Finite Group Databasebeta

← #10840266

Subgroups — #10840266

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74369C211yesmaximal, characteristic
2#74229C211yesmaximal, characteristic
3C3×#1836C211yesmaximal, characteristic, complemented
4#69837C211yesmaximal, characteristic
5#69813C211yesmaximal, characteristic, complemented
6#69567C211yesmaximal, characteristic, complemented
7#69559C211yesmaximal, characteristic, complemented
8#68890C211yesmaximal, characteristic, complemented
9#68889C211yesmaximal, characteristic, complemented
10#68879C211yesmaximal, characteristic, complemented
11#68878C211yesmaximal, characteristic, complemented
12#68873C211yesmaximal, characteristic, complemented
13#68871C211yesmaximal, characteristic, complemented
14#68825C211yesmaximal, characteristic, complemented
15#68825C211yesmaximal, characteristic, complemented
16#24984—31nomaximal, Sylow
17C2×#897C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×#384C2211yescharacteristic
25C3×#382C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#4994—21no—
33#4994—21no—
34#4988—21no—
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4984—21no—
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4669—21no—
47#4666—21no—
48#4665—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2538—31no—
61#2352—31no—
62#2076—31no—
63#2075—31no—
64#2036—31no—
65#2036—31no—
66#1955—31no—
67#1947—31no—
68#1873—31no—
69#1872—31no—
70#1836D311yescharacteristic, complemented
71#1825—31no—
72#1823—31no—
73#1805—31no—
74#1763—31no—
75C22×D12D411yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3C2311yescharacteristic
78C22×C4×D3C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
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×C4×Dic3—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#892—22no—
116#892—41no—
117D3×C4⋊C4—22no—
118D3×C4⋊C4—41no—
119C4×D12—22no—
120C4×D12—22no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#97—22no—
124C3×#95—21no—
125C3×#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131(C2×D3)⋊C8—41no—
132(C2×D3)⋊C8—41no—
133(C2×D3)⋊C8—41no—
134(C2×D3)⋊C8—41no—
135#816—22no—
136#816—22no—
137#810—41no—
138#810—41no—
139#807—41no—
140#807—41no—
141C4⋊(C3⋊C8)—22no—
142C2×C22⋊Q8—31no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×D4—31no—
146C2×#117—31no—
147C22×C4⋊C4—31no—
148C2×#107—31no—
149C2×#107—31no—
150C2×C4⋊C8—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159C2×#97C2×D311yescharacteristic
160#396—61no—
161#396—61no—
162#390—61no—
163#389—61no—
164#388—61no—
165#387—61no—
166#386—61no—
167#384C2×D311yescharacteristic
168#382C2×D311yescharacteristic
169#380—31no—
170#380—31no—
171#380—31no—
172#380—31no—
173C4×C4⋊C4—31no—
174C4×C22⋊C4—61no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#340C2×D311yescharacteristic
184#340C2×D311yescharacteristic
185C23×D3C2×D411yescharacteristic
186C23×D3C2×D411yescharacteristic
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—41no—
203C2×C4×D3—21no—
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—42no—
216C2×C4×D3—42no—
217C2×C4×D3—81no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6#4112yes—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C42#4112yes—
241C3×C42#4112yes—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—81no—
249(C2×D3)⋊C4—81no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—22no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—42no—
258C4⋊Dic3—42no—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264C4×Dic3—81no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C22×Q8—31no—
272C22×D4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—61no—
276C22×C8—31no—
277C22×C8—31no—
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280C22⋊Q8—62no—
281C22⋊Q8—62no—
282C22⋊Q8—121no—
283C22⋊Q8—121no—
284C4×Q8—62no—
285C4×Q8—62no—
286C4×D4—62no—
287C4×D4—62no—
288#117—62no—
289#117—121no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—62no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—121no—
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#107—121no—
320#107—121no—
321C4⋊C8—62no—
322#103—62no—
323#103—62no—
324#103—121no—
325#103—121no—
326C22⋊C8—121no—
327C22⋊C8—121no—
328C22⋊C8—121no—
329C22⋊C8—121no—
330#97—22no—
331#95—21no—
332#95—21no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338C23×C3C22×D411yescharacteristic
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
