Finite Group Databasebeta

← #10804112

Subgroups — #10804112

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5791C211yesmaximal, characteristic
2C2×#4727C212yesmaximal, complemented
3C3×#1779C211yesmaximal, characteristic, complemented
4#72126C211yesmaximal, characteristic, complemented
5#69958C212yesmaximal, complemented
6#69958C212yesmaximal, complemented
7#69957C212yesmaximal, complemented
8#69957C212yesmaximal, complemented
9#69911C212yesmaximal, complemented
10#21744—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C2×#953C412yes—
14#5791—28no—
15C2×#930C411yescharacteristic
16C2×#930C411yescharacteristic
17C2×#929C412yes—
18C2×C4×#67C2212yes—
19C2×C4×#67C2212yes—
20C2×#926C2211yescharacteristic
21C2×#924C411yescharacteristic
22C2×#924C411yescharacteristic
23C22×ΓL(1,𝔽25)C2212yescomplemented
24C22×D3×C8C2212yescomplemented
25C2×C3×C22⋊C8C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#830C2212yes—
30C2×#830C2211yescharacteristic
31C2×#830C2212yes—
32C2×#830C2211yescharacteristic
33C2×(C2×D3)⋊C8C2212yescomplemented
34C2×(C2×D3)⋊C8C2212yescomplemented
35#4727—28no—
36#4550C2212yescomplemented
37#4550C2212yescomplemented
38#4550C2212yescomplemented
39#4550C2212yescomplemented
40C2×#525—31no—
41C2×#408—32no—
42#1870—32no—
43#1870—32no—
44#1869—32no—
45#1869—32no—
46#1818—32no—
47#1779—31no—
48#1779D311yescharacteristic, complemented
49C25×C3C2×C411yescharacteristic
50C23×C3×C4—24no—
51C23×C3×C4—42no—
52C23×C3×C4—41no—
53C23×C3×C4—41no—
54C23×C3×C4C2×C411yescharacteristic
55C23×C3×C4C2×C411yescharacteristic
56C23×C3×C4C2×C411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C412yes—
61C22×#67C2×C412yes—
62C22×#67C2×C412yes—
63C2×#235—41no—
64C2×#235—42no—
65C2×#235—41no—
66C2×#235D418yes—
67C22×D12C2×C411yescharacteristic
68C22×C4×D3C2312yes—
69C22×Dic6C2×C411yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72#953—28no—
73C2×#217C2×C412yes—
74C2×#217C2×C412yes—
75C2×#217C2×C412yes—
76#930—24no—
77#930—24no—
78#929—28no—
79C4×#67—24no—
80C4×#67—24no—
81C4×#67—24no—
82C4×#67—24no—
83C2×(C2×D3)⋊C4C2×C412yes—
84C2×(C2×D3)⋊C4C2×C412yes—
85#926—24no—
86C2×C4⋊Dic3C2×C411yescharacteristic
87C2×C4⋊Dic3C2×C411yescharacteristic
88#924—24no—
89#924—24no—
90C2×Dic3⋊C4C2×C412yes—
91C2×Dic3⋊C4C2×C412yes—
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2312yes—
95C2×ΓL(1,𝔽25)—24no—
96C2×ΓL(1,𝔽25)—42no—
97C2×ΓL(1,𝔽25)—42no—
98C2×D3×C8—24no—
99C2×D3×C8—42no—
100C2×D3×C8—42no—
101C3×C22⋊C8—24no—
102C3×C22⋊C8—41no—
103C3×C22⋊C8—42no—
104C3×C22⋊C8—41no—
105#830—24no—
106#830—41no—
107#830—42no—
108#830—41no—
109(C2×D3)⋊C8—24no—
110(C2×D3)⋊C8—24no—
111C22×#41—31no—
112C24×C4C2×D311yescharacteristic
113C22×M16—32no—
114C23×C8—32no—
115#525—68no—
116C2×#123—32no—
117C2×C22⋊Q8—31no—
118C2×C22⋊Q8—31no—
119C2×#121—31no—
120C2×#121—31no—
121C2×#120—32no—
122C2×C4×D4—32no—
123C2×C4×D4—32no—
124C2×#117—31no—
125#408—68no—
126C2×C22⋊C8—32no—
127C2×C22⋊C8—32no—
128C2×C22⋊C8—32no—
129C2×C22⋊C8—31no—
130C2×C22⋊C8—32no—
131C2×C22⋊C8—31no—
132C2×C22⋊C8C2×D312yes—
133C2×C22⋊C8C2×D311yescharacteristic
134C2×C22⋊C8C2×D312yes—
135C2×C22⋊C8C2×D311yescharacteristic
136#336—32no—
137#336—32no—
138#336—32no—
139#336—32no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—22no—
143C24×C3—41no—
144C24×C3—41no—
145C24×C3—41no—
146C24×C3—41no—
147C24×C3C22×C411yescharacteristic
148C24×C3C22×C411yescharacteristic
149C24×C3C22×C411yescharacteristic
150C23×D3C22×C412yes—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—41no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—41no—
