
243#217—24no—
244#217—24no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—22no—
247C4⋊Dic3—22no—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—48no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—44no—
264C4×Dic3C22⋊C412yes—
265C4×Dic3C22⋊C412yes—
266C4×Dic3C22⋊C411yescharacteristic
267C4×Dic3C22⋊C411yescharacteristic
268C4×Dic3C22⋊C411yescharacteristic
269C4×Dic3C22⋊C411yescharacteristic
270C4×Dic3C22⋊C412yes—
271C4×Dic3C22⋊C412yes—
272C4×Dic3C22⋊C412yes—
273C4×Dic3C22⋊C412yes—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—64no—
277C2×#41—64no—
278C22×Q8—31no—
279C22×D4—31no—
280C22×D4—31no—
281C22×D4C22×D311yescharacteristic
282C23×C4—31no—
283C23×C4—31no—
284C23×C4—31no—
285C23×C4—31no—
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—31no—
289C23×C4C4×D311yescharacteristic
290C23×C4C22×D311yescharacteristic
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—62no—
298C4×D4—62no—
299C4×D4—62no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307#117—62no—
308#117—62no—
309#117—62no—
310#117—62no—
311#117—62no—
312#117—62no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—64no—
319C2×C4⋊C4C4×D311yescharacteristic
320C2×C22⋊C4—31no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—34no—
325C2×C22⋊C4C4×D311yescharacteristic
326C2×C22⋊C4C22×D314yes—
327C2×C22⋊C4C22×D311yescharacteristic
328C2×C42—22no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—32no—
334C2×C42—32no—
335C2×C42—32no—
336C2×C42—32no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42—61no—
343C2×C42—61no—
344C2×C42—61no—
345C2×C42—61no—
346C2×C42—61no—
347C2×C42—61no—
348C2×C42—61no—
349C2×C42—61no—
350C2×C42—62no—
351C2×C42C4×D311yescharacteristic
352#99—24no—
353#99—68no—
354#99—64no—
355#95—22no—
356#95—64no—
357#95—64no—
358#95—64no—
359#95—62no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3#11712yes—
367C23×C3C23×C411yescharacteristic
368C23×C3C22×D412yes—
369C23×C3C2×#4112yes—
370C23×C3C2×#4112yes—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3#11712yes—
376C3×D4—28no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—24no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—41no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—44no—
390C2×C3×C4—44no—
391C2×C3×C4—44no—
392C2×C3×C4C2×C22⋊C411yescharacteristic
393C2×C3×C4C2×C22⋊C412yes—
394C2×C3×C4C2×C22⋊C411yescharacteristic
395C2×C3×C4#11712yes—
396#67—28no—
397#67—28no—
398C2×Dic3—24no—
399C2×Dic3—24no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—24no—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—41no—
413C2×Dic3—42no—
414C2×Dic3—42no—
415C2×Dic3—42no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—44no—
419C2×Dic3—44no—
420C2×Dic3—44no—
421C2×Dic3C2×C22⋊C411yescharacteristic
422C2×Dic3C2×C22⋊C411yescharacteristic
423C2×Dic3C2×C22⋊C411yescharacteristic
424C2×Dic3C2×C22⋊C412yes—
425C2×Dic3C2×C22⋊C412yes—
426C2×Dic3C2×C22⋊C411yescharacteristic
427C2×Dic3#11712yes—
428C4×D3—24no—
429C4×D3—24no—
430C4×D3—42no—
431C4×D3—42no—
432C4×D3—42no—
433C4×D3—42no—
434Dic6—28no—
435C24—31no—
436C24C2×C4×D311yescharacteristic
437C24C23×D311yescharacteristic
438#41—616no—
439#41—616no—
440C2×Q8—62no—
441C2×Q8—62no—
442C2×Q8—62no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—34no—
446C2×D4—62no—
447C2×D4—62no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4#23514yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—32no—
462C22×C4—31no—
463C22×C4—32no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—62no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—62no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—64no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4C2×C4×D311yescharacteristic
490C22×C4C2×C4×D311yescharacteristic
