Finite Group Databasebeta

← #503035

Subgroups — #503035

Order 512.

816 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
1C2×#3170C211yesmaximal, characteristic
2C2×#2252C211yesmaximal, characteristic, complemented
3C2×#2241C211yesmaximal, characteristic, complemented
4#33209C211yesmaximal, characteristic
5#13383C214yesmaximal
6#13282C211yesmaximal, characteristic, complemented
7#13203C211yesmaximal, characteristic, complemented
8#12936C214yesmaximal, complemented
9C4×C22⋊C16C211yesmaximal, characteristic
10C2×#533—21no—
11#3170—24no—
12#3170—24no—
13C2×#500C2211yescharacteristic
14C2×#500C2211yescharacteristic
15C2×#494C2211yescharacteristic
16C2×#487—21no—
17C2×#467—21no—
18C2×#467C2212yes—
19C2×C4×Q16—21no—
20C2×C4×Q16C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2857—21no—
23#2252—24no—
24#2241—24no—
25C2×#366C2214yes—
26C2×#358C2214yes—
27C2×#358C2214yes—
28C2×C22⋊C16C2212yes—
29C2×C4×C16C2211yescharacteristic
30C2×C4×C16C2211yescharacteristic
31#1990—21no—
32#1942—21no—
33#1942C2212yes—
34#1845C2211yescharacteristic
35#1808—21no—
36#1808C2212yes—
37#1380C2214yes—
38#1374C2212yes—
39C22×Q16—21no—
40C22×Q16C2312yes—
41C23×C8D411yescharacteristic
42C23×C8C2311yescharacteristic
43#533—28no—
44#533—44no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×#125—41no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—41no—
51C2×C22⋊Q8—41no—
52C2×C4×Q8—22no—
53C2×C4×Q8—41no—
54C2×#117—41no—
55C22×C42D411yescharacteristic
56C22×C16C2312yes—
57C22×C16C2312yes—
58#500—22no—
59#500—22no—
60#500—22no—
61#500D414yes—
62#494—22no—
63#494D414yes—
64#487—28no—
65#467—28no—
66#467—24no—
67#467—24no—
68C4×Q16—28no—
69C4×Q16—24no—
70C4×Q16—24no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2312yes—
75C2×#107C2312yes—
76C2×#107C2312yes—
77C2×#103—22no—
78C2×#103—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—41no—
86#389—22no—
87#389—41no—
88#384—21no—
89#380—22no—
90#380—41no—
91#366—28no—
92#358—28no—
93#358—28no—
94C22⋊C16—24no—
95C4×C16—22no—
96C4×C16—22no—
97#340—22no—
98#340—22no—
99C22×Q8—22no—
100C22×Q8—41no—
101C22×Q8—41no—
102C23×C4D811yescharacteristic
103C23×C4D811yescharacteristic
104C23×C4C2×D411yescharacteristic
105C2×Q16—24no—
106C2×Q16—22no—
107C2×Q16—24no—
108C2×Q16—24no—
109C2×Q16—42no—
110C2×Q16—42no—
111C2×Q16C2×D414yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2411yescharacteristic
123C4⋊Q8—24no—
124C4⋊Q8—24no—
125C4⋊Q8—42no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8—42no—
129#125—48no—
130C22⋊Q8—28no—
131C22⋊Q8—44no—
132C22⋊Q8—48no—
133C22⋊Q8—84no—
134C4×Q8—28no—
135C4×Q8—44no—
136C4×Q8—48no—
137#117—48no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—42no—
155C2×C42D811yescharacteristic
156C2×C42D811yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16#4114yes—
165#107—24no—
166#107—22no—
167#107—22no—
168#107—22no—
169#107—24no—
170#107—42no—
171#107—42no—
172#107C2×D414yes—
173#103—28no—
174#103—28no—
175C4×C8—22no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—22no—
179C4×C8C2×D412yes—
180C4×C8C2×D412yes—
181#95—42no—
182#95—42no—
183C24C2×D811yescharacteristic
184C2×Q8—24no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—82no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4Q3214yes—
212C22×C4C2×D811yescharacteristic
213C22×C4C2×D811yescharacteristic
214C22×C4C2×D811yescharacteristic
215C22×C4C2×D811yescharacteristic
216C22×C4C2×D811yescharacteristic
217C22×C4C22×D411yescharacteristic
218Q16—28no—
219Q16—44no—
220Q16—48no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—44no—
230C2×C8#12012yes—
231C2×C8#13512yes—
232C2×C8#13712yes—
233C2×C8C22×D411yescharacteristic
234C2×C8C22×D411yescharacteristic
235C2×C8C2×#4112yes—
236C2×C82−1+412yes—
237C4⋊C4—24no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—84no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C22⋊C4—48no—
253C22⋊C4—84no—
254C42—21no—
255C42—21no—
256C42—22no—
257C42—22no—
258C42—48no—
259C42#12012yes—
260C42AGL(1,ℤ/8)12yes—
261C16—24no—
262C16—42no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23C2×Q3212yes—
270C23C22×D811yescharacteristic
271Q8—44no—
272Q8—44no—
273Q8—82no—
274Q8—88no—
275Q8—82no—
276Q8—164no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—28no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#44712yes—
304C2×C4#44712yes—
305C2×C4C2×Q3212yes—
306C2×C4C2×Q3212yes—
307C2×C4#50812yes—
308C2×C4#50912yes—
309C2×C4#51012yes—
310C2×C4C2×#12011yescharacteristic
311C2×C4#53211yescharacteristic
312C2×C4#53611yescharacteristic
313C2×C4C2×#13511yescharacteristic
314C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
315C2×C4C2×#13711yescharacteristic
316C8—41no—
317C8—41no—
318C8#45214yes—
319C8#54712yes—
320C8#54812yes—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22C2×#44711yescharacteristic
329C22#316911yescharacteristic
330C22#322611yescharacteristic
331C22C22×Q3211yescharacteristic
332C22C2×#50811yescharacteristic
333C22C2×#50911yescharacteristic
334C22C2×#51011yescharacteristic
335C4—22no—
336C4—41no—
337C4—41no—
338C4—84no—
339C4—84no—
340C4—162no—
341C4—162no—
342C4#218614yes—
343C4#218814yes—
344C4#243211yescharacteristic
345C4C2×#45211yescharacteristic
346C4#318211yescharacteristic
347C4#322811yescharacteristic
348C2—42no—
349C2#2567311yescharacteristic
350C2C2×#218811yescharacteristic
351C2C2×#218611yescharacteristic
352C2#3373411yescharacteristic
353C2#3375111yescharacteristic
354C2#3379611yescharacteristic
355C2#3379711yescharacteristic
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×#2252C2semidirect
3. C2×#2241C2semidirect
6. #13282C2semidirect
7. #13203C2semidirect
8. #12936C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
C2×#22522
C2×#22412
#332092
#133832
#132822
#132032
#129362
C4×C22⋊C162