Finite Group Databasebeta

← #876624

Subgroups — #876624

Order 512.

2358 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C212yesmaximal
2#17076C211yesmaximal, characteristic, complemented
3#17075C211yesmaximal, characteristic, complemented
4#17013C212yesmaximal, complemented
5#15485C214yesmaximal
6#15432C214yesmaximal
7#14929C214yesmaximal, complemented
8#14907C212yesmaximal
9#14697C212yesmaximal
10#14587C212yesmaximal, complemented
11#14441C212yesmaximal, complemented
12#14073C214yesmaximal, complemented
13#13756C211yesmaximal, characteristic
14C22×#125C2212yes—
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2611—216no—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#389C2214yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#385C2214yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#376C2212yes—
36#1445C2214yes—
37#1444C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1439C2212yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63#1434C2212yes—
64#1434C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C4×#95C2214yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C2×#125—28no—
71C2×#125—44no—
72C2×#125—44no—
73C2×#117—24no—
74C2×#117—42no—
75C2×#117—42no—
76C2×#117—42no—
77C2×#117—42no—
78C2×#117D418yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C42—24no—
100C22×C42—42no—
101C22×C42—42no—
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2314yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#400—28no—
109#399—28no—
110#398—28no—
111#398—28no—
112#397—28no—
113#397—44no—
114#389—216no—
115#389—28no—
116#389—44no—
117#389—44no—
118#388—216no—
119#387—216no—
120#387—216no—
121#385—216no—
122#380—28no—
123#380—44no—
124#376—28no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25C2411yescharacteristic
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C2×D414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177#125—432no—
178#117—416no—
179#117—416no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4#4118yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4C2×D418yes—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—28no—
242C2×C42—24no—
243C2×C42—28no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—42no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—42no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—44no—
255C2×C42—416no—
256C2×C42C2×D418yes—
257C2×C42#4114yes—
258#95—28no—
259#95—24no—
260#95—24no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—44no—
283C24—21no—
284C24—22no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24C22×D412yes—
292C24H(5,𝔽2)12yes—
293C242−1+412yes—
294C24C2511yescharacteristic
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—216no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
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—24no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C2×#4118yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4114yes—
380C22×C4C2×#4114yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C42−1+414yes—
384C22×C42−1+414yes—
385C4⋊C4—216no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—416no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—416no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—416no—
397C4⋊C4—416no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C22⋊C4—48no—
402C22⋊C4—416no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C42—216no—
406C42—44no—
407C42—416no—
408C42—48no—
409C42—48no—
410C42—44no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23#53714yes—
434C23C23×D411yescharacteristic
435C23C22×#4111yescharacteristic
436C23C22×#4111yescharacteristic
437C23C22×#4112yes—
438C23C2×H(5,𝔽2)11yescharacteristic
439C23C2×H(5,𝔽2)11yescharacteristic
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×2−1+411yescharacteristic
442C23C2×2−1+412yes—
443C23C2×2−1+412yes—
444C23#58512yes—
445C2×C4—232no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—432no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4#53214yes—
521C2×C4#53414yes—
522C2×C4#55218yes—
523C22—28no—
524C22—24no—
525C22—42no—
526C22—44no—
527C22—42no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22C2×#53211yescharacteristic
535C22C2×#53411yescharacteristic
536C22C2×#53711yescharacteristic
537C22#346012yes—
538C22D4×#4112yes—
539C22C2×#55212yes—
540C22#348212yes—
541C22#348311yescharacteristic
542C22#349011yescharacteristic
543C22#349011yescharacteristic
544C22#349212yes—
545C22#349211yescharacteristic
546C22#349211yescharacteristic
547C22#349512yes—
548C22#349912yes—
549C22#351512yes—
550C22#352312yes—
551C22#352911yescharacteristic
552C22#353512yes—
553C22#353812yes—
554C22#355011yescharacteristic
555C22#355011yescharacteristic
556C22#356312yes—
557C4—28no—
558C4—24no—
559C4—24no—
560C4—44no—
561C4—44no—
562C4—48no—
563C4—44no—
564C4—44no—
565C4—84no—
566C4—82no—
567C4—84no—
568C4—84no—
569C2—44no—
570C2#3505111yescharacteristic
571C2#3627212yes—
572C2#3738611yescharacteristic
573C2#3793312yes—
574C2#3842712yes—
575C2#3878312yes—
576C2#4623511yescharacteristic
577C2#4629311yescharacteristic
578C2#5216711yescharacteristic
579C2#5926211yescharacteristic
580C2#6004711yescharacteristic
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
2. #17076C2semidirect
3. #17075C2semidirect
4. #17013C2semidirect
7. #14929C2semidirect
10. #14587C2semidirect
11. #14441C2semidirect
12. #14073C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
#170762
#170752
#170132
#154852
#154322
#149292
#149072
#146972
#145872
#144412
#140732
#137562