Finite Group Databasebeta

← #369268

Subgroups — #369268

Order 512.

1544 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1959C211yesmaximal, characteristic
5#14273C211yesmaximal, characteristic
6#11067C212yesmaximal, complemented
7#11066C212yesmaximal
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10841C211yesmaximal, characteristic, complemented
11#10694C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C2×#488—24no—
15C2×#488—24no—
16C2×#488C418yescomplemented
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2330—24no—
24#2330—24no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389—22no—
30C2×#389C2212yes—
31C2×#387—22no—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×#376C2211yescharacteristic
36#1992—24no—
37#1959—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1439—22no—
45#1439—22no—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125C2×C414yes—
54C2×#125D414yes—
55C2×#125D414yes—
56C2×#124—22no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124C2×C414yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#488—232no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103C2×C418yes—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102C2×C418yes—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8C2×C414yes—
97#391—24no—
98#391—42no—
99#390—24no—
100#390—42no—
101#389—216no—
102#389—24no—
103#389—24no—
104#387—216no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384D414yes—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111#376—22no—
112#376—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95D412yes—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—42no—
123#340—42no—
124#336—24no—
125#336—24no—
126C25—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8C22×C412yes—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4C22×C412yes—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C22×C414yes—
170C22×C8#4114yes—
171#125—28no—
172#125—28no—
173#125—28no—
174#125C22⋊C4116yescomplemented
175#124—28no—
176#124—28no—
177#124—44no—
178#124—44no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—28no—
229C2×C42—82no—
230C2×C42—82no—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42#4114yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238#103—216no—
239#103—48no—
240#102—216no—
241#102—48no—
242#97—44no—
243C4×C8—28no—
244#95—28no—
245#95—28no—
246#95—24no—
247#95—24no—
248#95—42no—
249#95—42no—
250#95—44no—
251C24—22no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24C22×D411yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4C4×D414yes—
347C22×C4#12014yes—
348C22×C4#12114yes—
349C22×C4#12414yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—232no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—88no—
387C4⋊C4—88no—
388C4⋊C4—88no—
389C4⋊C4C2×C22⋊C418yes—
390C22⋊C4—416no—
391C22⋊C4—88no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—24no—
395C42—24no—
396C42—42no—
397C42—42no—
398C42—88no—
399C42C2×C22⋊C414yes—
400C42C4×D414yes—
401C42#12114yes—
402C42#12314yes—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—81no—
425C23—81no—
426C23C2×C4×D411yescharacteristic
427C23C2×#12011yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×#12411yescharacteristic
430C23#53211yescharacteristic
431C23#53511yescharacteristic
432C23#53611yescharacteristic
433C23#57612yes—
434C23#57712yes—
435C23#57712yes—
436C23#57812yes—
437Q8—44no—
438Q8—44no—
439Q8—82no—
440Q8—82no—
441D4—44no—
442D4—44no—
443D4—82no—
444D4—82no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4#38618yes—
500C2×C4C22×C22⋊C411yescharacteristic
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C4×#4112yes—
503C2×C4#52012yes—
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12311yescharacteristic
506C2×C4#52912yes—
507C2×C4#54212yes—
508C2×C4#54612yes—
509C2×C4#54812yes—
510C2×C4#54812yes—
511C2×C4#54812yes—
512C2×C4#55012yes—
513C2×C4#55012yes—
514C2×C4#55612yes—
515C2×C4#55912yes—
516C8—44no—
517C8—44no—
518C22—84no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—161no—
524C22—161no—
525C22—162no—
526C22—161no—
527C22—161no—
528C22—162no—
529C22C2×#38611yescharacteristic
530C22C4×#12411yescharacteristic
531C22#232611yescharacteristic
532C22#243111yescharacteristic
533C22#243611yescharacteristic
534C22#254511yescharacteristic
535C22#254811yescharacteristic
536C22#295112yes—
537C22#295212yes—
538C22#301212yes—
539C22#301312yes—
540C22#304512yes—
541C22#304612yes—
542C22#313612yes—
543C22#313712yes—
544C22#317812yes—
545C22#317912yes—
546C22#322011yescharacteristic
547C22#322111yescharacteristic
548C22#322211yescharacteristic
549C22#322311yescharacteristic
550C22#324312yes—
551C22#324412yes—
552C4—28no—
553C4—44no—
554C4—48no—
555C4—81no—
556C4—84no—
557C4—81no—
558C4—82no—
559C4—161no—
560C4—161no—
561C4#234112yes—
562C4#236712yes—
563C4#242612yes—
564C4#244812yes—
565C2—162no—
566C2#1402611yescharacteristic
567C2#2093712yes—
568C2#2164312yes—
569C2#2222711yescharacteristic
570C2#2222811yescharacteristic
571C2#2561111yescharacteristic
572C2#2561211yescharacteristic
573C2#2591612yes—
574C2#2866812yes—
575C2#2876011yescharacteristic
576C2#2876111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2330C2semidirect
3. C2×#1992C2semidirect
6. #11067C2semidirect
8. #10932C2semidirect
10. #10841C2semidirect
11. #10694C2semidirect
16. C2×#488C4semidirect
174. #125C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4882
C2×#23302
C2×#19922
C2×#19592
#142732
#110672
#110662
#109322
#109312
#108412
#106942