Finite Group Databasebeta

← #490648

Subgroups — #490648

Order 512.

451 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33736C211yesmaximal, characteristic, complemented
2C16×#41C211yesmaximal, characteristic, complemented
3#19126C211yesmaximal, characteristic, complemented
4#19125C211yesmaximal, characteristic
5#12478C211yesmaximal, characteristic, complemented
6#12476C211yesmaximal, characteristic, complemented
7#12470C211yesmaximal, characteristic, complemented
8#12468C211yesmaximal, characteristic, complemented
9#12459C211yesmaximal, characteristic, complemented
10#12453C211yesmaximal, characteristic, complemented
11#12441C211yesmaximal, characteristic, complemented
12#12432C211yesmaximal, characteristic, complemented
13C2×#1372C211yesmaximal, characteristic, complemented
14#11949C211yesmaximal, characteristic, complemented
15#11947C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3169C2211yescharacteristic
18#3076—21no—
19#3035—21no—
20C8×#41C2211yescharacteristic
21#2254C2211yescharacteristic
22#2248C2211yescharacteristic
23#2241C2211yescharacteristic
24#2240C2211yescharacteristic
25#2217—21no—
26#2217—21no—
27Q8×C16C2211yescharacteristic
28Q8×C16C2211yescharacteristic
29C4×SD32—21no—
30C4×SD32—21no—
31D4×C16—21no—
32D4×C16—21no—
33D4×C16C2211yescharacteristic
34D4×C16C2211yescharacteristic
35#2162—21no—
36#2162C2211yescharacteristic
37C2×C4×C16—21no—
38C2×C4×C16C2211yescharacteristic
39#1684—21no—
40#1683—21no—
41#1682—21no—
42#1681—21no—
43#1676C2211yescharacteristic
44#1675C2211yescharacteristic
45#1674C2211yescharacteristic
46#1673C2211yescharacteristic
47#1670C2211yescharacteristic
48#1669C2211yescharacteristic
49#1668C2211yescharacteristic
50#1667C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53#1542—21no—
54#1540—21no—
55#1372C2211yescharacteristic
56#1372C2211yescharacteristic
57#1372C2211yescharacteristic
58#1372C2211yescharacteristic
59#1365C2211yescharacteristic
60#1365C2211yescharacteristic
61#1364C2211yescharacteristic
62#1364C2211yescharacteristic
63#1362C2211yescharacteristic
64#1362C2211yescharacteristic
65#1361C2211yescharacteristic
66#1361C2211yescharacteristic
67C2×#135—41no—
68#533—21no—
69#532—21no—
70C4×#41D411yescharacteristic
71C2×SD32—21no—
72C2×SD32—21no—
73C22×C16—21no—
74C22×C16—21no—
75C22×C16D411yescharacteristic
76C22×C16D411yescharacteristic
77#500C2311yescharacteristic
78#500C2311yescharacteristic
79#494C2311yescharacteristic
80#493C2311yescharacteristic
81#487—21no—
82#486—21no—
83#479—41no—
84#478—41no—
85#477—21no—
86#476—21no—
87#475—21no—
88#474—21no—
89#467—21no—
90#467—21no—
91#466—21no—
92#466—21no—
93#464—41no—
94#463—41no—
95#462—21no—
96#461—21no—
97#460—21no—
98#459—21no—
99C8×Q8D411yescharacteristic
100C8×Q8C2311yescharacteristic
101C4×Q16—21no—
102C4×Q16—21no—
103C4×D8—21no—
104C4×D8—21no—
105C8×D4—21no—
106C8×D4—21no—
107C8×D4D411yescharacteristic
108C8×D4C2311yescharacteristic
109#431—21no—
110#431D411yescharacteristic
111#427—41no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114#416—41no—
115#416—41no—
116C2×C4×C8—21no—
117C2×C4×C8C2311yescharacteristic
118#367—21no—
119#367—21no—
120C4⋊C16—21no—
121C4⋊C16—21no—
122C4⋊C16C2311yescharacteristic
123C4⋊C16C2311yescharacteristic
124#358—21no—
125#358—21no—
126#357—21no—
127#357—21no—
128C22⋊C16—21no—
129C22⋊C16—21no—
130C22⋊C16—21no—
131C4×C16—21no—
132C4×C16—21no—
133C4×C16—21no—
134C4×C16D411yescharacteristic
135C4×C16D411yescharacteristic
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335—21no—
139#335—21no—
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#335C2311yescharacteristic
143#335C2311yescharacteristic
144#333—21no—
145#333—21no—
146#332—21no—
147#332—21no—
148#332—21no—
149#332—21no—
150#331—21no—
151#331—21no—
152C2×#41—41no—
153C2×#41D811yescharacteristic
154#135—84no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—41no—
158C2×SD16—41no—
159C2×SD16—41no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—41no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8C2×D411yescharacteristic
168C22×C8#4111yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172#127—21no—
