Finite Group Databasebeta

← #10694027

Subgroups — #10694027

Order 768.

920 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4957C211yesmaximal, characteristic
2C2×C3×#323C211yesmaximal, characteristic, complemented
3C2×#4525C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic, complemented
6#69908C211yesmaximal, characteristic, complemented
7#69906C211yesmaximal, characteristic, complemented
8#9653—31nomaximal, Sylow
9C23×#67C411yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217C411yescharacteristic
12C22×(C2×D3)⋊C4C411yescharacteristic
13C22×(C2×D3)⋊C4C2211yescharacteristic
14C22×Dic3⋊C4C411yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3×C22⋊C8C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#815—21no—
21C2×Dic3⋊C8—21no—
22#4957—21no—
23#4957—21no—
24#4957—21no—
25#4957—21no—
26C3×#323—22no—
27#4550—21no—
28#4550—21no—
29#4525—22no—
30#4468—22no—
31#4468—22no—
32C2×#390—31no—
33#1797—31no—
34#1749—31no—
35C2×#323—31no—
36C2×#323—31no—
37C2×#323—31no—
38C2×#323D311yescharacteristic, complemented
39C25×C3C2×C411yescharacteristic
40C24×D3C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—22no—
44C22×#67—22no—
45C22×#67—21no—
46C22×#67—21no—
47C22×#67—42no—
48C22×#67—42no—
49C23×Dic3C2×C411yescharacteristic
50C23×Dic3C2×C411yescharacteristic
51C23×Dic3C2×C411yescharacteristic
52C22×C3×C8—21no—
53C22×C3×C8—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—42no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—42no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×Dic3⋊C4—42no—
75C2×C4×Dic3—42no—
76C2×C4×Dic3D411yescharacteristic
77C2×C4×Dic3D411yescharacteristic
78C2×C4×Dic3D411yescharacteristic
79C2×C4×Dic3D411yescharacteristic
80C22×C3⋊C8—21no—
81C22×C3⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8D412yescomplemented
86#830—21no—
87#830—21no—
88#830—21no—
89#830D412yescomplemented
90#815—42no—
91Dic3⋊C8—42no—
92C23×D4—31no—
93C22×C4⋊C4—31no—
94C22×C22⋊C4—31no—
95C22×C22⋊C4—31no—
96C22×C22⋊C4—31no—
97C22×C22⋊C4C2×D311yescharacteristic
98C22×C42—31no—
99C2×C4⋊C8—61no—
100C2×C22⋊C8—31no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8C2×D311yescharacteristic
103C2×C22⋊C8C2×D311yescharacteristic
104C2×#97—61no—
105#390—61no—
106#390—61no—
107#390—61no—
108#390—61no—
109#336—61no—
110#336—61no—
111#323—22no—
112#323—62no—
113#323—62no—
114#323—62no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—41no—
122C24×C3—41no—
123C24×C3—41no—
124C24×C3—41no—
125C24×C3M1612yes—
126C24×C3C22×C411yescharacteristic
127C23×D3—21no—
128C23×D3—21no—
129C23×D3—21no—
130C23×D3—21no—
131C23×D3—21no—
132C23×D3—21no—
133C23×D3—41no—
134C23×D3—41no—
135C23×D3—41no—
136C23×D3—41no—
137C23×D3M1612yes—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4—41no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C2×#67—24no—
152C2×#67—24no—
153C2×#67—48no—
154C2×#67—48no—
155C2×#67—42no—
156C2×#67—42no—
157C2×#67—42no—
158C2×#67—42no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—41no—
170C22×Dic3—41no—
171C22×Dic3—41no—
172C22×Dic3C22⋊C411yescharacteristic
173C22×Dic3C22⋊C411yescharacteristic
174C22×Dic3C22⋊C411yescharacteristic
175C22×Dic3C22⋊C411yescharacteristic
176C2×C3×C8—22no—
177C2×C3×C8—41no—
178C2×C3×C8—41no—
179C2×C3×C8—41no—
180C2×C3×C8—41no—
181C2×C3×C8—41no—
182C3×C22⋊C4—41no—
183C3×C22⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186#217—42no—
