Finite Group Databasebeta

← #10840191

Subgroups — #10840191

Order 768.

828 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
1#74360C211yesmaximal, characteristic
2#74240C211yesmaximal, characteristic
3C3×#1836C211yesmaximal, characteristic, complemented
4#69837C211yesmaximal, characteristic
5#69813C211yesmaximal, characteristic, complemented
6#69588C211yesmaximal, characteristic, complemented
7#69540C211yesmaximal, characteristic, complemented
8#68891C211yesmaximal, characteristic, complemented
9#68889C211yesmaximal, characteristic, complemented
10#68878C211yesmaximal, characteristic, complemented
11#68878C211yesmaximal, characteristic, complemented
12#68872C211yesmaximal, characteristic, complemented
13#68871C211yesmaximal, characteristic, complemented
14#68825C211yesmaximal, characteristic, complemented
15#68824C211yesmaximal, characteristic, complemented
16#24979—31nomaximal, Sylow
17C2×#897C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#893—21no—
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×#384C2211yescharacteristic
26C3×#382C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33#4989—21no—
34#4989—21no—
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4972—21no—
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4672—21no—
47#4670—21no—
48#4670—21no—
49#4661—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2536—31no—
61D4×C4⋊C4—31no—
62#2077—31no—
63#2075—31no—
64#2036—31no—
65#2035—31no—
66#1955—31no—
67#1949—31no—
68#1872—31no—
69#1872—31no—
70#1836D311yescharacteristic, complemented
71#1824—31no—
72#1823—31no—
73#1803—31no—
74#1763—31no—
75C22×D12D411yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#897—22no—
112#897—22no—
113#897—41no—
114#897—41no—
115#893—44no—
116D3×C4⋊C4—22no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—41no—
119D3×C4⋊C4—41no—
120C4×D12—22no—
121C4×D12—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C3×#97—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132(C2×D3)⋊C8—41no—
133(C2×D3)⋊C8—41no—
134(C2×D3)⋊C8—41no—
135(C2×D3)⋊C8—41no—
136#816—22no—
137#816—22no—
138#810—41no—
139#810—41no—
140#808—41no—
141#807—41no—
142C4⋊(C3⋊C8)—22no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C22×C4⋊C4—31no—
149C22×C4⋊C4—31no—
150C2×#107—31no—
151C2×#106—31no—
152C2×C4⋊C8—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103—31no—
156C2×#103—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159C2×C22⋊C8—31no—
160C2×C22⋊C8—31no—
161C2×#97C2×D311yescharacteristic
162#398—61no—
163#394—61no—
164#389—61no—
165#388—61no—
166#388—61no—
167#385—61no—
168#385—61no—
169#384C2×D311yescharacteristic
170#382C2×D311yescharacteristic
171#380—31no—
172#380—31no—
173#380—31no—
174#380—31no—
175C4×C4⋊C4—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#340C2×D311yescharacteristic
185#340C2×D311yescharacteristic
186C23×D3C2×D411yescharacteristic
187C23×D3C2×D411yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×C4×D3—81no—
221C2×C4×D3—81no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6#4112yes—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—81no—
244C3×C42#4112yes—
245C3×C42#4112yes—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—22no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254C4⋊Dic3—22no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—42no—
261C4⋊Dic3—42no—
262C4⋊Dic3#4112yes—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—42no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C22×Q8—31no—
275C22×D4—31no—
276C23×C4—31no—
277C23×C4—31no—
278C23×C4—61no—
279C23×C4—61no—
280C22×C8—31no—
281C22×C8—31no—
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C22⋊Q8—62no—
285C22⋊Q8—62no—
286C22⋊Q8—121no—
287C22⋊Q8—121no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×D4—62no—
291C4×D4—62no—
292C4×D4—124no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—62no—
300C2×C4⋊C4—62no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—121no—
307C2×C4⋊C4—121no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—31no—
313C2×C22⋊C4—31no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—121no—
324#106—121no—
325C4⋊C8—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329#103—121no—
