Finite Group Databasebeta

← #413240

Subgroups — #413240

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2C2×#2061C212yesmaximal
3C2×#2060C212yesmaximal
4#21172C211yesmaximal, characteristic
5#20801C211yesmaximal, characteristic
6#14354C211yesmaximal, characteristic
7#11340C212yesmaximal
8#11339C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10840C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#544C2212yes—
13C2×#428C2211yescharacteristic
14C2×#428C2211yescharacteristic
15C2×#428C2212yes—
16C2×#428C2212yes—
17C2×#423C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2868—21no—
20#2859—21no—
21#2660—22no—
22#2552—22no—
23#2551—22no—
24#2440—24no—
25#2317C2212yes—
26#2285C2211yescharacteristic
27C2×#384C2212yes—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2212yes—
31C4×#117C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#2061—28no—
34#2060—28no—
35#1949—22no—
36#1949—22no—
37#1847—21no—
38#1847—21no—
39#1846—22no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42#1736C2211yescharacteristic
43#1736C2211yescharacteristic
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48#1444—24no—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2312yes—
52#544—24no—
53#544—24no—
54#544—42no—
55#544—42no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×C22⋊Q8—24no—
62C2×#117C2312yes—
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C4⋊C4—24no—
66C22×C4⋊C4D414yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#428—24no—
73#428—24no—
74#428—24no—
75#428—24no—
76#428—42no—
77#428—42no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—21no—
85C2×#106—21no—
86#423—22no—
87#423Q814yescomplemented
88C2×C4⋊C8—22no—
89C4×M16—22no—
90C4×M16Q814yescomplemented
91C2×#97—21no—
92C2×C4×C8—21no—
93#398—28no—
94#395—28no—
95#389—24no—
96#389—24no—
97#389—24no—
98#385—24no—
99#384—24no—
100#384—22no—
101#384—21no—
102#384—21no—
103#384—22no—
104#384—24no—
105#384—42no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#382—22no—
109#382—24no—
110#382—21no—
111#382—21no—
112#382—22no—
113#382—42no—
114#382D412yes—
115#380—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D412yes—
119C4×C22⋊C4—22no—
120#376—21no—
121#376—22no—
122#376Q814yescomplemented
123C2×#95—22no—
124C2×#95—24no—
125C2×#95—22no—
126C43—21no—
127#343—24no—
128#343—24no—
129#340—22no—
130#340—22no—
131#339—24no—
132#339—24no—
133#339D418yes—
134#339Q818yescomplemented
135C22×Q8—24no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D414yes—
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16C2×D414yes—
152C2×M16C2×Q812yes—
153C2×M16C2×Q812yes—
154C2×M16C2×Q814yes—
155C2×M16C2×Q814yes—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C4⋊Q8—44no—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8—88no—
164#125—44no—
165#125—44no—
166#125—44no—
167#125—44no—
168C22⋊Q8—48no—
169C22⋊Q8—48no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117C2×D414yes—
177#117C2×Q814yes—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4—88no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C2×D414yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×Q814yes—
224C2×C42C2×Q814yes—
225C2×C42C2×Q812yes—
226C2×C42C2×Q812yes—
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229#107—44no—
230#107—44no—
231#107—42no—
232#107—42no—
233#106—44no—
234#106—44no—
235#106—42no—
236#106—42no—
237C4⋊C8—28no—
238#97—24no—
239C4×C8—24no—
240#95—28no—
241#95—22no—
242#95—22no—
243#95—24no—
244#95—28no—
245#95—48no—
246#95—42no—
247#95—42no—
248C24C22×D411yescharacteristic
249C2×Q8—48no—
250C2×Q8—88no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4#12012yes—
274C22×C4#12012yes—
275C22×C4C22⋊Q814yes—
276C22×C4C4⋊Q814yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D412yes—
280C22×C4C22×Q811yescharacteristic
281C22×C4C22×Q811yescharacteristic
282C22×C4C22×Q811yescharacteristic
283C22×C4C22×Q812yes—
284C22×C4C2×#4111yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C42−1+412yes—
287M16—24no—
288M16—24no—
289M16—24no—
290M16—24no—
291M16—24no—
292M16—42no—
293M16—42no—
294M16C4⋊Q818yescomplemented
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—48no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—84no—
328C4⋊C4—88no—
329C4⋊C4—816no—
330C4⋊C4#12712yes—
331C4⋊C4#12712yes—
332C4⋊C4C4⋊Q814yes—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—44no—
343C42—44no—
344C42—42no—
345C42—42no—
346C42C22⋊Q812yes—
347C42C22⋊Q812yes—
348C42C22⋊Q814yes—
349C42C4⋊Q814yes—
350C42C4⋊Q812yes—
351C42C4⋊Q812yes—
352C42C4⋊Q818yescomplemented
353C23—22no—
354C23—22no—
355C23C2×#12011yescharacteristic
356C23C2×C22⋊Q811yescharacteristic
357C23C2×C4⋊Q811yescharacteristic
358C23D4212yes—
359C23D4×Q812yes—
360Q8—168no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—28no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4#39514yes—
394C2×C4#39514yes—
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12711yescharacteristic
398C2×C4C2×C4⋊Q811yescharacteristic
399C2×C4C2×C4⋊Q811yescharacteristic
400C2×C4C2×C4⋊Q812yes—
401C2×C4C2×C4⋊Q812yes—
402C2×C4C2×C4⋊Q811yescharacteristic
403C2×C4#53511yescharacteristic
404C2×C4#53611yescharacteristic
405C2×C4#54412yes—
406C2×C4#54412yes—
407C2×C4D4212yes—
408C2×C4#54712yes—
409C2×C4D4×Q812yes—
410C2×C4D4×Q812yes—
411C2×C4#55412yes—
412C2×C4Q8212yes—
413C8—28no—
414C8—42no—
415C8—42no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—24no—
421C22C2×#39511yescharacteristic
422C22#243311yescharacteristic
423C22#243411yescharacteristic
424C22#244012yes—
425C22#255112yes—
426C22#301912yes—
427C22#302012yes—
428C4—24no—
429C4—24no—
430C4—48no—
431C4—42no—
432C4—42no—
433C4—44no—
434C4—84no—
435C4—82no—
436C4—82no—
437C4—161no—
438C4—161no—
439C4#244012yes—
440C4#254211yescharacteristic
441C4#255112yes—
442C4#256711yescharacteristic
443C4#266011yescharacteristic
444C4#286811yescharacteristic
445C2—22no—
446C2—22no—
447C2#1514311yescharacteristic
448C2#2356112yes—
449C2#2574911yescharacteristic
450C2#2575011yescharacteristic
451C2#2583212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
87. #423Q8semidirect
90. C4×M16Q8semidirect
122. #376Q8semidirect
134. #339Q8semidirect
294. M16C4⋊Q8semidirect
352. C42C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
C2×#20612
C2×#20602
#211722
#208012
#143542
#113402
#113392
#108572
#108402
#107012