Finite Group Databasebeta

← #10845363

Subgroups — #10845363

Order 768.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2562C211yesmaximal, characteristic
2C2×#5063C211yesmaximal, characteristic, complemented
3#82375C211yesmaximal, characteristic
4C2×D4×C3⋊C8C211yesmaximal, characteristic
5#71740C211yesmaximal, characteristic, complemented
6#69527C211yesmaximal, characteristic, complemented
7#69526C211yesmaximal, characteristic, complemented
8#69518C211yesmaximal, characteristic, complemented
9#69514C211yesmaximal, characteristic, complemented
10#69514C211yesmaximal, characteristic, complemented
11#69512C211yesmaximal, characteristic, complemented
12#69511C211yesmaximal, characteristic, complemented
13#69507C211yesmaximal, characteristic, complemented
14C4×#810C211yesmaximal, characteristic, complemented
15#69468C211yesmaximal, characteristic, complemented
16#25343—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#924C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#400—21no—
25C3×#394—21no—
26C3×#388—21no—
27C3×#388—21no—
28C3×#380C2211yescharacteristic
29C3×#380C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33#5222—21no—
34#5212C2211yescharacteristic
35#5212C2211yescharacteristic
36C2×#830C2211yescharacteristic
37C2×#830C2211yescharacteristic
38#5063—24no—
39D4×C3⋊C8—24no—
40#4983—21no—
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4944—21no—
50#4944—21no—
51#4936C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4×C3⋊C8C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×C8×D4—31no—
64#2562D311yescharacteristic
65#2536—31no—
66#2077—31no—
67#2075—31no—
68#2036—31no—
69#2035—31no—
70#1991—31no—
71#1949—31no—
72#1892—31no—
73#1892—31no—
74#1808—31no—
75#1807—31no—
76C4×#103—31no—
77C22×C3×D4D411yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×#125—22no—
82C3×#125—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4D411yescharacteristic
97C2×C3×C22⋊C4D411yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107#924—22no—
108#924—22no—
109#924—41no—
110#924—41no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C22×C3⋊C8—22no—
116C22×C3⋊C8—22no—
117C22×C3⋊C8—22no—
118C22×C3⋊C8—22no—
119C22×C3⋊C8—41no—
120C22×C3⋊C8—41no—
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C4×Dic6—22no—
126C4×Dic6—22no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#830—22no—
137#830—22no—
138#810—22no—
139#810—22no—
140#810—22no—
141#810—22no—
142#808—22no—
143#808—41no—
144#807—22no—
145#807—41no—
146C4⋊(C3⋊C8)D414yes—
147C4×C3⋊C8—22no—
148C23×C8—31no—
149C23×C8—31no—
150C2×#125C2×D311yescharacteristic
151C2×C22⋊Q8—31no—
152C2×C22⋊Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4C2×D311yescharacteristic
155#479—64no—
156C8×D4—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×C4×C8—31no—
169#400—21no—
170#398—61no—
171#394—21no—
172#394—61no—
173#388—21no—
174#388—21no—
175#388—61no—
176#388—61no—
177#382—31no—
178#380—31no—
179#380—31no—
180#380C2×D311yescharacteristic
181#380C2×D311yescharacteristic
182#380C2×D311yescharacteristic
183#380C2×D311yescharacteristic
184C4×C4⋊C4C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3C2×D411yescharacteristic
194C24×C3C2×D411yescharacteristic
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
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×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C3×C4⋊C4—22no—
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—42no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—81no—
245C3×C22⋊C4—81no—
246C3×C42—81no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249#217—42no—
250#217—42no—
251#217—42no—
252#217—42no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—42no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—44no—
273C2×C3⋊C8—44no—
274C2×C3⋊C8C2×D412yes—
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8#4112yes—
277C22×Q8—31no—
278C22×D4#6711yescharacteristic
279C23×C4C22×D311yescharacteristic
280C23×C4C22×D311yescharacteristic
281C22×C8—31no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8—62no—
286C22×C8—62no—
287C22×C8—62no—
288C22×C8—62no—
289C22×C8—121no—
290C22×C8—121no—
291#125—22no—
292#125—22no—
293C22⋊Q8—62no—
294C22⋊Q8—62no—
295C22⋊Q8—121no—
296C22⋊Q8—121no—
297C4×Q8—62no—
298C4×Q8—62no—
299C4×D4—22no—
