Finite Group Databasebeta

← #363576

Subgroups — #363576

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2346C211yesmaximal, characteristic
2#21049C212yesmaximal
3#11204C212yesmaximal, complemented
4#11198C212yesmaximal
5#10913C212yesmaximal, complemented
6#10847C211yesmaximal, characteristic, complemented
7#10802C211yesmaximal, characteristic, complemented
8#10802C211yesmaximal, characteristic, complemented
9#10728C211yesmaximal, characteristic
10#10728C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C2×#423C2212yes—
14C2×#407C2212yes—
15C2×#407C2212yes—
16C2×C4×M16C2212yes—
17#2346—22no—
18#2346—22no—
19#2346—22no—
20#2346—22no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388C412yes—
24C2×#388C412yes—
25C2×#382C412yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×#376C412yes—
31#1924—22no—
32#1924—22no—
33#1923—22no—
34#1923—22no—
35C2×#343C2211yescharacteristic
36C2×#343C2211yescharacteristic
37C2×#343C2212yes—
38C2×#343C2211yescharacteristic
39C2×#343C2211yescharacteristic
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2212yes—
45#1728C2212yes—
46#1728C2212yes—
47C24×C4C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2312yes—
50C22×M16C2312yes—
51C22×M16C2312yes—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—24no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C42C2×C412yes—
68C22×C42C2×C412yes—
69C22×C42C2×C412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#423—24no—
73#423—42no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76#407—24no—
77#407—24no—
78#407—42no—
79#407—42no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C4×M16—24no—
85C4×M16—42no—
86C2×#97—22no—
87C2×C4×C8—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#388—24no—
93#388—24no—
94#388—24no—
95#388—24no—
96#382—24no—
97#382—24no—
98#380—22no—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C22⋊C4D412yes—
105C4×C22⋊C4D412yes—
106C4×C22⋊C4D412yes—
107C4×C22⋊C4D412yes—
108#376—22no—
109#376—22no—
110#376—22no—
111#376—22no—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119#343—24no—
120#343—24no—
121#343—22no—
122#343—22no—
123#343—24no—
124#343—24no—
125#343—22no—
126#343—22no—
127#343—42no—
128#343—42no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C25C22×C411yescharacteristic
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2411yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16#4114yes—
178C2×M16#4114yes—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183#117—28no—
184#117—28no—
185#117—28no—
186#117—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4#4112yes—
205C2×C4⋊C4#4112yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C4⋊C8—44no—
239C4⋊C8—44no—
240C22⋊C8—44no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—44no—
244#97—44no—
245C4×C8—44no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—24no—
253#95—24no—
254#95C22⋊C411yescharacteristic
255#95C22⋊C411yescharacteristic
256#95C22⋊C411yescharacteristic
257#95C22⋊C411yescharacteristic
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24C2×C22⋊C412yes—
271C24C23×C411yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4#11714yes—
343C22×C4#11714yes—
344C22×C4C22×D412yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C42−1+411yescharacteristic
353C22×C42−1+411yescharacteristic
354M16—24no—
355M16—24no—
356M16—24no—
357M16—24no—
358M16—42no—
359M16—42no—
360M16—42no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16—42no—
365M16—42no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16—42no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C22⋊C4—216no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—84no—
398C42—28no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—44no—
409C42—44no—
410C42—44no—
411C42—44no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23C22×C22⋊C411yescharacteristic
440C23C2×#11711yescharacteristic
441C23C2×#11711yescharacteristic
442C23C4×#4111yescharacteristic
443C23C4×#4111yescharacteristic
444C23#52011yescharacteristic
445C23#52011yescharacteristic
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#41312yes—
509C2×C4#41312yes—
510C2×C4#52512yes—
511C2×C4#52512yes—
512C2×C4#52912yes—
513C2×C4#52912yes—
514C2×C4#54011yescharacteristic
515C2×C4#54111yescharacteristic
516C2×C4#54211yescharacteristic
517C2×C4#54211yescharacteristic
518C2×C4#54612yes—
519C2×C4#54612yes—
520C2×C4#54712yes—
521C2×C4#54712yes—
522C2×C4#55212yes—
523C2×C4#55312yes—
524C2×C4#55512yes—
525C2×C4#55612yes—
526C8—44no—
527C8—44no—
528C8—82no—
529C8—82no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22#228011yescharacteristic
554C22#229811yescharacteristic
555C22#231411yescharacteristic
556C22#233911yescharacteristic
557C22#233911yescharacteristic
558C22#234611yescharacteristic
559C22#234711yescharacteristic
560C22C2×#41311yescharacteristic
561C22C2×#41311yescharacteristic
562C22#288811yescharacteristic
563C22#288911yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—44no—
574C4—82no—
575C4—82no—
576C4#247012yes—
577C4#247412yes—
578C4#247612yes—
579C4#247812yes—
580C2—21no—
581C2—21no—
582C2—21no—
583C2—21no—
584C2—82no—
585C2#1394311yescharacteristic
586C2#2072811yescharacteristic
587C2#2073111yescharacteristic
588C2#2248011yescharacteristic
589C2#2248011yescharacteristic
590C2#2248311yescharacteristic
591C2#2248311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11204C2semidirect
5. #10913C2semidirect
6. #10847C2semidirect
7. #10802C2semidirect
8. #10802C2semidirect
11. C4×#343C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23462
#210492
#112042
#111982
#109132
#108472
#108022
#107282
C4×#3432