Finite Group Databasebeta

← #374583

Subgroups — #374583

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#10850C211yesmaximal, characteristic
9#10834C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C2×#479—24no—
16C2×#479—24no—
17C2×#479C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2351—24no—
26#2351—24no—
27C2×#389—22no—
28C2×#389C2212yes—
29C2×#382—21no—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1991—24no—
35C4×#103—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C414yes—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C2×C4×Q8—42no—
67C2×C4×Q8C2×C414yes—
68C2×C4×Q8D414yes—
69C2×C4×Q8D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C42—21no—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#479—232no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107C2×C414yes—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106C2×C414yes—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yes—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103C2×C414yes—
103C2×#103C2×C414yes—
104C2×#103Q818yes—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108#389—216no—
109#389—24no—
110#389—24no—
111#382—28no—
112#382—22no—
113#382D414yes—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134#336—42no—
135#336—42no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C22×C412yes—
142C22×Q8C2×D412yes—
143C22×Q8C2×D412yes—
144C22×Q8C2×Q814yes—
145C22×Q8#4114yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C22×C412yes—
172C22×C8C22×C412yes—
173C22×C8C2×Q814yes—
174#125—28no—
175#125—28no—
176#125—44no—
177#125—44no—
178C4×Q8—28no—
179C4×Q8—28no—
180C4×Q8—28no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8—48no—
184C4×Q8—48no—
185C4×Q8C22⋊C4116yes—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×Q814yes—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—28no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—82no—
248C2×C42C22×C412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254#107—28no—
255#106—28no—
256C4⋊C8—28no—
257#103—28no—
258#103—28no—
259#103—216no—
260C4×C8—44no—
261#95—28no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—44no—
267#95—42no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8C2×C22⋊C414yes—
286C2×Q8C4×Q818yes—
287C2×Q8C22⋊Q818yes—
288C2×Q8#12414yes—
289C2×Q8C4⋊Q814yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12012yes—
344C22×C4#12012yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#13514yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q812yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4C2×#4111yescharacteristic
356C22×C4H(5,𝔽2)12yes—
357C22×C42−1+412yes—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—216no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—48no—
388C4⋊C4—416no—
389C4⋊C4—416no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4C2×C22⋊C414yes—
401C42—216no—
402C42—416no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—84no—
408C42C2×C22⋊C414yes—
409C42C4×D414yes—
410C42#12114yes—
411C42#12314yes—
412C23C2×C4×D411yescharacteristic
413C23C2×#12011yescharacteristic
414C23C2×#12111yescharacteristic
415C23#54612yes—
416C23D4×Q812yes—
417C23C2×#13512yes—
418C23#57512yes—
419C23#57712yes—
420C23#57812yes—
421Q8—24no—
422Q8—28no—
423Q8—28no—
424Q8—24no—
425Q8—24no—
426Q8#39118yes—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4#38614yes—
484C2×C4#38614yes—
485C2×C4C22×C22⋊C411yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×Q812yes—
488C2×C4#52012yes—
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×C22⋊Q812yes—
491C2×C4C2×#12311yescharacteristic
492C2×C4C2×#12411yescharacteristic
493C2×C4C2×C4⋊Q811yescharacteristic
494C2×C4#53511yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#53712yes—
497C2×C4#54612yes—
498C2×C4D4×Q812yes—
499C2×C4#55212yes—
500C2×C4#55212yes—
501C2×C4#55412yes—
502C2×C4#55412yes—
503C8—42no—
504C8—42no—
505C8—44no—
506C22C2×#38611yescharacteristic
507C22#236012yes—
508C22#241712yes—
509C22#243411yescharacteristic
510C22#243511yescharacteristic
511C22C4×#13511yescharacteristic
512C22#294311yescharacteristic
513C22#294712yes—
514C22#300212yes—
515C22#301312yes—
516C22#303512yes—
517C22#304311yescharacteristic
518C22#304411yescharacteristic
519C22#330012yes—
520C22#330112yes—
521C22#332612yes—
522C22#332712yes—
523C22#338112yes—
524C22#338312yes—
525C22#338812yes—
526C22#339012yes—
527C4—28no—
528C4—24no—
529C4—24no—
530C4—44no—
531C4—44no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—84no—
537C4C2×#39111yescharacteristic
538C4#231111yescharacteristic
539C4#236012yes—
540C4#241712yes—
541C4#250211yescharacteristic
542C4#250311yescharacteristic
543C2#1413911yescharacteristic
544C2#2092411yescharacteristic
545C2#2093011yescharacteristic
546C2#2296712yes—
547C2#2297312yes—
548C2#2481912yes—
549C2#2482112yes—
550C2#2584212yes—
551C2#2588912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
17. C2×#479C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
C2×#23512
C2×#19912
C2×C4×#1032
#141602
#112202
#112192
#108502
#108342
#107892