Finite Group Databasebeta

← #215564

Subgroups — #215564

Order 512.

1182 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
1#35317C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9705C211yesmaximal, characteristic, complemented
5#9704C211yesmaximal, characteristic, complemented
6#9698C212yesmaximal, complemented
7#9696C211yesmaximal, characteristic, complemented
8#9694C211yesmaximal, characteristic, complemented
9#9692C212yesmaximal, complemented
10#8341C212yesmaximal
11#8340C212yesmaximal, complemented
12C2×#536—21no—
13C2×#536—21no—
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#433—21no—
17#2868C2211yescharacteristic
18#2868C2211yescharacteristic
19#2858C2211yescharacteristic
20#2858C2211yescharacteristic
21#2658—21no—
22#2658—21no—
23#2542—22no—
24#2542C414yes—
25#2501—22no—
26#2501C414yes—
27C4×#117C2211yescharacteristic
28#1923—22no—
29#1916—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1832—21no—
33#1831C2211yescharacteristic
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41—21no—
50C23×Q8—21no—
51#536—48no—
52#536—44no—
53#536—44no—
54#532—42no—
55#532—42no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#124—21no—
70C2×#124—21no—
71C2×#124—21no—
72C2×#124—21no—
73C2×#124—21no—
74C2×#123—24no—
75C2×#123—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×#121—21no—
79C2×#121C2×C412yes—
80C2×C4×D4—21no—
81C2×C4×D4C2×C412yes—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4C2×C412yes—
86C22×C42C2311yescharacteristic
87#433—44no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8—22no—
100#399—24no—
101#399—24no—
102#399—24no—
103#399—24no—
104#390—22no—
105#390—22no—
106#390C2×C414yes—
107#390C2×C414yes—
108#384—22no—
109#384—22no—
110#384C2×C414yes—
111#384C2×C414yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4D412yes—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4D412yes—
116#376D411yescharacteristic
117#376D411yescharacteristic
118C43C2311yescharacteristic
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—21no—
123#336—21no—
124#336—22no—
125#336—22no—
126#332—42no—
127#332—42no—
128#332—42no—
129#332—42no—
130#332—42no—
131#332—42no—
132#331—42no—
133#331—42no—
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—42no—
137C2×#41—42no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—44no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—44no—
175#127—22no—
176#127—22no—
177#124—44no—
178#124—44no—
179#124—42no—
180#124—42no—
181#124—42no—
182#124—42no—
183#124—44no—
184#123—416no—
185#123—416no—
186C22⋊Q8—416no—
187C22⋊Q8—416no—
188#121—44no—
189#121—42no—
190#121—42no—
191C4×D4—44no—
192C4×D4—42no—
193C4×D4—42no—
194#117—24no—
195#117D814yes—
196#117SD1614yes—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2411yescharacteristic
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—44no—
245C4⋊C8—44no—
246C4⋊C8—82no—
247C22⋊C8—44no—
248C22⋊C8—82no—
249#97—42no—
250C4×C8—42no—
251C4×C8—82no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—22no—
257C24C2×C22⋊C411yescharacteristic
258#41—88no—
259#41—88no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—88no—
276C2×Q8—88no—
277C2×D4—24no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×D812yes—
315C22×C4C2×SD1612yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4#10218yes—
348C22⋊C4—24no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C22⋊C4—48no—
355C22⋊C4#10218yes—
356C42—22no—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—42no—
366C42—42no—
367C42C2×D812yes—
368C42C2×D812yes—
369C42C2×SD1612yes—
370C42C2×SD1612yes—
371C42AGL(1,ℤ/8)14yes—
372C42H(5,𝔽2)12yes—
373C42H(5,𝔽2)12yes—
374C23—21no—
375C23—22no—
376C23—21no—
377C23—22no—
378C23—42no—
379C23—44no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23C2×#10212yes—
384C23C22×C22⋊C411yescharacteristic
385Q8—88no—
386Q8—82no—
387Q8—82no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391Q8—82no—
392D4—84no—
393D4—88no—
394D4—82no—
395D4—82no—
396D4—81no—
397D4—81no—
398D4—82no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—88no—
431C2×C4—88no—
432C2×C4C2×#10214yes—
433C2×C4C2×#10214yes—
434C2×C4C2×#10212yes—
435C2×C4#41814yes—
436C2×C4#51812yes—
437C2×C4#53512yes—
438C2×C4#53511yescharacteristic
439C2×C4#53511yescharacteristic
440C2×C4C22×D811yescharacteristic
441C2×C4C22×SD1611yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)12yes—
443C2×C4#57712yes—
444C2×C4#57812yes—
445C8—82no—
446C8—82no—
447C8—82no—
448C8—82no—
449C22—22no—
450C22—24no—
451C22—41no—
452C22—41no—
453C22—42no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—84no—
458C22—84no—
459C22#230911yescharacteristic
460C22#288912yes—
461C22C22×#10211yescharacteristic
462C22C2×#41811yescharacteristic
463C22#289812yes—
464C4—22no—
465C4—22no—
466C4—24no—
467C4—24no—
468C4—24no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4#320512yes—
476C4#320812yes—
477C4#322111yescharacteristic
478C4#322311yescharacteristic
479C4#322512yes—
480C2—42no—
481C2—162no—
482C2#1772112yes—
483C2#1775912yes—
484C2#2180211yescharacteristic
485C2#2180311yescharacteristic
486C2#2181411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21922C2semidirect
3. #20812C2semidirect
4. #9705C2semidirect
5. #9704C2semidirect
6. #9698C2semidirect
7. #9696C2semidirect
8. #9694C2semidirect
9. #9692C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#353172
#219222
#208122
#97052
#97042
#96982
#96962
#96942
#96922
#83412
#83402