Finite Group Databasebeta

← #359185

Subgroups — #359185

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1763C212yesmaximal
4C4×#391C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11292C212yesmaximal
7#11267C211yesmaximal, characteristic
8#11067C212yesmaximal
9#11050C211yesmaximal, characteristic
10#10703C212yesmaximal
11#10693C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#97C2212yes—
18C4×C4⋊Q8—28no—
19C2×#391—21no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C43C2211yescharacteristic
29#2024C418yescomplemented
30#1763—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4×#95—21no—
45C4×#95—21no—
46C23×Q8—21no—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4×Q8—28no—
55C2×C4×Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—44no—
59C2×C4×Q8—44no—
60C2×C4×Q8Q814yescomplemented
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C42—22no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103C2×C418yes—
87C2×#97—24no—
88C2×#97—24no—
89C2×#97—24no—
90#391—28no—
91#391—22no—
92#391—22no—
93#391—22no—
94#391C2×C414yes—
95#389—28no—
96#389—24no—
97#389—24no—
98#384—24no—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4D418yes—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C43—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340C2×C418yes—
121#336—24no—
122#336—22no—
123#336C2×C414yes—
124C22×Q8—24no—
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8C22×C412yes—
139C22×Q8C2×Q812yes—
140C22×Q8#4112yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C22×C414yes—
167C22×C8#4114yes—
168C4⋊Q8—216no—
169C4⋊Q8—216no—
170C4×Q8—216no—
171C4×Q8—432no—
172C4×Q8—48no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C4⋊C4#4112yes—
211C2×C4⋊C4#4114yes—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—28no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C22×C412yes—
240C2×C42C2×D414yes—
241C2×C42C2×Q812yes—
242C2×C42#4112yes—
243C2×C42#4114yes—
244C2×C42#4112yes—
245#103—216no—
246#103—216no—
247#97—48no—
248#95—28no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—28no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259C24C22×D411yescharacteristic
260C2×Q8—216no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—48no—
282C2×Q8C4×Q814yescomplemented
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4C4×D412yes—
354C22×C4C4×D412yes—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4#12412yes—
360C22×C4#12412yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4AGL(1,ℤ/8)12yes—
363C22×C4#13712yes—
364C22×C4#13712yes—
365C22×C4#13712yes—
366C22×C4#13712yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—216no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—416no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4C4×D418yes—
410C42—28no—
411C42—28no—
412C42—24no—
413C42—24no—
414C42—44no—
415C42—44no—
416C42—416no—
417C42—42no—
418C42—42no—
419C42—48no—
420C42—84no—
421C42C22⋊Q818yes—
422C42#12418yes—
423C23C2×C4×D411yescharacteristic
424C23C2×#12011yescharacteristic
425C23C2×C22⋊Q811yescharacteristic
426C23#52511yescharacteristic
427C23#52511yescharacteristic
428C23C2×#12411yescharacteristic
429C23#53211yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432C23C2×#13711yescharacteristic
433C23C2×#13711yescharacteristic
434C23C2×#13711yescharacteristic
435C23C2×#13711yescharacteristic
436C23#57511yescharacteristic
437C23#57511yescharacteristic
438Q8—28no—
439Q8—432no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4#39314yes—
524C2×C4#39314yes—
525C2×C4#44014yes—
526C2×C4#44114yes—
527C2×C4C2×C4×D412yes—
528C2×C4C2×C4×Q811yescharacteristic
529C2×C4C4×#4112yes—
530C2×C4C4×#4111yescharacteristic
531C2×C4C2×C22⋊Q812yes—
532C2×C4#52512yes—
533C2×C4#52512yes—
534C2×C4C2×#12412yes—
535C2×C4#52912yes—
536C2×C4#52912yes—
537C2×C4#52911yescharacteristic
538C2×C4#52912yes—
539C2×C4#52912yes—
540C2×C4#53212yes—
541C2×C4#53311yescharacteristic
542C2×C4#53312yes—
543C8—44no—
544C8—44no—
545C22C4×#12011yescharacteristic
546C22C4×C22⋊Q811yescharacteristic
547C22C4×#12411yescharacteristic
548C22C2×#39311yescharacteristic
549C22#238811yescharacteristic
550C22#239611yescharacteristic
551C22#239811yescharacteristic
552C22C2×#44011yescharacteristic
553C22C2×#44111yescharacteristic
554C22C4×AGL(1,ℤ/8)11yescharacteristic
555C22C4×#13711yescharacteristic
556C22#300311yescharacteristic
557C22#300411yescharacteristic
558C22#300711yescharacteristic
559C22#300711yescharacteristic
560C22#307811yescharacteristic
561C22#307811yescharacteristic
562C22#307911yescharacteristic
563C22#307911yescharacteristic
564C22#311011yescharacteristic
565C22#311211yescharacteristic
566C22#311311yescharacteristic
567C22#311311yescharacteristic
568C22#311411yescharacteristic
569C22#311511yescharacteristic
570C22#311611yescharacteristic
571C22#311711yescharacteristic
572C22#313911yescharacteristic
573C22#314011yescharacteristic
574C22#314111yescharacteristic
575C22#314111yescharacteristic
576C22#318011yescharacteristic
577C22#318111yescharacteristic
578C22#318211yescharacteristic
579C22#318511yescharacteristic
580C4—24no—
581C4—24no—
582C4—24no—
583C4—24no—
584C4—42no—
585C4—42no—
586C4—44no—
587C4—48no—
588C4—82no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4C4×C22⋊Q812yes—
593C4C4×#12412yes—
594C4#239612yes—
595C4#239812yes—
596C2C4×#39311yescharacteristic
597C2#2145311yescharacteristic
598C2#2145511yescharacteristic
599C2#2161511yescharacteristic
600C2#2161611yescharacteristic
601C2#2164711yescharacteristic
602C2#2165511yescharacteristic
603C2#2345211yescharacteristic
604C2#2345511yescharacteristic
605C2#2357611yescharacteristic
606C2#2357711yescharacteristic
607C2#2378211yescharacteristic
608C2#2378311yescharacteristic
609C2#2382111yescharacteristic
610C2#2382311yescharacteristic
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
29. #2024C4semidirect
60. C2×C4×Q8Q8semidirect
282. C2×Q8C4×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×#20242
C2×#17632
C4×#3912
#113842
#112922
#112672
#110672
#110502
#107032
#106932