Finite Group Databasebeta

← #363604

Subgroups — #363604

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2#21023C212yesmaximal, complemented
3#11354C212yesmaximal, complemented
4#11044C212yesmaximal
5#10962C212yesmaximal
6#10847C211yesmaximal, characteristic, complemented
7#10802C211yesmaximal, characteristic, complemented
8#10802C211yesmaximal, characteristic, complemented
9#10762C211yesmaximal, characteristic
10#10762C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C412yes—
14C2×#423C2212yes—
15C22×#100C2212yes—
16C22×#100C2212yes—
17C2×C4×M16C2212yes—
18#2354—22no—
19#2354—22no—
20#2354—22no—
21#2354—22no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#388C412yes—
25C2×#388C412yes—
26C2×#384C2211yescharacteristic
27C2×#380C412yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1917—22no—
32#1917—22no—
33#1916—22no—
34#1916—22no—
35C2×#343C2211yescharacteristic
36C2×#343C2211yescharacteristic
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1728C2212yes—
44#1728C2212yes—
45#1728C2212yes—
46#1728C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4C2×C412yes—
49C22×M16C2312yes—
50C22×M16C2312yes—
51C22×M16C2312yes—
52C22×M16C2312yes—
53C2×C4×D4—24no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—48no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C42C2×C412yes—
76C22×C42C2×C412yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#423—24no—
81#423—42no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#100—24no—
85C2×#100—24no—
86C2×#100—24no—
87C2×#100—24no—
88C2×#100—42no—
89C2×#100—42no—
90C4×M16—24no—
91C4×M16—42no—
92C2×#97—22no—
93C2×C4×C8—22no—
94#390—22no—
95#390—22no—
96#390—22no—
97#390—22no—
98#388—24no—
99#388—24no—
100#388—24no—
101#388—24no—
102#384—22no—
103#384—22no—
104#380—24no—
105#380—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118#343—24no—
119#343—24no—
120#343—22no—
121#343—22no—
122#343—22no—
123#343—22no—
124#343—42no—
125#343—42no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134C25C22×C412yes—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4#4114yes—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C412yes—
160C23×C4C2411yescharacteristic
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—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16#4114yes—
184C2×M16#4114yes—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193#117—48no—
194#117—48no—
195#117—48no—
196#117—48no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4#4112yes—
221C2×C4⋊C4#4112yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C4⋊C8—44no—
257C4⋊C8—44no—
258#100—48no—
259#100—48no—
260#97—44no—
261C4×C8—44no—
262#95—24no—
263#95—24no—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24#11714yes—
279C24C23×C411yescharacteristic
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—44no—
331C22×C4—416no—
332C22×C4—416no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4#11714yes—
352C22×C4C22×D412yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C42−1+411yescharacteristic
361C22×C42−1+411yescharacteristic
362M16—24no—
363M16—24no—
364M16—24no—
365M16—24no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16—42no—
370M16—42no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375M16—42no—
376M16—42no—
377M16—42no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—84no—
404C4⋊C4#10014yescomplemented
405C22⋊C4—216no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—44no—
425C42—44no—
426C42—44no—
427C42—44no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23C22×C22⋊C411yescharacteristic
456C23C2×#11711yescharacteristic
457C23C2×#11711yescharacteristic
458C23C4×#4111yescharacteristic
459C23C4×#4111yescharacteristic
460C23#52011yescharacteristic
461C23#52011yescharacteristic
462D4—44no—
463D4—44no—
464D4—44no—
465D4—44no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—416no—
497C2×C4—416no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4C2×#10012yes—
519C2×C4C2×#10012yes—
520C2×C4C2×#10012yes—
521C2×C4#41312yes—
522C2×C4#52512yes—
523C2×C4#52512yes—
524C2×C4#52912yes—
525C2×C4#52912yes—
526C2×C4#54011yescharacteristic
527C2×C4#54111yescharacteristic
528C2×C4#54211yescharacteristic
529C2×C4#54211yescharacteristic
530C2×C4#54612yes—
531C2×C4#54712yes—
532C2×C4#54812yes—
533C2×C4#55012yes—
534C2×C4#55112yes—
535C2×C4#55112yes—
536C2×C4#55212yes—
537C2×C4#55212yes—
538C8—44no—
539C8—44no—
540C8—82no—
541C8—82no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22#228011yescharacteristic
566C22#229811yescharacteristic
567C22#231411yescharacteristic
568C22#233911yescharacteristic
569C22#234111yescharacteristic
570C22#234311yescharacteristic
571C22#234311yescharacteristic
572C22C22×#10011yescharacteristic
573C22C2×#41311yescharacteristic
574C22#288811yescharacteristic
575C22#288911yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—42no—
584C4—44no—
585C4—42no—
586C4—82no—
587C4—82no—
588C4#247112yes—
589C4#247512yes—
590C4#247712yes—
591C4#247912yes—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—84no—
597C2#1394411yescharacteristic
598C2#2072611yescharacteristic
599C2#2073111yescharacteristic
600C2#2247911yescharacteristic
601C2#2248311yescharacteristic
602C2#2253611yescharacteristic
603C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21023C2semidirect
3. #11354C2semidirect
6. #10847C2semidirect
7. #10802C2semidirect
8. #10802C2semidirect
11. C4×#343C2semidirect
404. C4⋊C4#100semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#210232
#113542
#110442
#109622
#108472
#108022
#107622
C4×#3432