Finite Group Databasebeta

← #382577

Subgroups — #382577

Order 512.

2950 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C2×#1819C212yesmaximal
3#14680C212yesmaximal
4#11203C218yesmaximal, complemented
5#10725C212yesmaximal
6C25×C4C2211yescharacteristic
7C23×C4⋊C4C412yescomplemented
8C23×C22⋊C4C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10C2×#385—24no—
11C2×#385—24no—
12C2×#385C414yescomplemented
13C2×#385C2214yes—
14C2×#380—24no—
15C2×#380C2214yes—
16C2×#379C2214yes—
17C22×#95C412yescomplemented
18#1819—28no—
19C2×#342C2218yes—
20C2×#342C2218yes—
21C22⋊#95—24no—
22C22⋊#95C2214yes—
23C26C2311yescharacteristic
24C24×C4—48no—
25C24×C4—44no—
26C24×C4C2×C412yes—
27C24×C4C2×C412yes—
28C24×C4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4C2×C412yes—
31C24×C4D414yes—
32C24×C4D414yes—
33C24×C4C2312yes—
34C22×C4⋊C4—24no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—48no—
39C22×C4⋊C4C2×C418yes—
40C22×C22⋊C4—24no—
41C22×C22⋊C4—24no—
42C22×C22⋊C4—24no—
43C22×C22⋊C4—28no—
44C22×C22⋊C4—44no—
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4D414yes—
48C22×C22⋊C4Q814yes—
49C22×C22⋊C4C2318yes—
50C22×C22⋊C4C2312yes—
51C22×C22⋊C4C2312yes—
52#385—24no—
53#385—28no—
54#385—24no—
55#385—24no—
56#385—28no—
57#385—24no—
58#380—48no—
59#380—416no—
60#379—28no—
61#379—28no—
62C2×#95—24no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—24no—
66C2×#95—28no—
67C2×#95C2×C418yes—
68#342—216no—
69#342—48no—
70C25—24no—
71C25—24no—
72C25—44no—
73C25—44no—
74C25C22×C411yescharacteristic
75C25C22×C411yescharacteristic
76C25C2×D414yes—
77C25C2×D412yes—
78C25C2×Q812yes—
79C25#4114yes—
80C25C2411yescharacteristic
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4—28no—
85C23×C4—28no—
86C23×C4—216no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—28no—
90C23×C4—48no—
91C23×C4—48no—
92C23×C4—48no—
93C23×C4—48no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4—44no—
97C23×C4—48no—
98C23×C4—48no—
99C23×C4—48no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—464no—
103C23×C4—44no—
104C23×C4—44no—
105C23×C4—44no—
106C23×C4—44no—
107C23×C4C22⋊C414yes—
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C414yes—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C4⋊C418yescomplemented
114C23×C4C22×C414yes—
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C414yes—
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C23×C4C2×D412yes—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—48no—
124C2×C4⋊C4—416no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—416no—
127C2×C4⋊C4—48no—
128C2×C4⋊C4—416no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—216no—
132C2×C22⋊C4—216no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—216no—
141C2×C22⋊C4—216no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—416no—
144C2×C22⋊C4—416no—
145C2×C22⋊C4—416no—
146C2×C22⋊C4—416no—
147C2×C22⋊C4C4⋊C418yescomplemented
148C2×C22⋊C4C2×D414yes—
149C2×C22⋊C4C2×Q814yes—
150#95—28no—
151#95—28no—
152#95—48no—
153#95—416no—
154C24—22no—
155C24—22no—
156C24—24no—
157C24—24no—
158C24—22no—
159C24—22no—
160C24—28no—
161C24—28no—
162C24—28no—
163C24—216no—
164C24—28no—
165C24—28no—
166C24—28no—
167C24—42no—
168C24—44no—
169C24—42no—
170C24—42no—
171C24—48no—
172C24—44no—
173C24—48no—
174C24—416no—
175C24—42no—
176C24—416no—
177C24—416no—
178C24—432no—
179C24C2×C22⋊C412yes—
180C24C2×C22⋊C412yes—
181C24C2×C4⋊C412yes—
182C24C2×C4⋊C412yes—
183C24#11712yes—
184C24#11712yes—
185C24#12118yes—
186C24C22⋊Q818yes—
187C24C23×C411yescharacteristic
