Finite Group Databasebeta

← #484144

Subgroups — #484144

Order 512.

402 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33750C211yesmaximal, characteristic
2#18093C211yesmaximal, characteristic, complemented
3#18075C211yesmaximal, characteristic, complemented
4#12089C211yesmaximal, characteristic, complemented
5#12088C211yesmaximal, characteristic, complemented
6#12080C211yesmaximal, characteristic, complemented
7#12080C211yesmaximal, characteristic, complemented
8#12077C211yesmaximal, characteristic, complemented
9#12077C211yesmaximal, characteristic, complemented
10#12067C211yesmaximal, characteristic, complemented
11#12066C211yesmaximal, characteristic, complemented
12#12000C211yesmaximal, characteristic, complemented
13#11656C211yesmaximal, characteristic, complemented
14#11651C211yesmaximal, characteristic, complemented
15#11414C211yesmaximal, characteristic, complemented
16#3168C2211yescharacteristic
17#3168C2211yescharacteristic
18#2994—21no—
19#2969—21no—
20#2253C411yescharacteristic
21#2249C411yescharacteristic
22#2242C411yescharacteristic
23#2242C411yescharacteristic
24#2220C411yescharacteristic
25#2220C411yescharacteristic
26#2219C411yescharacteristic
27#2218C411yescharacteristic
28#2177C2211yescharacteristic
29#2176C2211yescharacteristic
30#2175C2211yescharacteristic
31#2175C2211yescharacteristic
32#2163C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1594C2211yescharacteristic
35#1594C2211yescharacteristic
36#1593C2211yescharacteristic
37#1593C2211yescharacteristic
38#1588—21no—
39#1585—21no—
40#1583—21no—
41#1582—21no—
42#1580C2211yescharacteristic
43#1580C2211yescharacteristic
44#1579C2211yescharacteristic
45#1579C2211yescharacteristic
46#1564C2211yescharacteristic
47C2×#334C2211yescharacteristic
48#1483—21no—
49#1482—21no—
50#1451C2211yescharacteristic
51#1368C2211yescharacteristic
52#1368C2211yescharacteristic
53#1368C2211yescharacteristic
54#1368C2211yescharacteristic
55#1334C2211yescharacteristic
56#1334C2211yescharacteristic
57#1334C2211yescharacteristic
58#1334C2211yescharacteristic
59#1333C2211yescharacteristic
60#1333C2211yescharacteristic
61#1333C2211yescharacteristic
62#1333C2211yescharacteristic
63#1313C2211yescharacteristic
64#1313C2211yescharacteristic
65#1313C2211yescharacteristic
66#1313C2211yescharacteristic
67#529—21no—
68#529—21no—
69C2×Q32C2×C411yescharacteristic
70C2×SD32C2×C411yescharacteristic
71C2×SD32C2×C411yescharacteristic
72C2×D16C2×C411yescharacteristic
73C2×M32C2×C411yescharacteristic
74C2×M32C2×C411yescharacteristic
75#499C2×C411yescharacteristic
76#499C2×C411yescharacteristic
77#495C2×C411yescharacteristic
78#495C2×C411yescharacteristic
79#488—21no—
80#488—21no—
81#467C2×C411yescharacteristic
82#467C2×C411yescharacteristic
83#466C2×C411yescharacteristic
84#466C2×C411yescharacteristic
85#465—21no—
86#465—21no—
87#446—21no—
88#446—21no—
89C8×Q8—21no—
90C8×Q8—21no—
91C4×Q16C2311yescharacteristic
92C4×Q16C2311yescharacteristic
93C4×SD16—21no—
94C4×SD16—21no—
95C4×D8C2311yescharacteristic
96C4×D8C2311yescharacteristic
97#436—21no—
98#436—21no—
99#436—41no—
100#435—41no—
101#435—41no—
102C8×D4—21no—
103C8×D4—21no—
104C8×D4—41no—
105#433—41no—
106#433—41no—
107#431—21no—
108#431D411yescharacteristic
109#424—41no—
110C2×C4⋊C8—21no—
111#405—41no—
112C4×M16D411yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#367C2×C411yescharacteristic
115#367C2×C411yescharacteristic
116#366C2×C411yescharacteristic
117#366C2×C411yescharacteristic
118C4⋊C16—21no—
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#358C2×C411yescharacteristic
122#358C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16—21no—
128#346C2311yescharacteristic
129#346C2311yescharacteristic
130#346C2311yescharacteristic
131#346C2311yescharacteristic
132C4×C16C2×C411yescharacteristic
133C4×C16C2×C411yescharacteristic
134#335—21no—
135#334—21no—
136#334C2311yescharacteristic
137#334C2311yescharacteristic
138#334C2311yescharacteristic
139#334C2311yescharacteristic
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142Q8⋊C8—21no—
143Q8⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150#322C2311yescharacteristic
