Finite Group Databasebeta

← #356863

Subgroups — #356863

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×C8×C4⋊C4C211yesmaximal, characteristic
5C4⋊C42C211yesmaximal, characteristic
6#10884C212yesmaximal
7#10880C212yesmaximal
8#10849C211yesmaximal, characteristic
9#10806C212yesmaximal
10C4×#340C212yesmaximal
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2365—24no—
18#2365C418yescomplemented
19C2×#391—21no—
20C2×#389C2212yes—
21C2×#387—22no—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990C418yescomplemented
30#1988C418yescomplemented
31C8×C4⋊C4—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1436—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8C2×C414yes—
56C2×C4×Q8—28no—
57C2×C4×Q8—44no—
58C2×C4×Q8—44no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—22no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2×C414yes—
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—28no—
87#389—24no—
88#389C2×C418yes—
89#387—216no—
90#384—22no—
91#384C2×C414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—42no—
97C4×C4⋊C4—44no—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4C2×C418yes—
100C4×C4⋊C4C2×C414yes—
101C4×C4⋊C4C2×C414yes—
102C4×C4⋊C4D414yes—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D412yes—
111#340—22no—
112#340—22no—
113#340—24no—
114#340—22no—
115#340—42no—
116#340C2×C418yes—
117#340C2×C418yes—
118#340C2×C418yes—
119#340C2×C418yes—
120#340Q814yescomplemented
121#336—24no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8C22×C412yes—
157C22×C8C22×C414yes—
158C22×C8C22×C412yes—
159C22×C8C2×Q812yes—
160C22×C8#4112yes—
161C4⋊Q8—28no—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8C42116yescomplemented
165C4×Q8—432no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
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—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4C22⋊C414yes—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4D814yes—
207C2×C4⋊C4SD1614yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C414yes—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C22×C414yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C2×Q812yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4112yes—
219C2×C4⋊C4#4112yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—28no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—44no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—44no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—82no—
236C2×C42C22×C414yes—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C22×C412yes—
240C2×C42C2×D414yes—
241C2×C42C2×D412yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C4⋊C8—28no—
245C4×C8—28no—
246C4×C8—48no—
247#95—24no—
248#95—28no—
249#95—28no—
250#95—24no—
251#95—44no—
252#95—44no—
253#95—82no—
254#95C22⋊C418yes—
255C24C22×D411yescharacteristic
256C2×Q8—216no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—48no—
262C2×Q8—48no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
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—24no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4C2×C22⋊C414yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C414yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4C2×D814yes—
328C22×C4C2×D812yes—
329C22×C4C2×SD1614yes—
330C22×C4C2×SD1612yes—
331C22×C4#13512yes—
332C22×C4#13512yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C23×C411yescharacteristic
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C42−1+411yescharacteristic
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8C4×Q814yescomplemented
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—216no—
378C4⋊C4—28no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—84no—
394C4⋊C4#102116yes—
395C4⋊C4C2×C4218yes—
396C4⋊C4C4×Q814yes—
397C4⋊C4C4×Q814yescomplemented
398C42—24no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—416no—
403C42—48no—
404C42—48no—
405C42—88no—
406C42C2×C4214yes—
407C42#11714yes—
408C42C4×D414yes—
409C42C4×D414yes—
