Finite Group Databasebeta

← #355119

Subgroups — #355119

Order 512.

3094 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C2×#1819C212yesmaximal
3C4×#379C212yesmaximal
4#10912C218yesmaximal, 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×#379C2214yes—
15C2×C4×C22⋊C4—24no—
16C2×C4×C22⋊C4C2214yes—
17C22×#95C412yescomplemented
18#1819C4116yescomplemented
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—48no—
38C22×C4⋊C4C2×C418yes—
39C22×C22⋊C4—24no—
40C22×C22⋊C4—24no—
41C22×C22⋊C4—24no—
42C22×C22⋊C4—28no—
43C22×C22⋊C4—44no—
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2×C418yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2318yes—
49C22×C22⋊C4C2312yes—
50C22×C42—24no—
51#385—24no—
52#385—28no—
53#385—24no—
54#385—24no—
55#385—28no—
56#385—24no—
57#379—28no—
58#379C2×C4116yes—
59C4×C22⋊C4—48no—
60C4×C22⋊C4—416no—
61C2×#95—24no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—24no—
65C2×#95—28no—
66C2×#95C2×C418yes—
67#342—48no—
68#342C2×C4132yescomplemented
69C25—24no—
70C25—24no—
71C25—44no—
72C25—44no—
73C25C22×C411yescharacteristic
74C25C22×C411yescharacteristic
75C25C22×C414yes—
76C25C2×D414yes—
77C25#4114yes—
78C25C2411yescharacteristic
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—28no—
83C23×C4—28no—
84C23×C4—216no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—28no—
88C23×C4—48no—
89C23×C4—48no—
90C23×C4—48no—
91C23×C4—48no—
92C23×C4—44no—
93C23×C4—44no—
94C23×C4—44no—
95C23×C4—48no—
96C23×C4—48no—
97C23×C4—48no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—464no—
101C23×C4—44no—
102C23×C4—44no—
103C23×C4—44no—
104C23×C4—44no—
105C23×C4C4218yescomplemented
106C23×C4C22⋊C414yes—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C412yes—
109C23×C4C22⋊C414yes—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22×C414yes—
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C22×C414yes—
116C23×C4C2×D412yes—
117C23×C4C2×D414yes—
118C23×C4C2×D412yes—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—416no—
123C2×C4⋊C4—48no—
124C2×C4⋊C4—416no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—232no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—216no—
131C2×C22⋊C4—216no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—232no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—416no—
139C2×C22⋊C4—416no—
140C2×C22⋊C4—416no—
141C2×C22⋊C4C4218yescomplemented
142C2×C22⋊C4C22×C4116yes—
143C2×C22⋊C4C22×C418yes—
144C2×C42—48no—
145C2×C42—416no—
146#95—28no—
147#95—28no—
148#95—48no—
149#95—416no—
150C24—22no—
151C24—22no—
152C24—24no—
153C24—24no—
154C24—22no—
155C24—22no—
156C24—216no—
157C24—28no—
158C24—28no—
159C24—216no—
160C24—28no—
161C24—28no—
162C24—42no—
163C24—44no—
164C24—42no—
165C24—42no—
166C24—416no—
167C24—44no—
168C24—416no—
169C24—42no—
170C24—416no—
171C24—416no—
172C24—432no—
173C24C2×C4212yes—
174C24C2×C4212yes—
175C24C2×C22⋊C412yes—
176C24C2×C22⋊C412yes—
177C24#11712yes—
178C24#11712yes—
179C24C4×D4116yes—
180C24C23×C411yescharacteristic
181C24C23×C412yes—
182C24C22×D412yes—
183C24C2×#4112yes—
184C24H(5,𝔽2)14yes—
185C242−1+414yes—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—24no—
192C22×C4—28no—
193C22×C4—28no—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—28no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—28no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—216no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—464no—
221C22×C4—432no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—416no—
225C22×C4—432no—
226C22×C4—432no—
227C22×C4—464no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—416no—
232C22×C4—44no—
233C22×C4—416no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—416no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—88no—
246C22×C4—88no—
247C22×C4#9918yes—
248C22×C4C2×C4218yes—
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C414yes—
251C4⋊C4—416no—
252C4⋊C4—416no—
253C22⋊C4—216no—
254C22⋊C4—216no—
255C22⋊C4—232no—
256C22⋊C4—464no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—416no—
260C42—816no—
261C23—28no—
262C23—28no—
263C23—216no—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—22no—
268C23—22no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—28no—
274C23—28no—
275C23—24no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—232no—
280C23—24no—
281C23—24no—
282C23—232no—
283C23—24no—
284C23—44no—
285C23—44no—
286C23—44no—
287C23—416no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—48no—
292C23—416no—
293C23—416no—
294C23—416no—
295C23—416no—
296C23—44no—
297C23—464no—
298C23—464no—
299C23—48no—
300C23—44no—
301C23C4×C22⋊C414yes—
302C23C4×C22⋊C414yes—
303C23C2×#9914yes—
304C23#41014yes—
305C23C22×C4211yescharacteristic
306C23C22×C22⋊C411yescharacteristic
307C23C2×#11711yescharacteristic
308C23C2×C4×D414yes—
309C23#51812yes—
310C23#51912yes—
311C23#52018yes—
312C23#53414yes—
313C23#53614yes—
314C23#53918yes—
315C2×C4—216no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—416no—
325C2×C4—416no—
326C2×C4—416no—
327C2×C4—416no—
328C2×C4—48no—
329C2×C4—416no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—416no—
336C2×C4—48no—
337C2×C4—432no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—416no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—88no—
344C2×C4—816no—
345C2×C4—88no—
346C2×C4—88no—
347C2×C4—88no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4C2×#9918yes—
351C22—216no—
352C22—216no—
353C22—28no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—22no—
358C22—28no—
359C22—24no—
360C22—22no—
361C22—24no—
362C22—24no—
363C22—22no—
364C22—24no—
365C22—22no—
366C22—24no—
367C22—24no—
368C22—28no—
369C22—24no—
370C22—48no—
371C22—48no—
372C22—464no—
373C22—48no—
374C22—48no—
375C22—44no—
376C22—42no—
377C22—42no—
378C22C4×#9918yes—
379C22C2×C4×C22⋊C411yescharacteristic
380C22#227412yes—
381C22#230811yescharacteristic
382C22#231011yescharacteristic
383C22#231312yes—
384C22#232218yes—
385C22C22×#9912yes—
386C22C2×#41012yes—
387C4—28no—
388C4—48no—
389C4—44no—
390C4—816no—
391C4—82no—
392C4—82no—
393C2—28no—
394C2—22no—
395C2—28no—
396C2—22no—
397C2—44no—
398C2#1377611yescharacteristic
399C2C2×C4×#9912yes—
400C2#2174611yescharacteristic
401C2#2174711yescharacteristic
402C2#2182512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C23 ▷ 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. #10912C2semidirect
7. C23×C4⋊C4C4semidirect
12. C2×#385C4semidirect
17. C22×#95C4semidirect
18. #1819C4semidirect
68. #342C2×C4semidirect
105. C23×C4C42semidirect
141. C2×C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C2×#18192
C4×#3792
#109122
#107252