187#217—42no—
188(C2×D3)⋊C4—42no—
189(C2×D3)⋊C4—41no—
190(C2×D3)⋊C4—41no—
191(C2×D3)⋊C4—41no—
192(C2×D3)⋊C4—41no—
193(C2×D3)⋊C4—42no—
194Dic3⋊C4—42no—
195Dic3⋊C4—42no—
196C4×Dic3—22no—
197C4×Dic3—22no—
198C4×Dic3—41no—
199C4×Dic3—41no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—41no—
202C2×C3⋊C8—41no—
203C2×C3⋊C8—41no—
204C2×C3⋊C8—41no—
205C2×C3⋊C8—41no—
206C25—31no—
207C25C4×D311yescharacteristic
208C22×D4—61no—
209C22×D4—61no—
210C22×D4—62no—
211C22×D4—62no—
212C22×D4—122no—
213C22×D4—122no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4C4×D311yescharacteristic
218C23×C4C22×D311yescharacteristic
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—61no—
222C22×C8—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—122no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—61no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—61no—
242C2×C22⋊C4—122no—
243C2×C22⋊C4—122no—
244C2×C42—31no—
245C2×C42—31no—
246C2×C42—31no—
247C2×C42—31no—
248C2×C42—122no—
249C4⋊C8—122no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—32no—
254C22⋊C8—61no—
255C22⋊C8—61no—
256C22⋊C8—61no—
257C22⋊C8#6712yescomplemented
258#97—122no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—81no—
279C23×C3—81no—
280C23×C3—81no—
281C23×C3—81no—
282C23×C3—81no—
283C23×C3—81no—
284C23×C3—81no—
285C23×C3—81no—
286C23×C3C2×C22⋊C411yescharacteristic
287C23×C3C4×D411yescharacteristic
288C23×C3C4×D411yescharacteristic
289C23×C3C2×M1611yescharacteristic
290C23×C3C2×M1611yescharacteristic
291C23×C3#13111yescharacteristic
292C23×C3#13111yescharacteristic
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—81no—
309C22×D3—81no—
310C22×D3—81no—
311C22×D3—81no—
312C22×D3—81no—
313C22×D3—81no—
314C22×D3—81no—
315C22×D3—81no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—81no—
328C2×C3×C4—81no—
329C2×C3×C4#12111yescharacteristic
330C2×C3×C4#12111yescharacteristic
331C2×C3×C4#12411yescharacteristic
332C2×C3×C4#12711yescharacteristic
333#67—48no—
334#67—48no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—81no—
352C2×Dic3—81no—
353C2×Dic3#9912yes—
354C2×Dic3#10012yes—
355C2×Dic3#10012yes—
356C3×C8—42no—
357C3×C8—81no—
358C3⋊C8—42no—
359C3⋊C8—81no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—32no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24—41no—
371C24—61no—
372C24—61no—
373C24—61no—
374C24—61no—
375C24—61no—
376C24—61no—
377C24—121no—
378C24—121no—
379C24—121no—
380C24—121no—
381C24ΓL(1,𝔽25)12yes—
382C24C2×C4×D311yescharacteristic
383C2×D4—64no—
384C2×D4—64no—
385C2×D4—122no—
386C2×D4—122no—
387C2×D4—128no—
388C2×D4—128no—
389C2×D4—122no—
390C2×D4—122no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4—121no—
415C22×C4#23511yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C2×C8—22no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—62no—
428C2×C8—121no—
429C2×C8—121no—
430C2×C8—121no—
431C2×C8—121no—
432C2×C8—121no—
433C4⋊C4—122no—
434C4⋊C4—122no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—122no—
440C22⋊C4—122no—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—121no—
444C22⋊C4—121no—
445C22⋊C4—121no—
446C22⋊C4—121no—
447C42—62no—
448C42—62no—
449C42—121no—
450C42—121no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3#39011yescharacteristic
475C22×C3#40711yescharacteristic
476C22×C3#40811yescharacteristic
477C22×C3C2×#9911yescharacteristic
