Finite Group Databasebeta

← #401642

Subgroups — #401642

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C2×#2085C212yesmaximal, complemented
3C2×#1853C211yesmaximal, characteristic
4C2×#1852C211yesmaximal, characteristic
5#14340C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10731C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2316—28no—
20C2×#389—22no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380—22no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#2085—28no—
29#1853—24no—
30#1852—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1441—22no—
40#1439—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—24no—
50C2×#117—24no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—42no—
54C2×#117—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8Q814yes—
88C2×C22⋊C8Q814yes—
89#389—216no—
90#389—24no—
91#389—24no—
92#389—24no—
93#389D418yes—
94#384—42no—
95#382—24no—
96#380—216no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—42no—
113#340—42no—
114#336—24no—
115#336—42no—
116C25C2×D411yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—82no—
131C23×C4—82no—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×Q812yes—
137C23×C4C2×Q812yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C23×C4C2411yescharacteristic
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×C8—22no—
153C22×C8—22no—
154C22×C8C2×Q814yes—
155C22×C8C2×Q814yes—
156#117—48no—
157#117—48no—
158#117—48no—
159#117—48no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—216no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—216no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4#4114yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42C2×D414yes—
226#107—48no—
227#106—48no—
228C22⋊C8—28no—
229C22⋊C8—28no—
230#95—28no—
231#95—28no—
232#95—44no—
233#95—42no—
234#95—42no—
235#95—82no—
236#95—82no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24#12112yes—
246C24#12312yes—
247C24C4⋊Q812yes—
248C24C22×D411yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—216no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
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—84no—
314C22×C4—84no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4#12014yes—
318C22×C4#12112yes—
319C22×C4C22⋊Q814yes—
320C22×C4C22⋊Q814yes—
321C22×C4#12312yes—
322C22×C4#12514yes—
323C22×C4#12514yes—
324C22×C4C4⋊Q812yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×Q811yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+412yes—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—216no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—48no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C4⋊C4—88no—
369C4⋊C4#12118yes—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C42—24no—
375C42—24no—
376C42—48no—
377C42—42no—
378C42—42no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23#39814yes—
402C23C2×#12011yescharacteristic
403C23C2×#12111yescharacteristic
404C23C2×#12311yescharacteristic
405C23#52511yescharacteristic
406C23C2×C4⋊Q811yescharacteristic
407C23#53411yescharacteristic
408C23#53611yescharacteristic
409C23#57612yes—
410C23#57712yes—
411C23#57712yes—
412C23#57812yes—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—816no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4#39814yes—
473C2×C4C2×#12112yes—
474C2×C4C2×C22⋊Q811yescharacteristic
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4#52512yes—
477C2×C4C2×#12511yescharacteristic
478C2×C4C2×#12511yescharacteristic
479C2×C4#53712yes—
480C2×C4#53912yes—
481C2×C4#54612yes—
482C2×C4#54612yes—
483C2×C4D4×Q812yes—
484C2×C4D4×Q812yes—
485C2×C4#55212yes—
486C2×C4#55212yes—
487C2×C4#55412yes—
488C2×C4#55412yes—
489C8—44no—
490C8—44no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22C2×#39811yescharacteristic
503C22#240511yescharacteristic
504C22#243411yescharacteristic
505C22#248411yescharacteristic
506C22#258311yescharacteristic
507C22#261811yescharacteristic
508C22#284711yescharacteristic
509C22#301212yes—
510C22#301312yes—
511C22#304111yescharacteristic
512C22#304211yescharacteristic
513C22#304311yescharacteristic
514C22#304411yescharacteristic
515C22#309812yes—
516C22#309912yes—
517C22#311812yes—
518C22#312112yes—
519C22#316312yes—
520C22#316412yes—
521C22#319812yes—
522C22#320112yes—
523C22#324112yes—
524C22#324212yes—
525C4—24no—
526C4—48no—
527C4—44no—
528C4—82no—
529C4—84no—
530C4—161no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4#241712yes—
535C4#241712yes—
536C4#247312yes—
537C4#247312yes—
538C2—44no—
539C2#1482111yescharacteristic
540C2#2366011yescharacteristic
541C2#2366111yescharacteristic
542C2#2401712yes—
543C2#2582212yes—
544C2#2762312yes—
545C2#2930812yes—
546C2#2976311yescharacteristic
547C2#2976411yescharacteristic
548C2#3291711yescharacteristic
549C2#3291811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2085C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
8. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#20852
C2×#18532
C2×#18522
#143402
#112202
#112192
#112072
#107312
#107132