Finite Group Databasebeta

← #10804141

Subgroups — #10804141

Order 768.

1622 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5653C211yesmaximal, characteristic
2#82269C211yesmaximal, characteristic, complemented
3#74119C211yesmaximal, characteristic, complemented
4C3×#1899C211yesmaximal, characteristic, complemented
5#71705C211yesmaximal, characteristic, complemented
6#71595C211yesmaximal, characteristic, complemented
7#70036C212yesmaximal, complemented
8#70035C212yesmaximal, complemented
9#69535C211yesmaximal, characteristic, complemented
10#68970C212yesmaximal, complemented
11#68969C212yesmaximal, complemented
12#21749—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C23×D12C2211yescharacteristic
15C2×C3×#123C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic, complemented
17C2×#937C2211yescharacteristic
18C2×#929C2211yescharacteristic
19C2×#926C2211yescharacteristic, complemented
20#5653—24no—
21#5653—22no—
22#5653—22no—
23C2×#895C412yescomplemented
24C2×#894C412yescomplemented
25C2×#884C412yescomplemented
26C2×#882C412yescomplemented
27C2×#882C2211yescharacteristic
28C2×D3×C22⋊C4C2211yescharacteristic
29C2×D3×C22⋊C4C2212yescomplemented
30C2×C3×#99C2212yes—
31C3×#381C2211yescharacteristic, complemented
32C2×#834C2212yes—
33#5211C2211yescharacteristic, complemented
34#4988—22no—
35#4955C2212yes—
36#4955C2212yes—
37#4955C2212yescomplemented
38C2×#806C2212yes—
39C2×#806C2212yes—
40#4939—22no—
41#4665—24no—
42C3×#342C2212yescomplemented
43#4527C2212yescomplemented
44#4527C2212yescomplemented
45#4526C2212yescomplemented
46C2×#534—31no—
47#2309—31no—
48#2308—31no—
49#1899—31no—
50#1899—31no—
51#1899—31no—
52#1899D311yescharacteristic, complemented
53#1898—32no—
54#1898—32no—
55#1819—32no—
56#1781—32no—
57C24×D3C2×C411yescharacteristic, complemented
58C24×D3C2311yescharacteristic
59C24×D3C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67C2×C412yes—
63C22×#67C2×C411yescharacteristic, complemented
64C22×#67C2311yescharacteristic
65C2×D3×D4—22no—
66C2×D3×D4—22no—
67C2×D3×D4—42no—
68C2×D3×D4—42no—
69C22×D12—42no—
70C22×D12—42no—
71C22×D12C2×C412yes—
72C22×D12C2×C412yes—
73C22×D12C2×C412yes—
74C22×D12C2×C411yescharacteristic, complemented
75C22×D12D414yes—
76C22×D12D412yes—
77C22×D12D412yes—
78C22×C4×D3—22no—
79C22×C4×D3C2×C411yescharacteristic, complemented
80C22×C4×D3C2×C412yescomplemented
81C3×#123—22no—
82C3×#123—21no—
83C3×#123—21no—
84C3×#117—42no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2×C412yescomplemented
87C2×C3×C22⋊C4C2×C412yescomplemented
88C2×C3×C22⋊C4C2312yes—
89C2×C3×C22⋊C4C2312yes—
90C2×C3×C22⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×#217C2312yes—
93C2×#217C2312yes—
94C2×#217C2311yescharacteristic
95#937—22no—
96#937—22no—
97#929—22no—
98#929—21no—
99#929—21no—
100C2×(C2×D3)⋊C4—22no—
101C2×(C2×D3)⋊C4—24no—
102C2×(C2×D3)⋊C4—24no—
103C2×(C2×D3)⋊C4—24no—
104C2×(C2×D3)⋊C4C2×C412yes—
105C2×(C2×D3)⋊C4C2×C412yes—
106C2×(C2×D3)⋊C4C2×C412yescomplemented
107C2×(C2×D3)⋊C4C2×C412yescomplemented
108#926—42no—
109C2×C4⋊Dic3—21no—
110C2×Dic3⋊C4C2×C412yescomplemented
111C2×C4×Dic3—21no—
112#895—24no—
113#895—24no—
114#894—24no—
115#894—24no—
116#884—24no—
117#884—24no—
118#882—22no—
119#882—22no—
120#882—24no—
121#882—24no—
122D3×C22⋊C4—22no—
123D3×C22⋊C4—22no—
