Finite Group Databasebeta

← #10858741

Subgroups — #10858741

Order 768.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76206C211yesmaximal, characteristic
2#74326C211yesmaximal, characteristic, complemented
3C3×#2099C211yesmaximal, characteristic, complemented
4#69622C211yesmaximal, characteristic, complemented
5#69584C211yesmaximal, characteristic, complemented
6#68901C211yesmaximal, characteristic, complemented
7#68898C211yesmaximal, characteristic, complemented
8#68883C211yesmaximal, characteristic
9#68880C211yesmaximal, characteristic
10#68874C211yesmaximal, characteristic, complemented
11D3×#340C211yesmaximal, characteristic, complemented
12#68814C211yesmaximal, characteristic, complemented
13#68800C211yesmaximal, characteristic, complemented
14#68670C211yesmaximal, characteristic, complemented
15#68666C211yesmaximal, characteristic, complemented
16#26408—31nomaximal, Sylow
17C22×D3×C8C2211yescharacteristic
18C2×#896—21no—
19C2×#895C2211yescharacteristic
20C2×#893—21no—
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#384C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4989—21no—
33#4989—21no—
34#4988C2211yescharacteristic
35#4987C2211yescharacteristic
36#4983—21no—
37#4979—21no—
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C4×(C2×D3)⋊C4C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4669—21no—
47#4665—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2440—31no—
61#2361—31no—
62#2099D311yescharacteristic, complemented
63#2086—31no—
64#2067—31no—
65#2023—31no—
66#1988—31no—
67#1918—31no—
68#1893—31no—
69#1874—31no—
70#1862—31no—
71#1852—31no—
72#1850—31no—
73#1802—31no—
74C2×#340—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3C2311yescharacteristic
81C22×Dic6—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×D3×C8—22no—
112C2×D3×C8—41no—
113C2×D3×C8—41no—
114#896—44no—
115#895—22no—
116#895Q814yescomplemented
117#893—44no—
118D3×C4⋊C4—22no—
119D3×C4⋊C4—22no—
120D3×C4⋊C4—41no—
121D3×C4⋊C4—41no—
122#888—22no—
123#888—41no—
124C4×D12—22no—
125C4×D12—41no—
126C3×#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#821—22no—
134#821—41no—
135(C2×D3)⋊C8—22no—
136#818—22no—
137Dic3⋊C8—41no—
138#809—41no—
139#809—41no—
140#807—41no—
141C4⋊(C3⋊C8)—41no—
142C23×C8—31no—
143C2×C4⋊Q8—31no—
144C2×C22⋊Q8—61no—
145C2×#121—31no—
146C2×C4×D4—31no—
147C2×C4×D4—61no—
148C22×C4⋊C4—31no—
149C22×C4⋊C4—31no—
150C2×#107—31no—
151C2×#107—31no—
152C2×C4⋊C8—31no—
153C2×C4⋊C8—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×C22⋊C8—31no—
159#398—61no—
160#395—61no—
161#390—31no—
162#389—31no—
163#389—31no—
164#389C2×D311yescharacteristic
165#388—61no—
166#386—61no—
167#385—61no—
168#385—61no—
169#384C2×D311yescharacteristic
170#380—31no—
171C4×C4⋊C4—31no—
172C4×C22⋊C4—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#340C2×D311yescharacteristic
183#336—31no—
184#336C2×D311yescharacteristic
185C23×D3—21no—
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×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—24no—
205C2×D12—21no—
206C2×D12—42no—
207C2×D12—42no—
208C2×D12C2×Q812yes—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—42no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×C4×D3—42no—
219C2×C4×D3—81no—
220C2×C4×D3—81no—
221C2×C4×D3—81no—
222C2×C4×D3C2×Q812yes—
223C2×C4×D3#4112yes—
224C2×Dic6—42no—
225C2×Dic6—42no—
226C2×Dic6—81no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C3×C4⋊C4—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—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4C2×Q812yes—
244C3×C42—21no—
245C3×C42—21no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—81no—
252(C2×D3)⋊C4D814yes—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3#4112yes—
258C4⋊Dic3#4112yes—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273D3×C8—44no—
274C22×Q8—61no—
275C22×D4—31no—
276C22×D4—61no—
277C23×C4—31no—
278C23×C4—61no—
279C23×C4—61no—
280C23×C4—61no—
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—62no—
284C22×C8—121no—
285C22×C8—121no—
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288C4⋊Q8—62no—
289C4⋊Q8—121no—
290C22⋊Q8—124no—
291#121—34no—
292#121—62no—
293C4×D4—62no—
294C4×D4—121no—
295C4×D4—124no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—62no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—121no—
310C2×C4⋊C4—121no—
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C22⋊C4—31no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C42—31no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—62no—
326#107—121no—
327C4⋊C8—121no—
328C4⋊C8—121no—
329#102—62no—
330#102—121no—
331#102—121no—
332#102—121no—
333C22⋊C8—62no—
334#95—21no—
335#95—31no—
336#95—31no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341C23×C3C22×D411yescharacteristic
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—22no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3C22⋊Q812yes—
