Finite Group Databasebeta

← #10855131

Subgroups — #10855131

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#482C211yesmaximal, characteristic, complemented
2#82703C211yesmaximal, characteristic
3#82272C211yesmaximal, characteristic, complemented
4C2×#4881C211yesmaximal, characteristic, complemented
5#71679C211yesmaximal, characteristic, complemented
6#71576C211yesmaximal, characteristic, complemented
7#70609C211yesmaximal, characteristic, complemented
8#70580C211yesmaximal, characteristic, complemented
9#70571C211yesmaximal, characteristic, complemented
10#70203C211yesmaximal, characteristic, complemented
11#70200C211yesmaximal, characteristic, complemented
12#70182C211yesmaximal, characteristic, complemented
13#70174C211yesmaximal, characteristic, complemented
14#69958C211yesmaximal, characteristic, complemented
15#69940C211yesmaximal, characteristic, complemented
16#26156—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#938C2211yescharacteristic
20C2×C4×#67C2211yescharacteristic
21C22×ΓL(1,𝔽25)C2211yescharacteristic
22C2×#892C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C3×#482—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30#5223—21no—
31#5221—21no—
32#5221C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5212C2211yescharacteristic
36#5212C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×#817C2211yescharacteristic
40#4987C2211yescharacteristic
41#4984C2211yescharacteristic
42#4977C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4961—21no—
48#4956—21no—
49#4945—21no—
50#4944—21no—
51C4×(C2×D3)⋊C4—21no—
52#4933—21no—
53#4881—24no—
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#482D311yescharacteristic, complemented
63C2×#428—31no—
64#2414—31no—
65#2352—31no—
66#2092—31no—
67#2085—31no—
68#2077—31no—
69#2033—31no—
70#1926—31no—
71#1918—31no—
72#1895—31no—
73#1874—31no—
74#1870—31no—
75#1854—31no—
76#1854—31no—
77C22×C3×D4—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×#67D411yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—22no—
87C3×#121—41no—
88C3×#117—22no—
89C3×#117—41no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217D411yescharacteristic
102#938—22no—
103#938—22no—
104C4×#67—22no—
105C4×#67—22no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4D411yescharacteristic
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)—22no—
124#892—22no—
125#892—41no—
126D3×C4⋊C4—22no—
127D3×C4⋊C4—41no—
128C3×#107D414yescomplemented
129C3×#106—22no—
130C3×#102—22no—
131C3×#102—22no—
132C3×C22⋊C8—22no—
133#832—22no—
134#832—41no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#830—22no—
143(C2×D3)⋊C8—22no—
144#817—22no—
145#808—22no—
146#807—22no—
147#807—41no—
148C22×M16—31no—
149C2×#121—31no—
150C2×#121C2×D311yescharacteristic
151C2×C4×D4—31no—
152C2×#117—31no—
153C2×#117C2×D311yescharacteristic
154C22×C4⋊C4—31no—
155#482—24no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107C2×D311yescharacteristic
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102C2×D311yescharacteristic
166C2×#102C2×D311yescharacteristic
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170#394—61no—
171#392—61no—
172#390—31no—
173#390—61no—
174#389—31no—
175#388—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#382—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3C2×D411yescharacteristic
195C23×D3C2×D411yescharacteristic
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—42no—
203C2×C3×D4—42no—
204C2×C3×D4—81no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C2×#67—22no—
217C2×#67—41no—
218C2×#67—41no—
219C2×#67—42no—
220C2×#67—81no—
221C2×#67—81no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—81no—
236C2×C4×D3#4112yes—
237C2×C4×D3#4112yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2×D412yes—
249C3×C4⋊C4C2×D412yes—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—42no—
252C3×C42—42no—
253#217—22no—
254#217—42no—
255#217—42no—
256#217—81no—
257#217—81no—
258#217—81no—
259(C2×D3)⋊C4—22no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—81no—
271Dic3⋊C4—22no—
272Dic3⋊C4—42no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—42no—
276C4×Dic3—81no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8#4112yes—
283ΓL(1,𝔽25)—28no—
284C22×D4—21no—
285C22×D4—31no—
286C22×D4—61no—
287C22×D4C22×D311yescharacteristic
288C23×C4—31no—
289C23×C4—61no—
290C23×C4C22×D311yescharacteristic
291C2×M16—62no—
292C2×M16—62no—
293C2×M16—62no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298#121—22no—
299#121—41no—
300#121—62no—
301#121—62no—
302C4×D4—62no—
303C4×D4—62no—
304#117—22no—
305#117—41no—
306#117—62no—
307#117—121no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—62no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—121no—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—31no—
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×C22⋊C4—61no—
331C2×C42—21no—
332C2×C42—31no—
333C2×C42—61no—
334C2×C42—61no—
335#107—62no—
336#107—121no—
337#107#6714yescomplemented
338#106—22no—
339#106—62no—
340#106—62no—
341#102—22no—
342#102—22no—
