Finite Group Databasebeta

← #4869153

Subgroups — #4869153

Order 512.

2836 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C212yesmaximal
2#17186C214yesmaximal, complemented
3#16681C214yesmaximal, complemented
4#16573C214yesmaximal, complemented
5#16561C214yesmaximal, complemented
6#16264C214yesmaximal, complemented
7#16112C211yesmaximal, characteristic, complemented
8#16093C212yesmaximal, complemented
9#15762C211yesmaximal, characteristic, complemented
10#14997C214yesmaximal, complemented
11#14337C211yesmaximal, characteristic
12C23×#41C2211yescharacteristic
13C22×C4⋊Q8C2212yes—
14C22×#124C2212yes—
15C22×C22⋊Q8C2212yes—
16#2432—216no—
17C2×#398C2214yes—
18C2×#395C2214yes—
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#391C2212yes—
22C2×#390C2212yes—
23C2×#387C2214yes—
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#380C2214yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1446C2211yescharacteristic
30#1446C2211yescharacteristic
31#1446C2214yes—
32#1444C2214yes—
33#1443C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1441C2214yes—
37#1441C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1440C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1438C2212yes—
54#1438C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1434C2214yes—
58C22×#41—44no—
59C22×#41—48no—
60C22×#41D418yes—
61C23×Q8C2311yescharacteristic
62C23×Q8C2312yes—
63C23×D4C2312yes—
64C23×D4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C2×C4⋊Q8—28no—
68C2×C4⋊Q8—48no—
69C2×#124—28no—
70C2×#124—48no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8—44no—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C42C2312yes—
91C22×C42C2312yes—
92#398—48no—
93#395—48no—
94#394—216no—
95#394—216no—
96#391—28no—
97#390—28no—
98#387—48no—
99#385—216no—
100#385—216no—
101#380—48no—
102#380—44no—
103C4×C22⋊C4—28no—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C25C2412yes—
126C25C2411yescharacteristic
127C2×#41—416no—
128C2×#41—432no—
129C2×#41—464no—
130C22×Q8—24no—
131C22×Q8—28no—
132C22×Q8—28no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—44no—
137C22×Q8C2×D414yes—
138C22×D4—28no—
139C22×D4—44no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—44no—
144C22×D4C2×D418yes—
145C22×D4C2×D414yes—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—28no—
149C23×C4—24no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4C2×D418yes—
153C23×C4C2×D414yes—
154C23×C4C2414yes—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2411yescharacteristic
160C23×C4C2414yes—
161C23×C4C2411yescharacteristic
162C23×C4C2412yes—
163C4⋊Q8—432no—
164#124—432no—
165C22⋊Q8—432no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—28no—
217C2×C42—44no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239C24—22no—
240C24—22no—
241C24—24no—
242C24—22no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—24no—
248C24—24no—
249C24—24no—
250C24—24no—
251C24C22×D414yes—
252C24C22×D412yes—
253C24C2511yescharacteristic
254#41—4128no—
255C2×Q8—216no—
256C2×Q8—44no—
257C2×Q8—416no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×D4—216no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—48no—
267C2×D4—44no—
268C2×D4—416no—
269C2×D4—416no—
270C2×D4—416no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—216no—
284C22×C4—216no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
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—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—416no—
324C22×C4—416no—
325C22×C4—416no—
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D414yes—
329C22×C4H(5,𝔽2)18yes—
330C22×C4H(5,𝔽2)14yes—
331C22×C42−1+418yes—
332C4⋊C4—416no—
333C4⋊C4—48no—
334C4⋊C4—416no—
335C4⋊C4—416no—
336C4⋊C4—416no—
337C4⋊C4—88no—
338C4⋊C4—88no—
339C22⋊C4—416no—
340C22⋊C4—416no—
341C22⋊C4—416no—
342C22⋊C4—416no—
343C22⋊C4—416no—
344C22⋊C4—416no—
345C22⋊C4—88no—
346C22⋊C4—88no—
347C22⋊C4—88no—
348C42—48no—
349C42—48no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—22no—
354C23—24no—
355C23—28no—
356C23—28no—
357C23—28no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23C23×D411yescharacteristic
383C23C2×H(5,𝔽2)14yes—
384C23C2×H(5,𝔽2)12yes—
385C23C2×2−1+414yes—
386C23#58514yes—
387Q8—416no—
388Q8—816no—
389D4—432no—
390D4—416no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—432no—
403C2×C4—416no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4#53418yes—
445C2×C4#53414yes—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—42no—
451C22—48no—
452C22—48no—
453C22—48no—
454C22—84no—
455C22—84no—
456C22—84no—
457C22—84no—
458C22—84no—
459C22—84no—
460C22C2×#53411yescharacteristic
461C22C2×#53412yes—
462C22#348914yes—
463C22#353214yes—
464C22#354114yes—
465C22#354714yes—
466C22#354714yes—
467C22#355012yes—
468C22#355112yes—
469C22#355214yes—
470C22#356214yes—
471C4—24no—
472C4—28no—
473C4—42no—
474C4—48no—
475C4—84no—
476C4—84no—
477C4—84no—
478C4—84no—
479C4—84no—
480C2—84no—
481C2—82no—
482C2#3737612yes—
483C2#3743611yescharacteristic
484C2#4292614yes—
485C2#5513414yes—
486C2#5668714yes—
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
2. #17186C2semidirect
3. #16681C2semidirect
4. #16573C2semidirect
5. #16561C2semidirect
6. #16264C2semidirect
7. #16112C2semidirect
8. #16093C2semidirect
9. #15762C2semidirect
10. #14997C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
#171862
#166812
#165732
#165612
#162642
#161122
#160932
#157622
#149972
#143372