Finite Group Databasebeta

← #517417

Subgroups — #517417

Order 512.

14042 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C22⋊C4C211yesmaximal, characteristic
2C2×#2315C212yesmaximal, complemented
3C22×#379C212yesmaximal
4#14680C214yesmaximal, complemented
5#14338C214yesmaximal, complemented
6#13877C2116yesmaximal, complemented
7#13794C212yesmaximal, complemented
8C27C2211yescharacteristic
9C25×C4C2212yes—
10C23×C22⋊C4—216no—
11C23×C22⋊C4C4116yescomplemented
12C23×C22⋊C4C2214yes—
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C23×C22⋊C4C2218yes—
16C23×C22⋊C4C2218yes—
17C23×C22⋊C4C2211yescharacteristic
18#2315—216no—
19C2×#385C22116yescomplemented
20C2×#380C2218yescomplemented
21C2×#380C22116yescomplemented
22C2×#379—216no—
23C2×#379C2214yes—
24C2×#379C2218yes—
25C2×#379C2214yes—
26#1439C22132yescomplemented
27C22⋊#95C2218yescomplemented
28C22⋊#95C22116yescomplemented
29C22⋊#95C22116yescomplemented
30C26—28no—
31C26—28no—
32C26—416no—
33C26C2×C418yes—
34C26D418yes—
35C26Q818yes—
36C26C2311yescharacteristic
37C26C2314yes—
38C26C2311yescharacteristic
39C26C2311yescharacteristic
40C24×C4—216no—
41C24×C4—416no—
42C24×C4C2×C4116yescomplemented
43C24×C4C2318yes—
44C24×C4C2318yes—
45C24×C4C2312yes—
46C24×C4C2312yes—
47C22×C4⋊C4C23116yescomplemented
48C22×C22⋊C4—28no—
49C22×C22⋊C4—216no—
50C22×C22⋊C4—2128no—
51C22×C22⋊C4—264no—
52C22×C22⋊C4—216no—
53C22×C22⋊C4—232no—
54C22×C22⋊C4—216no—
55C22×C22⋊C4—216no—
56C22×C22⋊C4—432no—
57C22×C22⋊C4C2×C418yes—
58C22×C22⋊C4C2×C418yes—
59C22×C22⋊C4C2×C4116yescomplemented
60C22×C22⋊C4C2×C4164yescomplemented
61C22×C22⋊C4D4116yes—
62C22×C22⋊C4C2318yes—
63C22×C22⋊C4C23116yes—
64C22×C22⋊C4C2318yes—
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2318yes—
67C22×C22⋊C4C23116yes—
68C22×C22⋊C4C2314yes—
69#385—264no—
70#380—232no—
71#380—432no—
72#379—216no—
73#379—232no—
74#379—216no—
75C2×#95C23116yescomplemented
76C2×#95C23132yescomplemented
77C25—24no—
78C25—24no—
79C25—216no—
80C25—232no—
81C25—232no—
82C25—264no—
83C25—264no—
84C25—216no—
85C25—24no—
86C25—216no—
87C25—24no—
88C25—232no—
89C25—464no—
90C25—464no—
91C25—464no—
92C25—48no—
93C25—416no—
94C25—48no—
95C25—4128no—
96C25—4128no—
97C25C4⋊C4164yescomplemented
98C25C22×C414yes—
99C25C22×C4116yes—
100C25C22×C414yes—
101C25C22×C414yes—
102C25C2×D414yes—
103C25C2×D4116yes—
104C25C2×D414yes—
105C25C2×D418yes—
106C25C2×D414yes—
107C25C2×Q814yes—
108C25C2×Q8116yes—
109C25C2×Q814yes—
110C25C2×Q814yes—
111C25#4118yes—
112C25C2412yes—
113C25C2412yes—
114C25C2411yescharacteristic
115C25C2412yes—
116C23×C4—28no—
117C23×C4—232no—
118C23×C4—264no—
119C23×C4—232no—
120C23×C4—216no—
121C23×C4—216no—
122C23×C4—216no—
123C23×C4—464no—
124C23×C4—48no—
125C23×C4—416no—
126C23×C4—48no—
127C23×C4—432no—
128C23×C4—4128no—
129C23×C4C22×C4132yes—
130C23×C4C22×C418yes—
131C23×C4C22×C418yes—
132C23×C4C2×D4116yes—
