Finite Group Databasebeta

← #193245

Subgroups — #193245

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34232C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8924C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8920C212yesmaximal, complemented
10#7894C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2366C2212yes—
17#2365C2212yes—
18#2363—22no—
19#2353—22no—
20#2341—21no—
21#2339—21no—
22#2333—22no—
23#2319—21no—
24C4×#124—22no—
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2275—21no—
28C4×#117C2211yescharacteristic
29#1990C2212yes—
30#1957C2212yes—
31#1949—22no—
32#1948C2212yes—
33#1948C2212yes—
34#1947C2212yes—
35#1947C2212yes—
36#1875—22no—
37#1854—22no—
38#1808—21no—
39#1807—21no—
40#1804—22no—
41C4×#107—21no—
42C4×#106—21no—
43#1540C418yescomplemented
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#331C2211yescharacteristic
47C2×#331C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41—21no—
53C23×C8—21no—
54#532—22no—
55#532—22no—
56#532C2×C414yes—
57#532Q814yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#127C2311yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121D412yes—
63C2×C4×Q8—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4D412yes—
68C2×#117—21no—
69C2×#117D412yes—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#431—22no—
74#431C2×C414yescomplemented
75C2×#107—22no—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2×C414yescomplemented
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#102—24no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92#391—22no—
93#390—24no—
94#390—22no—
95#390—22no—
96#389—22no—
97#388—24no—
98#388—24no—
99#387—24no—
100#386—24no—
101#386—24no—
102#384C2312yes—
103#383—21no—
104#382—21no—
105#382D412yes—
106#381—22no—
107#381—21no—
108#380—22no—
109#380—21no—
110#380—21no—
111#380D411yescharacteristic
112#380D411yescharacteristic
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4C2312yes—
119C4×C4⋊C4C2312yes—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126#376—21no—
127#376—21no—
128C2×#95—22no—
129C43—21no—
130#340—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#332C2×C414yescomplemented
136#332C2×C414yescomplemented
137#331C2×C414yes—
138#331C2×C414yes—
139C2×#41—24no—
140C2×#41—24no—
141C2×#41—82no—
142C2×#41—82no—
143C22×Q8—21no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C2×D412yes—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—82no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8C22×C412yes—
163C4⋊Q8C2×Q812yes—
164#127—21no—
165#127—21no—
166#127C22×C412yes—
167#127C2×Q812yes—
168#124—24no—
169#124—24no—
170#121—24no—
171#121—24no—
172#121—24no—
173#121C4⋊C418yes—
174C4×Q8—82no—
175C4×Q8—82no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—82no—
182C4×D4—82no—
183C4×D4C4⋊C418yes—
184#117—24no—
185#117—42no—
186#117—82no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—41no—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42C2411yescharacteristic
221#107—44no—
222#107—82no—
223#106—44no—
224#106—82no—
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8C22×C412yes—
232C4⋊C8C22×C412yes—
233C4⋊C8C22×C412yes—
234C4⋊C8C22×C412yes—
235#102—84no—
236C22⋊C8—24no—
237C22⋊C8—42no—
238C4×C8—24no—
239C4×C8—22no—
240C4×C8—22no—
241#95—24no—
242#95—24no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—41no—
252#95—41no—
253C24—22no—
254C24#12011yescharacteristic
255#41—88no—
256#41—84no—
257#41—84no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×D4—28no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—84no—
275C2×D4C2×C4⋊C414yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—48no—
289C22×C4—42no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4C2×C4⋊C412yes—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4C22⋊Q812yes—
300C22×C4C22⋊Q812yes—
301C22×C4#12312yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—84no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—84no—
333C4⋊C4C2×C4⋊C414yes—
334C4⋊C4#13514yes—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—44no—
338C22⋊C4—48no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C22⋊C4—82no—
344C42—44no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—81no—
350C42—81no—
351C42—82no—
352C42—84no—
353C42C2×C4⋊C412yes—
354C42C4×D412yes—
355C42C4×D412yes—
356C42C22⋊Q812yes—
357C42C22⋊Q812yes—
358C42#12312yes—
359C42C23×C411yescharacteristic
360C42C22×Q811yescharacteristic
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C23—22no—
364C23—22no—
365C23—21no—
366C23—21no—
367C23—24no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23#38512yes—
373C23C2×#12011yescharacteristic
374Q8—44no—
375Q8—42no—
376Q8—42no—
377D4—44no—
378D4—44no—
379D4—44no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—82no—
385D4—88no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—88no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—168no—
419C2×C4#38512yes—
420C2×C4#38512yes—
421C2×C4#38512yes—
422C2×C4C22×C4⋊C411yescharacteristic
423C2×C4C2×C4×D411yescharacteristic
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×C22⋊Q811yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C2×C4C2×#13512yes—
429C2×C4#57512yes—
430C2×C4#57612yes—
431C2×C4#57712yes—
432C8—42no—
433C8—42no—
434C8—42no—
435C8—42no—
436C8—82no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—42no—
441C22—42no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—84no—
446C22—84no—
447C22C2×#38511yescharacteristic
448C22#300212yes—
449C22#301212yes—
450C22#302111yescharacteristic
451C22#302211yescharacteristic
452C4—21no—
453C4—21no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—44no—
458C4—44no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—162no—
463C4C4×#13511yescharacteristic
464C4#294311yescharacteristic
465C4#294811yescharacteristic
466C4#294911yescharacteristic
467C4#307612yes—
468C4#308112yes—
469C2—42no—
470C2—84no—
471C2#1889012yes—
472C2#1892012yes—
473C2#2085811yescharacteristic
474C2#2087211yescharacteristic
475C2#2087911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21962C2semidirect
4. #21362C2semidirect
6. #8926C2semidirect
7. #8924C2semidirect
8. #8923C2semidirect
9. #8920C2semidirect
10. #7894C2semidirect
11. #7891C2semidirect
43. #1540C4semidirect
74. #431C2×C4semidirect
82. C2×C4⋊C8C2×C4semidirect
135. #332C2×C4semidirect
136. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342322
#219622
#218632
#213622
C2×#15402
#89262
#89242
#89232
#89202
#78942
#78912