Finite Group Databasebeta

← #10845204

Subgroups — #10845204

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2565C211yesmaximal, characteristic, complemented
2C2×#5067C211yesmaximal, characteristic
3C2×#5027C211yesmaximal, characteristic
4#73653C211yesmaximal, characteristic
5#71738C211yesmaximal, characteristic, complemented
6#71631C211yesmaximal, characteristic, complemented
7#69687C211yesmaximal, characteristic, complemented
8#69687C211yesmaximal, characteristic, complemented
9#69680C211yesmaximal, characteristic, complemented
10#69680C211yesmaximal, characteristic, complemented
11#69585C211yesmaximal, characteristic, complemented
12#69584C211yesmaximal, characteristic, complemented
13#69557C211yesmaximal, characteristic, complemented
14#69556C211yesmaximal, characteristic, complemented
15#69485C211yesmaximal, characteristic, complemented
16#25332—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#893—21no—
22C2×#874C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#398—21no—
26C3×#398—21no—
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#387—21no—
30C3×#387—21no—
31C3×#384C2211yescharacteristic
32C3×#382C2211yescharacteristic
33C3×C4×C4⋊C4C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36#5067—24no—
37#5027—24no—
38#4988—21no—
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45#4939C2211yescharacteristic
46#4939C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4665—21no—
53#4665—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#490—31no—
63C2×#460—31no—
64#2565D311yescharacteristic, complemented
65#2366—31no—
66#2073—31no—
67#2071—31no—
68#2055—31no—
69#2055—31no—
70#1989—31no—
71#1940—31no—
72#1877—31no—
73#1877—31no—
74#1804—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C3×#125—22no—
85C3×#125—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#215—22no—
103C2×#215—22no—
104C2×#215—22no—
105C2×#215—22no—
106C2×#215—41no—
107C2×#215—41no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#893—44no—
122#874—22no—
123#874D414yes—
124C4×D12—22no—
125C4×D12—22no—
126C4×D12—41no—
127C4×D12—41no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136#835—22no—
137#835—22no—
138C3⋊#95—21no—
139#809—22no—
140#809—22no—
141#809—22no—
142#809—22no—
143#809—22no—
144#809—22no—
145C4⋊(C3⋊C8)—22no—
146C4⋊(C3⋊C8)—41no—
147C4⋊(C3⋊C8)—41no—
148#803—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#125C2×D311yescharacteristic
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C2×C4×D4—61no—
157#490—64no—
158#460—64no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171#398—21no—
172#398—21no—
173#391C2×D311yescharacteristic
174#391C2×D311yescharacteristic
175#390—31no—
176#390—31no—
177#390—61no—
178#387—21no—
179#387—21no—
180#386—61no—
181#386—61no—
182#384C2×D311yescharacteristic
183#382C2×D311yescharacteristic
184C4×C4⋊C4—31no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—24no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—24no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—81no—
230C2×D12—81no—
231C2×D12C2×D412yes—
232C2×D12C2×D412yes—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—81no—
238C2×C4×D3—81no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—42no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—42no—
257C3×C42C2×D412yes—
258C3×C42#4112yes—
259#215—44no—
260#215—44no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3#4112yes—
271C4×Dic3—42no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C23×C4—61no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×SD16—121no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301#127—34no—
302#127—62no—
303#125—22no—
304#125—22no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4—121no—
310C4×D4—121no—
311C4×D4—124no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4#6711yescharacteristic
320C2×C4⋊C4#6711yescharacteristic
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C42—21no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335C4⋊C8—62no—
336C4⋊C8—121no—
337C4⋊C8—121no—
338#103—62no—
339#103—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346#97—62no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—61no—
354#95#6711yescharacteristic
355#95#6711yescharacteristic
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×Q8—22no—
372C3×Q8—22no—
373C3×Q8—41no—
374C3×Q8—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4#12112yes—
403C2×C3×C4#12112yes—
404C2×C3×C4AGL(1,ℤ/8)12yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4H(5,𝔽2)11yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411C2×Dic3—22no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417D12—22no—
418D12—22no—
419D12—22no—
420D12—22no—
421D12—48no—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—41no—
426D12—84no—
427C4×D3—84no—
428C4×D3—84no—
429C3⋊C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—61no—
434C24—61no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8#23912yes—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—241no—
458C2×D4—241no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—241no—
470C22×C4—241no—
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479SD16—124no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C22⋊C4—122no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—241no—
514C22⋊C4—241no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—122no—
521C42D3×D412yes—
522C42#23712yes—
523C22×C3C2×#12111yescharacteristic
524C22×C3#54611yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57711yescharacteristic
529C22×C3#57711yescharacteristic
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—42no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—81no—
552C3×C4—81no—
553C3×C4—81no—
554C3×C4#52911yescharacteristic
555C3×C4#54211yescharacteristic
556C3×C4#54611yescharacteristic
557C3×C4#54711yescharacteristic
558Dic3—42no—
559Dic3—81no—
560C23—62no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C22×#6711yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579D4—62no—
580D4—62no—
581D4—62no—
582D4—62no—
583D4—128no—
584D4—121no—
585D4—121no—
586D4—121no—
587D4—121no—
588D4—244no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—62no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—244no—
623C2×C4—244no—
624C2×C4#93812yes—
625C2×C4#93812yes—
626C2×C4#94912yes—
627C2×C4C2×D3×D411yescharacteristic
628C2×C4C2×#23711yescharacteristic
629C2×C4C2×#23911yescharacteristic
630C2×C4#100711yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4#100911yescharacteristic
633C8—121no—
634C8—121no—
635C8—121no—
636C8—121no—
637C2×C3#242711yescharacteristic
638C2×C3#303611yescharacteristic
639C2×C3#304211yescharacteristic
640C2×C3#331011yescharacteristic
641C2×C3#331711yescharacteristic
642C2×C3#336811yescharacteristic
643C2×C3#337111yescharacteristic
644D3—82no—
645D3—82no—
646C22—122no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22—241no—
653C22—241no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×#93811yescharacteristic
661C22#581711yescharacteristic
662C22C2×#94911yescharacteristic
663C22#582211yescharacteristic
664C22#582911yescharacteristic
665C22#583711yescharacteristic
666C22#583711yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4—122no—
678C4—241no—
679C4#562411yescharacteristic
680C4#563011yescharacteristic
681C4#563111yescharacteristic
682C4#564311yescharacteristic
683C3#2533211yescharacteristic, complemented, Sylow
684C2—242no—
685C2—242no—
686C2#8275611yescharacteristic
687C2#8329711yescharacteristic
688C2#8340511yescharacteristic
689C2#8359911yescharacteristic
690C2#8360411yescharacteristic
691C2#8369011yescharacteristic
692C2#8369111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2565C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2565D3
hypocenterC3#25332
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2565

Complements

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

NH ≅ G/Ntype
1. C3×#2565C2semidirect
5. #71738C2semidirect
6. #71631C2semidirect
7. #69687C2semidirect
8. #69687C2semidirect
9. #69680C2semidirect
10. #69680C2semidirect
11. #69585C2semidirect
12. #69584C2semidirect
13. #69557C2semidirect
14. #69556C2semidirect
15. #69485C2semidirect
64. #2565D3semidirect
683. C3#25332semidirect

Maximal subgroups

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

subgroupindex
C3×#25652
C2×#50672
C2×#50272
#736532
#717382
#716312
#696872
#696802
#695852
#695842
#695572
#695562
#694852
#253323