Finite Group Databasebeta

← #215580

Subgroups — #215580

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35317C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#9710C212yesmaximal, complemented
5#9705C211yesmaximal, characteristic, complemented
6#9704C211yesmaximal, characteristic, complemented
7#9703C212yesmaximal, complemented
8#9696C211yesmaximal, characteristic, complemented
9#9694C211yesmaximal, characteristic, complemented
10#8341C212yesmaximal
11#8340C212yesmaximal, complemented
12C2×#536—21no—
13C2×#536—21no—
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18#2858C2211yescharacteristic
19#2858C2211yescharacteristic
20#2658—21no—
21#2658—21no—
22#2542—22no—
23#2542C414yes—
24#2501—22no—
25#2501C414yes—
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1917—22no—
29#1842C2211yescharacteristic
30#1840—21no—
31#1831C2212yes—
32#1831C2211yescharacteristic
33#1818—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2211yescharacteristic
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39#1750—21no—
40#1746—21no—
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
87C2×C4⋊C8—22no—
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×#97—21no—
95C2×#97—22no—
96C2×C4×C8—21no—
97#399—24no—
98#399—24no—
99#399—24no—
100#399—24no—
101#390—22no—
102#390—22no—
103#390C2×C414yes—
104#390C2×C414yes—
105#384—22no—
106#384—22no—
107#384C2×C414yes—
108#384C2×C414yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113#376D411yescharacteristic
114#376D411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—22no—
120#336—21no—
121#336—21no—
122#336—22no—
123#336—22no—
124#336—22no—
125#332—42no—
126#332—42no—
127#332—42no—
128#332—42no—
129#332—42no—
130#332—42no—
131#331—42no—
132#331—42no—
133C2×#41—44no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—44no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—44no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—44no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—44no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174#127—22no—
175#127—22no—
176#124—44no—
177#124—44no—
178#124—44no—
179#124—42no—
180#124—42no—
181#124—42no—
182#124—42no—
183#123—416no—
184#123—416no—
185C22⋊Q8—416no—
186C22⋊Q8—416no—
187#121—44no—
188#121—42no—
189#121—42no—
190C4×D4—44no—
191C4×D4—42no—
192C4×D4—42no—
193#117—24no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—82no—
244C22⋊C8—44no—
245C22⋊C8—82no—
246#97—42no—
247#97—82no—
248C4×C8—42no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24—22no—
254C24C2×C22⋊C411yescharacteristic
255#41—88no—
256#41—88no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—44no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×D4—24no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4AGL(1,ℤ/8)14yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
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—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—48no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C42—22no—
354C42—21no—
355C42—21no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—21no—
360C42—21no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—42no—
366C42—42no—
367C42AGL(1,ℤ/8)12yes—
368C42AGL(1,ℤ/8)12yes—
369C42H(5,𝔽2)12yes—
370C42H(5,𝔽2)12yes—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—22no—
375C23—42no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23#41812yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—88no—
383Q8—82no—
384Q8—82no—
385Q8—82no—
386Q8—82no—
387Q8—82no—
388Q8—82no—
389D4—84no—
390D4—88no—
391D4—82no—
392D4—82no—
393D4—81no—
394D4—81no—
395D4—82no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
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—88no—
430C2×C4—88no—
431C2×C4#41814yes—
432C2×C4#41812yes—
433C2×C4#51812yes—
434C2×C4#53512yes—
435C2×C4#53511yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)12yes—
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4#57712yes—
441C2×C4#57812yes—
442C8—82no—
443C8—82no—
444C8—82no—
445C8—82no—
446C22—22no—
447C22—24no—
448C22—41no—
449C22—41no—
450C22—42no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—84no—
455C22—84no—
456C22#230911yescharacteristic
457C22#288912yes—
458C22C2×#41811yescharacteristic
459C22C2×#41811yescharacteristic
460C22#289812yes—
461C4—22no—
462C4—22no—
463C4—24no—
464C4—41no—
465C4—41no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4#321712yes—
473C4#321712yes—
474C4#322111yescharacteristic
475C4#322311yescharacteristic
476C4#322512yes—
477C2—42no—
478C2—162no—
479C2#1775912yes—
480C2#1775912yes—
481C2#2180211yescharacteristic
482C2#2181011yescharacteristic
483C2#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. #21926C2semidirect
3. #20786C2semidirect
4. #9710C2semidirect
5. #9705C2semidirect
6. #9704C2semidirect
7. #9703C2semidirect
8. #9696C2semidirect
9. #9694C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#353172
#219262
#207862
#97102
#97052
#97042
#97032
#96962
#96942
#83412
#83402