410C23C22×C22⋊C411yescharacteristic
411C23C2×C4×D411yescharacteristic
412C23C2×C4×D411yescharacteristic
413C23#54611yescharacteristic
414C23#54711yescharacteristic
415C23#54811yescharacteristic
416C23D4×Q811yescharacteristic
417C23C22×D811yescharacteristic
418C23C22×SD1611yescharacteristic
419C23C2×#13511yescharacteristic
420C23C2×#13511yescharacteristic
421C23#57711yescharacteristic
422C23#57711yescharacteristic
423C23#57811yescharacteristic
424C23#57811yescharacteristic
425Q8—432no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4C4×C22⋊C414yes—
488C2×C4C4×C22⋊C414yes—
489C2×C4C2×#10218yes—
490C2×C4C2×#10214yes—
491C2×C4#41614yes—
492C2×C4C4×D814yes—
493C2×C4C4×D814yes—
494C2×C4C4×SD1614yes—
495C2×C4C4×SD1614yes—
496C2×C4C22×C4211yescharacteristic
497C2×C4C2×#11711yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×C4×Q811yescharacteristic
501C2×C4C2×C4×Q811yescharacteristic
502C2×C4C2×C4×Q811yescharacteristic
503C2×C4C4×#4111yescharacteristic
504C2×C4C4×#4111yescharacteristic
505C2×C4C4×#4111yescharacteristic
506C2×C4#51911yescharacteristic
507C2×C4#51911yescharacteristic
508C2×C4#51911yescharacteristic
509C2×C4#52011yescharacteristic
510C2×C4#52011yescharacteristic
511C2×C4#52011yescharacteristic
512C2×C4#54611yescharacteristic
513C2×C4#54711yescharacteristic
514C2×C4#54811yescharacteristic
515C2×C4D4×Q811yescharacteristic
516C2×C4#55211yescharacteristic
517C2×C4#55211yescharacteristic
518C2×C4#55411yescharacteristic
519C2×C4#55611yescharacteristic
520C2×C4#55611yescharacteristic
521C2×C4#55611yescharacteristic
522C2×C4#55711yescharacteristic
523C2×C4#55911yescharacteristic
524C8—24no—
525C8—42no—
526C8—44no—
527C22C2×C4×C22⋊C411yescharacteristic
528C22#233911yescharacteristic
529C22Q8×C22⋊C411yescharacteristic
530C22#236011yescharacteristic
531C22#236211yescharacteristic
532C22#236511yescharacteristic
533C22#236711yescharacteristic
534C22C22×#10211yescharacteristic
535C22C2×#41611yescharacteristic
536C22#289811yescharacteristic
537C22#289811yescharacteristic
538C22C2×C4×D811yescharacteristic
539C22C2×C4×D811yescharacteristic
540C22C2×C4×SD1611yescharacteristic
541C22C2×C4×SD1611yescharacteristic
542C22#294711yescharacteristic
543C22#294711yescharacteristic
544C22#294911yescharacteristic
545C22#295011yescharacteristic
546C22#329411yescharacteristic
547C22#329811yescharacteristic
548C22#330211yescharacteristic
549C22#330411yescharacteristic
550C22#333411yescharacteristic
551C22#333511yescharacteristic
552C22#334011yescharacteristic
553C22#334211yescharacteristic
554C22#335811yescharacteristic
555C22#335911yescharacteristic
556C22#336411yescharacteristic
557C22#336511yescharacteristic
558C22Q8×D811yescharacteristic
559C22Q8×SD1611yescharacteristic
560C22#338111yescharacteristic
561C22#338311yescharacteristic
562C4—24no—
563C4—28no—
564C4—44no—
565C4—48no—
566C4—44no—
567C4—44no—
568C4—82no—
569C4—84no—
570C4—82no—
571C4C4×#10218yes—
572C4C42×Q811yescharacteristic
573C4#227611yescharacteristic
574C4#235511yescharacteristic
575C4#235611yescharacteristic
576C4#236011yescharacteristic
577C4#236211yescharacteristic
578C4#236511yescharacteristic
579C4#236711yescharacteristic
580C2#1381711yescharacteristic
581C2C2×C4×#10211yescharacteristic
582C2#2041711yescharacteristic
583C2#2248511yescharacteristic
584C2#2249011yescharacteristic
585C2Q8×#10211yescharacteristic
586C2#2252111yescharacteristic
587C2#2296211yescharacteristic
588C2#2296411yescharacteristic
589C2#2306011yescharacteristic
590C2#2306211yescharacteristic
591C2#2315611yescharacteristic
592C2#2315711yescharacteristic
593C2#2321411yescharacteristic
594C2#2321611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×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
18. #2365C4semidirect
29. #1990C4semidirect
30. #1988C4semidirect
120. #340Q8semidirect
164. C4⋊Q8C42semidirect
367. C2×C8C4×Q8semidirect
397. C4⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#19902
C2×#19882
C2×C8×C4⋊C42
C4⋊C422
#108842
#108802
#108492
#108062
C4×#3402