Finite Group Databasebeta

← #223221

Subgroups — #223221

Order 512.

1253 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#22749C212yesmaximal, complemented
3#9903C212yesmaximal, complemented
4#9141C212yesmaximal, complemented
5#9134C212yesmaximal, complemented
6#8724C212yesmaximal, complemented
7#8592C211yesmaximal, characteristic, complemented
8#8231C211yesmaximal, characteristic, complemented
9#8043C212yesmaximal, complemented
10C2×#547—22no—
11C2×#547C412yes—
12C2×#431C2212yes—
13C2×#423C2212yes—
14#2868C2211yescharacteristic
15#2856C2212yes—
16#2542—22no—
17#2542C412yes—
18#2501—22no—
19#2501—22no—
20#2501C414yes—
21C4×#127C2211yescharacteristic
22C2×#384C2212yes—
23C42×D4C2211yescharacteristic
24#1924—22no—
25#1916—22no—
26#1870—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31#1753—22no—
32C2×#331C2212yes—
33C2×#331C2211yescharacteristic
34C2×#331C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C22×#41—22no—
43C22×M16—22no—
44#547—48no—
45#547—48no—
46#547—48no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2×C412yes—
49C2×#127C2311yescharacteristic
50C2×#123—24no—
51C2×#123—24no—
52C2×#123—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121C2×C414yes—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yes—
65C22×C42C2312yes—
66#431—24no—
67#431—42no—
68#423—24no—
69#423—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—22no—
79C2×C4×C8—22no—
80#399—22no—
81#399—22no—
82#399—24no—
83#399—24no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390C2×C412yes—
88#390C2×C414yes—
89#384—22no—
90#384—22no—
91#384—22no—
92#384C2×C412yes—
93#384C2×C414yes—
94#384C2311yescharacteristic
95#384C2312yes—
96C4×C4⋊C4C2×C411yescharacteristic
97C4×C4⋊C4D412yes—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4D412yes—
100C4×C22⋊C4D412yes—
101C43C2311yescharacteristic
102#336—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#331—24no—
108#331—41no—
109#331—41no—
110C2×#41—416no—
111C2×#41—48no—
112C22×Q8—22no—
113C22×D4—24no—
114C22×D4—22no—
115C22×D4C22⋊C411yescharacteristic
116C22×D4C22×C412yes—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4C22×C412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C2×M16—44no—
123C2×M16—44no—
124C2×M16—82no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132#127—22no—
133#127—21no—
134#127—21no—
135#123—416no—
136#123—416no—
137#123—416no—
138C22⋊Q8—416no—
139C22⋊Q8—48no—
140#121—44no—
141#121—44no—
142#121—44no—
143#121—44no—
144#121—48no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—44no—
149C4×D4—44no—
150C4×D4—44no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C414yes—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2411yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—44no—
196C4⋊C8#4114yescomplemented
197C22⋊C8—44no—
198C22⋊C8—44no—
199C22⋊C8—82no—
200#97—82no—
201C4×C8—44no—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204#95C22⋊C411yescharacteristic
205#95C22⋊C411yescharacteristic
206C24—22no—
207C24C2×C22⋊C412yes—
208#41—832no—
209C2×Q8—48no—
210C2×Q8—44no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—48no—
218C2×D4—48no—
219C2×D4—48no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4AGL(1,ℤ/8)14yes—
254C22×C4AGL(1,ℤ/8)14yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D412yes—
257M16—88no—
258C2×C8—24no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C4⋊C4—24no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4#10014yes—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42C2×#4112yes—
306C42H(5,𝔽2)11yescharacteristic
307C422−1+411yescharacteristic
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23#41814yes—
320C23C22×C22⋊C411yescharacteristic
321Q8—88no—
322D4—28no—
323D4—82no—
324D4—82no—
325D4—816no—
326D4—82no—
327D4—82no—
328D4—82no—
329D4—82no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—88no—
361C2×C4—816no—
362C2×C4C2×#10012yes—
363C2×C4C2×#10014yes—
364C2×C4#41814yes—
365C2×C4C4×#4111yescharacteristic
366C2×C4#52011yescharacteristic
367C2×C4#54612yes—
368C2×C4#54712yes—
369C2×C4C2×AGL(1,ℤ/8)12yes—
370C2×C4C2×AGL(1,ℤ/8)12yes—
371C2×C4#57712yes—
372C2×C4#57812yes—
373C8—44no—
374C8—82no—
375C8—82no—
376C8—82no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—82no—
385C22—84no—
386C22—82no—
387C22—82no—
388C22—84no—
389C22#233911yescharacteristic
390C22C22×#10011yescharacteristic
391C22#288911yescharacteristic
392C22C2×#41812yes—
393C22#289812yes—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—24no—
398C4—24no—
399C4—42no—
400C4—41no—
401C4—42no—
402C4—82no—
403C4—84no—
404C4#330612yes—
405C4#330812yes—
406C4#334312yes—
407C4#334512yes—
408C2—44no—
409C2—162no—
410C2#1774111yescharacteristic
411C2#1775912yes—
412C2#1777311yescharacteristic
413C2#2247911yescharacteristic
414C2#2249512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22749C2semidirect
3. #9903C2semidirect
4. #9141C2semidirect
5. #9134C2semidirect
6. #8724C2semidirect
7. #8592C2semidirect
8. #8231C2semidirect
9. #8043C2semidirect
196. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#361792
#227492
#99032
#91412
#91342
#87242
#85922
#82312
#80432