300C4×D4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4#6711yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4#6711yescharacteristic
323C2×C22⋊C4#6711yescharacteristic
324C2×C42—21no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—121no—
329#106—62no—
330#106—121no—
331C4⋊C8—34no—
332#103—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336C22⋊C8—62no—
337C22⋊C8—62no—
338C4×C8—62no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3#13512yes—
361C23×C3#13512yes—
362C23×C3C22×D411yescharacteristic
363C3×D4—28no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
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#12112yes—
395C2×C3×C4#12112yes—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402C2×Dic3—22no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414Dic6—22no—
415Dic6—22no—
416Dic6—41no—
417Dic6—41no—
418C3⋊C8—22no—
419C3⋊C8—22no—
420C3⋊C8—22no—
421C3⋊C8—41no—
422C24C2×#6711yescharacteristic
423C24C2×#6711yescharacteristic
424C2×Q8—32no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C23×D311yescharacteristic
458C2×C8—32no—
459C2×C8—32no—
460C2×C8—32no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—124no—
471C2×C8—124no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—122no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C42—81no—
503C42—122no—
504C42D3×D412yes—
505C42#23712yes—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3C2×#12111yescharacteristic
521C22×C3#54611yescharacteristic
522C22×C3#54711yescharacteristic
523C22×C3C2×#13511yescharacteristic
524C22×C3C2×#13511yescharacteristic
525C22×C3#57711yescharacteristic
526C22×C3#57811yescharacteristic
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4—81no—
535C3×C4#52911yescharacteristic
536C3×C4#54211yescharacteristic
537C3×C4#54611yescharacteristic
538C3×C4#54811yescharacteristic
539Dic3—42no—
540Dic3—81no—
541Dic3—81no—
542Dic3—81no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23#95012yes—
556C23#95012yes—
557C23C22×#6711yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—121no—
561Q8—121no—
562D4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
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—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—62no—
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#93912yes—
606C2×C4#93912yes—
607C2×C4C2×D3×D411yescharacteristic
608C2×C4C2×#23711yescharacteristic
609C2×C4D3×#4111yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C2×C4#100911yescharacteristic
612C2×C4#101011yescharacteristic
613C8—62no—
614C8—62no—
615C8—62no—
616C8—121no—
617C2×C3—42no—
618C2×C3—42no—
619C2×C3#242711yescharacteristic
620C2×C3#305011yescharacteristic
621C2×C3#306311yescharacteristic
622C2×C3#330011yescharacteristic
623C2×C3#330111yescharacteristic
624C2×C3#333611yescharacteristic
625C2×C3#333711yescharacteristic
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22C2×#93911yescharacteristic
641C22C2×#95011yescharacteristic
642C22C2×#95011yescharacteristic
643C22#582411yescharacteristic
644C22#582911yescharacteristic
645C22#583611yescharacteristic
646C22#583811yescharacteristic
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—122no—
656C4—241no—
657C4—241no—
658C4—241no—
659C4#567611yescharacteristic
660C4#567711yescharacteristic
661C4#568011yescharacteristic
662C4#568211yescharacteristic
663C3#2534311yescharacteristic, complemented, Sylow
664C2—42no—
665C2—42no—
666C2#8292211yescharacteristic
667C2#8299611yescharacteristic
668C2#8342311yescharacteristic
669C2#8359011yescharacteristic
670C2#8360211yescharacteristic
671C2#8360611yescharacteristic
672C2#8361811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2562C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2562D3
hypocenterC3#25343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2562

Complements

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

NH ≅ G/Ntype
2. C2×#5063C2semidirect
5. #71740C2semidirect
6. #69527C2semidirect
7. #69526C2semidirect
8. #69518C2semidirect
9. #69514C2semidirect
10. #69514C2semidirect
11. #69512C2semidirect
12. #69511C2semidirect
13. #69507C2semidirect
14. C4×#810C2semidirect
15. #69468C2semidirect
663. C3#25343semidirect

Maximal subgroups

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

subgroupindex
C3×#25622
C2×#50632
#823752
C2×D4×C3⋊C82
#717402
#695272
#695262
#695182
#695142
#695122
#695112
#695072
C4×#8102
#694682
#253433