Finite Group Databasebeta

← #193068

Subgroups — #193068

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34374C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3#20800C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5C2×#1526C211yesmaximal, characteristic
6#8912C218yesmaximal, complemented
7#8028C211yesmaximal, characteristic
8#8028C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#428—22no—
11C2×C4×M16C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2382C2214yes—
15#2365—21no—
16#2365—21no—
17#2355—21no—
18#2355—21no—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4—21no—
21#2323—22no—
22#2323—22no—
23#2314—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#1992C2214yes—
28#1990—21no—
29#1990—21no—
30#1847C2214yes—
31#1845—21no—
32#1844—21no—
33#1836C2214yes—
34#1834—21no—
35#1834—21no—
36#1736—22no—
37#1736—22no—
38#1526C418yescomplemented
39C2×#330C2214yes—
40#1295C2214yes—
41#1295C2214yes—
42C22×M16—21no—
43C22×M16—21no—
44C22×M16—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533Q814yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8D411yescharacteristic
51C2×#125C2312yes—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×#117D412yes—
59C2×#117D412yes—
60C2×#117D412yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#428—48no—
66C2×#107—24no—
67C2×#106—24no—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16—22no—
71C4×M16—22no—
72C4×M16C2×C414yes—
73C4×M16C2×C414yes—
74C2×#97C2312yes—
75C2×#97C2312yes—
76C2×#97C2312yes—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#391—22no—
81#391—22no—
82#389—22no—
83#389—22no—
84#389—24no—
85#389—24no—
86#388—24no—
87#388—24no—
88#387—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#384—22no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384D411yescharacteristic
97#384D411yescharacteristic
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2312yes—
101#382—22no—
102#382C2312yes—
103#382C2312yes—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#330C2×C4116yescomplemented
121C22×Q8—21no—
122C22×Q8—21no—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—42no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8C4⋊C414yes—
142C4⋊Q8C4⋊C414yes—
143#125—22no—
144#125—22no—
145#125C22×C414yes—
146#125C2×Q814yes—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—84no—
156C4×Q8—84no—
157#117—24no—
158#117—44no—
159#117—44no—
160#117C4⋊C418yes—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×Q812yes—
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42C2411yescharacteristic
206#107—48no—
207#106—48no—
208#97—22no—
209#97—22no—
210#97—22no—
211#97—22no—
212#97C22×C414yes—
213#97C22×C414yes—
214C4×C8—24no—
215C4×C8—24no—
216C4×C8—42no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225C24#12011yescharacteristic
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4C2×C4⋊C412yes—
249C22×C4C4×D412yes—
250C22×C4C4×D412yes—
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4#12312yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260M16—216no—
261M16—48no—
262M16—48no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4C2×C4⋊C414yes—
301C4⋊C4C2×C4⋊C414yes—
302C22⋊C4—28no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C42—28no—
306C42—44no—
307C42—44no—
308C42—84no—
309C42—84no—
310C42C2×C4⋊C412yes—
311C42C4×D412yes—
312C42C4×D412yes—
313C42C22⋊Q812yes—
314C42C22⋊Q812yes—
315C42#12312yes—
316C42C23×C411yescharacteristic
317C42C22×Q811yescharacteristic
318C42C2×#4111yescharacteristic
319C42C2×#4111yescharacteristic
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23#38512yes—
327C23C2×#12011yescharacteristic
328Q8—48no—
329Q8—48no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—164no—
364C2×C4#38512yes—
365C2×C4#38512yes—
366C2×C4#38512yes—
367C2×C4#45312yes—
368C2×C4#45412yes—
369C2×C4#45412yes—
370C2×C4#45512yes—
371C2×C4#45612yes—
372C2×C4#45612yes—
373C2×C4C22×C4⋊C411yescharacteristic
374C2×C4C2×C4×D411yescharacteristic
375C2×C4C2×C4×D411yescharacteristic
376C2×C4C2×C22⋊Q811yescharacteristic
377C2×C4C2×C22⋊Q811yescharacteristic
378C2×C4C2×#12311yescharacteristic
379C8—28no—
380C8—44no—
381C8—44no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22C2×#38511yescharacteristic
390C22C2×#45311yescharacteristic
391C22C2×#45511yescharacteristic
392C22C2×#45411yescharacteristic
393C22C2×#45411yescharacteristic
394C22C2×#45611yescharacteristic
395C22C2×#45611yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—44no—
403C4—44no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—82no—
408C4—162no—
409C4#185714yes—
410C4#185914yes—
411C4#186014yes—
412C2—42no—
413C2#1907611yescharacteristic
414C2#1907711yescharacteristic
415C2#1908111yescharacteristic
416C2#1908411yescharacteristic
417C2C2×#185711yescharacteristic
418C2C2×#185911yescharacteristic
419C2C2×#186011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #8912C2semidirect
38. #1526C4semidirect
120. #330C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#213222
#208002
#207642
C2×#15262
#89122
#80282