Finite Group Databasebeta

← #415030

Subgroups — #415030

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2C2×#1994C211yesmaximal, characteristic, complemented
3#21322C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×C4×#104C211yesmaximal, characteristic
6#14687C211yesmaximal, characteristic, complemented
7#11211C214yesmaximal, complemented
8#11079C214yesmaximal
9#10857C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×#104C2212yes—
14C2×#417C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2857—22no—
17#2629—22no—
18#2582—22no—
19#2538—22no—
20#2417—24no—
21#2316C2212yes—
22#2311C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2214yes—
26C2×C43C2211yescharacteristic
27#1994—24no—
28#1992—22no—
29#1990—22no—
30#1803—22no—
31#1802—22no—
32C4×#104—24no—
33C2×#343C2214yes—
34#1736C2214yes—
35#1736C2212yes—
36C2×#339C2214yes—
37C2×#339C2212yes—
38C4⋊#95—24no—
39C22×#41C2311yescharacteristic
40C22×M16C2312yes—
41C22×M16C2312yes—
42#533—22no—
43#533—22no—
44#533—22no—
45#533—22no—
46C2×C4⋊Q8D412yes—
47C2×#125D412yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—24no—
50C4×#41—22no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8D412yes—
56C2×C4×D4—22no—
57C2×C4×D4D412yes—
58C2×#117—21no—
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—22no—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C42—22no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#423—22no—
70#423—22no—
71C2×C4⋊C8—22no—
72C2×#104—24no—
73C2×#104—24no—
74C2×#104Q818yescomplemented
75#417—44no—
76C2×#103—22no—
77C2×#102—22no—
78C4×M16—22no—
79C4×M16—22no—
80C2×#97—21no—
81C2×C4×C8—21no—
82#398—28no—
83#396—28no—
84#391—24no—
85#391—22no—
86#390—22no—
87#389—24no—
88#389—24no—
89#389—24no—
90#386—24no—
91#384—22no—
92#384—21no—
93#384—21no—
94#384—22no—
95#384—21no—
96#384—21no—
97#382—28no—
98#382—44no—
99#380—24no—
100#380—24no—
101#380—24no—
102C2×#95—28no—
103C43—22no—
104C43D414yes—
105#343—44no—
106#340—24no—
107#340—28no—
108#339—28no—
109#339—24no—
110C2×#41—22no—
111C2×#41—21no—
112C2×#41—21no—
113C2×#41—21no—
114C2×#41—21no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41C2×Q814yes—
118C22×Q8—22no—
119C22×Q8C2×D411yescharacteristic
120C22×D4—21no—
121C22×D4C2×D411yescharacteristic
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D412yes—
127C23×C4C2×D414yes—
128C23×C4C2411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—24no—
137C2×M16—24no—
138C2×M16C2×Q814yes—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143#125—24no—
144#125—24no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C22⋊Q8—88no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—44no—
151C4×Q8—44no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—44no—
155C4×D4—44no—
156#117—21no—
157#117—22no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—22no—
162#117—22no—
163#117—22no—
164#117—42no—
165#117—42no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—28no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—28no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C2×D414yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q814yes—
207C2×C42#4114yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C4⋊C8—44no—
212#104—216no—
213#103—44no—
214#102—44no—
215#97—42no—
216C4×C8—42no—
217#95—28no—
218#95—28no—
219#95—416no—
220C24—21no—
221C24C22×D411yescharacteristic
222#41—24no—
223#41—24no—
224#41—24no—
225#41—24no—
226#41—42no—
227#41—42no—
228#41—42no—
229#41—42no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8C4⋊Q812yes—
239C2×D4—24no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4C4⋊Q812yes—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4#12012yes—
273C22×C4#12012yes—
274C22×C4#12114yes—
275C22×C4#12414yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×Q812yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4111yescharacteristic
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+412yes—
286M16—24no—
287M16—24no—
288M16—42no—
289M16—42no—
290M16—44no—
291M16—44no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4—48no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—88no—
315C4⋊C4—84no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—88no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—416no—
326C42—41no—
327C42—42no—
328C42—41no—
329C42—42no—
330C42—44no—
331C42#12112yes—
332C42#12112yes—
333C42#12114yes—
334C42C22⋊Q818yescomplemented
335C42#12312yes—
336C42#12312yes—
337C42#12314yes—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23C2×#12011yescharacteristic
346C23C2×#12111yescharacteristic
347C23C2×#12411yescharacteristic
348C23#54612yes—
349C23D4×Q812yes—
350Q8—22no—
351Q8—22no—
352Q8—41no—
353Q8—41no—
354Q8—88no—
355D4—22no—
356D4—22no—
357D4—44no—
358D4—44no—
359D4—41no—
360D4—41no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—22no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—88no—
384C2×C4—88no—
385C2×C4—88no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4#39414yes—
389C2×C4#39414yes—
390C2×C4#45414yes—
391C2×C4#45514yes—
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×C22⋊Q812yes—
395C2×C4C2×#12311yescharacteristic
396C2×C4C2×#12311yescharacteristic
397C2×C4C2×C4⋊Q811yescharacteristic
398C2×C4#53211yescharacteristic
399C2×C4#53312yes—
400C2×C4#53511yescharacteristic
401C2×C4#53611yescharacteristic
402C2×C4#53712yes—
403C2×C4#54212yes—
404C2×C4D4×Q812yes—
405C2×C4#55012yes—
406C2×C4#55212yes—
407C2×C4#55412yes—
408C2×C4#55412yes—
409C2×C4#55612yes—
410C8—44no—
411C8—82no—
412C22—22no—
413C22—22no—
414C22—24no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—44no—
419C22—82no—
420C22—81no—
421C22—81no—
422C22C2×#39411yescharacteristic
423C22#241712yes—
424C22#243111yescharacteristic
425C22#243411yescharacteristic
426C22#249212yes—
427C22C2×#45512yes—
428C22C2×#45412yes—
429C4—24no—
430C4—48no—
431C4—42no—
432C4—82no—
433C4—84no—
434C4—84no—
435C4—162no—
436C4#204018yes—
437C4#242312yes—
438C4#249911yescharacteristic
439C4#250211yescharacteristic
440C4#253812yes—
441C4#262011yescharacteristic
442C4#285711yescharacteristic
443C2—22no—
444C2—22no—
445C2—82no—
446C2#1518211yescharacteristic
447C2C2×#204012yes—
448C2#2559011yescharacteristic
449C2#2559111yescharacteristic
450C2#2582912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1994C2semidirect
3. #21322C2semidirect
6. #14687C2semidirect
7. #11211C2semidirect
9. #10857C2semidirect
74. C2×#104Q8semidirect
334. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#352802
C2×#19942
#213222
#206482
C2×C4×#1042
#146872
#112112
#110792
#108572