133C23×C4C24116yes—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C2×C4⋊C4—232no—
137C2×C4⋊C4—432no—
138C2×C22⋊C4—232no—
139C2×C22⋊C4—216no—
140C2×C22⋊C4—264no—
141C2×C22⋊C4—232no—
142C2×C22⋊C4—2256no—
143C2×C22⋊C4—232no—
144C2×C22⋊C4—216no—
145C2×C22⋊C4—216no—
146C2×C22⋊C4—232no—
147C2×C22⋊C4—216no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—216no—
150C2×C22⋊C4—216no—
151C2×C22⋊C4—264no—
152C2×C22⋊C4—432no—
153C2×C22⋊C4—4128no—
154C2×C22⋊C4—464no—
155C2×C22⋊C4C22⋊C4164yescomplemented
156C2×C22⋊C4C22×C4116yes—
157C2×C22⋊C4C22×C4132yes—
158C2×C22⋊C4C22×C4116yes—
159C2×C22⋊C4C2×D4132yes—
160#95—232no—
161#95—432no—
162C24—264no—
163C24—264no—
164C24—28no—
165C24—28no—
166C24—232no—
167C24—216no—
168C24—216no—
169C24—232no—
170C24—232no—
171C24—24no—
172C24—24no—
173C24—28no—
174C24—28no—
175C24—216no—
176C24—232no—
177C24—28no—
178C24—232no—
179C24—28no—
180C24—264no—
181C24—264no—
182C24—216no—
183C24—264no—
184C24—216no—
185C24—264no—
186C24—264no—
187C24—24no—
188C24—24no—
189C24—216no—
190C24—2256no—
191C24—4256no—
192C24—416no—
193C24—432no—
194C24—416no—
195C24—416no—
196C24—432no—
197C24—416no—
198C24—432no—
199C24—416no—
200C24—432no—
201C24—416no—
202C24—432no—
203C24—4128no—
204C24—4128no—
205C24—4256no—
206C24—4256no—
207C24—4512no—
208C24—4512no—
209C24—432no—
210C24—432no—
211C24C2×C22⋊C4116yes—
212C24C2×C4⋊C4116yes—
213C24C2×C4⋊C4164yes—
214C24C2×C4⋊C4116yes—
215C24C2×C4⋊C4116yes—
216C24#117116yes—
217C24#121132yes—
218C24C22⋊Q8132yes—
219C24C23×C414yes—
220C24C23×C414yes—
221C24C23×C412yes—
222C24C23×C414yes—
223C24C22×D414yes—
224C24C22×D418yes—
225C24C22×D414yes—
226C24C22×D412yes—
227C24C22×D414yes—
228C24C22×D414yes—
229C24C22×Q814yes—
230C24C22×Q812yes—
231C24C22×Q814yes—
232C24C22×Q814yes—
233C24C2×#4114yes—
234C24C2×#4118yes—
235C24H(5,𝔽2)116yes—
236C24C2511yescharacteristic
237C22×C4—216no—
238C22×C4—216no—
239C22×C4—264no—
240C22×C4—28no—
241C22×C4—216no—
242C22×C4—264no—
243C22×C4—216no—
244C22×C4—216no—
245C22×C4—216no—
246C22×C4—264no—
247C22×C4—416no—
248C22×C4—432no—
249C22×C4—416no—
250C22×C4—432no—
251C22×C4—4128no—
252C22×C4—4256no—
253C22×C4—432no—
254C22×C4—464no—
255C22×C4—432no—
256C22×C4—832no—
257C22×C4—832no—
258C22×C4C2×C22⋊C4132yescomplemented
259C22×C4C23×C418yes—
260C22×C4C23×C418yes—
261C22×C4C22×D4116yes—
262C4⋊C4—832no—
263C22⋊C4—232no—
264C22⋊C4—264no—
265C22⋊C4—464no—
266C22⋊C4—432no—
267C22⋊C4—432no—
268C22⋊C4—464no—
269C22⋊C4C2×C22⋊C4164yescomplemented
270C23—264no—
271C23—264no—
272C23—2128no—
273C23—2128no—
274C23—264no—
275C23—24no—
276C23—24no—
277C23—28no—
278C23—28no—
279C23—216no—
280C23—28no—
281C23—28no—
282C23—216no—
283C23—28no—
284C23—28no—
285C23—28no—
286C23—28no—
287C23—216no—
288C23—216no—
289C23—232no—
290C23—232no—
291C23—232no—
292C23—216no—
293C23—232no—
294C23—232no—
295C23—2128no—
296C23—24no—
297C23—24no—
298C23—216no—
