Finite Group Databasebeta

← #10690587

Subgroups — #10690587

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4726C211yesmaximal, characteristic
2C2×#4668C211yesmaximal, characteristic
3#69955C211yesmaximal, characteristic
4C2×#4469C211yesmaximal, characteristic, complemented
5C2×#4469C211yesmaximal, characteristic, complemented
6C3×#1280C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9085—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C22×(C2×D3)⋊C4C411yescharacteristic
11C22×(C2×D3)⋊C4C411yescharacteristic
12C22×C4⋊Dic3C411yescharacteristic
13C22×Dic3⋊C4C2211yescharacteristic
14C22×ΓL(1,𝔽25)—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3×#95C2211yescharacteristic
18C2×C3⋊#95C411yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—21no—
22C2×(C2×D3)⋊C8—21no—
23C2×(C2×D3)⋊C8—21no—
24#4726—22no—
25#4726—22no—
26#4668—21no—
27#4668—21no—
28#4668—21no—
29#4668—21no—
30#4469—22no—
31#4469—22no—
32C2×#407—31no—
33C2×#385—31no—
34#1779—31no—
35C2×#324—31no—
36C2×#324—31no—
37#1280—31no—
38#1280D311yescharacteristic, complemented
39C24×D3C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2×C411yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C22×C4×D3—21no—
47C22×C4×D3—21no—
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×C4×D3—42no—
51C22×C4×D3—42no—
52C22×C4×D3D411yescharacteristic
53C22×C4×D3D411yescharacteristic
54C22×C4×D3D411yescharacteristic
55C22×C4×D3D411yescharacteristic
56C22×C3×C8—21no—
57C22×C3×C8—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—42no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—42no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C22×C3⋊C8—21no—
75C22×C3⋊C8—21no—
76C2×ΓL(1,𝔽25)—24no—
77C2×ΓL(1,𝔽25)—42no—
78C2×ΓL(1,𝔽25)—42no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8D412yes—
83C3×#95—21no—
84C3×#95—21no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830D412yes—
91(C2×D3)⋊C8—24no—
92(C2×D3)⋊C8—42no—
93(C2×D3)⋊C8—42no—
94C24×C4—31no—
95C22×M16—61no—
96C22×C4⋊C4—31no—
97C22×C4⋊C4—31no—
98C22×C22⋊C4—31no—
99C22×C22⋊C4—31no—
100#407—62no—
101#407—62no—
102C2×C22⋊C8—31no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8—61no—
105C2×C22⋊C8—61no—
106C2×C22⋊C8—61no—
107C2×C22⋊C8C2×D311yescharacteristic
108C2×C22⋊C8C2×D311yescharacteristic
109#385—61no—
110#385—61no—
111#385—61no—
112#385—61no—
113C2×#95—31no—
114C2×#95C2×D311yescharacteristic
115#324—62no—
116#324—62no—
117C24×C3C22×C411yescharacteristic
118C23×D3—21no—
119C23×D3—21no—
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—21no—
123C23×D3—21no—
124C23×D3—21no—
125C23×D3—21no—
126C23×D3—41no—
127C23×D3—41no—
128C23×D3C22⋊C411yescharacteristic
129C23×D3C22⋊C411yescharacteristic
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4M1612yescomplemented
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3C22⋊C411yescharacteristic
164C22×Dic3C22⋊C411yescharacteristic
165C22×Dic3M1612yescomplemented
166C2×C4×D3—22no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—21no—
172C2×C4×D3—21no—
173C2×C4×D3—21no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×C4×D3—22no—
178C2×C4×D3—44no—
179C2×C4×D3—44no—
180C2×C4×D3—44no—
181C2×C4×D3—44no—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—41no—
185C2×C4×D3—41no—
186C2×C3×C8—22no—
187C2×C3×C8—41no—
188C2×C3×C8—41no—
189C2×C3×C8—41no—
190C2×C3×C8—41no—
191C2×C3×C8—41no—
192(C2×D3)⋊C4—24no—
193(C2×D3)⋊C4—24no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—42no—
196C4⋊Dic3—42no—
197C4⋊Dic3—42no—
198Dic3⋊C4—41no—
199Dic3⋊C4—41no—
200Dic3⋊C4—41no—
201Dic3⋊C4—41no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—41no—
204C2×C3⋊C8—41no—
205C2×C3⋊C8—41no—
206C2×C3⋊C8—41no—
207C2×C3⋊C8—41no—
208ΓL(1,𝔽25)—48no—
209C25—31no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—31no—
217C23×C4—61no—
218C23×C4—61no—
219C23×C4—61no—
220C23×C4—61no—
221C23×C4—122no—
222C23×C4—122no—
223C23×C4C4×D311yescharacteristic
224C23×C4C4×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C2×M16—64no—
227C2×M16—122no—
228C2×M16—122no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—61no—
232C22×C8—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—122no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—122no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—32no—
253C22⋊C8—64no—
254C22⋊C8—61no—
255C22⋊C8—61no—
256C22⋊C8—61no—
257C22⋊C8—122no—
258C22⋊C8—122no—
259C22⋊C8D1212yes—
260#95—21no—
261#95—21no—
262#95—61no—
263#95—61no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3C2×C22⋊C411yescharacteristic
269C23×C3C4×D411yescharacteristic
270C23×C3C4×D411yescharacteristic
271C23×C3C2×M1611yescharacteristic
272C23×C3C2×M1611yescharacteristic
273C23×C3#13111yescharacteristic
274C23×C3#13111yescharacteristic
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—21no—
282C22×D3—21no—
283C22×D3—21no—
284C22×D3—21no—
285C22×D3—21no—
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3#9912yes—