124D3×C22⋊C4—42no—
125D3×C22⋊C4—42no—
126C3×#99—42no—
127C3×#95—22no—
128#834—42no—
129C3⋊#95—22no—
130#806—42no—
131#806—42no—
132C23×D4—31no—
133C23×D4—31no—
134#534—64no—
135#534—62no—
136#534—62no—
137C2×#123—31no—
138C2×#123—32no—
139C2×#123C2×D311yescharacteristic
140C2×#121—32no—
141C2×#121—32no—
142C2×#120—31no—
143C2×#120—32no—
144C2×#120—31no—
145C2×#117—31no—
146C2×#117C2×D311yescharacteristic, complemented
147C22×C22⋊C4—31no—
148C22×C22⋊C4—32no—
149C2×#99—32no—
150C2×#99—32no—
151C2×#99—32no—
152C2×#99C2×D312yes—
153#390—62no—
154#390—62no—
155#386—64no—
156#381—31no—
157#381C2×D311yescharacteristic, complemented
158#379—32no—
159#379—32no—
160#379—32no—
161#342—32no—
162#342—32no—
163#342—32no—
164#342C2×D312yescomplemented
165C24×C3C22×C411yescharacteristic
166C24×C3C2411yescharacteristic
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—22no—
171C23×D3—22no—
172C23×D3—42no—
173C23×D3—82no—
174C23×D3—82no—
175C23×D3—82no—
176C23×D3C22×C411yescharacteristic
177C23×D3C22×C412yes—
178C23×D3C22×C411yescharacteristic
179C23×D3C22×C411yescharacteristic
180C23×D3C2×D412yes—
181C23×D3C2×D412yes—
182C23×D3C2×D412yes—
183C23×D3C2×D412yes—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C22×C412yes—
194C22×C3×C4C22×C412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C2×#67—21no—
199C2×#67—21no—
200C2×#67—22no—
201C2×#67—22no—
202C2×#67—22no—
203C2×#67—24no—
204C2×#67—42no—
205C2×#67—82no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3C22×C411yescharacteristic
209C22×Dic3C22×C412yes—
210D3×D4—44no—
211D3×D4—44no—
212D3×D4—48no—
213C2×D12—22no—
214C2×D12—24no—
215C2×D12—44no—
216C2×D12—44no—
217C2×D12—42no—
218C2×D12—42no—
219C2×D12—42no—
220C2×D12—42no—
221C2×D12—44no—
222C2×D12—44no—
223C2×D12—42no—
224C2×D12—82no—
225C2×D12C22⋊C414yes—
226C2×D12C22⋊C414yes—
227C2×D12C22⋊C414yes—
228C2×D12C22⋊C414yes—
229C2×C4×D3—22no—
230C2×C4×D3—24no—
231C2×C4×D3—42no—
232C2×C4×D3—44no—
233C2×C4×D3—84no—
234C2×C4×D3—82no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—42no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C42—42no—
247#217—21no—
248#217—21no—
249#217—22no—
250#217—22no—
251#217—22no—
252#217—22no—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—24no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—22no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—44no—
259(C2×D3)⋊C4—84no—
260(C2×D3)⋊C4—84no—
261(C2×D3)⋊C4—84no—
262C4⋊Dic3—42no—
263Dic3⋊C4—22no—
264Dic3⋊C4—22no—
265C4×Dic3—42no—
266C25—31no—
267C25—31no—
268C25—31no—
269C22×D4—31no—
270C22×D4—31no—
271C22×D4—32no—
272C22×D4—32no—
273C22×D4—32no—
274C22×D4—31no—
275C22×D4—32no—
276C22×D4—34no—
277C22×D4—32no—
278C22×D4—32no—
279C22×D4—62no—
280C22×D4—62no—
281C22×D4—122no—
282C22×D4—122no—
283C22×D4—122no—
284C22×D4—122no—
285C22×D4C22×D311yescharacteristic
286C23×C4—31no—
287C23×C4—32no—
288C23×C4—62no—
289C23×C4C22×D311yescharacteristic
290#123—22no—
291#123—21no—
292#123—21no—
293#123—64no—
294#123—64no—
295#123—62no—
296#123—61no—
297#123—61no—
298#121—64no—
299#121—64no—
300#121—64no—
301#121—64no—
302#120—64no—
303#120—64no—
304#120—62no—
305#120—62no—
306#120—62no—
307#120—62no—
308#117—42no—
309#117—122no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C4×D312yescomplemented
314C2×C22⋊C4—31no—
