Finite Group Databasebeta

← #10824852

Subgroups — #10824852

Order 768.

876 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2323C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic, complemented
3#82236C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic, complemented
5#71699C211yesmaximal, characteristic
6#71694C211yesmaximal, characteristic, complemented
7C4×#835C211yesmaximal, characteristic, complemented
8#69683C211yesmaximal, characteristic, complemented
9#69680C211yesmaximal, characteristic, complemented
10#69629C211yesmaximal, characteristic, complemented
11#69622C211yesmaximal, characteristic, complemented
12#69523C211yesmaximal, characteristic
13#69519C211yesmaximal, characteristic, complemented
14#69481C211yesmaximal, characteristic, complemented
15C4×#809C211yesmaximal, characteristic, complemented
16#23715—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#930C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#391C2211yescharacteristic
24C3×#389—21no—
25C3×#389C2211yescharacteristic
26C3×#388—21no—
27C3×#388—21no—
28C3×#387—21no—
29C3×#380C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5212C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5026—24no—
38#5000—24no—
39#4990—21no—
40#4989—21no—
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4957—21no—
48#4956—21no—
49#4939C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×#803C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#440—31no—
63C2×#433—31no—
64#2324—31no—
65#2323D311yescharacteristic, complemented
66#2086—31no—
67#2076—31no—
68#2068—31no—
69#2065—31no—
70#2055—31no—
71#2035—31no—
72#2033—31no—
73#2000—31no—
74#1776—31no—
75C4×#103—31no—
76C4×#102—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×#67—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#215—22no—
106C2×#215—22no—
107C2×#215—22no—
108#930—22no—
109#930—22no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C4×D12—22no—
124C4×D12—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131#835—22no—
132#835—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#830—22no—
138#809—22no—
139#809—22no—
140#809—22no—
141#808—22no—
142#807—22no—
143#807—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—22no—
146#803D414yescomplemented
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C2×C22⋊Q8C2×D311yescharacteristic
150C2×#121—31no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×D4—31no—
153C2×#117C2×D311yescharacteristic
154#440—64no—
155#433—64no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#391C2×D311yescharacteristic
172#390—31no—
173#390—61no—
174#389—21no—
175#389—31no—
176#389C2×D311yescharacteristic
177#388—21no—
178#388—21no—
179#388—61no—
180#387—21no—
181#386—61no—
182#385—61no—
183#380—31no—
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186C4×C4⋊C4C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3C2×D411yescharacteristic
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2411yescharacteristic
217C2×#67—24no—
218C2×#67—81no—
219C2×#67—81no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12#4112yes—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
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—81no—
242C3×C4⋊C4#4112yes—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—81no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250C3×C42—81no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#217—81no—
254#215—28no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3#4112yes—
265Dic3⋊C4—81no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8C2×D412yes—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4—61no—
282C23×C4—61no—
283C23×C4C22×D311yescharacteristic
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—62no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8—31no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293#121—62no—
294#121—62no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×D4—62no—
298C4×D4—62no—
299#117—22no—
300#117—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4#6711yescharacteristic
309C2×C4⋊C4#6711yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4#6711yescharacteristic
321C2×C22⋊C4#6711yescharacteristic
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—61no—
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—62no—
329#106—62no—
330C4⋊C8—62no—
331C4⋊C8—62no—
332#103—62no—
333#103—62no—
334#103—62no—
335#102—62no—
336#102—62no—
337#102—62no—
338C22⋊C8—62no—
339C22⋊C8—62no—
340#97—34no—
341C4×C8—62no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
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×C4#12112yes—
399C2×C3×C4#12112yes—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406#67—84no—
407C2×Dic3—22no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416D12—22no—
417D12—22no—
418D12—41no—
419D12—41no—
420C4×D3—84no—
421C3⋊C8—22no—
422C3⋊C8—22no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C24—61no—
426C24C2×#6711yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8#23512yes—
433C2×D4—32no—
434C2×D4—64no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—241no—
441C2×D4—241no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—122no—
456C22×C4—241no—
457C22×C4—241no—
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—68no—
465C2×C8—32no—
466C2×C8—32no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—32no—
486C4⋊C4—42no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—81no—
493C4⋊C4—241no—
494C4⋊C4#23512yes—
495C22⋊C4—24no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—64no—
499C22⋊C4—81no—
500C22⋊C4—122no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—42no—
507C42—81no—
508C42—241no—
509C42D3×D412yes—
510C42#23712yes—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#52511yescharacteristic
520C22×C3#52511yescharacteristic
521C22×C3#57611yescharacteristic
522C22×C3#57711yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57811yescharacteristic
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—41no—
536C3×C4—42no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—81no—
540C3×C4—81no—
541C3×C4#52511yescharacteristic
542C3×C4#52911yescharacteristic
543C3×C4#52911yescharacteristic
544C3×C4#53211yescharacteristic
545Dic3—42no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—62no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C22×#6711yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567D4—62no—
568D4—62no—
569D4—121no—
570D4—121no—
571D4—244no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—62no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—244no—
611C2×C4#93912yes—
612C2×C4#93912yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×#23511yescharacteristic
615C2×C4C2×D3×D411yescharacteristic
616C2×C4C2×#23711yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4D3×#4111yescharacteristic
619C8—62no—
620C8—62no—
621C8—121no—
622C8—121no—
623C2×C3—81no—
624C2×C3#239211yescharacteristic
625C2×C3#306211yescharacteristic
626C2×C3#306311yescharacteristic
627C2×C3#311811yescharacteristic
628C2×C3#312011yescharacteristic
629C2×C3#312411yescharacteristic
630C2×C3#312611yescharacteristic
631D3—82no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22#579111yescharacteristic
647C22C2×#93911yescharacteristic
648C22#582811yescharacteristic
649C22#583511yescharacteristic
650C22#583711yescharacteristic
651C22#583711yescharacteristic
652C22#583811yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—41no—
660C4—81no—
661C4—81no—
662C4—122no—
663C4—241no—
664C4—241no—
665C4#553811yescharacteristic
666C4#554711yescharacteristic
667C4#554811yescharacteristic
668C4#555711yescharacteristic
669C3#2371511yescharacteristic, complemented, Sylow
670C2—81no—
671C2—242no—
672C2#8234511yescharacteristic
673C2#8255311yescharacteristic
674C2#8256911yescharacteristic
675C2#8340011yescharacteristic
676C2#8342211yescharacteristic
677C2#8344611yescharacteristic
678C2#8346011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2323D3
hypocenterC3#23715
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C3×#2323C2semidirect
2. C2×#5026C2semidirect
4. C2×#5000C2semidirect
6. #71694C2semidirect
7. C4×#835C2semidirect
8. #69683C2semidirect
9. #69680C2semidirect
10. #69629C2semidirect
11. #69622C2semidirect
13. #69519C2semidirect
14. #69481C2semidirect
15. C4×#809C2semidirect
65. #2323D3semidirect
146. #803D4semidirect
669. C3#23715semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
C2×#50262
#822362
C2×#50002
#716992
#716942
C4×#8352
#696832
#696802
#696292
#696222
#695232
#695192
#694812
C4×#8092
#237153