171C22×C3×C4—48no—
172C22×C3×C4—48no—
173C22×C3×C4C22×C411yescharacteristic
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—24no—
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—24no—
182C2×#67—42no—
183C2×#67—42no—
184C2×#67—42no—
185C2×#67—42no—
186C2×#67C22⋊C418yes—
187C22×Dic3C22×C412yes—
188C22×Dic3C22×C412yes—
189C22×Dic3C22×C412yes—
190#235—48no—
191#235—48no—
192C2×D12—41no—
193C2×D12—41no—
194C2×D12C22⋊C414yes—
195C2×C4×D3—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3C2×D414yes—
200C2×C4×D3C2×D414yes—
201C2×Dic6—41no—
202C2×Dic6—41no—
203C2×Dic6C22⋊C414yes—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210#217—24no—
211#217—24no—
212#217—24no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—24no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227ΓL(1,𝔽25)—48no—
228D3×C8—48no—
229C25C4×D311yescharacteristic
230C2×#41—38no—
231C2×#41—121no—
232C2×#41—122no—
233C2×#41—121no—
234C22×Q8—31no—
235C22×D4—32no—
236C22×D4—32no—
237C22×D4—32no—
238C22×D4—31no—
239C23×C4—24no—
240C23×C4—32no—
241C23×C4—42no—
242C23×C4—41no—
243C23×C4—41no—
244C23×C4C4×D311yescharacteristic
245C23×C4C4×D311yescharacteristic
246C23×C4C4×D311yescharacteristic
247C23×C4C22×D311yescharacteristic
248C23×C4C22×D311yescharacteristic
249C23×C4C22×D311yescharacteristic
250C2×M16—64no—
251C2×M16—122no—
252C2×M16—122no—
253C22×C8—32no—
254C22×C8—32no—
255C22×C8—64no—
256C22×C8—122no—
257C22×C8—122no—
258C22×C8C22×D312yes—
259C22×C8C22×D312yes—
260#123—68no—
261C22⋊Q8—64no—
262C22⋊Q8—64no—
263#121—64no—
264#121—64no—
265#120—68no—
266C4×D4—64no—
267C4×D4—64no—
268C4×D4—64no—
269C4×D4—64no—
270#117—64no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—32no—
275C2×C22⋊C4—32no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—32no—
279C2×C22⋊C4—32no—
280C2×C42—32no—
281C22⋊C8—24no—
282C22⋊C8—41no—
283C22⋊C8—42no—
284C22⋊C8—41no—
285C22⋊C8—64no—
286C22⋊C8—64no—
287C22⋊C8—64no—
288C22⋊C8—121no—
289C22⋊C8—122no—
290C22⋊C8—121no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3—44no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3C2×C22⋊C412yes—
317C23×C3#13112yes—
318C23×C3#13112yes—
319C23×C3C23×C411yescharacteristic
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3C2×C22⋊C414yes—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4C2×C22⋊C412yes—
345C2×C3×C4C22×D412yes—
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C4H(5,𝔽2)11yescharacteristic
348C2×C3×C4H(5,𝔽2)12yes—
349#67—48no—
350#67—48no—
351#67—48no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3C2×C22⋊C414yes—
361D12—44no—
362C4×D3—24no—
363C4×D3—24no—
364C4×D3—24no—
365C4×D3—24no—
366Dic6—44no—
367C3×C8—42no—
368C3×C8—42no—
369C3⋊C8—42no—
370C3⋊C8—42no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—32no—
375C24—41no—
376C24—41no—
377C24—41no—
378C24—41no—
379C24C2×C4×D311yescharacteristic
380C24C2×C4×D311yescharacteristic
381C24C2×C4×D311yescharacteristic
382#41—128no—
383#41—128no—
384C2×Q8—34no—
385C2×Q8—121no—
386C2×Q8—121no—
387C2×D4—38no—
388C2×D4—34no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—64no—
392C2×D4—64no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4—122no—
396C2×D4—122no—
397C2×D4—122no—
398C2×D4—122no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—32no—
414C22×C4—34no—
415C22×C4—34no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4C2×C4×D311yescharacteristic
431C22×C4C2×C4×D311yescharacteristic