315C2×C22⋊C4—32no—
316C2×C22⋊C4—32no—
317C2×C22⋊C4—32no—
318C2×C22⋊C4—32no—
319C2×C22⋊C4—32no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—62no—
322C2×C22⋊C4—62no—
323C2×C22⋊C4—62no—
324C2×C22⋊C4—64no—
325C2×C22⋊C4—64no—
326C2×C22⋊C4—64no—
327C2×C22⋊C4—122no—
328C2×C22⋊C4—122no—
329C2×C22⋊C4C4×D312yescomplemented
330C2×C22⋊C4C22×D312yes—
331C2×C22⋊C4C22×D312yes—
332C2×C22⋊C4C22×D311yescharacteristic
333C2×C42—21no—
334C2×C42—61no—
335#99—42no—
336#99—122no—
337#99—122no—
338#99—122no—
339#95—22no—
340#95—62no—
341C23×C3—21no—
342C23×C3—22no—
343C23×C3—42no—
344C23×C3—42no—
345C23×C3C23×C411yescharacteristic
346C23×C3C22×D411yescharacteristic
347C23×C3C22×D411yescharacteristic
348C23×C3H(5,𝔽2)12yes—
349C23×C3H(5,𝔽2)12yes—
350C22×D3—22no—
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—24no—
355C22×D3—22no—
356C22×D3—22no—
357C22×D3—24no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—21no—
361C22×D3—22no—
362C22×D3—42no—
363C22×D3—44no—
364C22×D3—42no—
365C22×D3—44no—
366C22×D3—44no—
367C22×D3—44no—
368C22×D3—42no—
369C22×D3—42no—
370C22×D3—44no—
371C22×D3—42no—
372C22×D3—84no—
373C22×D3—84no—
374C22×D3—84no—
375C22×D3—84no—
376C22×D3—84no—
377C22×D3—88no—
378C22×D3C2×C22⋊C412yes—
379C22×D3C2×C22⋊C414yes—
380C22×D3C2×C22⋊C412yes—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—42no—
384C2×C3×C4—24no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—24no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—42no—
394C2×C3×C4—44no—
395C2×C3×C4—44no—
396C2×C3×C4—42no—
397C2×C3×C4—42no—
398C2×C3×C4C2×C22⋊C412yes—
399C2×C3×C4C2×C22⋊C412yes—
400#67—44no—
401#67—41no—
402#67—41no—
403#67—42no—
404#67—84no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—42no—
409C2×Dic3—44no—
410C2×Dic3—44no—
411C2×Dic3—42no—
412C2×Dic3—42no—
413D12—44no—
414D12—44no—
415D12—44no—
416D12—44no—
417D12—44no—
418D12—84no—
419C4×D3—44no—
420C4×D3—84no—
421C4×D3—88no—
422C24—31no—
423C24—32no—
424C24—31no—
425C24—32no—
426C24—32no—
427C24—32no—
428C24—32no—
429C24—31no—
430C24—62no—
431C24—62no—
432C24—62no—
433C24—62no—
434C24—62no—
435C24—122no—
436C24—242no—
437C24—242no—
438C24—242no—
439C24C2×C4×D311yescharacteristic
440C24C23×D311yescharacteristic
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—34no—
446C2×D4—34no—
447C2×D4—34no—
448C2×D4—34no—
449C2×D4—62no—
450C2×D4—64no—
451C2×D4—62no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—62no—
455C2×D4—62no—
456C2×D4—64no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—124no—
460C2×D4—124no—
461C2×D4—122no—
462C2×D4—122no—
463C2×D4—124no—
464C2×D4—124no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—124no—
468C2×D4—124no—
469C2×D4—128no—
470C2×D4—242no—
471C2×D4—242no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—31no—
477C22×C4—32no—
478C22×C4—62no—
479C22×C4—64no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—122no—
483C22×C4—124no—
484C22×C4—244no—
485C22×C4—242no—
486C22×C4C2×C4×D311yescharacteristic
487C22×C4C2×C4×D312yes—
488C22×C4C2×C4×D312yes—
489C22×C4D3×D412yes—
490C22×C4D3×D411yescharacteristic
491C22×C4D3×D411yescharacteristic
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—122no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—64no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—61no—
