Finite Group Databasebeta

← #2811200

Subgroups — #2811200

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2499C211yesmaximal, characteristic
3#16653C212yesmaximal, complemented
4#16528C214yesmaximal
5#16069C214yesmaximal, complemented
6#15946C212yesmaximal, complemented
7#15739C214yesmaximal
8#15670C212yesmaximal
9#15104C212yesmaximal, complemented
10#14893C212yesmaximal
11#14424C211yesmaximal, characteristic, complemented
12#14404C214yesmaximal, complemented
13#14240C212yesmaximal
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2571—28no—
17#2499—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#399C2212yes—
22C2×#397C2214yes—
23C2×#397C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2214yes—
33C2×#382C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1446C2214yes—
37#1445C2214yes—
38#1445C2214yes—
39#1445C2212yes—
40#1445C2214yes—
41#1445C2212yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2212yes—
57#1437C2212yes—
58#1437C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
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⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C42C2312yes—
96C22×C42C2312yes—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2314yes—
100#400—28no—
101#400—28no—
102#399—28no—
103#399—44no—
104#397—216no—
105#397—44no—
106#391—24no—
107#391—24no—
108#389—216no—
109#389—44no—
110#387—216no—
111#387—216no—
112#384D418yes—
113#382—216no—
114#382—24no—
115#382—44no—
116C4×C4⋊C4—28no—
117C4×C22⋊C4—24no—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C25C2411yescharacteristic
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—42no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4C2412yes—
158C23×C4C2414yes—
159C23×C4C2412yes—
160C23×C4C2414yes—
161C23×C4C2412yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2414yes—
166C23×C4C2411yescharacteristic
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C22⋊Q8—416no—
170#117—416no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C2×D4116yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—28no—
227C2×C42—28no—
228C2×C42—24no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42—42no—
234C2×C42C2×D414yes—
235#95—24no—
236#95—28no—
237#95—24no—
238#95—24no—
239#95—28no—
240#95—28no—
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—24no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—42no—
260#95—44no—
261#95—42no—
262#95#4118yes—
263C24—21no—
264C24—22no—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—24no—
269C24—22no—
270C24H(5,𝔽2)12yes—
271C24C2511yescharacteristic
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—84no—
276C2×Q8—84no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
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—48no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4C22×D414yes—
342C22×C4C22×D414yes—
343C22×C4C22×D412yes—
344C22×C4C22×D414yes—
345C22×C4C2×#4118yes—
346C22×C4C2×#4114yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+414yes—
349C22×C42−1+414yes—
350C4⋊C4—232no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—44no—
357C4⋊C4—48no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C4⋊C4—88no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—88no—
367C22⋊C4—88no—
368C42—28no—
369C42—48no—
370C42—48no—
371C42—416no—
372C42—44no—
373C23—22no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—41no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—41no—
386C23—42no—
387C23—44no—
388C23—44no—
389C23C23×D411yescharacteristic
390C23C22×#4112yes—
391C23C2×H(5,𝔽2)11yescharacteristic
392C23C2×H(5,𝔽2)11yescharacteristic
393C23C2×2−1+412yes—
394C23C2×2−1+412yes—
395C23#58512yes—
396C23#58514yes—
397Q8—88no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—416no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
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#53614yes—
466C2×C4#54718yes—
467C22—42no—
468C22—42no—
469C22—48no—
470C22—42no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22C2×#53611yescharacteristic
476C22C2×#54712yes—
477C22#348111yescharacteristic
478C22#348411yescharacteristic
479C22#348712yes—
480C22#348711yescharacteristic
481C22#351014yes—
482C22#351514yes—
483C22#352212yes—
484C22#353512yes—
485C22#353512yes—
486C22#354512yes—
487C22#355011yescharacteristic
488C22#355112yes—
489C22#355714yes—
490C22#356314yes—
491C4—28no—
492C4—24no—
493C4—48no—
494C4—42no—
495C4—44no—
496C4—44no—
497C4—82no—
498C4—82no—
499C4—84no—
500C4—84no—
501C4—82no—
502C4—84no—
503C2—82no—
504C2#3671111yescharacteristic
505C2#3891912yes—
506C2#4483112yes—
507C2#4526612yes—
508C2#4904014yes—
509C2#5208311yescharacteristic
510C2#5866412yes—
511C2#5907011yescharacteristic
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×#2571C2semidirect
3. #16653C2semidirect
5. #16069C2semidirect
6. #15946C2semidirect
9. #15104C2semidirect
11. #14424C2semidirect
12. #14404C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#24992
#166532
#165282
#160692
#159462
#157392
#156702
#151042
#148932
#144242
#144042
#142402