Finite Group Databasebeta

← #10845232

Subgroups — #10845232

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2367C211yesmaximal, characteristic, complemented
2C2×#5057C211yesmaximal, characteristic
3C2×#5055C211yesmaximal, characteristic, complemented
4#73800C211yesmaximal, characteristic, complemented
5#71633C211yesmaximal, characteristic, complemented
6#71626C211yesmaximal, characteristic, complemented
7#71613C211yesmaximal, characteristic
8#71604C211yesmaximal, characteristic, complemented
9#69680C211yesmaximal, characteristic, complemented
10#69670C211yesmaximal, characteristic, complemented
11#69650C211yesmaximal, characteristic
12#69640C211yesmaximal, characteristic, complemented
13#69489C211yesmaximal, characteristic, complemented
14#69485C211yesmaximal, characteristic, complemented
15#69474C211yesmaximal, characteristic, complemented
16#25334—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×#897—21no—
23C2×#897—21no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#388—21no—
29C3×#388—21no—
30C3×#385—21no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40#5057—24no—
41#5055—24no—
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44#4939C2211yescharacteristic
45#4939C2211yescharacteristic
46#4935C2211yescharacteristic
47#4935C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4677—21no—
53#4677—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#463—31no—
64#2542—31no—
65#2367D311yescharacteristic, complemented
66#2072—31no—
67#2071—31no—
68#2055—31no—
69#2054—31no—
70#1987—31no—
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1803—31no—
75#1802—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6—21no—
83C3×#124—22no—
84C3×#124D414yes—
85C3×C4×Q8—22no—
86C3×C4×Q8—41no—
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#215—22no—
105C2×#215—22no—
106C2×#215—41no—
107C2×#213—22no—
108C2×#213—22no—
109C2×#213—41no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#897—44no—
128#897—44no—
129C4×D12—22no—
130C4×D12—22no—
131#869—22no—
132#869—22no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95D411yescharacteristic
137C3×#95D411yescharacteristic
138#835—22no—
139#835—22no—
140#835—22no—
141#832—22no—
142#832—22no—
143#832—22no—
144#809—22no—
145#809—22no—
146C4⋊(C3⋊C8)—22no—
147C4⋊(C3⋊C8)—41no—
148C4⋊(C3⋊C8)—41no—
149#803—22no—
150C22×SD16—31no—
151C22×D8—31no—
152C2×C4⋊Q8—31no—
153C2×#124C2×D311yescharacteristic
154C2×C22⋊Q8—61no—
155C2×C22⋊Q8—61no—
156C2×C4×Q8C2×D311yescharacteristic
157C2×C4×D4—31no—
158C2×C4×D4C2×D311yescharacteristic
159#489—64no—
160#463—64no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#97—31no—
173#399—61no—
174#399—61no—
175#391C2×D311yescharacteristic
176#390—31no—
177#390—31no—
178#390C2×D311yescharacteristic
179#388—21no—
180#388—21no—
181#385—21no—
182#384—31no—
183#384—31no—
184C4×C4⋊C4C2×D311yescharacteristic
185C4×C4⋊C4C2×D311yescharacteristic
186C4×C22⋊C4—21no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#336—31no—
194#336—31no—
195C24×C3—21no—
196C23×D3—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D412yes—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4C2×D412yes—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—42no—
213C22×C3×C4—42no—
214C22×C3×C4—81no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12#4112yes—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×Dic6—42no—
235C2×Dic6—42no—
236C2×Dic6—42no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4#4112yes—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—81no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—42no—
254C3×C42—81no—
255C3×C42C2×D412yes—
256C3×C42#4112yes—
257#215—44no—
258#213—44no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—42no—
271C4⋊Dic3—42no—
272C4⋊Dic3—42no—
273C4⋊Dic3—42no—
274C4⋊Dic3—42no—
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—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C22×Q8—61no—
288C22×Q8C22×D311yescharacteristic
289C22×D4—31no—
290C22×D4C22×D311yescharacteristic
291C23×C4—21no—
292C23×C4—61no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×D8—62no—
297C2×D8—62no—
298C2×D8—121no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—31no—
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305#124—22no—
306#124D1214yes—
307C22⋊Q8—124no—
308C22⋊Q8—124no—
309C4×Q8—22no—
310C4×Q8—41no—
311C4×D4—22no—
312C4×D4—41no—
313C4×D4—62no—
314C4×D4—62no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4#6711yescharacteristic
325C2×C4⋊C4#6711yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42C22×D311yescharacteristic
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343C4⋊C8—62no—
344C4⋊C8—121no—
345C4⋊C8—121no—
346#103—62no—
347#103—62no—
348#103—62no—
349#102—62no—
350#102—62no—
351#102—62no—
352#102—62no—
353#102—62no—
354#97—62no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95#6711yescharacteristic
359#95#6711yescharacteristic