510C22⋊C4—61no—
511C22⋊C4—62no—
512C22⋊C4—62no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—124no—
518C22⋊C4—244no—
519C22⋊C4—244no—
520C22⋊C4—244no—
521C42—42no—
522C42—122no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—41no—
529C22×C3—42no—
530C22×C3—82no—
531C22×C3C22×C22⋊C411yescharacteristic
532C22×C3#51812yes—
533C22×C3#53411yescharacteristic
534C22×C3#53411yescharacteristic
535C22×C3#53412yes—
536C2×D3—22no—
537C2×D3—22no—
538C2×D3—24no—
539C2×D3—24no—
540C2×D3—24no—
541C2×D3—24no—
542C2×D3—24no—
543C2×D3—44no—
544C2×D3—44no—
545C2×D3—44no—
546C2×D3—44no—
547C2×D3—41no—
548C2×D3—42no—
549C2×D3—82no—
550C2×D3—82no—
551C2×D3—84no—
552C2×D3—82no—
553C2×D3—84no—
554C2×D3—82no—
555C2×D3—88no—
556C2×D3—82no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—41no—
560C3×C4—42no—
561C3×C4—82no—
562C3×C4—82no—
563Dic3—41no—
564Dic3—42no—
565Dic3—82no—
566Dic3—82no—
567C23—21no—
568C23—22no—
569C23—32no—
570C23—34no—
571C23—32no—
572C23—42no—
573C23—42no—
574C23—62no—
575C23—62no—
576C23—62no—
577C23—62no—
578C23—62no—
579C23—62no—
580C23—64no—
581C23—62no—
582C23—62no—
583C23—64no—
584C23—61no—
585C23—62no—
586C23—124no—
587C23—122no—
588C23—122no—
589C23—124no—
590C23—124no—
591C23—124no—
592C23—122no—
593C23—122no—
594C23—122no—
595C23—124no—
596C23—244no—
597C23—244no—
598C23—244no—
599C23—244no—
600C23—244no—
601C23—248no—
602C23C22×C4×D311yescharacteristic
603C23C2×D3×D411yescharacteristic
604C23C2×D3×D411yescharacteristic
605C23#100412yes—
606C23#100912yes—
607D4—41no—
608D4—41no—
609D4—42no—
610D4—124no—
611D4—124no—
612D4—124no—
613D4—124no—
614D4—124no—
615D4—124no—
616D4—121no—
617D4—121no—
618D4—122no—
619D4—244no—
620D4—244no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—24no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—62no—
636C2×C4—61no—
637C2×C4—62no—
638C2×C4—124no—
639C2×C4—122no—
640C2×C4—124no—
641C2×C4—124no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—244no—
645C2×C4—248no—
646C2×C4D3×C22⋊C412yes—
647C2×C4D3×C22⋊C412yes—
648C2×C3—21no—
649C2×C3—22no—
650C2×C3—21no—
651C2×C3—22no—
652C2×C3—81no—
653C2×C3#230811yescharacteristic
654C2×C3#288312yes—
655D3—42no—
656D3—44no—
657D3—42no—
658D3—81no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—41no—
666C22—64no—
667C22—64no—
668C22—64no—
669C22—64no—
670C22—62no—
671C22—62no—
672C22—64no—
673C22—82no—
674C22—124no—
675C22—124no—
676C22—124no—
677C22—124no—
678C22—122no—
679C22—121no—
680C22—242no—
681C22—242no—
682C22—244no—
683C22—242no—
684C22—242no—
685C22—244no—
686C22—242no—
687C22—248no—
688C22C2×D3×C22⋊C411yescharacteristic
689C22#554512yes—
690C22#558911yescharacteristic
691C22#560512yes—
692C22#565311yescharacteristic
693C4—22no—
694C4—22no—
695C4—42no—
696C4—41no—
697C4—82no—
698C4—82no—
699C4—122no—
700C4—121no—
701C4—242no—
702C4—242no—
703C3#2174911yescharacteristic, complemented, Sylow
704C2—21no—
705C2—22no—
706C2—21no—
707C2—22no—
708C2—81no—
709C2—122no—
710C2—122no—
711C2—124no—
712C2—241no—
713C2#7487411yescharacteristic
714C2#7869412yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1899C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5653C2
cosocleC24×C3C24
hypercenter#1899D3
hypocenterC3#21749
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1899

