Finite Group Databasebeta

← #398666

Subgroups — #398666

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2087C211yesmaximal, characteristic, complemented
2C2×#1850C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13907C212yesmaximal
6#11377C211yesmaximal, characteristic, complemented
7#11328C211yesmaximal, characteristic, complemented
8#11293C212yesmaximal, complemented
9#11232C211yesmaximal, characteristic, complemented
10#11228C212yesmaximal, complemented
11#10730C212yesmaximal
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#380—22no—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#2087—24no—
25#1850—24no—
26#1808—24no—
27C2×#340—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1441—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—22no—
43C24×C4—22no—
44C24×C4C2311yescharacteristic
45C23×C8—24no—
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—42no—
49C23×C8D414yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C22×C4⋊C4—24no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—42no—
61C22×C4⋊C4—42no—
62C22×C4⋊C4Q818yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—42no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—24no—
84#389—24no—
85#389—24no—
86#389—42no—
87#389—42no—
88#380—24no—
89#380—24no—
90#380—48no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D411yescharacteristic
101C2×#95D411yescharacteristic
102#340—24no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#340D414yes—
110#336—22no—
111#336—22no—
112C25C2×D411yescharacteristic
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—28no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4—82no—
129C23×C4—82no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×Q814yes—
137C23×C4#4112yes—
138C23×C4#4114yes—
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—28no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—28no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—44no—
159C22×C8—48no—
160C22×C8—48no—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8#4112yes—
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—216no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
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—48no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×Q814yes—
200C2×C4⋊C4C2×Q814yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4Q1614yes—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220#107—44no—
221#107—44no—
222C4⋊C8—44no—
223C4⋊C8—44no—
224C22⋊C8—44no—
225#95—22no—
226#95—22no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—82no—
233#95—82no—
234C24—21no—
235C24—22no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—21no—
240C24#12112yes—
241C24C22⋊Q814yes—
242C24C2×Q1612yes—
243C24#13512yes—
244C24AGL(1,ℤ/8)14yes—
245C24C22×D411yescharacteristic
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—416no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—416no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—88no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4—88no—
318C22×C4—88no—
319C22×C4#12112yes—
320C22×C4C22⋊Q814yes—
321C22×C4#12314yes—
322C22×C4#12518yes—
323C22×C4C4⋊Q814yes—
324C22×C4C2×Q1612yes—
325C22×C4C2×Q1612yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q812yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C42−1+411yescharacteristic
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—416no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8#12114yes—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—416no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—88no—
374C4⋊C4—88no—
375C4⋊C4C22⋊Q818yes—
376C22⋊C4—28no—
377C22⋊C4—44no—
378C22⋊C4—88no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—42no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—22no—
390C23—24no—
391C23—22no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—24no—
404C23#39814yes—
405C23#46714yes—
406C23#46914yes—
407C23#47718yes—
408C23#47818yes—
409C23C2×#12111yescharacteristic
410C23C2×C22⋊Q812yes—
411C23D4211yescharacteristic
412C23#54612yes—
413C23#54711yescharacteristic
414C23C22×Q1611yescharacteristic
415C23C2×#13511yescharacteristic
416C23C2×AGL(1,ℤ/8)12yes—
417C23#57611yescharacteristic
418C23#57712yes—
419C23#57811yescharacteristic
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—832no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
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#39814yes—
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×C22⋊Q812yes—
484C2×C4C2×#12311yescharacteristic
485C2×C4C2×#12512yes—
486C2×C4C2×C4⋊Q811yescharacteristic
487C2×C4#53211yescharacteristic
488C2×C4#53312yes—
489C2×C4#53411yescharacteristic
490C2×C4#53711yescharacteristic
491C2×C4#54312yes—
492C2×C4#54412yes—
493C2×C4#54612yes—
494C2×C4#54811yescharacteristic
495C2×C4#55112yes—
496C2×C4#55112yes—
497C2×C4#55212yes—
498C2×C4#55311yescharacteristic
499C2×C4#55312yes—
500C8—24no—
501C8—42no—
502C8—81no—
503C8—81no—
504C22—28no—
505C22—28no—
506C22—24no—
507C22—24no—
508C22—28no—
509C22—28no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22#208718yescomplemented
515C22C2×#39811yescharacteristic
516C22#241611yescharacteristic
517C22#243712yes—
518C22#243812yes—
519C22#254311yescharacteristic
520C22C2×#46711yescharacteristic
521C22C2×#46911yescharacteristic
522C22#306011yescharacteristic
523C22#306111yescharacteristic
524C22C2×#47812yes—
525C22C2×#47712yes—
526C22#308112yes—
527C22#308212yes—
528C22#326512yes—
529C22#328011yescharacteristic
530C22D4×Q1611yescharacteristic
531C22#329612yes—
532C22#329912yes—
533C22#330812yes—
534C22#331812yes—
535C22#333611yescharacteristic
536C22#333811yescharacteristic
537C22#334512yes—
538C4—24no—
539C4—48no—
540C4—42no—
541C4—42no—
542C4—84no—
543C4—81no—
544C4—81no—
545C4—162no—
546C4—162no—
547C4#241211yescharacteristic
548C4#244212yes—
549C4#249811yescharacteristic
550C4#256912yes—
551C4#265911yescharacteristic
552C4#285711yescharacteristic
553C2—28no—
554C2#1476611yescharacteristic
555C2C2×#208711yescharacteristic
556C2#2366911yescharacteristic
557C2#2478111yescharacteristic
558C2#2478211yescharacteristic
559C2#2595212yes—
560C2#2596312yes—
561C2#2599412yes—
562C2#2599612yes—
563C2#2855311yescharacteristic
564C2#2856111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2087C2semidirect
6. #11377C2semidirect
7. #11328C2semidirect
8. #11293C2semidirect
9. #11232C2semidirect
10. #11228C2semidirect
514. C22#2087semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
C2×#18502
C2×#18082
C22×#3402
#139072
#113772
#113282
#112932
#112322
#112282
#107302