Finite Group Databasebeta

← #4474597

Subgroups — #4474597

Order 512.

2290 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17189C214yesmaximal, complemented
2#16936C214yesmaximal, complemented
3#16904C211yesmaximal, characteristic, complemented
4#16296C214yesmaximal, complemented
5#16254C211yesmaximal, characteristic, complemented
6#15995C214yesmaximal, complemented
7#15427C214yesmaximal, complemented
8#15297C214yesmaximal
9#14717C212yesmaximal
10#14449C212yesmaximal, complemented
11#14338C211yesmaximal, characteristic
12C22×C22⋊Q8C2212yes—
13C23×C22⋊C4C2211yescharacteristic
14C2×#397C2214yes—
15C2×#396C2214yes—
16C2×#395C2214yescomplemented
17C2×#393C2212yes—
18C2×#393C2212yes—
19C2×#391C2214yescomplemented
20C2×#388C2214yes—
21C2×#387C2214yescomplemented
22C2×#380C2212yes—
23C2×#379C2212yes—
24C2×#379C2214yes—
25#1445C2214yescomplemented
26#1444C2214yescomplemented
27#1443C2214yescomplemented
28#1443C2214yescomplemented
29#1443C2214yescomplemented
30#1442C2214yescomplemented
31#1442C2214yescomplemented
32#1441C2214yescomplemented
33#1441C2214yescomplemented
34#1441C2214yescomplemented
35#1441C2214yescomplemented
36#1440C2214yes—
37#1440C2214yes—
38#1440C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2214yes—
42#1439C2212yes—
43#1439C2214yes—
44#1439C2214yes—
45#1439C2212yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1437C2214yes—
50#1437C2214yes—
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2214yes—
57C26C2311yescharacteristic
58C23×Q8C2312yes—
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—48no—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—44no—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C42C2314yes—
84#397—48no—
85#396—48no—
86#395—48no—
87#393—44no—
88#393—44no—
89#391—48no—
90#388—48no—
91#387—48no—
92#380—44no—
93#379—48no—
94#379—44no—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2312yes—
101C2×#95C2312yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
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—
117C25—22no—
118C25—22no—
119C25—22no—
120C25—24no—
121C25—42no—
122C25—44no—
123C25—44no—
124C25C2411yescharacteristic
125C25C2411yescharacteristic
126C25C2411yescharacteristic
127C22×Q8—24no—
128C22×Q8—28no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—44no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—42no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4#4114yes—
138C23×C4C2414yes—
139C23×C4C2414yes—
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2412yes—
144C23×C4C2414yes—
145C23×C4C2412yes—
146C22⋊Q8—432no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—416no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—44no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—42no—
219#95—42no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226C24—24no—
227C24—21no—
228C24—22no—
229C24—21no—
230C24—24no—
231C24—28no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—24no—
237C24—24no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—24no—
243C24—24no—
244C24—48no—
245C24—48no—
246C24—44no—
247C24—416no—
248C24—44no—
249C24—84no—
250C24—84no—
251C24—88no—
252C24—88no—
253C24—88no—
254C24—88no—
255C24—88no—
256C24C2×#4112yes—
257C24H(5,𝔽2)12yes—
258C242−1+412yes—
259C242−1+414yes—
260C24C2511yescharacteristic
261C2×Q8—216no—
262C2×Q8—84no—
263C2×Q8—84no—
264C2×Q8—88no—
265C2×Q8—88no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
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—44no—
317C22×C4—48no—
318C22×C4—88no—
319C22×C4—84no—
320C22×C4—88no—
321C22×C4C2×#4114yes—
322C22×C4H(5,𝔽2)14yes—
323C22×C4H(5,𝔽2)18yes—
324C4⋊C4—416no—
325C4⋊C4—416no—
326C4⋊C4—48no—
327C4⋊C4—88no—
328C4⋊C4—88no—
329C4⋊C4—88no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—416no—
333C22⋊C4—416no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C22⋊C4—88no—
339C22⋊C4—88no—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C42—88no—
343C23—24no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—28no—
350C23—28no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—28no—
356C23—28no—
357C23—416no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—816no—
383C23—816no—
384C23—832no—
385C23—832no—
386C23—88no—
387C23—88no—
388C23#54114yes—
389C23C22×#4111yescharacteristic
390C23C2×H(5,𝔽2)12yes—
391C23C2×H(5,𝔽2)14yes—
392C23C2×H(5,𝔽2)11yescharacteristic
393C23C2×2−1+412yes—
394C23C2×2−1+411yescharacteristic
395C23#58514yes—
396Q8—816no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
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—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—88no—
450C2×C4—88no—
451C22—28no—
452C22—24no—
453C22—44no—
454C22—42no—
455C22—48no—
456C22—48no—
457C22—42no—
458C22—44no—
459C22—44no—
460C22—48no—
461C22—48no—
462C22—42no—
463C22—42no—
464C22—832no—
465C22—81no—
466C22—81no—
467C22—82no—
468C22—82no—
469C22—84no—
470C22—84no—
471C22—81no—
472C22—81no—
473C22C2×#54111yescharacteristic
474C22#349012yes—
475C22#349011yescharacteristic
476C22#349211yescharacteristic
477C22#351012yes—
478C22#353612yes—
479C22#353714yes—
480C22#353812yes—
481C22#354114yes—
482C22#354214yes—
483C22#356014yes—
484C22#356114yes—
485C22#356114yes—
486C4—44no—
487C4—48no—
488C4—44no—
489C4—84no—
490C4—84no—
491C4—84no—
492C4—84no—
493C4—84no—
494C2—44no—
495C2—82no—
496C2—82no—
497C2#3672811yescharacteristic
498C2#4054112yes—
499C2#4638411yescharacteristic
500C2#4645711yescharacteristic
501C2#4788212yes—
502C2#5362014yes—
503C2#5641814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17189C2semidirect
2. #16936C2semidirect
3. #16904C2semidirect
4. #16296C2semidirect
5. #16254C2semidirect
6. #15995C2semidirect
7. #15427C2semidirect
10. #14449C2semidirect
16. C2×#395C22semidirect
19. C2×#391C22semidirect
21. C2×#387C22semidirect
25. #1445C22semidirect
26. #1444C22semidirect
27. #1443C22semidirect
28. #1443C22semidirect
29. #1443C22semidirect
30. #1442C22semidirect
31. #1442C22semidirect
32. #1441C22semidirect
33. #1441C22semidirect
34. #1441C22semidirect
35. #1441C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#171892
#169362
#169042
#162962
#162542
#159952
#154272
#152972
#147172
#144492
#143382