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#13712yes—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
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#12312yes—
370C2×C3×C4#12312yes—
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—21no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395D12—44no—
396C4×D3—41no—
397C4×D3—41no—
398C4×D3—41no—
399C4×D3—41no—
400C4×D3—41no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—84no—
405Dic6—22no—
406Dic6—22no—
407Dic6—41no—
408Dic6—41no—
409C3×C8—41no—
410C3×C8—41no—
411C3⋊C8—41no—
412C3⋊C8—41no—
413C24—31no—
414C24—31no—
415C2×Q8—32no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—121no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
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—122no—
447C22×C4—122no—
448C22×C4—241no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—32no—
473C4⋊C4—42no—
474C4⋊C4—62no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—81no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C22⋊C4—62no—
486C22⋊C4—62no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—241no—
492C22⋊C4—241no—
493C22⋊C4—241no—
494C42—122no—
495C42—122no—
496C42—241no—
497C42#23512yes—
498C42#23912yes—
499C22×C3C2×#12311yescharacteristic
500C22×C3#54611yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3C2×#13711yescharacteristic
503C22×C3#57511yescharacteristic
504C22×C3#57611yescharacteristic
505C22×C3#57811yescharacteristic
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—41no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—41no—
514C2×D3—41no—
515C2×D3—41no—
516C2×D3—41no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C3×C4—22no—
521C3×C4—22no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4#52911yescharacteristic
527C3×C4#54011yescharacteristic
528C3×C4#55311yescharacteristic
529C3×C4#55511yescharacteristic
530Dic3—42no—
531Dic3—41no—
532Dic3—41no—
533Dic3—42no—
534Dic3—81no—
535Dic3—81no—
536C23—32no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23—61no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23C2×D3×D411yescharacteristic
551Q8—62no—
552Q8—62no—
553Q8—121no—
554Q8—121no—
555D4—124no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
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—244no—
599C2×C4#89412yes—
600C2×C4#89412yes—
601C2×C4C2×#23511yescharacteristic
602C2×C4C2×#23911yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C2×C4#100911yescharacteristic
606C2×C4#100911yescharacteristic
607C8—41no—
608C8—41no—
609C8—121no—
610C8—121no—
611C2×C3#242011yescharacteristic
612C2×C3#309511yescharacteristic
613C2×C3#309811yescharacteristic
614C2×C3#330911yescharacteristic
615C2×C3#331211yescharacteristic
616C2×C3#331311yescharacteristic
617C2×C3#331411yescharacteristic
618D3—42no—
619D3—81no—
620C22—62no—
621C22—62no—
622C22—62no—
623C22—121no—
624C22—121no—
625C22—121no—
626C22—121no—
627C22—121no—
628C22—121no—
629C22—121no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22C2×#89411yescharacteristic
635C22#567611yescharacteristic
636C22#567611yescharacteristic
637C22#577311yescharacteristic
638C22#577411yescharacteristic
639C22D3×#13711yescharacteristic
640C22#577911yescharacteristic
641C4—22no—
642C4—22no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—122no—
648C4—121no—
649C4—121no—
650C4—122no—
651C4—241no—
652C4—241no—
653C4#553011yescharacteristic
654C4#553211yescharacteristic
655C4#569111yescharacteristic
656C4#569711yescharacteristic
657C3#2498411yescharacteristic, complemented, Sylow
658C2—122no—
659C2—241no—
660C2#7423111yescharacteristic
661C2#7921811yescharacteristic
662C2#7922111yescharacteristic
663C2#8169911yescharacteristic
664C2#8170011yescharacteristic
665C2#8172411yescharacteristic
666C2#8174911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1836C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1836D3
hypocenterC3#24984
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. #69567C2semidirect
7. #69559C2semidirect
8. #68890C2semidirect
9. #68889C2semidirect
10. #68879C2semidirect
11. #68878C2semidirect
12. #68873C2semidirect
13. #68871C2semidirect
14. #68825C2semidirect
15. #68825C2semidirect
70. #1836D3semidirect
657. C3#24984semidirect

Maximal subgroups

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

subgroupindex
#743692
#742292
C3×#18362
#698372
#698132
#695672
#695592
#688902
#688892
#688792
#688782
#688732
#688712
#688252
#249843