478C22×C3#41011yescharacteristic
479C22×C3C2×#10011yescharacteristic
480C22×C3C2×#10011yescharacteristic
481C22×C3#43511yescharacteristic
482C22×C3#43511yescharacteristic
483C22×C3#43511yescharacteristic
484C22×C3#43511yescharacteristic
485C2×D3—42no—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—82no—
489C2×D3—82no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3—82no—
495C2×D3—82no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—81no—
501Dic3—22no—
502Dic3—22no—
503Dic3—22no—
504Dic3—41no—
505Dic3—81no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—81no—
529C23—81no—
530C23—81no—
531C23—81no—
532C23—81no—
533C23—81no—
534C23—81no—
535C23—81no—
536C23—122no—
537C23—122no—
538C23—122no—
539C23—122no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23—241no—
549C23—241no—
550C23—241no—
551C23—241no—
552C23—241no—
553C23—241no—
554C23—241no—
555C23—241no—
556C23D3×C22⋊C411yescharacteristic
557C23#67⋊C411yescharacteristic
558C23C2×ΓL(1,𝔽25)11yescharacteristic
559C23#90111yescharacteristic
560C23D3×M1611yescharacteristic
561C23#90711yescharacteristic
562C23C4×#6711yescharacteristic
563D4—128no—
564D4—128no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—241no—
598C2×C4—241no—
599C2×C4#88411yescharacteristic
600C2×C4#88511yescharacteristic
601C2×C4#93911yescharacteristic
602C2×C4#94011yescharacteristic
603C8—42no—
604C8—81no—
605C8—122no—
606C8—241no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—82no—
612C2×C3#146211yescharacteristic
613C2×C3#146611yescharacteristic
614C2×C3#146911yescharacteristic
615C2×C3#147111yescharacteristic
616C2×C3#195511yescharacteristic
617C2×C3#196411yescharacteristic
618C2×C3#196511yescharacteristic
619D3—82no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—122no—
644C22—122no—
645C22—122no—
646C22—242no—
647C22—242no—
648C22—242no—
649C22—242no—
650C22—241no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22#472611yescharacteristic
655C22#472711yescharacteristic
656C22#472811yescharacteristic
657C22#473011yescharacteristic
658C22#474011yescharacteristic
659C22D3×#9911yescharacteristic
660C22D3×#10011yescharacteristic
661C22D3×#10011yescharacteristic
662C22#495711yescharacteristic
663C22#511111yescharacteristic
664C22#512711yescharacteristic
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—62no—
670C4—62no—
671C4—62no—
672C4—81no—
673C4—121no—
674C4—241no—
675C3#965311yescharacteristic, complemented, Sylow
676C2—21no—
677C2—21no—
678C2—21no—
679C2—21no—
680C2—82no—
681C2—242no—
682C2#6810311yescharacteristic
683C2#6810411yescharacteristic
684C2#6812011yescharacteristic
685C2#6812311yescharacteristic
686C2#6991711yescharacteristic
687C2#7003011yescharacteristic
688C2#7006411yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#323C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#67C4
cosocleC23×C3×C4C23
hypercenterC2×#323D3
hypocenterC3#9653
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×C3×#323C2semidirect
3. C2×#4525C2semidirect
4. C2×#4468C2semidirect
5. C2×#4468C2semidirect
6. #69908C2semidirect
7. #69906C2semidirect
38. C2×#323D3semidirect
85. C3×C22⋊C8D4semidirect
89. #830D4semidirect
257. C22⋊C8#67semidirect
675. C3#9653semidirect

Maximal subgroups

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

subgroupindex
C2×#49572
C2×C3×#3232
C2×#45252
C2×#44682
#699082
#699062
#96533