432C22×C4C2×C4×D311yescharacteristic
433C22×C4D3×D414yes—
434C22×C4C23×D311yescharacteristic
435M16—128no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—62no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—128no—
449C4⋊C4—64no—
450C4⋊C4—64no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C22⋊C4—64no—
454C22⋊C4—64no—
455C22⋊C4—64no—
456C22⋊C4—64no—
457C22⋊C4—64no—
458C42—62no—
459C42—62no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—416no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3C22×C22⋊C411yescharacteristic
483C22×C3#51811yescharacteristic
484C22×C3#51811yescharacteristic
485C22×C3C2×#13111yescharacteristic
486C22×C3C2×#13111yescharacteristic
487C22×C3#56811yescharacteristic
488C22×C3#56811yescharacteristic
489C2×D3—24no—
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C3×C4—22no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4#53412yes—
500C3×C4#53412yes—
501Dic3—24no—
502Dic3—42no—
503Dic3—42no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—34no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—62no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23D3×C22⋊C412yes—
537C23#90112yes—
538C23#90712yes—
539C23C22×C4×D311yescharacteristic
540Q8—124no—
541D4—124no—
542D4—128no—
543D4—128no—
544D4—128no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—34no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4D3×C22⋊C412yes—
576C2×C4C2×D3×D412yes—
577C2×C4#100411yescharacteristic
578C2×C4#100411yescharacteristic
579C2×C4#100412yes—
580C8—42no—
581C8—42no—
582C8—122no—
583C8—122no—
584C2×C3—42no—
585C2×C3—42no—
586C2×C3—42no—
587C2×C3#230811yescharacteristic
588C2×C3#288011yescharacteristic
589C2×C3#288011yescharacteristic
590C2×C3#297611yescharacteristic
591C2×C3#297611yescharacteristic
592C2×C3#297711yescharacteristic
593C2×C3#297811yescharacteristic
594D3—44no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—416no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—64no—
618C22—64no—
619C22—64no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—122no—
624C22C2×D3×C22⋊C411yescharacteristic
625C22#548611yescharacteristic
626C22#548611yescharacteristic
627C22C2×#90111yescharacteristic
628C22C2×#90711yescharacteristic
629C22#574511yescharacteristic
630C22#574511yescharacteristic
631C4—22no—
632C4—41no—
633C4—41no—
634C4—64no—
635C4—122no—
636C4—122no—
637C4#558912yes—
638C4#558912yes—
639C3#2174411yescharacteristic, complemented, Sylow
640C2—42no—
641C2—42no—
642C2—42no—
643C2—124no—
644C2#7384911yescharacteristic
645C2#7860011yescharacteristic
646C2#7860411yescharacteristic
647C2#7860511yescharacteristic
648C2#7863111yescharacteristic
649C2#7863511yescharacteristic
650C2#7863611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1779C2
prime radicalC2×#5791C2
cosocleC22×C3×C4C24
hypercenter#1779D3
hypocenterC3#21744
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#4727C2semidirect
3. C3×#1779C2semidirect
4. #72126C2semidirect
5. #69958C2semidirect
6. #69958C2semidirect
7. #69957C2semidirect
8. #69957C2semidirect
9. #69911C2semidirect
23. C22×ΓL(1,𝔽25)C22semidirect
24. C22×D3×C8C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
34. C2×(C2×D3)⋊C8C22semidirect
36. #4550C22semidirect
37. #4550C22semidirect
38. #4550C22semidirect
39. #4550C22semidirect
48. #1779D3semidirect
639. C3#21744semidirect

Maximal subgroups

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

subgroupindex
C2×#57912
C2×#47272
C3×#17792
#721262
#699582
#699572
#699112
#217443