354C22×D3C2×D812yes—
355C22×D3#13712yes—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
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×C4C2×D812yes—
374C2×C3×C4C22×Q811yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
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×Dic3C22⋊Q812yes—
397C2×Dic3C2×D812yes—
398D12—22no—
399D12—22no—
400D12—48no—
401D12—41no—
402D12—41no—
403C4×D3—22no—
404C4×D3—22no—
405C4×D3—22no—
406C4×D3—22no—
407C4×D3—84no—
408C4×D3—84no—
409C4×D3—84no—
410Dic6—84no—
411C3×C8—41no—
412C3×C8—41no—
413C3⋊C8—41no—
414C3⋊C8—41no—
415C24—31no—
416C24—61no—
417C2×Q8—122no—
418C2×Q8—122no—
419C2×Q8—241no—
420C2×D4—32no—
421C2×D4—64no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—122no—
428C2×D4—122no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—31no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—241no—
452C22×C4—241no—
453C22×C4—241no—
454C22×C4D3×D411yescharacteristic
455C22×C4D3×D411yescharacteristic
456C22×C4C23×D311yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—124no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—32no—
478C4⋊C4—32no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—81no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4D3×Q812yes—
492C22⋊C4—34no—
493C22⋊C4—64no—
494C22⋊C4—62no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—241no—
499C42—21no—
500C42—21no—
501C42—61no—
502C42—61no—
503C42—122no—
504C22×C3C2×C22⋊Q811yescharacteristic
505C22×C3#54611yescharacteristic
506C22×C3#54611yescharacteristic
507C22×C3C22×D811yescharacteristic
508C22×C3C2×#13711yescharacteristic
509C22×C3#57711yescharacteristic
510C22×C3#57711yescharacteristic
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3#47414yes—
525C3×C4—22no—
526C3×C4—41no—
527C3×C4—42no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4#52911yescharacteristic
532C3×C4#53811yescharacteristic
533C3×C4#55411yescharacteristic
534C3×C4#55411yescharacteristic
535Dic3—22no—
536Dic3—22no—
537Dic3—42no—
538Dic3—42no—
539Dic3—41no—
540Dic3—81no—
541C23—32no—
542C23—32no—
543C23—32no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—62no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23C2×D3×D411yescharacteristic
556Q8—244no—
557D4—62no—
558D4—62no—
559D4—128no—
560D4—121no—
561D4—121no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—32no—
569C2×C4—32no—
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—41no—
580C2×C4—41no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—61no—
591C2×C4—62no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—244no—
602C2×C4—244no—
603C2×C4—244no—
604C2×C4D3×D812yes—
605C2×C4C2×D3×Q811yescharacteristic
606C2×C4D3×#4111yescharacteristic
607C2×C4D3×#4111yescharacteristic
608C2×C4D3×#4111yescharacteristic
609C2×C4#100911yescharacteristic
610C2×C4#100911yescharacteristic
611C8—41no—
612C8—41no—
613C8—121no—
614C8—121no—
615C2×C3#245011yescharacteristic
616C2×C3C2×#47411yescharacteristic
617C2×C3#308411yescharacteristic
618C2×C3#329411yescharacteristic
619C2×C3#329411yescharacteristic
620C2×C3#331611yescharacteristic
621C2×C3#332011yescharacteristic
622D3—24no—
623D3—82no—
624C22—34no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—62no—
629C22—62no—
630C22—62no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—241no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22D3×C22⋊Q811yescharacteristic
639C22#565411yescharacteristic
640C22#567611yescharacteristic
641C22C2×D3×D811yescharacteristic
642C22#576211yescharacteristic
643C22D3×#13711yescharacteristic
644C22#577811yescharacteristic
645C4—22no—
646C4—41no—
647C4—42no—
648C4—62no—
649C4—62no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—121no—
654C4—122no—
655C4—122no—
656C4—241no—
657C4#569011yescharacteristic
658C4#569011yescharacteristic
659C4#570411yescharacteristic
660C4#570611yescharacteristic
661C3#2640811yescharacteristic, complemented, Sylow
662C2—64no—
663C2—242no—
664C2#7834411yescharacteristic
665C2D3×#47411yescharacteristic
666C2#8101411yescharacteristic
667C2#8130211yescharacteristic
668C2#8143311yescharacteristic
669C2#8157611yescharacteristic
670C2#8175111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2099C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2099D3
hypocenterC3#26408
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
2. #74326C2semidirect
3. C3×#2099C2semidirect
4. #69622C2semidirect
5. #69584C2semidirect
6. #68901C2semidirect
7. #68898C2semidirect
10. #68874C2semidirect
11. D3×#340C2semidirect
12. #68814C2semidirect
13. #68800C2semidirect
14. #68670C2semidirect
15. #68666C2semidirect
62. #2099D3semidirect
116. #895Q8semidirect
661. C3#26408semidirect

Maximal subgroups

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

subgroupindex
#762062
#743262
C3×#20992
#696222
#695842
#689012
#688982
#688832
#688802
#688742
D3×#3402
#688142
#688002
#686702
#686662
#264083