304C22×D3#10112yes—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—22no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—81no—
322C2×C3×C4—81no—
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12311yescharacteristic
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—22no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—81no—
345C2×Dic3—81no—
346C2×Dic3#9912yes—
347C4×D3—24no—
348C4×D3—24no—
349C4×D3—42no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—42no—
355C3×C8—42no—
356C3×C8—81no—
357C3⋊C8—42no—
358C3⋊C8—81no—
359C24—31no—
360C24—31no—
361C24—62no—
362C24—62no—
363C24—61no—
364C24—61no—
365C24—61no—
366C24—61no—
367C24—61no—
368C24—61no—
369C24—121no—
370C24—121no—
371C24C2×C4×D311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—32no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4—121no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4—124no—
417C22×C4—124no—
418C22×C4—121no—
419C22×C4—121no—
420C22×C4—121no—
421C22×C4ΓL(1,𝔽25)12yescomplemented
422C22×C4C2×D1211yescharacteristic
423C22×C4#23511yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4D3×D411yescharacteristic
426C22×C4D3×D411yescharacteristic
427C22×C4#23911yescharacteristic
428M16—128no—
429C2×C8—22no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—62no—
436C2×C8—121no—
437C2×C8—121no—
438C2×C8—121no—
439C2×C8—121no—
440C2×C8—121no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—121no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C4⋊C4—121no—
447C22⋊C4—64no—
448C22⋊C4—64no—
449C22⋊C4—122no—
450C22⋊C4—122no—
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#38611yescharacteristic
464C22×C3#40711yescharacteristic
465C22×C3#40811yescharacteristic
466C22×C3C2×#9911yescharacteristic
467C22×C3C2×#9911yescharacteristic
468C22×C3C2×#10111yescharacteristic
469C22×C3#41311yescharacteristic
470C22×C3#43511yescharacteristic
471C22×C3#43511yescharacteristic
472C22×C3#43611yescharacteristic
473C22×C3#43611yescharacteristic
474C2×D3—22no—
475C2×D3—22no—
476C2×D3—22no—
477C2×D3—22no—
478C2×D3—22no—
479C2×D3—22no—
480C2×D3—44no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—41no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—42no—
497C3×C4—81no—
498Dic3—22no—
499Dic3—42no—
500Dic3—41no—
501Dic3—81no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—32no—
507C23—32no—
508C23—64no—
509C23—64no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23—122no—
523C23—122no—
524C23—122no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23—122no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C4×D1211yescharacteristic
537C23D3×C22⋊C411yescharacteristic
538C23#89311yescharacteristic
539C23C2×ΓL(1,𝔽25)11yescharacteristic
540C23#90111yescharacteristic
541C23D3×M1611yescharacteristic
542C23#90711yescharacteristic
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—32no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—241no—
588C2×C4—241no—
589C2×C4#88211yescharacteristic
590C2×C4#88411yescharacteristic
591C2×C4#89411yescharacteristic
592C2×C4#89511yescharacteristic
593C8—42no—
594C8—81no—
595C8—122no—
596C8—241no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3#146311yescharacteristic
602C2×C3#146511yescharacteristic
603C2×C3#146811yescharacteristic
604C2×C3#147211yescharacteristic
605C2×C3#187011yescharacteristic
606C2×C3#189811yescharacteristic
607C2×C3#190111yescharacteristic
608D3—42no—
609D3—42no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—62no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—124no—
629C22—124no—
630C22—124no—
631C22—124no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22#466511yescharacteristic
641C22#468811yescharacteristic
642C22#470611yescharacteristic
643C22#472611yescharacteristic
644C22#472711yescharacteristic
645C22D3×#9911yescharacteristic
646C22D3×#9911yescharacteristic
647C22#474511yescharacteristic
648C22D3×#10111yescharacteristic
649C22#483611yescharacteristic
650C22#483811yescharacteristic
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—42no—
656C4—62no—
657C4—81no—
658C4—122no—
659C4—121no—
660C4—241no—
661C3#908511yescharacteristic, complemented, Sylow
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—122no—
667C2—122no—
668C2#6812111yescharacteristic
669C2#6812211yescharacteristic
670C2#6814411yescharacteristic
671C2#6814711yescharacteristic
672C2#6866911yescharacteristic
673C2#6897011yescharacteristic
674C2#6899311yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3C2×C4
cosocleC23×C3×C4C23
hypercenter#1280D3
hypocenterC3#9085
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
4. C2×#4469C2semidirect
5. C2×#4469C2semidirect
6. C3×#1280C2semidirect
7. #67613C2semidirect
38. #1280D3semidirect
141. C22×C3×C4M16semidirect
165. C22×Dic3M16semidirect
421. C22×C4ΓL(1,𝔽25)semidirect
661. C3#9085semidirect

Maximal subgroups

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

subgroupindex
C2×#47262
C2×#46682
#699552
C2×#44692
C3×#12802
#676132
#90853