Finite Group Databasebeta

← #2811498

Subgroups — #2811498

Order 512.

2034 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2502C211yesmaximal, characteristic
3#16687C214yesmaximal, complemented
4#16653C212yesmaximal, complemented
5#15798C212yesmaximal, complemented
6#15767C214yesmaximal, complemented
7#15670C212yesmaximal
8#15290C214yesmaximal
9#15089C212yesmaximal, complemented
10#14822C212yesmaximal
11#14454C211yesmaximal, characteristic, complemented
12#14393C214yesmaximal
13#14225C212yesmaximal
14C22×C22⋊Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#2571—28no—
17#2502—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#398C2214yes—
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2214yes—
27C2×#388C2214yes—
28C2×#385C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2214yes—
32C2×#382C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1446C2214yes—
37#1445C2214yes—
38#1445C2212yes—
39#1445C2212yes—
40#1445C2214yes—
41#1444C2212yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1439C2212yes—
57#1437C2212yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1434C2214yes—
63#1434C2214yes—
64#1434C2214yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—42no—
74C2×C22⋊Q8—42no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C42C2314yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99#400—28no—
100#400—28no—
101#400—44no—
102#398—216no—
103#397—28no—
104#397—44no—
105#391—24no—
106#391—24no—
107#389—216no—
108#388—216no—
109#385—216no—
110#384—24no—
111#384D418yes—
112#382—216no—
113#382—44no—
114#380—44no—
115C4×C4⋊C4—28no—
116C4×C22⋊C4—24no—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
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—24no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174C22⋊Q8—416no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—416no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D4116yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4#4118yes—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—28no—
228C2×C42—28no—
229C2×C42—28no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—44no—
233C2×C42C2×D414yes—
234#95—24no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—24no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—44no—
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—44no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—24no—
269C24C2×#4114yes—
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—22no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
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—48no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—44no—
342C22×C4—48no—
343C22×C4—88no—
344C22×C4—84no—
345C22×C4—88no—
346C22×C4C22×D414yes—
347C22×C4C22×D414yes—
348C22×C4C22×D412yes—
349C22×C4C22×D414yes—
350C22×C4C2×#4114yes—
351C22×C4C2×#4114yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+414yes—
354C22×C42−1+414yes—
355C4⋊C4—44no—
356C4⋊C4—416no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—84no—
368C22⋊C4—216no—
369C22⋊C4—48no—
370C22⋊C4—416no—
371C22⋊C4—88no—
372C42—28no—
373C42—44no—
374C42—48no—
375C42—416no—
376C23—24no—
377C23—22no—
378C23—24no—
379C23—28no—
380C23—28no—
381C23—24no—
382C23—24no—
383C23—22no—
384C23—41no—
385C23—42no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23#55514yes—
392C23C23×D411yescharacteristic
393C23C22×#4112yes—
394C23C2×H(5,𝔽2)11yescharacteristic
395C23C2×H(5,𝔽2)11yescharacteristic
396C23C2×2−1+412yes—
397C23C2×2−1+412yes—
398C23#58514yes—
399C23#58512yes—
400Q8—88no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—416no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
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—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
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—816no—
469C2×C4#53614yes—
470C2×C4#54818yes—
471C22—28no—
472C22—24no—
473C22—44no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—44no—
478C22—48no—
479C22C2×#53611yescharacteristic
480C22C2×#54812yes—
481C22C2×#55511yescharacteristic
482C22#348411yescharacteristic
483C22#348512yes—
484C22#348711yescharacteristic
485C22#351112yes—
486C22#351514yes—
487C22#352114yes—
488C22#353312yes—
489C22#353512yes—
490C22#353512yes—
491C22#354012yes—
492C22#354411yescharacteristic
493C22#356314yes—
494C22#356514yes—
495C4—24no—
496C4—28no—
497C4—44no—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—84no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4—84no—
506C4—84no—
507C2—44no—
508C2#3671111yescharacteristic
509C2#3912212yes—
510C2#4368812yes—
511C2#4514212yes—
512C2#5119414yes—
513C2#5205411yescharacteristic
514C2#5875112yes—
515C2#5913511yescharacteristic
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. #16687C2semidirect
4. #16653C2semidirect
5. #15798C2semidirect
6. #15767C2semidirect
9. #15089C2semidirect
11. #14454C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#25022
#166872
#166532
#157982
#157672
#156702
#152902
#150892
#148222
#144542
#143932
#142252