299C23—28no—
300C23—28no—
301C23—216no—
302C23—216no—
303C23—2128no—
304C23—232no—
305C23—432no—
306C23—432no—
307C23—464no—
308C23—432no—
309C23—432no—
310C23—464no—
311C23—464no—
312C23—4128no—
313C23—4128no—
314C23—4256no—
315C23—4256no—
316C23—41024no—
317C23—432no—
318C23—432no—
319C23—464no—
320C23—464no—
321C23—416no—
322C23—416no—
323C23#380164yescomplemented
324C23C22×C22⋊C418yes—
325C23C22×C22⋊C414yes—
326C23C22×C4⋊C418yes—
327C23C22×C4⋊C418yes—
328C23C22×C4⋊C414yes—
329C23C22×C4⋊C418yes—
330C23C2×#11714yes—
331C23C2×#11718yes—
332C23#518116yes—
333C23C2×#12118yes—
334C23C2×#121116yes—
335C23C2×C22⋊Q818yes—
336C23C2×C22⋊Q8116yes—
337C23#534116yes—
338C23#543116yes—
339C23C24×C411yescharacteristic
340C23C23×D412yes—
341C23C23×D411yescharacteristic
342C23C23×Q811yescharacteristic
343C23C22×#4112yes—
344C23C2×H(5,𝔽2)18yes—
345C2×C4—216no—
346C2×C4—232no—
347C2×C4—28no—
348C2×C4—216no—
349C2×C4—432no—
350C2×C4—4128no—
351C2×C4—432no—
352C2×C4—416no—
353C2×C4—416no—
354C2×C4—432no—
355C2×C4—432no—
356C2×C4—432no—
357C2×C4—864no—
358C2×C4C22×C22⋊C4116yes—
359C22—2128no—
360C22—28no—
361C22—28no—
362C22—216no—
363C22—28no—
364C22—216no—
365C22—216no—
366C22—28no—
367C22—216no—
368C22—216no—
369C22—232no—
370C22—216no—
371C22—232no—
372C22—216no—
373C22—232no—
374C22—232no—
375C22—28no—
376C22—216no—
377C22—28no—
378C22—464no—
379C22—4256no—
380C22—464no—
381C22—48no—
382C22—48no—
383C22—416no—
384C22—416no—
385C22C2×#38018yes—
386C22C2×#380116yes—
387C22#2315116yescomplemented
388C22C23×C22⋊C411yescharacteristic
389C22C23×C4⋊C411yescharacteristic
390C22C22×#11711yescharacteristic
391C22C2×#51814yes—
392C22C22×#12112yes—
393C22C22×C22⋊Q812yes—
394C22C2×#53414yes—
395C22C2×#54314yes—
396C22#348418yes—
397C22#349018yes—
398C4—216no—
399C4—48no—
400C4—816no—
401C2—216no—
402C2—28no—
403C2—216no—
404C2—48no—
405C2C22×#38011yescharacteristic
406C2C2×#231512yes—
407C2#3417112yes—
408C2#3417612yes—
409C2#3498114yes—
410C2#3498214yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC27C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2315C2semidirect
4. #14680C2semidirect
5. #14338C2semidirect
6. #13877C2semidirect
7. #13794C2semidirect
11. C23×C22⋊C4C4semidirect
19. C2×#385C22semidirect
20. C2×#380C22semidirect
21. C2×#380C22semidirect
26. #1439C22semidirect
27. C22⋊#95C22semidirect
28. C22⋊#95C22semidirect
29. C22⋊#95C22semidirect
42. C24×C4C2×C4semidirect
47. C22×C4⋊C4C23semidirect
59. C22×C22⋊C4C2×C4semidirect
60. C22×C22⋊C4C2×C4semidirect
75. C2×#95C23semidirect
76. C2×#95C23semidirect
97. C25C4⋊C4semidirect
155. C2×C22⋊C4C22⋊C4semidirect
258. C22×C4C2×C22⋊C4semidirect
269. C22⋊C4C2×C22⋊C4semidirect
323. C23#380semidirect
387. C22#2315semidirect

Maximal subgroups

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

subgroupindex
C24×C22⋊C42
C2×#23152
C22×#3792
#146802
#143382
#138772
#137942