491C22×C4C2×C4×D311yescharacteristic
492C22×C4C2×C4×D311yescharacteristic
493C22×C4C2×C4×D311yescharacteristic
494C22×C4D3×D412yes—
495C22×C4D3×D412yes—
496C4⋊C4—22no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—62no—
502C4⋊C4—128no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—62no—
508C22⋊C4—62no—
509C22⋊C4—62no—
510C22⋊C4—62no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C42—21no—
514C42—21no—
515C42—31no—
516C42—31no—
517C42—32no—
518C42—32no—
519C42—32no—
520C42—32no—
521C42—32no—
522C42—31no—
523C42—31no—
524C42—32no—
525C42—44no—
526C42—61no—
527C42—61no—
528C42—61no—
529C42—61no—
530C42—61no—
531C42—61no—
532C42—61no—
533C42—61no—
534C42—61no—
535C42—61no—
536C42—124no—
537C22×C3—22no—
538C22×C3—21no—
539C22×C3—22no—
540C22×C3—21no—
541C22×C3—22no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3C22×C22⋊C411yescharacteristic
549C22×C3C2×#11711yescharacteristic
550C22×C3C2×#11711yescharacteristic
551C22×C3#52514yes—
552C2×D3—22no—
553C2×D3—42no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—42no—
557C3×C4—21no—
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—84no—
563C3×C4#41012yes—
564Dic3—21no—
565Dic3—22no—
566Dic3—22no—
567Dic3—22no—
568Dic3—21no—
569Dic3—21no—
570Dic3—41no—
571Dic3—41no—
572Dic3—84no—
573Dic3#41012yes—
574C23—21no—
575C23—21no—
576C23—32no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—61no—
582C23—61no—
583C23—121no—
584C23—121no—
585C23#87212yes—
586C23C22×C4×D311yescharacteristic
587C23C2×#23512yes—
588C23C2×D3×D412yes—
589C23D3×#4112yes—
590Q8—68no—
591D4—28no—
592D4—68no—
593D4—68no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—31no—
604C2×C4—32no—
605C2×C4—32no—
606C2×C4—32no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—62no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—121no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—124no—
643C2×C4—124no—
644C2×C4—124no—
645C2×C4#87212yes—
646C2×C4D3×C22⋊C411yescharacteristic
647C2×C4D3×C22⋊C412yes—
648C2×C4D3×C22⋊C411yescharacteristic
649C2×C3—24no—
650C2×C3—21no—
651C2×C3—21no—
652C2×C3—42no—
653C2×C3#228011yescharacteristic
654C2×C3C2×#41011yescharacteristic
655C2×C3C2×#41011yescharacteristic
656D3—42no—
657C22—22no—
658C22—21no—
659C22—22no—
660C22—21no—
661C22—22no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—41no—
667C22—41no—
668C22—62no—
669C22—122no—
670C22—122no—
671C22—121no—
672C22—121no—
673C22C2×#87211yescharacteristic
674C22C2×D3×C22⋊C411yescharacteristic
675C22D3×#11711yescharacteristic
676C22#554714yes—
677C4—22no—
678C4—21no—
679C4—22no—
680C4—32no—
681C4—41no—
682C4—41no—
683C4—61no—
684C4—61no—
685C4—62no—
686C4—62no—
687C4—61no—
688C4—62no—
689C4—84no—
690C4—121no—
691C4—121no—
692C4—244no—
693C4#474012yes—
694C3#2072311yescharacteristic, complemented, Sylow
695C2—24no—
696C2—21no—
697C2—21no—
698C2—42no—
699C2—122no—
700C2#7356111yescharacteristic
701C2C2×#474011yescharacteristic
702C2D3×#41011yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×C4×#99C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#237C22
cosocleC24×C3C24
hypercenterC4×#99D3
hypocenterC3#20723
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #73437C2semidirect
3. C3×C4×#99C2semidirect
4. C4×#834C2semidirect
5. #70028C2semidirect
7. C4×#806C2semidirect
8. #68959C2semidirect
9. #68953C2semidirect
14. C2×C4×#67C4semidirect
16. C22×C4×Dic3C4semidirect
22. C2×#892C4semidirect
23. C2×#887C4semidirect
24. C2×#67⋊C4C4semidirect
26. C2×#877C4semidirect
28. C2×C42×D3C4semidirect
29. C2×C4×Dic6C4semidirect
52. C4×#99D3semidirect
694. C3#20723semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2372
#734372
C3×C4×#992
C4×#8342
#700282
#700272
C4×#8062
#689592
#689532
#207233