Finite Group Databasebeta

← #950388

Subgroups — #950388

Order 512.

2644 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17272C212yesmaximal, complemented
2#17137C211yesmaximal, characteristic, complemented
3#17135C211yesmaximal, characteristic, complemented
4#16822C214yesmaximal, complemented
5#16556C214yesmaximal, complemented
6#15965C212yesmaximal, complemented
7#15510C212yesmaximal, complemented
8#15165C212yesmaximal, complemented
9#14770C214yesmaximal, complemented
10#14319C212yesmaximal, complemented
11#14210C214yesmaximal, complemented
12#13937C212yesmaximal
13#13851C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C2×#396C2214yes—
16C2×#396C2214yes—
17C2×#396C2214yes—
18C2×#395C2214yes—
19C2×#393C2212yes—
20C2×#393C2214yes—
21C2×#391C2214yes—
22C2×#391C2212yes—
23C2×#390C2214yes—
24C2×#389C2212yes—
25C2×#387C2214yes—
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2214yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31#1446C2211yescharacteristic
32#1446C2211yescharacteristic
33#1446C2212yes—
34#1444C2212yes—
35#1444C2214yes—
36#1444C2212yes—
37#1443C2214yes—
38#1442C2212yes—
39#1442C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1439C2212yes—
43#1439C2212yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2212yes—
51#1438C2211yescharacteristic
52#1438C2214yes—
53#1438C2211yescharacteristic
54#1438C2212yes—
55#1438C2212yes—
56#1438C2212yes—
57#1438C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22×#41—42no—
68C22×#41—42no—
69C22×#41—48no—
70C22×#41Q818yescomplemented
71C23×Q8C2311yescharacteristic
72C23×Q8C2312yes—
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C42C2312yes—
98C22×C42C2312yes—
99C22×C42C2314yes—
100#396—216no—
101#396—48no—
102#396—48no—
103#395—48no—
104#393—28no—
105#393—48no—
106#391—44no—
107#391—48no—
108#390—48no—
109#389—44no—
110#387—48no—
111#380—44no—
112C4×C22⋊C4—48no—
113C4×C22⋊C4—44no—
114C4×C22⋊C4—42no—
115C4×C22⋊C4—42no—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2314yes—
139C2×#95C2312yes—
140C25C2412yes—
141C25C2411yescharacteristic
142C2×#41—416no—
143C2×#41—432no—
144C2×#41—432no—
145C2×#41—432no—
146C22×Q8—24no—
147C22×Q8—28no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8C2×Q814yes—
155C22×D4—28no—
156C22×D4—42no—
157C22×D4—44no—
158C22×D4—42no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—44no—
162C22×D4C2×Q818yes—
163C22×D4C2×Q814yes—
164C23×C4—28no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4C2×Q818yes—
172C23×C4C2×Q814yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
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—28no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—24no—
252#95—28no—
253#95—24no—
254#95—44no—
255#95—42no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95#4118yescomplemented
268C24—22no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—21no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—24no—
281C24—22no—
282C24—24no—
283C24C22×Q814yes—
284C24C22×Q812yes—
285C24C2511yescharacteristic
286#41—4128no—
287C2×Q8—216no—
288C2×Q8—44no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—88no—
293C2×Q8—84no—
294C2×Q8—88no—
295C2×Q8—84no—
296C2×D4—216no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—48no—
301C2×D4—416no—
302C2×D4—416no—
303C2×D4—48no—
304C2×D4—48no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
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—21no—
317C22×C4—21no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—216no—
331C22×C4—216no—
332C22×C4—28no—
333C22×C4—22no—
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—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—416no—
362C22×C4—416no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4C22×Q812yes—
366C22×C4C22×Q812yes—
367C22×C4C22×Q814yes—
368C22×C4C2×#4118yes—
369C22×C4C2×#4114yes—
370C22×C4H(5,𝔽2)14yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)18yes—
374C22×C42−1+414yes—
375C4⋊C4—48no—
376C4⋊C4—416no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C4⋊C4—88no—
380C4⋊C4—84no—
381C4⋊C4—88no—
382C4⋊C4—84no—
383C22⋊C4—48no—
384C22⋊C4—416no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—416no—
388C22⋊C4—48no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—84no—
396C42—84no—
397C42—88no—
398C23—24no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—28no—
404C23—24no—
405C23—28no—
406C23—24no—
407C23—42no—
408C23—44no—
409C23—42no—
410C23—41no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—41no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23C23×Q811yescharacteristic
436C23C22×#4112yes—
437C23C2×H(5,𝔽2)11yescharacteristic
438C23C2×H(5,𝔽2)11yescharacteristic
439C23C2×H(5,𝔽2)14yes—
440C23C2×H(5,𝔽2)12yes—
441C23C2×2−1+412yes—
442C23#58512yes—
443Q8—416no—
444Q8—816no—
445D4—432no—
446D4—416no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—432no—
466C2×C4—416no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
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—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4#54314yes—
509C2×C4#54318yes—
510C22—48no—
511C22—48no—
512C22—48no—
513C22—42no—
514C22—44no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—44no—
519C22—82no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22C2×#54312yes—
528C22C2×#54311yescharacteristic
529C22#346012yes—
530C22Q8×#4112yes—
531C22#348111yescharacteristic
532C22#350312yes—
533C22#351114yes—
534C22#351914yes—
535C22#352911yescharacteristic
536C22#353212yes—
537C22#353212yes—
538C22#353612yes—
539C22#354114yes—
540C22#354614yes—
541C22#354911yescharacteristic
542C22#355011yescharacteristic
543C4—24no—
544C4—28no—
545C4—44no—
546C4—48no—
547C4—44no—
548C4—42no—
549C4—44no—
550C4—84no—
551C4—84no—
552C4—82no—
553C4—84no—
554C2—84no—
555C2—82no—
556C2#3730611yescharacteristic
557C2#3747511yescharacteristic
558C2#3998114yes—
559C2#4123412yes—
560C2#4204312yes—
561C2#4432512yes—
562C2#5228811yescharacteristic
563C2#5242012yes—
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. #17272C2semidirect
2. #17137C2semidirect
3. #17135C2semidirect
4. #16822C2semidirect
5. #16556C2semidirect
6. #15965C2semidirect
7. #15510C2semidirect
8. #15165C2semidirect
9. #14770C2semidirect
10. #14319C2semidirect
11. #14210C2semidirect
70. C22×#41Q8semidirect
267. #95#41semidirect

Maximal subgroups

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

subgroupindex
#172722
#171372
#171352
#168222
#165562
#159652
#155102
#151652
#147702
#143192
#142102
#139372
#138512