Finite Group Databasebeta

← #508479

Subgroups — #508479

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32930C211yesmaximal, characteristic
2C2×#1808C211yesmaximal, characteristic, complemented
3#13494C212yesmaximal, complemented
4#13493C212yesmaximal, complemented
5#13381C212yesmaximal, complemented
6#13380C212yesmaximal, complemented
7#12991C211yesmaximal, characteristic, complemented
8#12991C211yesmaximal, characteristic, complemented
9#12920C211yesmaximal, characteristic, complemented
10#12919C211yesmaximal, characteristic, complemented
11#12679C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C2×#466C2211yescharacteristic
14C2×#466C2211yescharacteristic
15C2×#466C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18#2847—21no—
19C2×#380—21no—
20C2×#367C2212yescomplemented
21C2×#366C2212yescomplemented
22C2×#357C2212yescomplemented
23C2×#357C2212yescomplemented
24C2×C22⋊C16C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26C2×C22⋊C16C2212yes—
27C2×C22⋊C16C2212yes—
28#1893—21no—
29#1893—21no—
30#1893—21no—
31#1808—21no—
32#1808—21no—
33#1808C2212yes—
34#1808C2212yes—
35#1808C2211yescharacteristic
36#1808C2211yescharacteristic
37#1808C2211yescharacteristic
38C2×#340—21no—
39#1380C2212yescomplemented
40#1380C2212yescomplemented
41#1380C2212yescomplemented
42#1380C2212yescomplemented
43C24×C4D411yescharacteristic
44C22×D8C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—42no—
48C23×C8—42no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#120—41no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C22⋊C4—41no—
62C22×C16C2312yes—
63C22×C16C2312yes—
64#466—22no—
65#466—22no—
66#466—42no—
67#466—42no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—42no—
71C2×#107—42no—
72C2×#107C2312yes—
73C2×#107C2312yes—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#102—21no—
78C2×#102—21no—
79C2×#102—21no—
80#386—41no—
81#386—41no—
82#386—41no—
83#380—22no—
84#380—22no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380—41no—
89#380—41no—
90C2×#95—41no—
91#367—42no—
92#366—42no—
93#357—42no—
94#357—42no—
95C22⋊C16—22no—
96C22⋊C16—22no—
97C22⋊C16—42no—
98C22⋊C16—42no—
99#340—22no—
100#340—22no—
101#340—21no—
102#340—21no—
103#340—21no—
104C25D811yescharacteristic
105C22×D4—21no—
106C22×D4—41no—
107C22×D4—41no—
108C22×D4—41no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4—41no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—41no—
117C23×C4D811yescharacteristic
118C23×C4D811yescharacteristic
119C23×C4D811yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×D8—21no—
124C2×D8—21no—
125C2×D8—22no—
126C2×D8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—48no—
147C22×C8—48no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8#4112yes—
152C22×C8#4112yes—
153C22×C8C2411yescharacteristic
154#121—42no—
155#121—42no—
156#121—42no—
157#121—42no—
158#121—42no—
159#121—42no—
160#120—84no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—84no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187#107—21no—
188#107—21no—
189#107—22no—
190#107—22no—
191#107—22no—
192#107—22no—
193#107—22no—
194#107—22no—
195#107—42no—
196#107—42no—
197#102—42no—
198#102—42no—
199#102—42no—
200#95—42no—
201#95—42no—
202#95—41no—
203#95—41no—
204#95—41no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24C2×D811yescharacteristic
214C24C2×D811yescharacteristic
215C24C2×D811yescharacteristic
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C2×D4—82no—
226C2×D4—82no—
227C2×D4—82no—
228C2×D4—82no—
229C2×D4—82no—
230C2×D4—82no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4#12312yes—
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277D8—42no—
278D8—42no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8#12312yes—
297C2×C8C2×#4111yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8H(5,𝔽2)12yes—
300C2×C8H(5,𝔽2)12yes—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—84no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—84no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C16—42no—
322C16—42no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—28no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—44no—
341C23—44no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23#48012yes—
354C23#50812yes—
355C23#50912yes—
356C23#50912yes—
357C23#51012yes—
358C23C22×D811yescharacteristic
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—164no—
364D4—164no—
365D4—164no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—164no—
405C2×C4—164no—
406C2×C4#48012yes—
407C2×C4C2×#12311yescharacteristic
408C2×C4#53411yescharacteristic
409C2×C4#53411yescharacteristic
410C2×C4C2×#13711yescharacteristic
411C2×C4#57511yescharacteristic
412C2×C4#57511yescharacteristic
413C8—22no—
414C8—41no—
415C8—41no—
416C8#53812yes—
417C8#53812yes—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—416no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22#223414yes—
447C22C2×#48011yescharacteristic
448C22#318611yescharacteristic
449C22#318611yescharacteristic
450C22C2×#50811yescharacteristic
451C22C2×#50911yescharacteristic
452C22C2×#50911yescharacteristic
453C22C2×#51011yescharacteristic
454C4—22no—
455C4—41no—
456C4—41no—
457C4—161no—
458C4—162no—
459C4—162no—
460C4—161no—
461C4—161no—
462C4#262211yescharacteristic
463C4#309311yescharacteristic
464C4#319611yescharacteristic
465C4#319611yescharacteristic
466C2—24no—
467C2—42no—
468C2—42no—
469C2—321no—
470C2#2989011yescharacteristic
471C2C2×#223411yescharacteristic
472C2#3364611yescharacteristic
473C2#3375411yescharacteristic
474C2#3375611yescharacteristic
475C2#3375811yescharacteristic
476C2#3375911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#32930C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1808C2semidirect
3. #13494C2semidirect
4. #13493C2semidirect
5. #13381C2semidirect
6. #13380C2semidirect
7. #12991C2semidirect
8. #12991C2semidirect
9. #12920C2semidirect
10. #12919C2semidirect
11. #12679C2semidirect
20. C2×#367C22semidirect
21. C2×#366C22semidirect
22. C2×#357C22semidirect
23. C2×#357C22semidirect
39. #1380C22semidirect
40. #1380C22semidirect
41. #1380C22semidirect
42. #1380C22semidirect

Maximal subgroups

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

subgroupindex
#329302
C2×#18082
#134942
#134932
#133812
#133802
#129912
#129202
#129192
#126792