Complements

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

NH ≅ G/Ntype
2. #82269C2semidirect
3. #74119C2semidirect
4. C3×#1899C2semidirect
5. #71705C2semidirect
6. #71595C2semidirect
7. #70036C2semidirect
8. #70035C2semidirect
9. #69535C2semidirect
10. #68970C2semidirect
11. #68969C2semidirect
16. C2×C3×#117C22semidirect
19. C2×#926C22semidirect
23. C2×#895C4semidirect
24. C2×#894C4semidirect
25. C2×#884C4semidirect
26. C2×#882C4semidirect
29. C2×D3×C22⋊C4C22semidirect
31. C3×#381C22semidirect
33. #5211C22semidirect
37. #4955C22semidirect
42. C3×#342C22semidirect
43. #4527C22semidirect
44. #4527C22semidirect
45. #4526C22semidirect
52. #1899D3semidirect
57. C24×D3C2×C4semidirect
63. C22×#67C2×C4semidirect
74. C22×D12C2×C4semidirect
79. C22×C4×D3C2×C4semidirect
80. C22×C4×D3C2×C4semidirect
86. C2×C3×C4⋊C4C2×C4semidirect
87. C2×C3×C22⋊C4C2×C4semidirect
106. C2×(C2×D3)⋊C4C2×C4semidirect
107. C2×(C2×D3)⋊C4C2×C4semidirect
110. C2×Dic3⋊C4C2×C4semidirect
146. C2×#117C2×D3semidirect
157. #381C2×D3semidirect
164. #342C2×D3semidirect
313. C2×C4⋊C4C4×D3semidirect
329. C2×C22⋊C4C4×D3semidirect
703. C3#21749semidirect

Maximal subgroups

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

subgroupindex
C2×#56532
#822692
#741192
C3×#18992
#717052
#715952
#700362
#700352
#695352
#689702
#689692
#217493