151#322C2311yescharacteristic
152C2×Q16C22×C411yescharacteristic
153C2×Q16C22×C411yescharacteristic
154C2×SD16—21no—
155C2×SD16—21no—
156C2×D8C22×C411yescharacteristic
157C2×D8C22×C411yescharacteristic
158C2×M16—21no—
159C2×M16—41no—
160C2×M16—41no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8C22×C411yescharacteristic
165C22×C8#4111yescharacteristic
166#126—41no—
167#126—41no—
168#125—21no—
169#125—21no—
170#124—21no—
171#124—21no—
172#123—41no—
173#123—41no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176#121—21no—
177#121—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—41no—
183C4×D4—41no—
184#117—41no—
185#117—41no—
186C2×C42C2×D411yescharacteristic
187Q32—22no—
188SD32—41no—
189SD32—41no—
190D16—22no—
191M32—22no—
192M32—41no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16C22×C411yescharacteristic
196C2×C16C22×C411yescharacteristic
197C2×C16C22×C411yescharacteristic
198C2×C16C22×C411yescharacteristic
199#107C22×C411yescharacteristic
200#107C22×C411yescharacteristic
201#107C22×C411yescharacteristic
202#107C22×C411yescharacteristic
203#106—21no—
204#106—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215#103—21no—
216#103—21no—
217#103—21no—
218#103—21no—
219#102—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223C22⋊C8—21no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229#97—21no—
230#97—41no—
231#97—41no—
232C4×C8—41no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8C22×C411yescharacteristic
236C4×C8C2×D411yescharacteristic
237C4×C8C2×D411yescharacteristic
238C4×C8C2×D411yescharacteristic
239C4×C8C2×D411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8#4111yescharacteristic
242C4×C8C2411yescharacteristic
243C2×Q8—21no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C4×D411yescharacteristic
252C22×C4AGL(1,ℤ/8)11yescharacteristic
253C22×C4AGL(1,ℤ/8)11yescharacteristic
254Q16—21no—
255Q16—21no—
256Q16—21no—
257Q16—21no—
258SD16—81no—
259SD16—81no—
260D8—21no—
261D8—21no—
262D8—21no—
263D8—21no—
264M16—81no—
265M16—82no—
266M16—161no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—82no—
279C2×C8—161no—
280C2×C8—161no—
281C2×C8C4×D411yescharacteristic
282C2×C8C4×D411yescharacteristic
283C2×C8C23×C411yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—41no—
308C42—41no—
309C42C4×D411yescharacteristic
310C42AGL(1,ℤ/8)11yescharacteristic
311C42AGL(1,ℤ/8)11yescharacteristic
312C42C22×D411yescharacteristic
313C16—21no—
314C16—21no—
315C16—21no—
316C16—41no—
317C16—41no—
318C16#13112yes—
319C23—41no—
320C23—41no—
321C23#44211yescharacteristic
322Q8—41no—
323Q8—41no—
324Q8—41no—
325Q8—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—82no—
331D4—161no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4#44211yescharacteristic
347C2×C4#44211yescharacteristic
348C2×C4#44211yescharacteristic
349C2×C4C2×C4×D411yescharacteristic
350C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
351C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
352C8—21no—
353C8—21no—
354C8—21no—
355C8—41no—
356C8—41no—
357C8—41no—
358C8—41no—
359C8—81no—
360C8—81no—
361C8—81no—
362C8—81no—
363C8C2×#13111yescharacteristic
364C8#56811yescharacteristic
365C22—21no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22C2×#44211yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4#293511yescharacteristic
387C4#295711yescharacteristic
388C2—41no—
389C2—161no—
390C2—161no—
391C2#1807311yescharacteristic
392C2#3346811yescharacteristic
393C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#2218C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18093C2semidirect
3. #18075C2semidirect
4. #12089C2semidirect
5. #12088C2semidirect
6. #12080C2semidirect
7. #12080C2semidirect
8. #12077C2semidirect
9. #12077C2semidirect
10. #12067C2semidirect
11. #12066C2semidirect
12. #12000C2semidirect
13. #11656C2semidirect
14. #11651C2semidirect
15. #11414C2semidirect

Maximal subgroups

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

subgroupindex
#337502
#180932
#180752
#120892
#120882
#120802
#120772
#120672
#120662
#120002
#116562
#116512
#114142