Finite Group Databasebeta

← #4414806

Subgroups — #4414806

Order 512.

2470 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C212yesmaximal
2#17075C211yesmaximal, characteristic, complemented
3#17074C212yesmaximal, complemented
4#17070C211yesmaximal, characteristic, complemented
5#16705C214yesmaximal, complemented
6#16605C214yesmaximal, complemented
7#14826C214yesmaximal
8#14821C214yesmaximal
9#14766C214yesmaximal
10#14634C214yesmaximal, complemented
11C2×C4⋊#95C211yesmaximal, characteristic
12C22×#125C2212yes—
13C23×C4⋊C4C2212yes—
14C23×C42C2211yescharacteristic
15#2851—216no—
16C2×#400C2214yes—
17C2×#398C2214yes—
18C2×#398C2214yes—
19C2×#398C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#387C2214yes—
23C2×#387C2214yes—
24C2×#385C2214yes—
25C2×#385C2214yes—
26C2×#382C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2214yes—
30C22×#95C2214yes—
31#1445C2214yes—
32#1445C2214yes—
33#1444C2214yes—
34#1443C2218yes—
35#1442C2218yes—
36#1442C2218yes—
37#1442C2214yes—
38#1441C2218yes—
39#1441C2214yes—
40#1441C2218yes—
41#1441C2214yes—
42#1441C2218yes—
43#1441C2214yes—
44#1441C2214yes—
45C4⋊#95C2212yes—
46C4⋊#95C2214yes—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2211yescharacteristic
49C22⋊#95C2218yes—
50C22⋊#95C2214yes—
51C24×C4C2311yescharacteristic
52C24×C4C2314yes—
53C24×C4C2312yes—
54C2×#125—28no—
55C2×#125—48no—
56C22×C4⋊C4—28no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4—44no—
59C22×C4⋊C4C2314yes—
60C22×C4⋊C4C2314yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2314yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C42—44no—
75C22×C42D418yes—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#400—48no—
80#398—216no—
81#398—48no—
82#398—48no—
83#389—28no—
84#389—28no—
85#387—216no—
86#387—216no—
87#385—216no—
88#385—216no—
89#382—28no—
90#380—28no—
91#380—28no—
92#380—48no—
93C2×#95—48no—
94C2×#95C2318yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2318yes—
105C2×#95C2318yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2318yes—
109C2×#95C2318yes—
110C2×#95C2318yes—
111C25C2411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—28no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—28no—
120C23×C4—216no—
121C23×C4—28no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—48no—
125C23×C4C2×D414yes—
126C23×C4C2×D418yes—
127C23×C4C2414yes—
128C23×C4C2414yes—
129C23×C4C2414yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2412yes—
133C23×C4C2414yes—
134C23×C4C2411yescharacteristic
135C23×C4C2411yescharacteristic
136C23×C4C2412yes—
137#125—432no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—216no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—432no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4#4118yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—216no—
176C2×C22⋊C4—28no—
177C2×C42—28no—
178C2×C42—416no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C2×D418yes—
182C2×C42C2×D414yes—
183C2×C42C2×D414yes—
184#95—28no—
185#95—28no—
186#95—28no—
187#95—28no—
188#95—28no—
189#95—28no—
190#95—28no—
191#95—216no—
192#95—28no—
193#95—28no—
194#95—48no—
195#95—48no—
196#95—48no—
197#95—44no—
198#95—48no—
199#95—48no—
200#95—48no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24C22×D412yes—
206C24H(5,𝔽2)12yes—
207C242−1+418yes—
208C242−1+412yes—
209C24C2511yescharacteristic
210C22×C4—216no—
211C22×C4—216no—
212C22×C4—28no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—28no—
222C22×C4—28no—
223C22×C4—216no—
224C22×C4—28no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—28no—
232C22×C4—216no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—48no—
261C22×C4—416no—
262C22×C4—416no—
263C22×C4—416no—
264C22×C4—432no—
265C22×C4—416no—
266C22×C4—816no—
267C22×C4—88no—
268C22×C4C22×D414yes—
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C22×D414yes—
272C22×C4C2×#4118yes—
273C22×C4C2×#4114yes—
274C22×C4H(5,𝔽2)14yes—
275C22×C4H(5,𝔽2)14yes—
276C22×C42−1+414yes—
277C4⋊C4—48no—
278C4⋊C4—416no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—416no—
283C4⋊C4—416no—
284C4⋊C4—416no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—416no—
288C4⋊C4—88no—
289C4⋊C4—88no—
290C22⋊C4—416no—
291C22⋊C4—416no—
292C42—216no—
293C42—44no—
294C42—44no—
295C23—22no—
296C23—22no—
297C23—24no—
298C23—24no—
299C23—24no—
300C23—28no—
301C23—28no—
302C23—28no—
303C23—28no—
304C23—22no—
305C23—22no—
306C23—24no—
307C23#53618yes—
308C23#53714yes—
309C23#56318yes—
310C23#56418yes—
311C23C23×D411yescharacteristic
312C23C22×#4112yes—
313C23C2×H(5,𝔽2)12yes—
314C23C2×H(5,𝔽2)11yescharacteristic
315C23C2×H(5,𝔽2)12yes—
316C23C2×2−1+414yes—
317C23C2×2−1+411yescharacteristic
318C23C2×2−1+412yes—
319C2×C4—232no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
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—24no—
335C2×C4—28no—
336C2×C4—216no—
337C2×C4—432no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—832no—
371C2×C4#53214yes—
372C2×C4#53414yes—
373C2×C4#53718yes—
374C22—28no—
375C22—28no—
376C22—216no—
377C22—28no—
378C22—28no—
379C22—24no—
380C22—24no—
381C22#282218yes—
382C22C2×#53211yescharacteristic
383C22C2×#53411yescharacteristic
384C22C2×#53612yes—
385C22C2×#53712yes—
386C22C2×#53711yescharacteristic
387C22#349012yes—
388C22#349212yes—
389C22#352014yes—
390C22#352314yes—
391C22C2×#56312yes—
392C22C2×#56412yes—
393C22#353614yes—
394C22#353814yes—
395C22#354311yescharacteristic
396C22#354411yescharacteristic
397C22#355212yes—
398C4—28no—
399C4—24no—
400C4—24no—
401C4—44no—
402C4—48no—
403C4—84no—
404C4—84no—
405C4—84no—
406C4—84no—
407C4—84no—
408C2—28no—
409C2#3505111yescharacteristic
410C2#3524112yes—
411C2C2×#282211yescharacteristic
412C2#3744212yes—
413C2#3762011yescharacteristic
414C2#4619912yes—
415C2#4620014yes—
416C2#4620512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17075C2semidirect
3. #17074C2semidirect
4. #17070C2semidirect
5. #16705C2semidirect
6. #16605C2semidirect
10. #14634C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#170752
#170742
#170702
#167052
#166052
#148262
#148212
#147662
#146342
C2×C4⋊#952