330C22⋊C8—121no—
331C22⋊C8—121no—
332C22⋊C8—121no—
333C22⋊C8—121no—
334#97—22no—
335#95—21no—
336#95—21no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3C22×D411yescharacteristic
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3—21no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3AGL(1,ℤ/8)12yes—
356C22×D3#13712yes—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4#12312yes—
374C2×C3×C4#12312yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399D12—28no—
400C4×D3—41no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—41no—
405C4×D3—41no—
406C4×D3—41no—
407C4×D3—41no—
408C4×D3—84no—
409C4×D3—84no—
410Dic6—22no—
411Dic6—22no—
412Dic6—41no—
413Dic6—41no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—31no—
419C24—31no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—62no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—241no—
456C22×C4—241no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—32no—
481C4⋊C4—42no—
482C4⋊C4—62no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—81no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C4⋊C4—122no—
493C4⋊C4—122no—
494C22⋊C4—62no—
495C22⋊C4—62no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C42—122no—
503C42—122no—
504C42#23512yes—
505C42#23912yes—
506C22×C3C2×#12311yescharacteristic
507C22×C3#54611yescharacteristic
508C22×C3#54611yescharacteristic
509C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
510C22×C3C2×#13711yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57811yescharacteristic
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#52911yescharacteristic
534C3×C4#54011yescharacteristic
535C3×C4#55311yescharacteristic
536C3×C4#55511yescharacteristic
537Dic3—42no—
538Dic3—41no—
539Dic3—41no—
540Dic3—42no—
541Dic3—81no—
542Dic3—81no—
543C23—32no—
544C23—32no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23C2×D3×D411yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—121no—
561Q8—121no—
562D4—68no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—244no—
606C2×C4—244no—
607C2×C4#89412yes—
608C2×C4#89412yes—
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×#23911yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C2×C4#100911yescharacteristic
614C2×C4#100911yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3#242011yescharacteristic
620C2×C3#309211yescharacteristic
621C2×C3#309711yescharacteristic
622C2×C3#330811yescharacteristic
623C2×C3#331011yescharacteristic
624C2×C3#331211yescharacteristic
625C2×C3#331611yescharacteristic
626D3—42no—
627D3—42no—
628C22—62no—
629C22—62no—
630C22—62no—
631C22—62no—
632C22—62no—
633C22—62no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22C2×#89411yescharacteristic
643C22#567611yescharacteristic
644C22#567611yescharacteristic
645C22D3×AGL(1,ℤ/8)11yescharacteristic
646C22#577511yescharacteristic
647C22D3×#13711yescharacteristic
648C22#577911yescharacteristic
649C4—22no—
650C4—22no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—122no—
656C4—121no—
657C4—121no—
658C4—122no—
659C4—241no—
660C4—241no—
661C4#553011yescharacteristic
662C4#553211yescharacteristic
663C4#569111yescharacteristic
664C4#569711yescharacteristic
665C3#2497911yescharacteristic, complemented, Sylow
666C2—122no—
667C2—122no—
668C2#7423111yescharacteristic
669C2#7921511yescharacteristic
670C2#7922011yescharacteristic
671C2#8169711yescharacteristic
672C2#8170111yescharacteristic
673C2#8172511yescharacteristic
674C2#8174911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1836C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1836D3
hypocenterC3#24979
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
3. C3×#1836C2semidirect
5. #69813C2semidirect
6. #69588C2semidirect
7. #69540C2semidirect
8. #68891C2semidirect
9. #68889C2semidirect
10. #68878C2semidirect
11. #68878C2semidirect
12. #68872C2semidirect
13. #68871C2semidirect
14. #68825C2semidirect
15. #68824C2semidirect
70. #1836D3semidirect
665. C3#24979semidirect

Maximal subgroups

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

subgroupindex
#743602
#742402
C3×#18362
#698372
#698132
#695882
#695402
#688912
#688892
#688782
#688722
#688712
#688252
#688242
#249793