Finite Group Databasebeta

← #2040828

Subgroups — #2040828

Order 512.

2858 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2706C214yesmaximal, complemented
2#17073C211yesmaximal, characteristic, complemented
3#17068C211yesmaximal, characteristic
4#16701C214yesmaximal, complemented
5#15498C214yesmaximal, complemented
6#15384C214yesmaximal, complemented
7#15305C214yesmaximal, complemented
8#15303C214yesmaximal, complemented
9#14700C212yesmaximal, complemented
10#14687C211yesmaximal, characteristic, complemented
11#14440C212yesmaximal, complemented
12C22×#123C2214yes—
13#2706—232no—
14C2×#399C2214yes—
15C2×#398C2214yes—
16C2×#397C2214yes—
17C2×#396C2214yes—
18C2×#394C2214yes—
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#392C2214yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#384C2212yes—
34C2×#382C2214yes—
35C2×#380C2214yes—
36C2×C43C2211yescharacteristic
37#1445C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52C4⋊#95C2212yes—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2214yes—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C23×D4C2314yes—
59C23×D4C2312yes—
60C24×C4C2314yes—
61C2×#123—216no—
62C2×#123—216no—
63C2×#123—48no—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#399—216no—
92#398—216no—
93#397—216no—
94#396—216no—
95#394—216no—
96#394—48no—
97#394—48no—
98#392—216no—
99#390—28no—
100#390—44no—
101#390D4116yescomplemented
102#389—44no—
103#388—216no—
104#388—48no—
105#386—216no—
106#386—216no—
107#385—216no—
108#385—48no—
109#385—48no—
110#384—44no—
111#382—216no—
112#380—216no—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C43—24no—
129C25C2414yes—
130C22×D4—28no—
131C22×D4—28no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—44no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4—42no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4C2×D418yes—
144C23×C4—28no—
145C23×C4—28no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2412yes—
156C23×C4C2414yes—
157C23×C4C2411yescharacteristic
158C23×C4C2412yes—
159C23×C4C2411yescharacteristic
160C23×C4C2411yescharacteristic
161#123—464no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C2×D418yes—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4C2×D4116yes—
227C2×C22⋊C4C2×D4116yes—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C2×D418yes—
240C2×C42#4114yes—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24C22×D418yes—
271C24C2511yescharacteristic
272C2×D4—216no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—416no—
276C2×D4—416no—
277C2×D4—84no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—84no—
281C2×D4—88no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4—88no—
285C2×D4—88no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
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—24no—
339C22×C4—416no—
340C22×C4—416no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4C22×D414yes—
347C22×C4C22×D418yes—
348C22×C4C22×D414yes—
349C22×C4C22×D414yes—
350C22×C4C2×#4112yes—
351C22×C4C2×#4114yes—
352C22×C4H(5,𝔽2)14yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)14yes—
355C22×C42−1+414yes—
356C22×C42−1+412yes—
357C4⋊C4—48no—
358C4⋊C4—416no—
359C4⋊C4—416no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—88no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C22⋊C4—232no—
369C22⋊C4—416no—
370C22⋊C4—416no—
371C22⋊C4—416no—
372C22⋊C4—416no—
373C22⋊C4—416no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C42—28no—
380C42—216no—
381C42—44no—
382C42—48no—
383C42—44no—
384C23—28no—
385C23—28no—
386C23—28no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23C23×D412yes—
420C23C22×#4111yescharacteristic
421C23C2×H(5,𝔽2)12yes—
422C23C2×H(5,𝔽2)11yescharacteristic
423C23C2×H(5,𝔽2)12yes—
424C23C2×2−1+412yes—
425C23C2×2−1+411yescharacteristic
426C23#58512yes—
427C23#58512yes—
428D4—816no—
429D4—832no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
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—44no—
479C2×C4—44no—
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—832no—
491C2×C4#54014yes—
492C2×C4D4214yes—
493C2×C4#54718yes—
494C2×C4#56214yes—
495C22—48no—
496C22—48no—
497C22—48no—
498C22—48no—
499C22—48no—
500C22—48no—
501C22—48no—
502C22—84no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22—84no—
507C22—84no—
508C22—84no—
509C22—84no—
510C22#344912yes—
511C22C2×#54011yescharacteristic
512C22C2×D4211yescharacteristic
513C22C2×#54712yes—
514C22#349111yescharacteristic
515C22#349211yescharacteristic
516C22#349412yes—
517C22#349612yes—
518C22#349612yes—
519C22#349812yes—
520C22#349912yes—
521C22C2×#56211yescharacteristic
522C22#354411yescharacteristic
523C22#354812yes—
524C22#355311yescharacteristic
525C22#355412yes—
526C22#355812yes—
527C22#355812yes—
528C22#355912yes—
529C22#356112yes—
530C22#356412yes—
531C4—24no—
532C4—24no—
533C4—24no—
534C4—28no—
535C4—44no—
536C4—48no—
537C4—84no—
538C4—84no—
539C4—84no—
540C4—84no—
541C2—88no—
542C2#3697611yescharacteristic
543C2#3713112yes—
544C2#3786612yes—
545C2#3935212yes—
546C2#3935512yes—
547C2#4598811yescharacteristic
548C2#4599512yes—
549C2#5335712yes—
550C2#5931311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17068C2
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×#2706C2semidirect
2. #17073C2semidirect
4. #16701C2semidirect
5. #15498C2semidirect
6. #15384C2semidirect
7. #15305C2semidirect
8. #15303C2semidirect
9. #14700C2semidirect
10. #14687C2semidirect
11. #14440C2semidirect
101. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27062
#170732
#170682
#167012
#154982
#153842
#153052
#153032
#147002
#146872
#144402