360C23×C3—22no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3C22×D411yescharacteristic
368C22×D3—22no—
369C22×D3—41no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C3×Q8—22no—
376C3×Q8—22no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×D4—22no—
380C3×D4—22no—
381C3×D4—41no—
382C3×D4—41no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
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—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—84no—
407C2×C3×C4#12112yes—
408C2×C3×C4#12112yes—
409C2×C3×C4AGL(1,ℤ/8)12yes—
410C2×C3×C4AGL(1,ℤ/8)12yes—
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4H(5,𝔽2)11yescharacteristic
416C2×C3×C42−1+411yescharacteristic
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426D12—22no—
427D12—22no—
428D12—41no—
429D12—41no—
430C4×D3—84no—
431Dic6—84no—
432C3⋊C8—41no—
433C3⋊C8—41no—
434C3⋊C8—41no—
435C3⋊C8—41no—
436C24—21no—
437C24—61no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—122no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×Q8C2×D1212yes—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—32no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4C2×D1212yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—81no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479D8—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—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4#23712yes—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—81no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—81no—
526C42C2×D1212yes—
527C42#23512yes—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3C2×#12111yescharacteristic
536C22×C3#54611yescharacteristic
537C22×C3#54811yescharacteristic
538C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
539C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
540C22×C3#57711yescharacteristic
541C22×C3#57811yescharacteristic
542C2×D3—42no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—81no—
546C2×D3—81no—
547C2×D3—81no—
548C2×D3—81no—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—22no—
552C3×C4—22no—
553C3×C4—42no—
554C3×C4—41no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—81no—
558C3×C4#52911yescharacteristic
559C3×C4#54211yescharacteristic
560C3×C4#54611yescharacteristic
561C3×C4#54711yescharacteristic
562Dic3—81no—
563Dic3—81no—
564Dic3—81no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—62no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23—121no—
578C23—121no—
579C23C22×#6711yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—41no—
583Q8—41no—
584Q8—244no—
585D4—22no—
586D4—22no—
587D4—41no—
588D4—41no—
589D4—62no—
590D4—62no—
591D4—121no—
592D4—121no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—244no—
627C2×C4#93012yes—
628C2×C4#93012yes—
629C2×C4#94912yes—
630C2×C4#94912yes—
631C2×C4C22×D1211yescharacteristic
632C2×C4C2×#23511yescharacteristic
633C2×C4C2×#23711yescharacteristic
634C2×C4#100411yescharacteristic
635C2×C4D3×#4111yescharacteristic
636C2×C4#101011yescharacteristic
637C8—121no—
638C8—121no—
639C8—121no—
640C8—121no—
641C2×C3—82no—
642C2×C3#242711yescharacteristic
643C2×C3#303811yescharacteristic
644C2×C3#304611yescharacteristic
645C2×C3#330611yescharacteristic
646C2×C3#330811yescharacteristic
647C2×C3#336711yescharacteristic
648C2×C3#337311yescharacteristic
649D3—82no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—122no—
658C22—122no—
659C22—122no—
660C22—241no—
661C22—241no—
662C22—241no—
663C22—241no—
664C22C2×#93011yescharacteristic
665C22#580611yescharacteristic
666C22C2×#94911yescharacteristic
667C22C2×#94911yescharacteristic
668C22#583111yescharacteristic
669C22#583611yescharacteristic
670C22#583811yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—41no—
676C4—42no—
677C4—41no—
678C4—42no—
679C4—81no—
680C4—241no—
681C4—241no—
682C4—241no—
683C4#555411yescharacteristic
684C4#555511yescharacteristic
685C4#555711yescharacteristic
686C4#556111yescharacteristic
687C3#2533411yescharacteristic, complemented, Sylow
688C2—82no—
689C2—242no—
690C2#8240811yescharacteristic
691C2#8252611yescharacteristic
692C2#8257211yescharacteristic
693C2#8349611yescharacteristic
694C2#8353611yescharacteristic
695C2#8366811yescharacteristic
696C2#8371511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2367C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2367D3
hypocenterC3#25334
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. C3×#2367C2semidirect
3. C2×#5055C2semidirect
4. #73800C2semidirect
5. #71633C2semidirect
6. #71626C2semidirect
8. #71604C2semidirect
9. #69680C2semidirect
10. #69670C2semidirect
12. #69640C2semidirect
13. #69489C2semidirect
14. #69485C2semidirect
15. #69474C2semidirect
65. #2367D3semidirect
687. C3#25334semidirect

Maximal subgroups

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

subgroupindex
C3×#23672
C2×#50572
C2×#50552
#738002
#716332
#716262
#716132
#716042
#696802
#696702
#696502
#696402
#694892
#694852
#694742
#253343