Finite Group Databasebeta

← #373792

Subgroups — #373792

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#14150C211yesmaximal, characteristic
6#11124C212yesmaximal
7#11067C212yesmaximal
8#10907C212yesmaximal
9#10884C212yesmaximal
10#10843C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#97C2211yescharacteristic
19#2355—24no—
20#2355—24no—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1947—24no—
32#1805C418yescomplemented
33#1763—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1442—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×C4×Q8D414yes—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—22no—
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×#107—24no—
73C2×#107—24no—
74C2×#107—42no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2×C414yes—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103C2×C418yes—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97—22no—
90#391—24no—
91#391—24no—
92#389—24no—
93#389—42no—
94#387—216no—
95#384—22no—
96#384D414yes—
97#382—28no—
98#382—22no—
99#382C2×C414yes—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#340—42no—
122#340C2×C418yes—
123#336—24no—
124#336—42no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8C22×C412yes—
137C22×Q8C2×D412yes—
138C22×Q8C2×Q812yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C22×C414yes—
169C22×C8#4112yes—
170C22×C8#4112yes—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8C4⋊C4116yescomplemented
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×Q812yes—
217C2×C4⋊C4#4112yes—
218C2×C4⋊C4#4112yes—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—28no—
229C2×C42—42no—
230C2×C42—82no—
231C2×C42—82no—
232C2×C42—82no—
233C2×C42C22×C412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×Q812yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239#107—48no—
240C4⋊C8—28no—
241#103—28no—
242#103—28no—
243#103—48no—
244#97—44no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—82no—
255C24C22×D411yescharacteristic
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C2×C4⋊C414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
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—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
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×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4C4×D412yes—
338C22×C4C4×D412yes—
339C22×C4#12012yes—
340C22×C4#12012yes—
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4#13714yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×Q811yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357C22×C42−1+411yescharacteristic
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—216no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4C2×C4⋊C414yes—
405C42—216no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—84no—
411C42—88no—
412C42C2×C4⋊C414yes—
413C42C4×D414yes—
414C42C22⋊Q814yes—
415C42#12314yes—
416C23C2×C4×D411yescharacteristic
417C23C2×#12011yescharacteristic
418C23C2×C22⋊Q811yescharacteristic
419C23#54611yescharacteristic
420C23#54711yescharacteristic
421C23#54811yescharacteristic
422C23#55011yescharacteristic
423C23C2×#13712yes—
424C23#57512yes—
425C23#57612yes—
426C23#57812yes—
427Q8—24no—
428Q8—24no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433Q8—42no—
434Q8—42no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#38514yes—
494C2×C4#38514yes—
495C2×C4C22×C4⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C4×#4111yescharacteristic
498C2×C4C4×#4111yescharacteristic
499C2×C4#51811yescharacteristic
500C2×C4#51911yescharacteristic
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4#53311yescharacteristic
506C2×C4#53311yescharacteristic
507C2×C4#53711yescharacteristic
508C2×C4#53711yescharacteristic
509C2×C4#54411yescharacteristic
510C2×C4#54411yescharacteristic
511C2×C4#54611yescharacteristic
512C2×C4#54711yescharacteristic
513C2×C4#54811yescharacteristic
514C2×C4#55011yescharacteristic
515C2×C4#55211yescharacteristic
516C2×C4#55211yescharacteristic
517C2×C4#55311yescharacteristic
518C2×C4#55311yescharacteristic
519C2×C4#55511yescharacteristic
520C2×C4#55511yescharacteristic
521C2×C4#55611yescharacteristic
522C2×C4#55611yescharacteristic
523C8—44no—
524C8—42no—
525C8—42no—
526C22C2×#38511yescharacteristic
527C22#236211yescharacteristic
528C22#236311yescharacteristic
529C22#240711yescharacteristic
530C22#240711yescharacteristic
531C22#243911yescharacteristic
532C22#244111yescharacteristic
533C22C4×#13712yes—
534C22#295112yes—
535C22#300411yescharacteristic
536C22#300511yescharacteristic
537C22#300811yescharacteristic
538C22#301111yescharacteristic
539C22#307812yes—
540C22#308312yes—
541C22#331312yes—
542C22#331412yes—
543C22#334612yes—
544C22#334912yes—
545C22#337012yes—
546C22#337412yes—
547C22#339812yes—
548C22#340312yes—
549C4—24no—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—82no—
558C4—84no—
559C4#229611yescharacteristic
560C4#231611yescharacteristic
561C4#236211yescharacteristic
562C4#236311yescharacteristic
563C4#243911yescharacteristic
564C4#244111yescharacteristic
565C4#248511yescharacteristic
566C4#248511yescharacteristic
567C2#1412911yescharacteristic
568C2#2087411yescharacteristic
569C2#2088111yescharacteristic
570C2#2307412yes—
571C2#2311112yes—
572C2#2413011yescharacteristic
573C2#2413511yescharacteristic
574C2#2419811yescharacteristic
575C2#2420311yescharacteristic
576C2#2603412yes—
577C2#2610112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #1805C4semidirect
174. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#19472
C2×#18052
C2×#17632
#141502
#111242
#110672
#109072
#108842
#108432
#108352