188C24C22×D412yes—
189C24C22×D411yescharacteristic
190C24C22×Q811yescharacteristic
191C24C2×#4112yes—
192C24H(5,𝔽2)14yes—
193C242−1+414yes—
194C22×C4—24no—
195C22×C4—28no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—28no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—28no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—28no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—464no—
228C22×C4—432no—
229C22×C4—44no—
230C22×C4—416no—
231C22×C4—44no—
232C22×C4—432no—
233C22×C4—432no—
234C22×C4—464no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—416no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—416no—
249C22×C4—416no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—88no—
254C22×C4—88no—
255C22×C4#9918yes—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C4⋊C418yes—
259C4⋊C4—416no—
260C4⋊C4—416no—
261C4⋊C4—816no—
262C22⋊C4—28no—
263C22⋊C4—28no—
264C22⋊C4—216no—
265C22⋊C4—48no—
266C22⋊C4—48no—
267C22⋊C4—416no—
268C22⋊C4—416no—
269C22⋊C4—832no—
270C23—28no—
271C23—28no—
272C23—216no—
273C23—22no—
274C23—22no—
275C23—28no—
276C23—28no—
277C23—24no—
278C23—24no—
279C23—22no—
280C23—22no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—22no—
286C23—22no—
287C23—216no—
288C23—216no—
289C23—24no—
290C23—24no—
291C23—216no—
292C23—24no—
293C23—24no—
294C23—216no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—48no—
299C23—48no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23—48no—
305C23—416no—
306C23—416no—
307C23—416no—
308C23—416no—
309C23—44no—
310C23—432no—
311C23—464no—
312C23—432no—
313C23—48no—
314C23—44no—
315C23#38014yes—
316C23#38014yes—
317C23C2×#9914yes—
318C23#41014yes—
319C23C22×C22⋊C411yescharacteristic
320C23C22×C4⋊C411yescharacteristic
321C23C2×#11711yescharacteristic
322C23#51812yes—
323C23#51912yes—
324C23C2×#12112yes—
325C23C2×C22⋊Q812yes—
326C23#53414yes—
327C23#53614yes—
328C23#53714yes—
329C23#53918yes—
330C23#54414yes—
331C2×C4—216no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—416no—
341C2×C4—416no—
342C2×C4—416no—
343C2×C4—416no—
344C2×C4—48no—
345C2×C4—416no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—416no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—88no—
358C2×C4—88no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—816no—
362C2×C4—88no—
363C2×C4—816no—
364C2×C4—88no—
365C2×C4—88no—
366C2×C4C2×#9918yes—
367C22—216no—
368C22—216no—
369C22—24no—
370C22—28no—
371C22—24no—
372C22—24no—
373C22—24no—
374C22—22no—
375C22—22no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—22no—
381C22—24no—
382C22—22no—
383C22—24no—
384C22—24no—
385C22—24no—
386C22—28no—
387C22—44no—
388C22—48no—
389C22—44no—
390C22—432no—
391C22—432no—
392C22—48no—
393C22—48no—
394C22—44no—
395C22—42no—
396C22—42no—
397C22#179918yes—
398C22C2×#38011yescharacteristic
399C22#230811yescharacteristic
400C22#231011yescharacteristic
401C22#231312yes—
402C22#231612yes—
403C22#261814yes—
404C22#262714yes—
405C22C22×#9912yes—
406C22C2×#41012yes—
407C4—28no—
408C4—48no—
409C4—44no—
410C4—82no—
411C4—82no—
412C4—168no—
413C2—24no—
414C2—28no—
415C2—22no—
416C2—22no—
417C2—24no—
418C2—44no—
419C2#1433911yescharacteristic
420C2C2×#179912yes—
421C2#2174611yescharacteristic
422C2#2174711yescharacteristic
423C2#2182512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11203C2semidirect
7. C23×C4⋊C4C4semidirect
12. C2×#385C4semidirect
17. C22×#95C4semidirect
113. C23×C4C4⋊C4semidirect
147. C2×C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C2×#18192
#146802
#112032
#107252