343#102—62no—
344#102—121no—
345C22⋊C8—22no—
346C22⋊C8—62no—
347C22⋊C8—62no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3C22×D411yescharacteristic
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3#12312yes—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—84no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
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×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404#67—44no—
405#67—84no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
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—
426C2×Dic3#12312yes—
427C4×D3—22no—
428C4×D3—22no—
429C4×D3—22no—
430C4×D3—22no—
431C4×D3—84no—
432C3×C8—22no—
433C3×C8—41no—
434C3⋊C8—22no—
435C3⋊C8—41no—
436C24—21no—
437C24—31no—
438C24D3×D411yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—62no—
448C2×D4—81no—
449C2×D4—121no—
450C2×D4—121no—
451C2×D4—122no—
452C2×D4—241no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—32no—
465C22×C4—32no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—122no—
477C22×C4—122no—
478C22×C4—241no—
479C22×C4D3×D411yescharacteristic
480C22×C4C23×D311yescharacteristic
481M16—68no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—32no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8C2×#6712yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—62no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—122no—
508C4⋊C4—241no—
509C4⋊C4C2×#6712yes—
510C4⋊C4C2×#6712yes—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—62no—
514C22⋊C4—62no—
515C22⋊C4—122no—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C22⋊C4—241no—
522C42—42no—
523C42—61no—
524C42—61no—
525C42—122no—
526C42—241no—
527C22×C3—41no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3—81no—
541C22×C3C2×#12311yescharacteristic
542C22×C3D4211yescharacteristic
543C22×C3#54711yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57511yescharacteristic
546C22×C3#57611yescharacteristic
547C22×C3#57711yescharacteristic
548C2×D3—22no—
549C2×D3—22no—
550C2×D3—22no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C3×C4—41no—
556C3×C4—42no—
557C3×C4—42no—
558C3×C4—81no—
559C3×C4#52511yescharacteristic
560C3×C4#53911yescharacteristic
561C3×C4#54811yescharacteristic
562C3×C4#55311yescharacteristic
563Dic3—22no—
564Dic3—41no—
565Dic3—41no—
566Dic3—81no—
567Dic3—81no—
568Dic3—81no—
569Dic3—81no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—32no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23C2×D3×D411yescharacteristic
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—84no—
597D4—124no—
598D4—244no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—32no—
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—61no—
617C2×C4—61no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—244no—
641C2×C4#100411yescharacteristic
642C2×C4D3×#4111yescharacteristic
643C2×C4D3×#4111yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C2×C4#101011yescharacteristic
646C2×C4C22×#6711yescharacteristic
647C8—22no—
648C8—41no—
649C8—62no—
650C8—121no—
651C2×C3—81no—
652C2×C3—161no—
653C2×C3#244311yescharacteristic
654C2×C3#309311yescharacteristic
655C2×C3#309611yescharacteristic
656C2×C3#327111yescharacteristic
657C2×C3#327311yescharacteristic
658C2×C3#334711yescharacteristic
659C2×C3#335111yescharacteristic
660D3—42no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—62no—
669C22—62no—
670C22—62no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—121no—
679C22—121no—
680C22—121no—
681C22—121no—
682C22D3×#12311yescharacteristic
683C22#565511yescharacteristic
684C22#576811yescharacteristic
685C22#577011yescharacteristic
686C22#577311yescharacteristic
687C22#577411yescharacteristic
688C22D4×#6711yescharacteristic
689C4—41no—
690C4—42no—
691C4—42no—
692C4—62no—
693C4—81no—
694C4—121no—
695C4—121no—
696C4—241no—
697C4—241no—
698C4—241no—
699C4—241no—
700C4#566011yescharacteristic
701C4#566111yescharacteristic
702C4#582811yescharacteristic
703C4#583111yescharacteristic
704C3#2615611yescharacteristic, complemented, Sylow
705C2—81no—
706C2—122no—
707C2—161no—
708C2#8136511yescharacteristic
709C2#8137211yescharacteristic
710C2#8137511yescharacteristic
711C2#8139611yescharacteristic
712C2#8289711yescharacteristic
713C2#8392311yescharacteristic
714C2#8396111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#482C2
socleC23×C3C22×D4
prime radical#82703C2
hypercenterC2×#482D3
hypocenterC3#26156
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
1. C2×C3×#482C2semidirect
3. #82272C2semidirect
4. C2×#4881C2semidirect
5. #71679C2semidirect
6. #71576C2semidirect
7. #70609C2semidirect
8. #70580C2semidirect
9. #70571C2semidirect
10. #70203C2semidirect
11. #70200C2semidirect
12. #70182C2semidirect
13. #70174C2semidirect
14. #69958C2semidirect
15. #69940C2semidirect
62. C2×#482D3semidirect
128. C3×#107D4semidirect
337. #107#67semidirect
704. C3#26156semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4822
#827032
#822722
C2×#48812
#716792
#715762
#706092
#705802
#705712
#702032
#702002
#701822
#701742
#699582
#699402
#261563