Finite Group Databasebeta

← #500866

Subgroups — #500866

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#500C211yesmaximal, characteristic
2C2×#2225C212yesmaximal, complemented
3C4×#467C211yesmaximal, characteristic
4#13452C212yesmaximal, complemented
5#13364C212yesmaximal, complemented
6#13275C212yesmaximal, complemented
7#12993C212yesmaximal
8#12920C212yesmaximal
9#12783C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C2×#500—28no—
12C2×#500—21no—
13C2×#500—21no—
14C2×#500C2212yes—
15C2×#500C2212yes—
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C22×#107—21no—
19C22×#107C2212yes—
20C22×C4×C8C2211yescharacteristic
21C4×C22⋊Q8—21no—
22#2225—28no—
23C2×#367C2214yes—
24C2×C4⋊C16C2212yes—
25C2×#358C2214yes—
26C2×C22⋊C16C2212yes—
27#1969C2211yescharacteristic
28#1942C2211yescharacteristic
29#1933—21no—
30#1892—21no—
31#1808C2211yescharacteristic
32C4×#107C2211yescharacteristic
33C4×#103—21no—
34#1380C2214yes—
35#1380C2214yes—
36#1374C2212yes—
37C23×Q8—41no—
38C22×Q16C2311yescharacteristic
39C23×C8D411yescharacteristic
40C23×C8C2311yescharacteristic
41C2×C4⋊Q8—28no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—44no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—41no—
51C22×C42D411yescharacteristic
52C22×C16C2314yes—
53#500—28no—
54#500—28no—
55#500—28no—
56#500—24no—
57#500D418yes—
58#467—22no—
59#467Q814yescomplemented
60C4×Q16—24no—
61C2×#107—28no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—28no—
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#107C2314yes—
69C2×#107C2311yescharacteristic
70C2×#103—21no—
71C2×#103—21no—
72C2×#103—21no—
73C2×C4×C8—24no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—21no—
79#389—21no—
80#387—41no—
81#385—41no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—41no—
85C4×C22⋊C4—41no—
86#367—28no—
87C4⋊C16—24no—
88#358—28no—
89C22⋊C16—24no—
90#340—21no—
91#340—21no—
92#340—21no—
93C22×Q8—21no—
94C22×Q8—48no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8—84no—
98C22×Q8—84no—
99C23×C4—42no—
100C23×C4D811yescharacteristic
101C23×C4D811yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×Q16—22no—
104C2×Q16—21no—
105C2×Q16—22no—
106C2×Q16C2×Q812yescomplemented
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×Q812yes—
116C22×C8#4112yes—
117C22×C8C2411yescharacteristic
118C4⋊Q8—28no—
119C4⋊Q8—28no—
120C4⋊Q8—44no—
121C4⋊Q8—48no—
122C4⋊Q8—48no—
123C4⋊Q8—44no—
124C22⋊Q8—24no—
125C22⋊Q8—42no—
126C4×Q8—42no—
127C4×Q8—42no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—48no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42D811yescharacteristic
148C2×C42D811yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C16—24no—
152C2×C16—24no—
153C2×C16—24no—
154#107—28no—
155#107—28no—
156#107—28no—
157#107—24no—
158#107—21no—
159#107—22no—
160#107—22no—
161#107C2×D418yes—
162#107C2×Q812yescomplemented
163#103—24no—
164#103—42no—
165#103—42no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8C2×D414yes—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175C24C2×D811yescharacteristic
176C2×Q8—22no—
177C2×Q8—48no—
178C2×Q8—41no—
179C2×Q8—48no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—84no—
185C2×Q8—832no—
186C2×Q8—84no—
187C2×Q8—84no—
188C2×Q8—84no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—48no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—44no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4SD3214yes—
207C22×C4C22⋊Q812yes—
208C22×C4C2×D811yescharacteristic
209C22×C4C2×D811yescharacteristic
210C22×C4C2×D811yescharacteristic
211C22×C4C2×D811yescharacteristic
212C22×C4C2×D811yescharacteristic
213C22×C4#13712yes—
214C22×C4C22×D411yescharacteristic
215Q16—24no—
216Q16—42no—
217C2×C8—28no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8#12012yes—
227C2×C8C22⋊Q812yes—
228C2×C8#13712yes—
229C2×C8C22×D412yes—
230C2×C8C22×Q811yescharacteristic
231C2×C8C2×#4111yescharacteristic
232C2×C82−1+412yes—
233C4⋊C4—28no—
234C4⋊C4—22no—
235C4⋊C4—48no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—41no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C22⋊C4—44no—
248C22⋊C4—82no—
249C42—21no—
250C42—21no—
251C42—22no—
252C42—22no—
253C42—82no—
254C42—82no—
255C42#12012yes—
256C42AGL(1,ℤ/8)12yes—
257C16—44no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23#47412yes—
263C23C2×SD3212yes—
264C23#51012yes—
265C23C22×D811yescharacteristic
266Q8—42no—
267Q8—42no—
268Q8—816no—
269Q8—816no—
270Q8—81no—
271Q8—81no—
272C2×C4—28no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—816no—
290C2×C4—84no—
291C2×C4—81no—
292C2×C4—84no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#44712yes—
303C2×C4#44712yes—
304C2×C4#47412yes—
305C2×C4C2×SD3212yes—
306C2×C4C2×SD3212yes—
307C2×C4#50912yes—
308C2×C4#51012yes—
309C2×C4C2×#12011yescharacteristic
310C2×C4C2×C22⋊Q811yescharacteristic
311C2×C4#53611yescharacteristic
312C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
313C2×C4C2×#13711yescharacteristic
314C2×C4C2×#13711yescharacteristic
315C8—22no—
316C8—22no—
317C8—22no—
318C8#54812yes—
319C8D4×Q812yes—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22#222514yescomplemented
327C22C2×#44711yescharacteristic
328C22C2×#47411yescharacteristic
329C22#322611yescharacteristic
330C22C22×SD3211yescharacteristic
331C22C2×#50911yescharacteristic
332C22C2×#51011yescharacteristic
333C22C2×#51011yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—82no—
339C4—88no—
340C4—82no—
341C4—161no—
342C4—161no—
343C4#218714yes—
344C4#240911yescharacteristic
345C4#300411yescharacteristic
346C4#307911yescharacteristic
347C4#323911yescharacteristic
348C2—24no—
349C2#2429811yescharacteristic
350C2C2×#218711yescharacteristic
351C2#3351811yescharacteristic
352C2C2×#222511yescharacteristic
353C2#3362911yescharacteristic
354C2#3379511yescharacteristic
355C2#3380511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2225C2semidirect
4. #13452C2semidirect
5. #13364C2semidirect
6. #13275C2semidirect
59. #467Q8semidirect
106. C2×Q16C2×Q8semidirect
162. #107C2×Q8semidirect
326. C22#2225semidirect

Maximal subgroups

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

subgroupindex
C22×#5002
C2×#22252
C4×#4672
#134522
#133642
#132752
#129932
#129202
#127832