Finite Group Databasebeta

← #376037

Subgroups — #376037

Order 512.

1480 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#1920C212yesmaximal
4C4×#389C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11267C211yesmaximal, characteristic
7#11207C211yesmaximal, characteristic
8#10908C212yesmaximal
9#10885C212yesmaximal
10#10884C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#106C2212yes—
15C4×C4⋊Q8—28no—
16C2×#391—21no—
17C2×#389—21no—
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4—21no—
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#2058C418yescomplemented
29#1920—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4×#95—21no—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8C2×C414yes—
53C2×C4×Q8—28no—
54C2×C4×Q8—44no—
55C2×C4×Q8—44no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#106—24no—
79C2×#106—24no—
80C2×#106C2×C418yes—
81#391—28no—
82#389—28no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389C2×C414yes—
87#389D418yes—
88#389Q814yescomplemented
89#384—24no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—44no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C43—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340C2×C418yes—
115#336—24no—
116#336—22no—
117#336—24no—
118#336C2×C414yes—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8C22×C414yes—
153C4⋊Q8—216no—
154C4⋊Q8—216no—
155C4×Q8—432no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×Q812yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4#4112yes—
202C2×C4⋊C4#4114yes—
203C2×C4⋊C4#4112yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—28no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42—44no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—44no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42SD1614yes—
232C2×C42C22×C412yes—
233C2×C42C2×D414yes—
234C2×C42C2×Q812yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237#106—216no—
238#95—24no—
239#95—28no—
240#95—24no—
241#95—28no—
242#95—24no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248C24C22×D411yescharacteristic
249C2×Q8—216no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—48no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
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—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4C4×D412yes—
327C22×C4C4×D412yes—
328C22×C4#12012yes—
329C22×C4#12012yes—
330C22×C4C22⋊Q812yes—
331C22×C4C22⋊Q812yes—
332C22×C4C2×SD1612yes—
333C22×C4C2×SD1612yes—
334C22×C4C2×SD1612yes—
335C22×C4#13512yes—
336C22×C4AGL(1,ℤ/8)12yes—
337C22×C4AGL(1,ℤ/8)12yes—
338C22×C4#13712yes—
339C22×C4#13712yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C42−1+412yes—
349C22×C42−1+412yes—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—216no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—416no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4C4×D418yes—
387C4⋊C4C4×Q814yescomplemented
388C42—28no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—24no—
395C42—28no—
396C42—44no—
397C42—44no—
398C42—48no—
399C42—416no—
400C42—84no—
401C23C2×C4×D411yescharacteristic
402C23C2×#12011yescharacteristic
403C23C2×C22⋊Q811yescharacteristic
404C23#52511yescharacteristic
405C23#52511yescharacteristic
406C23#53611yescharacteristic
407C23#53611yescharacteristic
408C23C22×SD1611yescharacteristic
409C23C2×#13511yescharacteristic
410C23C2×AGL(1,ℤ/8)11yescharacteristic
411C23C2×AGL(1,ℤ/8)11yescharacteristic
412C23C2×#13711yescharacteristic
413C23C2×#13711yescharacteristic
414C23#57511yescharacteristic
415C23#57511yescharacteristic
416Q8—432no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4C4×SD1614yes—
498C2×C4#44014yes—
499C2×C4#44814yes—
500C2×C4#47614yes—
501C2×C4C2×C4×D412yes—
502C2×C4C2×C4×Q811yescharacteristic
503C2×C4C4×#4111yescharacteristic
504C2×C4#51912yes—
505C2×C4#52512yes—
506C2×C4#52911yescharacteristic
507C2×C4#53311yescharacteristic
508C2×C4#54112yes—
509C2×C4#54712yes—
510C2×C4#54712yes—
511C2×C4#54812yes—
512C2×C4D4×Q812yes—
513C2×C4#55212yes—
514C2×C4#55612yes—
515C2×C4#55612yes—
516C2×C4#55712yes—
517C8—44no—
518C8—82no—
519C22C4×#12011yescharacteristic
520C22C4×C22⋊Q811yescharacteristic
521C22#232711yescharacteristic
522C22#240611yescharacteristic
523C22#240911yescharacteristic
524C22#258111yescharacteristic
525C22#259911yescharacteristic
526C22C2×C4×SD1611yescharacteristic
527C22C2×#44011yescharacteristic
528C22C4×AGL(1,ℤ/8)11yescharacteristic
529C22C4×#13711yescharacteristic
530C22C2×#44811yescharacteristic
531C22#300211yescharacteristic
532C22#300311yescharacteristic
533C22#300711yescharacteristic
534C22C2×#47611yescharacteristic
535C22#307611yescharacteristic
536C22#307811yescharacteristic
537C22#307911yescharacteristic
538C22#310111yescharacteristic
539C22#310311yescharacteristic
540C22#310611yescharacteristic
541C22#310811yescharacteristic
542C22#311011yescharacteristic
543C22#311311yescharacteristic
544C22#311411yescharacteristic
545C22#311511yescharacteristic
546C22#322711yescharacteristic
547C22#322811yescharacteristic
548C22#322911yescharacteristic
549C22#323111yescharacteristic
550C22#323511yescharacteristic
551C22#323611yescharacteristic
552C22#323811yescharacteristic
553C22#324011yescharacteristic
554C4—24no—
555C4—24no—
556C4—24no—
557C4—24no—
558C4—44no—
559C4—48no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—84no—
566C4#236212yes—
567C4#236512yes—
568C4#246212yes—
569C4#246912yes—
570C2#1417411yescharacteristic
571C2C4×#44811yescharacteristic
572C2#2145311yescharacteristic
573C2C4×#47611yescharacteristic
574C2#2161511yescharacteristic
575C2#2224611yescharacteristic
576C2#2225111yescharacteristic
577C2#2406211yescharacteristic
578C2#2406311yescharacteristic
579C2#2429911yescharacteristic
580C2#2430611yescharacteristic
581C2#2927611yescharacteristic
582C2#2928011yescharacteristic
583C2#2939211yescharacteristic
584C2#2939511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #2058C4semidirect
88. #389Q8semidirect
387. C4⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20582
C2×#19202
C4×#3892
#113842
#112672
#112072
#109082
#108852
#108842
#107952