Finite Group Databasebeta

← #950618

Subgroups — #950618

Order 512.

2744 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17308C211yesmaximal, characteristic, complemented
2#17306C212yesmaximal, complemented
3#17306C211yesmaximal, characteristic, complemented
4#16562C214yesmaximal, complemented
5#16323C214yesmaximal, complemented
6#15961C212yesmaximal, complemented
7#15429C212yesmaximal, complemented
8#15165C212yesmaximal, complemented
9#14769C214yesmaximal, complemented
10#14317C212yesmaximal, complemented
11#14004C214yesmaximal, complemented
12#13956C212yesmaximal
13#13851C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C2×#399C2212yes—
16C2×#396C2214yes—
17C2×#396C2214yes—
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#386C2214yes—
21C2×#386C2214yes—
22C2×#385C2214yes—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#381C2212yes—
26C2×#380C2214yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2214yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C22×#95C2212yes—
31#1446C2211yescharacteristic
32#1446C2211yescharacteristic
33#1446C2212yes—
34#1446C2212yes—
35#1445C2212yes—
36#1445C2214yes—
37#1444C2212yes—
38#1444C2214yes—
39#1442C2212yes—
40#1442C2212yes—
41#1442C2214yes—
42#1442C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1438C2211yescharacteristic
57#1438C2211yescharacteristic
58#1438C2214yes—
59#1438C2212yes—
60#1438C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22×#41—42no—
68C22×#41—42no—
69C22×#41—48no—
70C22×#41Q818yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C23×D4C2312yes—
74C24×C4C2312yes—
75C24×C4C2314yes—
76C24×C4C2311yescharacteristic
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C42C2312yes—
96C22×C42C2312yes—
97#399—28no—
98#396—48no—
99#396—48no—
100#389—44no—
101#389—44no—
102#386—48no—
103#386—48no—
104#385—216no—
105#385—48no—
106#385—48no—
107#381—44no—
108#380—48no—
109C4×C22⋊C4—42no—
110C4×C22⋊C4—42no—
111C4×C22⋊C4—48no—
112C2×#95—44no—
113C2×#95C2314yes—
114C2×#95C2312yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140C25C2411yescharacteristic
141C25C2412yes—
142C2×#41—416no—
143C2×#41—432no—
144C2×#41—432no—
145C2×#41—432no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—44no—
149C22×Q8C2×Q814yes—
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—42no—
155C22×D4—44no—
156C22×D4—42no—
157C22×D4C2×Q814yes—
158C22×D4C2×Q818yes—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C2×Q818yes—
174C23×C4C2×Q814yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4#4118yescomplemented
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—44no—
240C2×C42—44no—
241#95—28no—
242#95—28no—
243#95—24no—
244#95—28no—
245#95—24no—
246#95—28no—
247#95—24no—
248#95—28no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—44no—
253#95—44no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—44no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268C24—24no—
269C24—24no—
270C24—22no—
271C24—24no—
272C24—22no—
273C24—24no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—24no—
280C24—22no—
281C24—22no—
282C24C22×Q812yes—
283C24C22×Q814yes—
284C24H(5,𝔽2)18yes—
285C24C2511yescharacteristic
286#41—4128no—
287C2×Q8—44no—
288C2×Q8—48no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×D4—216no—
292C2×D4—216no—
293C2×D4—44no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—416no—
299C2×D4—416no—
300C22×C4—24no—
301C22×C4—216no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—216no—
328C22×C4—28no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—416no—
358C22×C4—416no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—416no—
362C22×C4—416no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4C22×Q814yes—
369C22×C4C22×Q812yes—
370C22×C4C22×Q812yes—
371C22×C4C2×#4114yes—
372C22×C4C2×#4118yes—
373C22×C4H(5,𝔽2)14yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+414yes—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—88no—
380C4⋊C4—88no—
381C4⋊C4—88no—
382C4⋊C4—88no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—88no—
386C22⋊C4—232no—
387C22⋊C4—48no—
388C22⋊C4—416no—
389C22⋊C4—416no—
390C22⋊C4—48no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C42—84no—
396C42—84no—
397C23—24no—
398C23—24no—
399C23—28no—
400C23—28no—
401C23—28no—
402C23—28no—
403C23—28no—
404C23—28no—
405C23—28no—
406C23—28no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—44no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—42no—
422C23—41no—
423C23—44no—
424C23—44no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—44no—
433C23#54318yes—
434C23C23×Q811yescharacteristic
435C23C22×#4112yes—
436C23C2×H(5,𝔽2)11yescharacteristic
437C23C2×H(5,𝔽2)11yescharacteristic
438C23C2×H(5,𝔽2)14yes—
439C23C2×H(5,𝔽2)12yes—
440C23C2×2−1+412yes—
441C23#58512yes—
442Q8—416no—
443D4—416no—
444D4—432no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
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—24no—
461C2×C4—24no—
462C2×C4—432no—
463C2×C4—416no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
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—42no—
493C2×C4—44no—
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—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—832no—
508C2×C4#54314yes—
509C22—216no—
510C22—28no—
511C22—48no—
512C22—48no—
513C22—48no—
514C22—44no—
515C22—48no—
516C22—48no—
517C22—44no—
518C22—48no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22C2×#54312yes—
527C22C2×#54311yescharacteristic
528C22#345912yes—
529C22Q8×#4112yes—
530C22#348311yescharacteristic
531C22#349214yes—
532C22#350312yes—
533C22#351914yes—
534C22#352911yescharacteristic
535C22#353212yes—
536C22#353212yes—
537C22#353414yes—
538C22#353612yes—
539C22#354911yescharacteristic
540C22#355011yescharacteristic
541C22#355914yes—
542C4—24no—
543C4—44no—
544C4—44no—
545C4—44no—
546C4—42no—
547C4—48no—
548C4—44no—
549C4—84no—
550C4—82no—
551C4—84no—
552C4—84no—
553C2—48no—
554C2—82no—
555C2#3730611yescharacteristic
556C2#3747511yescharacteristic
557C2#3844514yes—
558C2#4097412yes—
559C2#4209112yes—
560C2#4430512yes—
561C2#5229111yescharacteristic
562C2#5240812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17308C2semidirect
2. #17306C2semidirect
3. #17306C2semidirect
4. #16562C2semidirect
5. #16323C2semidirect
6. #15961C2semidirect
7. #15429C2semidirect
8. #15165C2semidirect
9. #14769C2semidirect
10. #14317C2semidirect
11. #14004C2semidirect
209. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
#173082
#173062
#165622
#163232
#159612
#154292
#151652
#147692
#143172
#140042
#139562
#138512