Finite Group Databasebeta

← #409263

Subgroups — #409263

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#13795C211yesmaximal, characteristic
5#11357C212yesmaximal, complemented
6#11325C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#11256C212yesmaximal, 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
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1853—24no—
24#1852—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1441—22no—
34#1439—22no—
35C4⋊#95—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—22no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—42no—
52C2×#117—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—42no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D412yes—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—42no—
75C2×#107—42no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#106—42no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8Q814yes—
84C2×C22⋊C8Q814yes—
85#384—24no—
86#382—42no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#336—24no—
107#336—42no—
108C25C2×D411yescharacteristic
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—82no—
123C23×C4—82no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×Q812yes—
129C23×C4C2×Q812yes—
130C23×C4#4112yes—
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8C2×Q814yes—
147C22×C8C2×Q814yes—
148#117—48no—
149#117—48no—
150#117—48no—
151#117—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
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—48no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4#4114yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42#4114yes—
212#107—48no—
213#106—48no—
214C22⋊C8—28no—
215C22⋊C8—28no—
216#95—44no—
217#95—42no—
218#95—42no—
219#95—82no—
220#95—82no—
221#95—82no—
222#95—82no—
223#95—82no—
224#95—82no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24#12112yes—
234C24#12312yes—
235C24C4⋊Q812yes—
236C24C22×D411yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—48no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4#12112yes—
306C22×C4C22⋊Q814yes—
307C22×C4C22⋊Q814yes—
308C22×C4#12314yes—
309C22×C4#12312yes—
310C22×C4#12514yes—
311C22×C4#12514yes—
312C22×C4C4⋊Q812yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×Q811yescharacteristic
315C22×C4C22×Q811yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4112yes—
321C22×C4H(5,𝔽2)12yes—
322C22×C4H(5,𝔽2)12yes—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—88no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C42—24no—
361C42—24no—
362C42—48no—
363C42—42no—
364C42—42no—
365C23—22no—
366C23—24no—
367C23—24no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23#39814yes—
388C23C2×#12111yescharacteristic
389C23C2×#12311yescharacteristic
390C23C2×#12311yescharacteristic
391C23#52511yescharacteristic
392C23C2×C4⋊Q811yescharacteristic
393C23#53411yescharacteristic
394C23#53511yescharacteristic
395C23#57612yes—
396C23#57712yes—
397C23#57712yes—
398C23#57812yes—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—816no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4#39814yes—
459C2×C4C2×C22⋊Q811yescharacteristic
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4C2×#12511yescharacteristic
462C2×C4C2×#12511yescharacteristic
463C2×C4#52912yes—
464C2×C4#53212yes—
465C2×C4#53412yes—
466C2×C4#53812yes—
467C2×C4#55112yes—
468C2×C4#55112yes—
469C2×C4#55112yes—
470C2×C4#55112yes—
471C2×C4#55412yes—
472C2×C4#55412yes—
473C2×C4#55412yes—
474C2×C4#55412yes—
475C8—44no—
476C8—44no—
477C22—24no—
478C22—24no—
479C22—24no—
480C22—42no—
481C22—42no—
482C22—44no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—44no—
487C22—44no—
488C22C2×#39811yescharacteristic
489C22#248211yescharacteristic
490C22#248311yescharacteristic
491C22#250211yescharacteristic
492C22#258311yescharacteristic
493C22#261711yescharacteristic
494C22#285411yescharacteristic
495C22#304111yescharacteristic
496C22#304211yescharacteristic
497C22#304311yescharacteristic
498C22#304411yescharacteristic
499C22#309612yes—
500C22#309712yes—
501C22#309812yes—
502C22#309912yes—
503C22#311912yes—
504C22#312012yes—
505C22#316312yes—
506C22#316412yes—
507C22#319812yes—
508C22#320112yes—
509C22#322412yes—
510C22#322512yes—
511C4—24no—
512C4—44no—
513C4—84no—
514C4—84no—
515C4—82no—
516C4—161no—
517C4—161no—
518C4—161no—
519C4—161no—
520C4#250512yes—
521C4#250512yes—
522C4#256412yes—
523C4#256412yes—
524C2—44no—
525C2#1501411yescharacteristic
526C2#2366011yescharacteristic
527C2#2366111yescharacteristic
528C2#2759312yes—
529C2#2760612yes—
530C2#2795912yes—
531C2#2930912yes—
532C2#2972711yescharacteristic
533C2#2972811yescharacteristic
534C2#3311811yescharacteristic
535C2#3311911yescharacteristic
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
5. #11357C2semidirect
6. #11325C2semidirect
7. #11257C2semidirect
8. #11256C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C2×#18522
#143402
#137952
#113572
#113252
#112572
#112562
#107312
#107132