Finite Group Databasebeta

← #4386581

Subgroups — #4386581

Order 512.

2442 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2611C212yesmaximal
3C2×#2317C211yesmaximal, characteristic
4#16688C212yesmaximal, complemented
5#16563C212yesmaximal, complemented
6#15678C214yesmaximal, complemented
7#15618C214yesmaximal, complemented
8#15585C214yesmaximal
9#15472C214yesmaximal
10#14907C212yesmaximal
11#14821C212yesmaximal
12#14283C212yesmaximal, complemented
13#14168C211yesmaximal, characteristic, complemented
14C22×#125C2212yes—
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2822—28no—
19#2611—216no—
20C2×#400C2212yes—
21C2×#400C2214yes—
22C2×#399C2212yes—
23C2×#398C2212yes—
24C2×#398C2214yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#397C2212yes—
29C2×#397C2212yes—
30#2317—28no—
31C2×#389C2214yes—
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×#376C2212yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1440C2214yes—
61#1439C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C2×#125—28no—
73C2×#125—28no—
74C2×#125—28no—
75C2×#117—24no—
76C2×#117—24no—
77C2×#117—24no—
78C2×#117—42no—
79C2×#117—42no—
80C2×#117D418yes—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#400—28no—
111#400—216no—
112#399—28no—
113#398—28no—
114#398—28no—
115#398—28no—
116#398—216no—
117#398—44no—
118#397—28no—
119#397—44no—
120#389—216no—
121#389—216no—
122#389—44no—
123#388—216no—
124#387—216no—
125#385—216no—
126#385—216no—
127#384—24no—
128#384—24no—
129#382—24no—
130#382D418yes—
131#380—44no—
132#376—28no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C25C2411yescharacteristic
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×D414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183#125—432no—
184#117—232no—
185#117—416no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—416no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C2×D418yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C2×D418yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—28no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42C2×D414yes—
260C2×C42C2×D414yes—
261C2×C42C2×D414yes—
262C2×C42#4114yes—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—28no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—44no—
285#95—42no—
286#95—44no—
287#95C2×D4116yes—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—24no—
292C24—21no—
293C24—24no—
294C24C22×D412yes—
295C24H(5,𝔽2)12yes—
296C242−1+412yes—
297C24C2511yescharacteristic
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—88no—
367C22×C4—84no—
368C22×C4—88no—
369C22×C4C22×D414yes—
370C22×C4C22×D414yes—
371C22×C4C22×D414yes—
372C22×C4C22×D412yes—
373C22×C4C22×D414yes—
374C22×C4C22×D414yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4114yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C42−1+418yes—
381C22×C42−1+412yes—
382C4⋊C4—216no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—416no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—416no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—416no—
395C4⋊C4—88no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C4⋊C4—84no—
400C22⋊C4—216no—
401C22⋊C4—416no—
402C22⋊C4—48no—
403C22⋊C4—416no—
404C42—28no—
405C42—416no—
406C42—48no—
407C42#121116yes—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—28no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—28no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23#53714yes—
425C23C23×D411yescharacteristic
426C23C23×D411yescharacteristic
427C23C22×#4111yescharacteristic
428C23C2×H(5,𝔽2)11yescharacteristic
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×2−1+411yescharacteristic
431C23C2×2−1+411yescharacteristic
432C23C2×2−1+414yes—
433C23#58514yes—
434C2×C4—232no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—816no—
505C2×C4C2×#12114yes—
506C2×C4C2×#12118yes—
507C2×C4#53714yes—
508C22—24no—
509C22—28no—
510C22—42no—
511C22—44no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22C22×#12111yescharacteristic
518C22C2×#53711yescharacteristic
519C22C2×#53711yescharacteristic
520C22#348411yescharacteristic
521C22#348611yescharacteristic
522C22#348914yes—
523C22#349211yescharacteristic
524C22#349211yescharacteristic
525C22#349912yes—
526C22#350012yes—
527C22#350714yes—
528C22#353312yes—
529C22#353812yes—
530C22#354412yes—
531C22#355112yes—
532C22#356314yes—
533C22#356314yes—
534C4—28no—
535C4—24no—
536C4—24no—
537C4—44no—
538C4—44no—
539C4—44no—
540C4—48no—
541C4—82no—
542C4—82no—
543C4—84no—
544C4—84no—
545C4—84no—
546C2—44no—
547C2#3522311yescharacteristic
548C2#3562112yes—
549C2#4216712yes—
550C2#4298012yes—
551C2#4782012yes—
552C2#4782112yes—
553C2#5154211yescharacteristic
554C2#5154412yes—
555C2#5197811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2822C2semidirect
4. #16688C2semidirect
5. #16563C2semidirect
6. #15678C2semidirect
7. #15618C2semidirect
12. #14283C2semidirect
13. #14168C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#26112
C2×#23172
#166882
#165632
#156782
#156182
#155852
#154722
#149072
#148212
#142832
#141682