Finite Group Databasebeta

← #394886

Subgroups — #394886

Order 512.

1608 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
1C22×#477C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic
4C2×#1865C212yesmaximal
5#15128C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11325C212yesmaximal
8#11227C211yesmaximal, characteristic
9#11202C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#477—28no—
17C2×#477—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C2×#395—21no—
24#2351—24no—
25#2351—24no—
26C2×#389—21no—
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#382—21no—
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#2058—24no—
36#1865—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C2×C4×Q8—42no—
69C2×C4×Q8D414yes—
70C2×C4×Q8Q814yescomplemented
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#477—232no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—24no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#106—42no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103D418yes—
105C2×#103Q818yescomplemented
106#395—44no—
107#389—28no—
108#389—28no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—24no—
113#389—42no—
114#389D414yes—
115#382—28no—
116#382—22no—
117#382—22no—
118C4×C4⋊C4—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#336—24no—
137#336—22no—
138#336—24no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—44no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—82no—
149C22×Q8C2×D412yes—
150C22×Q8C2×D414yes—
151C22×Q8C2×Q812yes—
152C22×Q8C2×Q814yes—
153C22×Q8#4112yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C2×D414yes—
180C22×C8C2×Q814yes—
181C4⋊Q8—28no—
182C4⋊Q8—48no—
183C4⋊Q8—44no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—28no—
187C4×Q8—28no—
188C4×Q8—28no—
189C4×Q8—44no—
190C4×Q8—48no—
191C4×Q8—48no—
192C4×Q8—44no—
193C4×Q8—44no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×Q814yes—
242C2×C4⋊C4C2×Q812yes—
243C2×C4⋊C4#4112yes—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×Q812yes—
264C2×C42#4112yes—
265#107—216no—
266#106—48no—
267C4⋊C8—28no—
268#103—216no—
269#103—216no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—22no—
275#95—22no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—82no—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—48no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—88no—
299C2×Q8—88no—
300C2×Q8#12014yes—
301C2×Q8C22⋊Q818yes—
302C2×Q8C22⋊Q814yes—
303C2×Q8C4⋊Q814yes—
304C2×Q8C4⋊Q818yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4C22⋊Q812yes—
369C22×C4C22⋊Q812yes—
370C22×C4#13512yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4#13712yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×Q811yescharacteristic
377C22×C4C22×Q812yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—216no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—88no—
423C4⋊C4C22⋊Q814yes—
424C42—216no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—24no—
432C42—48no—
433C42—48no—
434C42—41no—
435C42—41no—
436C42—42no—
437C23C2×#12011yescharacteristic
438C23C2×C22⋊Q811yescharacteristic
439C23#52511yescharacteristic
440C23D4212yes—
441C23D4×Q812yes—
442C23C2×#13511yescharacteristic
443C23C2×AGL(1,ℤ/8)11yescharacteristic
444C23C2×#13711yescharacteristic
445C23#57511yescharacteristic
446C23#57712yes—
447C23#57812yes—
448Q8—24no—
449Q8—24no—
450Q8—24no—
451Q8—28no—
452Q8—28no—
453Q8—816no—
454Q8#39518yescomplemented
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4#45214yes—
523C2×C4C2×#12011yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×C22⋊Q812yes—
526C2×C4C2×C22⋊Q811yescharacteristic
527C2×C4#52511yescharacteristic
528C2×C4#52511yescharacteristic
529C2×C4C2×C4⋊Q811yescharacteristic
530C2×C4C2×C4⋊Q812yes—
531C2×C4#53411yescharacteristic
532C2×C4#53611yescharacteristic
533C2×C4#53912yes—
534C2×C4#54412yes—
535C2×C4D4212yes—
536C2×C4#54712yes—
537C2×C4#54712yes—
538C2×C4D4×Q812yes—
539C2×C4#55412yes—
540C2×C4#55412yes—
541C8—44no—
542C8—44no—
543C22#240311yescharacteristic
544C22#243411yescharacteristic
545C22#244012yes—
546C22#244912yes—
547C22#284811yescharacteristic
548C22C2×#45211yescharacteristic
549C22#300611yescharacteristic
550C22#301011yescharacteristic
551C22#301111yescharacteristic
552C22#307611yescharacteristic
553C22#307911yescharacteristic
554C22#308212yes—
555C22#310611yescharacteristic
556C22#311611yescharacteristic
557C22#312112yes—
558C22#326512yes—
559C22#327012yes—
560C22#327512yes—
561C22#328512yes—
562C22#338112yes—
563C22#338612yes—
564C22#338712yes—
565C22#339012yes—
566C4—24no—
567C4—28no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—44no—
572C4—42no—
573C4—84no—
574C4—81no—
575C4—82no—
576C4—82no—
577C4—84no—
578C4C2×#39511yescharacteristic
579C4#240611yescharacteristic
580C4#244012yes—
581C4#244912yes—
582C4#258311yescharacteristic
583C4#262711yescharacteristic
584C2#1472011yescharacteristic
585C2#2385911yescharacteristic
586C2#2386611yescharacteristic
587C2#2583611yescharacteristic
588C2#2584911yescharacteristic
589C2#2605012yes—
590C2#2605612yes—
591C2#2637712yes—
592C2#2639312yes—
593C2#3295811yescharacteristic
594C2#3296411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
70. C2×C4×Q8Q8semidirect
105. C2×#103Q8semidirect
454. Q8#395semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#23512
C2×#20582
C2×#18652
#151282
#113282
#113252
#112272
#112022
#107892