Finite Group Databasebeta

← #360611

Subgroups — #360611

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2#20661C212yesmaximal
3C22×#343C211yesmaximal, characteristic
4#13924C211yesmaximal, characteristic
5#11342C212yesmaximal, complemented
6#11296C212yesmaximal, complemented
7#10899C214yesmaximal, complemented
8#10732C212yesmaximal
9C23×M16C2212yes—
10C23×C4⋊C4C2211yescharacteristic
11C2×#428C2214yes—
12#2345—22no—
13#2345—22no—
14#2344—24no—
15#2316C2212yes—
16#2313—22no—
17#2280—22no—
18#2280C2212yes—
19C2×#380C2211yescharacteristic
20#2274C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1809—24no—
23#1808—22no—
24#1807—22no—
25C2×#343C418yescomplemented
26C2×#343C2211yescharacteristic
27C2×#343C2211yescharacteristic
28C2×#343C2212yes—
29#1736C2214yes—
30C2×#339C2214yes—
31C2×#339C2214yes—
32#1445—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C24×C4D412yes—
37C24×C4C2311yescharacteristic
38C22×M16—22no—
39C22×M16—24no—
40C22×M16—28no—
41C22×M16—44no—
42C22×M16C2×C418yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8—22no—
46C2×#117—24no—
47C2×#117C2314yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4Q814yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42C2312yes—
60#428—28no—
61#428—44no—
62C2×#107—24no—
63C2×#106—24no—
64#389—24no—
65#389—24no—
66#387—28no—
67#387—28no—
68#381—24no—
69#381—22no—
70#380—22no—
71#380—22no—
72#380—24no—
73#380—22no—
74#380—22no—
75#380D412yes—
76#380D412yes—
77#380D414yes—
78#379—22no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C22⋊C4—24no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—24no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—24no—
90#343C2×C414yes—
91#343C2×C418yes—
92#343C2×C414yes—
93#340—24no—
94#340—24no—
95#339—28no—
96#339—44no—
97C25C2×D411yescharacteristic
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—84no—
107C23×C4C4⋊C418yes—
108C23×C4C22×C412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×Q812yes—
115C23×C4#4114yes—
116C23×C4C2411yescharacteristic
117C2×M16—22no—
118C2×M16—24no—
119C2×M16—28no—
120C2×M16—22no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—216no—
124C2×M16—48no—
125C2×M16—48no—
126C2×M16—416no—
127C2×M16C22×C414yes—
128C2×M16C22×C414yes—
129C2×M16#4118yes—
130C22×C8—22no—
131C22×C8—28no—
132C22×C8—22no—
133C22×C8—48no—
134#117—24no—
135#117—24no—
136#117—24no—
137#117—24no—
138#117—88no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4C4⋊C418yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×Q812yes—
161C2×C4⋊C4C2×Q812yes—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—28no—
178#107—48no—
179#106—48no—
180#95—24no—
181#95—24no—
182#95—28no—
183#95—22no—
184#95—22no—
185#95—24no—
186#95—42no—
187#95—42no—
188#95—84no—
189#95—84no—
190C24—22no—
191C24—22no—
192C24—22no—
193C24—22no—
194C24—42no—
195C24C2×C4⋊C412yes—
196C24#12011yescharacteristic
197C24#12011yescharacteristic
198C24#12312yes—
199C24C22×D411yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—28no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—48no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—88no—
237C22×C4—84no—
238C22×C4—88no—
239C22×C4C2×C4⋊C412yes—
240C22×C4C2×C4⋊C414yes—
241C22×C4C2×C4⋊C414yes—
242C22×C4C4×D414yes—
243C22×C4C4×D414yes—
244C22×C4#12012yes—
245C22×C4C22⋊Q814yes—
246C22×C4C22⋊Q814yes—
247C22×C4#12312yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C22×Q811yescharacteristic
251C22×C4C2×#4114yes—
252C22×C4C2×#4112yes—
253C22×C4H(5,𝔽2)12yes—
254C22×C42−1+412yes—
255M16—24no—
256M16—28no—
257M16—24no—
258M16—24no—
259M16—416no—
260M16—44no—
261M16#11718yescomplemented
262C2×C8—24no—
263C2×C8—28no—
264C2×C8—24no—
265C2×C8—416no—
266C2×C8—44no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—216no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C42—44no—
289C42—44no—
290C42—44no—
291C42—44no—
292C42—88no—
293C23—22no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—24no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23#38512yes—
314C23#38512yes—
315C23#45412yes—
316C23#45612yes—
317C23C22×C4⋊C411yescharacteristic
318C23C2×#12011yescharacteristic
319C23C2×#12311yescharacteristic
320C23#53212yes—
321C23#53712yes—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—416no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—88no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#38514yes—
361C2×C4C2×#11712yes—
362C2×C4C2×C4×D412yes—
363C2×C4#51811yescharacteristic
364C2×C4#51911yescharacteristic
365C2×C4C2×C22⋊Q812yes—
366C2×C4#53312yes—
367C2×C4#54412yes—
368C2×C4#54614yes—
369C2×C4#54714yes—
370C2×C4#55114yes—
371C2×C4#55214yes—
372C8—28no—
373C8—44no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—24no—
383C22—22no—
384C22—22no—
385C22—24no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—48no—
390C22—44no—
391C22—44no—
392C22—42no—
393C22—42no—
394C22#185714yes—
395C22#228511yescharacteristic
396C22C2×#38511yescharacteristic
397C22#231611yescharacteristic
398C22#243012yes—
399C22#249812yes—
400C22C2×#45411yescharacteristic
401C22C2×#45611yescharacteristic
402C22#301912yes—
403C4—24no—
404C4—44no—
405C4—44no—
406C4—84no—
407C4—84no—
408C4—84no—
409C4—162no—
410C4#234212yes—
411C4#234412yes—
412C4#243814yes—
413C2—22no—
414C2—21no—
415C2—24no—
416C2—21no—
417C2—42no—
418C2#1387811yescharacteristic
419C2C2×#185711yescharacteristic
420C2#2085611yescharacteristic
421C2#2551912yes—
422C2#2552112yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11342C2semidirect
6. #11296C2semidirect
7. #10899C2semidirect
25. C2×#343C4semidirect
261. M16#117semidirect

Maximal subgroups

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

subgroupindex
#341782
#206612
C22×#3432
#139242
#113422
#112962
#108992
#107322