Finite Group Databasebeta

← #360452

Subgroups — #360452

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1933C212yesmaximal
2C2×#1805C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic
6#13942C211yesmaximal, characteristic
7#11292C212yesmaximal
8#11067C212yesmaximal
9#10880C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10842C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×C4×C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1933—28no—
27#1805—24no—
28C8×C4⋊C4—24no—
29C4×#103C418yescomplemented
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#336C2212yes—
38#1444—22no—
39#1441—22no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2212yes—
43#1434—21no—
44#1434—21no—
45C23×Q8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8D414yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103C2×C414yes—
77C2×#103C2×C414yes—
78C2×#103D414yes—
79C2×#103Q814yes—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—24no—
83C2×C4×C8—24no—
84C2×C4×C8—42no—
85C2×C4×C8C2×C414yes—
86#391—24no—
87#391—42no—
88#389—24no—
89#389—24no—
90#389—42no—
91#389—42no—
92#382—22no—
93#382—22no—
94#382—22no—
95#382—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C414yes—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340C2×C414yes—
119#340C2×C414yes—
120#336—24no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8C22×C412yes—
133C22×Q8C2×D412yes—
134C22×Q8C2×Q812yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8C22×C412yes—
163C22×C8C22×C412yes—
164C22×C8C2×D414yes—
165C22×C8C2×D412yes—
166C22×C8C2×Q812yes—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—44no—
170C4×Q8—44no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C4⋊C4#4112yes—
212C2×C4⋊C4#4112yes—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—82no—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230C4⋊C8—28no—
231#103—28no—
232#103—48no—
233#103C4⋊C4116yescomplemented
234C4×C8—28no—
235C4×C8—48no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—44no—
248#95—44no—
249C24C22×D411yescharacteristic
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
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—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8C2×C4⋊C414yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
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—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
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×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12412yes—
335C22×C4#12412yes—
336C22×C4#13514yes—
337C22×C4#13514yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C42−1+411yescharacteristic
351C22×C42−1+411yescharacteristic
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8C2×C4⋊C414yes—
373C4⋊C4—216no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4C2×C4⋊C414yes—
396C4⋊C4#11714yes—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—48no—
402C42—84no—
403C42#11714yes—
404C42C4×D414yes—
405C42#12314yes—
406C42#12614yes—
407C23C2×C4×D411yescharacteristic
408C23C2×#12111yescharacteristic
409C23C2×#12411yescharacteristic
410C23D4211yescharacteristic
411C23#54611yescharacteristic
412C23#54811yescharacteristic
413C23D4×Q811yescharacteristic
414C23C2×#13512yes—
415C23C2×#13512yes—
416C23#57712yes—
417C23#57812yes—
418Q8—24no—
419Q8—24no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
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—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#38814yes—
485C2×C4#38814yes—
486C2×C4C22×C4⋊C411yescharacteristic
487C2×C4C2×#11711yescharacteristic
488C2×C4C2×#11711yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4#52011yescharacteristic
491C2×C4#52011yescharacteristic
492C2×C4C2×#12311yescharacteristic
493C2×C4C2×#12611yescharacteristic
494C2×C4#53211yescharacteristic
495C2×C4#53211yescharacteristic
496C2×C4#53311yescharacteristic
497C2×C4#53311yescharacteristic
498C2×C4#54011yescharacteristic
499C2×C4#54111yescharacteristic
500C2×C4#54211yescharacteristic
501C2×C4#54211yescharacteristic
502C2×C4D4211yescharacteristic
503C2×C4#54711yescharacteristic
504C2×C4#54711yescharacteristic
505C2×C4#54711yescharacteristic
506C2×C4D4×Q811yescharacteristic
507C2×C4#55011yescharacteristic
508C2×C4#55211yescharacteristic
509C2×C4#55311yescharacteristic
510C2×C4#55411yescharacteristic
511C2×C4#55411yescharacteristic
512C2×C4#55511yescharacteristic
513C2×C4#55611yescharacteristic
514C8—24no—
515C8—42no—
516C8—44no—
517C22C2×#38811yescharacteristic
518C22D4×C4⋊C411yescharacteristic
519C22#235211yescharacteristic
520C22#242811yescharacteristic
521C22#242911yescharacteristic
522C22#249111yescharacteristic
523C22#249211yescharacteristic
524C22C4×#13511yescharacteristic
525C22C4×#13511yescharacteristic
526C22#294911yescharacteristic
527C22#295011yescharacteristic
528C22#305012yes—
529C22#306312yes—
530C22#313112yes—
531C22#313712yes—
532C22#328212yes—
533C22#328512yes—
534C22#329612yes—
535C22#330012yes—
536C22#336012yes—
537C22#336412yes—
538C22#338112yes—
539C22#338312yes—
540C4—24no—
541C4—24no—
542C4—44no—
543C4—44no—
544C4—44no—
545C4—44no—
546C4—44no—
547C4—82no—
548C4—162no—
549C4—162no—
550C4#228511yescharacteristic
551C4#231411yescharacteristic
552C4D4×C4⋊C411yescharacteristic
553C4#235411yescharacteristic
554C4#242211yescharacteristic
555C4#242311yescharacteristic
556C4#249611yescharacteristic
557C4#249711yescharacteristic
558C2#1387611yescharacteristic
559C2#2108311yescharacteristic
560C2#2108811yescharacteristic
561C2#2268311yescharacteristic
562C2#2268511yescharacteristic
563C2#2276811yescharacteristic
564C2#2276911yescharacteristic
565C2#2538912yes—
566C2#2547412yes—
567C2#2777212yes—
568C2#2781812yes—
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
29. C4×#103C4semidirect
233. #103C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19332
C2×#18052
C2×C8×C4⋊C42
C2×C4×#1032
#139542
#139422
#112922
#110672
#108802
#108502
#108422