Finite Group Databasebeta

← #375911

Subgroups — #375911

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60193C211yesmaximal, characteristic, complemented
2#23467C211yesmaximal, characteristic
3#20834C212yesmaximal
4#20385C211yesmaximal, characteristic
5#11391C211yesmaximal, characteristic, complemented
6#11345C211yesmaximal, characteristic, complemented
7#11071C211yesmaximal, characteristic
8#10937C212yesmaximal
9#10920C212yesmaximal, complemented
10#10914C212yesmaximal, complemented
11#10803C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13#3458C2211yescharacteristic
14#2962—28no—
15#2961—22no—
16#2960—22no—
17#2905C2211yescharacteristic
18C2×#424C2212yes—
19C2×#424C2212yes—
20C2×#407C2212yes—
21C2×#407C2211yescharacteristic
22#2873C2212yes—
23C2×#405C2211yescharacteristic
24C2×#405C2211yescharacteristic
25#2280C2211yescharacteristic
26#2280C2212yes—
27#2030C414yes—
28#2029C414yes—
29#1850—28no—
30#1752—22no—
31C8×C22⋊C4—22no—
32C2×#344C2212yes—
33C2×#344C2212yes—
34C2×#344C2212yes—
35#1742C2211yescharacteristic
36#1742C2212yes—
37#1740C2211yescharacteristic
38#1740C2212yes—
39#1729C2211yescharacteristic
40#1729C2212yes—
41#1729C2211yescharacteristic
42#1729C2212yes—
43C24×C4C2311yescharacteristic
44C22×M16C2312yes—
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C23×C8—22no—
49C23×C8—21no—
50C23×C8—21no—
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—42no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56#544—22no—
57#544—24no—
58#543C2×C414yes—
59#533—22no—
60#533—24no—
61C2×#117C2311yescharacteristic
62C2×#117C2312yes—
63C2×#117C2311yescharacteristic
64C2×#117C2312yes—
65#446—28no—
66#446—216no—
67C8×Q8—28no—
68#424—22no—
69#424—22no—
70#424—24no—
71#424—24no—
72#424—24no—
73#424—42no—
74#423—22no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8D418yes—
77#407—22no—
78#407—24no—
79#407—42no—
80#407Q814yescomplemented
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83#405—24no—
84#405—24no—
85#405—24no—
86#405—42no—
87C4×M16—24no—
88C2×#97—21no—
89C2×#97D412yes—
90C2×C4×C8—21no—
91C2×C4×C8D412yes—
92#381C2×C412yes—
93#381C2×C414yes—
94#380—22no—
95#380—24no—
96#379C2×C412yes—
97#379C2×C414yes—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—22no—
100#344—24no—
101#344—42no—
102#344—42no—
103#343—24no—
104#343—22no—
105#342C2×C414yes—
106#342C2×C418yes—
107#336—24no—
108#336—22no—
109#336—24no—
110#336—22no—
111C25C22×C411yescharacteristic
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×Q812yes—
122C23×C4#4112yes—
123C23×C4C2411yescharacteristic
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—21no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16C2×Q812yes—
142C2×M16C2×Q812yes—
143C2×M16#4112yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—48no—
156C22×C8—48no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8—42no—
161C22×C8C2×D414yes—
162C22×C8C2×D414yes—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D412yes—
166C22×C8#4112yes—
167C4⋊Q8—24no—
168C4⋊Q8—28no—
169#125—24no—
170#125—28no—
171C22⋊Q8—28no—
172C22⋊Q8—216no—
173C4×Q8—28no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4#4112yes—
189C2×C4⋊C4#4114yes—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4C22×C414yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C22×C412yes—
195C2×C22⋊C4C22×C414yes—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42C2×D412yes—
199C2×C42C2×D414yes—
200C4⋊C8—28no—
201C4⋊C8—28no—
202C4⋊C8—28no—
203C4⋊C8—48no—
204C22⋊C8—28no—
205C22⋊C8—48no—
206#97—28no—
207#97—24no—
208#97—42no—
209C4×C8—28no—
210C4×C8—24no—
211C4×C8—42no—
212#95—28no—
213#95—24no—
214C24—22no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—44no—
219C24C4×Q812yes—
220C24C23×C411yescharacteristic
221C2×Q8—28no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4C4×D414yes—
247C22×C4C4×D418yes—
248C22×C4C4×Q812yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4112yes—
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4H(5,𝔽2)12yes—
258C22×C42−1+412yes—
259M16—24no—
260M16—24no—
261M16—24no—
262M16—42no—
263M16—42no—
264M16—42no—
265M16—42no—
266M16—42no—
267M16—42no—
268M16—42no—
269M16—42no—
270M16—42no—
271C2×C8—22no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—48no—
286C2×C8—48no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8#12012yes—
296C2×C8#12012yes—
297C2×C8C22⋊Q814yes—
298C4⋊C4—216no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—416no—
313C22⋊C4—48no—
314C42—24no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—41no—
319C42—41no—
320C42—42no—
321C42—42no—
322C23—24no—
323C23—28no—
324C23—24no—
325C23—22no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—48no—
332C23—41no—
333C23—42no—
334C23—41no—
335C23C2×C4×D411yescharacteristic
336C23C2×C4×D412yes—
337C23C2×C4×Q811yescharacteristic
338C23C4×#4111yescharacteristic
339C23#52012yes—
340Q8—88no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4C2×#12011yescharacteristic
361C2×C4C2×C22⋊Q811yescharacteristic
362C2×C4#52511yescharacteristic
363C2×C4#52511yescharacteristic
364C2×C4#52512yes—
365C2×C4#52911yescharacteristic
366C2×C4#53311yescharacteristic
367C2×C4#53511yescharacteristic
368C2×C4#53611yescharacteristic
369C2×C4#54212yes—
370C2×C4#54612yes—
371C2×C4#54612yes—
372C2×C4#54812yes—
373C2×C4D4×Q812yes—
374C2×C4#55212yes—
375C2×C4#55412yes—
376C2×C4#55612yes—
377C2×C4#55912yes—
378C8—22no—
379C8—22no—
380C8—24no—
381C8—42no—
382C8—42no—
383C8—44no—
384C8—82no—
385C22—22no—
386C22—24no—
387C22—28no—
388C22—24no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—44no—
393C22—48no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22—44no—
398C22C4×#12011yescharacteristic
399C22C4×C22⋊Q811yescharacteristic
400C22#232611yescharacteristic
401C22#236012yes—
402C22#236712yes—
403C4—42no—
404C4—48no—
405C4—44no—
406C4—82no—
407C4—84no—
408C4#240411yescharacteristic
409C4#240911yescharacteristic
410C4#245812yes—
411C4#246512yes—
412C4#258011yescharacteristic
413C4#259911yescharacteristic
414C2—21no—
415C2—22no—
416C2—22no—
417C2—21no—
418C2—44no—
419C2#1417211yescharacteristic
420C2#2148312yes—
subgroupG/H
centerC2×C4C2×#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C4×#120
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60193C2semidirect
5. #11391C2semidirect
6. #11345C2semidirect
9. #10920C2semidirect
10. #10914C2semidirect
11. #10803C2semidirect
80. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#601932
#234672
#208342
#203852
#113912
#113452
#110712
#109372
#109202
#109142
#108032