Finite Group Databasebeta

← #2924567

Subgroups — #2924567

Order 512.

2076 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic, complemented
2C2×#2499C211yesmaximal, characteristic
3#17062C214yesmaximal
4#16689C212yesmaximal, complemented
5#16110C212yesmaximal, complemented
6#15629C214yesmaximal, complemented
7#15583C214yesmaximal
8#15466C212yesmaximal
9#15145C212yesmaximal, complemented
10#14893C212yesmaximal
11#14583C214yesmaximal, complemented
12#14229C212yesmaximal
13#14162C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2809—28no—
17#2499—28no—
18C2×#400C2212yes—
19C2×#399C2212yes—
20C2×#399C2212yes—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#398C2212yes—
24C2×#398C2214yes—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2214yes—
37C2×#382C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39#1446C2212yes—
40#1445C2212yes—
41#1445C2214yes—
42#1445C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2212yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2212yes—
57#1437C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62C4⋊#95C2214yes—
63#1434C2212yes—
64#1434C2214yes—
65#1434C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—42no—
73C2×C22⋊Q8—42no—
74C2×#117—24no—
75C2×#117—42no—
76C2×#117—42no—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2314yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102#400—28no—
103#399—44no—
104#399—44no—
105#398—216no—
106#398—216no—
107#398—44no—
108#398—48no—
109#397—216no—
110#397—28no—
111#391—24no—
112#391—28no—
113#391D418yes—
114#389—28no—
115#389—44no—
116#388—48no—
117#387—216no—
118#387—216no—
119#384—24no—
120#384—48no—
121#382—24no—
122C4×C22⋊C4—24no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—41no—
152C22×Q8—41no—
153C22×Q8—42no—
154C22×Q8C2×D414yes—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4C2414yes—
161C23×C4C2412yes—
162C23×C4C2412yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2411yescharacteristic
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C22⋊Q8—416no—
173#117—416no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D414yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—28no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42C2×D414yes—
244#95—24no—
245#95—28no—
246#95—24no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95C2×D4116yes—
269#95#4118yes—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—21no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24H(5,𝔽2)12yes—
278C24C2511yescharacteristic
279C2×Q8—28no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—84no—
283C2×Q8—84no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4C22×D414yes—
348C22×C4C22×D414yes—
349C22×C4C22×D412yes—
350C22×C4C22×D414yes—
351C22×C4C2×#4118yes—
352C22×C4C2×#4114yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)14yes—
355C22×C42−1+418yes—
356C22×C42−1+414yes—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—416no—
363C4⋊C4—416no—
364C4⋊C4—48no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—88no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C22⋊C4—416no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—88no—
377C42—44no—
378C42—416no—
379C42—48no—
380C42—44no—
381C42—84no—
382C23—22no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—41no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—44no—
398C23C23×D411yescharacteristic
399C23C22×#4112yes—
400C23C2×H(5,𝔽2)12yes—
401C23C2×H(5,𝔽2)11yescharacteristic
402C23C2×H(5,𝔽2)11yescharacteristic
403C23C2×2−1+412yes—
404C23C2×2−1+414yes—
405C23#58512yes—
406Q8—88no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—416no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
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—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4#53614yes—
475C2×C4#55018yes—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22C2×#53611yescharacteristic
485C22C2×#55012yes—
486C22#348311yescharacteristic
487C22#348411yescharacteristic
488C22#348511yescharacteristic
489C22#349912yes—
490C22#350914yes—
491C22#351514yes—
492C22#351912yes—
493C22#353512yes—
494C22#353514yes—
495C22#353512yes—
496C22#353812yes—
497C22#354412yes—
498C22#354911yescharacteristic
499C22#355214yes—
500C4—24no—
501C4—28no—
502C4—48no—
503C4—44no—
504C4—42no—
505C4—44no—
506C4—48no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—84no—
511C4—84no—
512C2—82no—
513C2#3672011yescharacteristic
514C2#3974412yes—
515C2#4349712yes—
516C2#4388312yes—
517C2#4840114yes—
518C2#5154211yescharacteristic
519C2#5519012yes—
520C2#5905611yescharacteristic
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×#2809C2semidirect
4. #16689C2semidirect
5. #16110C2semidirect
6. #15629C2semidirect
9. #15145C2semidirect
11. #14583C2semidirect
13. #14162C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#24992
#170622
#166892
#161102
#156292
#155832
#154662
#151452
#148932
#145832
#142292
#141622