Finite Group Databasebeta

← #503034

Subgroups — #503034

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic
2C2×#2252C211yesmaximal, characteristic, complemented
3C2×#2240C211yesmaximal, characteristic, complemented
4#33209C211yesmaximal, characteristic, complemented
5#13380C214yesmaximal, complemented
6#13282C211yesmaximal, characteristic, complemented
7#13202C211yesmaximal, characteristic, complemented
8#12935C214yesmaximal, complemented
9C4×C22⋊C16C211yesmaximal, characteristic, complemented
10C2×#532—21no—
11#3169—24no—
12#3169—24no—
13C2×#500C2211yescharacteristic
14C2×#500C2211yescharacteristic
15C2×#493C2211yescharacteristic
16C2×#486—21no—
17C2×#466—21no—
18C2×#466C2212yes—
19C2×C4×D8—21no—
20C2×C4×D8C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2857—21no—
23#2252—24no—
24#2240—24no—
25C2×#366C2214yes—
26C2×#357C2214yes—
27C2×#357C2214yes—
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×#41—41no—
40C22×D8—21no—
41C22×D8C2312yes—
42C23×C8D411yescharacteristic
43C23×C8C2311yescharacteristic
44#532—28no—
45#532—44no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×#127—21no—
49C2×#124—41no—
50C2×C22⋊Q8—41no—
51C2×#121—22no—
52C2×#121—41no—
53C2×C4×D4—22no—
54C2×C4×D4—41no—
55C22×C42D411yescharacteristic
56C22×C16C2312yes—
57C22×C16C2312yes—
58#500—22no—
59#500—22no—
60#500—22no—
61#500D414yes—
62#493—22no—
63#493D414yes—
64#486—28no—
65#466—28no—
66#466—24no—
67#466—24no—
68C4×D8—28no—
69C4×D8—24no—
70C4×D8—24no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2312yes—
75C2×#107C2312yes—
76C2×#107C2312yes—
77C2×#102—22no—
78C2×#102—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#357—28no—
93#357—28no—
94C22⋊C16—24no—
95C4×C16—22no—
96C4×C16—22no—
97#340—22no—
98#340—22no—
99C2×#41—48no—
100C2×#41—84no—
101C2×#41—84no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×D4—22no—
105C22×D4—42no—
106C22×D4—41no—
107C22×D4—41no—
108C23×C4—42no—
109C23×C4D811yescharacteristic
110C23×C4D811yescharacteristic
111C23×C4C2×D411yescharacteristic
112C2×D8—24no—
113C2×D8—22no—
114C2×D8—24no—
115C2×D8—24no—
116C2×D8—42no—
117C2×D8—42no—
118C2×D8C2×D414yes—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2411yescharacteristic
130C4⋊Q8—24no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134#127—24no—
135#127—42no—
136#124—48no—
137C22⋊Q8—84no—
138#121—28no—
139#121—44no—
140#121—48no—
141C4×D4—28no—
142C4×D4—44no—
143C4×D4—48no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42D811yescharacteristic
158C2×C42D811yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16—22no—
166C2×C16#4114yescomplemented
167#107—24no—
168#107—22no—
169#107—22no—
170#107—22no—
171#107—24no—
172#107—42no—
173#107—42no—
174#107C2×D414yes—
175#102—28no—
176#102—28no—
177C4×C8—22no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—22no—
181C4×C8C2×D412yes—
182C4×C8C2×D412yes—
183#95—42no—
184#95—42no—
185C24—42no—
186C24C2×D811yescharacteristic
187#41—832no—
188C2×Q8—44no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×D4—24no—
195C2×D4—48no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—84no—
204C2×D4—84no—
205C2×D4—82no—
206C2×D4—82no—
207C2×D4—82no—
208C2×D4—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—48no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4D1614yes—
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C22×D411yescharacteristic
233D8—28no—
234D8—44no—
235D8—48no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—44no—
245C2×C8#12012yes—
246C2×C8#13512yes—
247C2×C8#13712yes—
248C2×C8C22×D411yescharacteristic
249C2×C8C22×D411yescharacteristic
250C2×C8C2×#4112yes—
251C2×C82−1+412yes—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—84no—
262C4⋊C4—82no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C22⋊C4—84no—
266C42—21no—
267C42—21no—
268C42—22no—
269C42—22no—
270C42#12012yes—
271C42AGL(1,ℤ/8)12yes—
272C16—24no—
273C16—42no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—44no—
281C23—82no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23C2×D1612yes—
288C23C22×D811yescharacteristic
289Q8—88no—
290Q8—164no—
291D4—44no—
292D4—44no—
293D4—816no—
294D4—82no—
295D4—88no—
296D4—88no—
297D4—82no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—28no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—816no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#44712yes—
323C2×C4#44712yes—
324C2×C4C2×D1612yes—
325C2×C4C2×D1612yes—
326C2×C4#50812yes—
327C2×C4#50912yes—
328C2×C4#51012yes—
329C2×C4C2×#12011yescharacteristic
330C2×C4#53211yescharacteristic
331C2×C4#53611yescharacteristic
332C2×C4C2×#13511yescharacteristic
333C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
334C2×C4C2×#13711yescharacteristic
335C8—41no—
336C8—41no—
337C8#45214yes—
338C8#54712yes—
339C8#54812yes—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—84no—
349C22—84no—
350C22—162no—
351C22—162no—
352C22—162no—
353C22—162no—
354C22C2×#44711yescharacteristic
355C22#316911yescharacteristic
356C22#322611yescharacteristic
357C22C22×D1611yescharacteristic
358C22C2×#50811yescharacteristic
359C22C2×#50911yescharacteristic
360C22C2×#51011yescharacteristic
361C4—22no—
362C4—41no—
363C4—41no—
364C4—84no—
365C4—162no—
366C4—162no—
367C4#218414yes—
368C4#218914yes—
369C4#243211yescharacteristic
370C4C2×#45211yescharacteristic
371C4#318211yescharacteristic
372C4#322811yescharacteristic
373C2—42no—
374C2—164no—
375C2#2567311yescharacteristic
376C2C2×#218411yescharacteristic
377C2C2×#218911yescharacteristic
378C2#3373311yescharacteristic
379C2#3375111yescharacteristic
380C2#3379211yescharacteristic
381C2#3379411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#3169C2
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×#2240C2semidirect
4. #33209C2semidirect
5. #13380C2semidirect
6. #13282C2semidirect
7. #13202C2semidirect
8. #12935C2semidirect
9. C4×C22⋊C16C2semidirect
166. C2×C16#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#22522
C2×#22402
#332092
#133802
#132822
#132022
#129352
C4×C22⋊C162