173#125—41no—
174#124—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177#121—41no—
178#121—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8D811yescharacteristic
182C4×Q8C2×D411yescharacteristic
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4D811yescharacteristic
188C4×D4C2×D411yescharacteristic
189#117—21no—
190#117—41no—
191#117D811yescharacteristic
192C2×C42—21no—
193C2×C42C2×D411yescharacteristic
194SD32—42no—
195SD32—42no—
196C2×C16—22no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—41no—
200C2×C16—42no—
201C2×C16—42no—
202C2×C16C2×D411yescharacteristic
203C2×C16C2×D411yescharacteristic
204C2×C16C2×D411yescharacteristic
205C2×C16C2×D411yescharacteristic
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—41no—
211#107—41no—
212#106—41no—
213#106—41no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C4⋊C8C2×D411yescharacteristic
223C4⋊C8C2×D411yescharacteristic
224#103—41no—
225#103—41no—
226#103—41no—
227#103—41no—
228#103—41no—
229#103—41no—
230#102—41no—
231#102—41no—
232#102—41no—
233#102—41no—
234#102—41no—
235#102—41no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8#4111yescharacteristic
245C4×C8C2411yescharacteristic
246#41—22no—
247#41—82no—
248#41—82no—
249#41SD3214yescomplemented
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8C2×D811yescharacteristic
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—81no—
259C2×D4C2×D811yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—81no—
264C22×C4#12111yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4#13511yescharacteristic
267Q16—41no—
268Q16—41no—
269Q16—81no—
270Q16—82no—
271SD16—82no—
272SD16—82no—
273D8—41no—
274D8—41no—
275D8—81no—
276D8—82no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—82no—
290C2×C8—82no—
291C2×C8#12111yescharacteristic
292C2×C8#12111yescharacteristic
293C2×C8C22×D411yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4C2×D811yescharacteristic
306C4⋊C4C2×D811yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—81no—
317C42#12111yescharacteristic
318C42C2×D811yescharacteristic
319C42#13511yescharacteristic
320C42C22×D411yescharacteristic
321C16—21no—
322C16—21no—
323C16—21no—
324C16—41no—
325C16—41no—
326C16#13512yescomplemented
327C23—21no—
328C23—81no—
329C23#46611yescharacteristic
330Q8—21no—
331Q8—81no—
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—161no—
336Q8C2×SD3212yes—
337D4—24no—
338D4—21no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—161no—
344D4—162no—
345D4C2×SD3212yes—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—162no—
360C2×C4#46611yescharacteristic
361C2×C4#46611yescharacteristic
362C2×C4#46611yescharacteristic
363C2×C4C2×SD3212yes—
364C2×C4#50812yes—
365C2×C4C2×#12111yescharacteristic
366C2×C4C22×D811yescharacteristic
367C2×C4C2×#13511yescharacteristic
368C8—21no—
369C8—21no—
370C8—21no—
371C8—41no—
372C8—41no—
373C8—81no—
374C8—81no—
375C8—81no—
376C8—81no—
377C8C2×#13511yescharacteristic
378C8#57511yescharacteristic
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22#221712yes—
388C22C2×#46611yescharacteristic
389C4—22no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—161no—
396C4—161no—
397C4—161no—
398C4#221712yes—
399C4#303511yescharacteristic
400C4#306711yescharacteristic
401C4C22×SD3211yescharacteristic
402C4C2×#50811yescharacteristic
403C2—22no—
404C2—42no—
405C2—321no—
406C2#1912511yescharacteristic
407C2C2×#221711yescharacteristic
408C2#3361711yescharacteristic
subgroupG/H
centerC2×C4#466
commutator, Frattini, cosocleC4×C8C24
socleC22C2×#466
prime radical#19125C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33736C2semidirect
2. C16×#41C2semidirect
3. #19126C2semidirect
5. #12478C2semidirect
6. #12476C2semidirect
7. #12470C2semidirect
8. #12468C2semidirect
9. #12459C2semidirect
10. #12453C2semidirect
11. #12441C2semidirect
12. #12432C2semidirect
13. C2×#1372C2semidirect
14. #11949C2semidirect
15. #11947C2semidirect
249. #41SD32semidirect
326. C16#135semidirect

Maximal subgroups

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

subgroupindex
#337362
C16×#412
#191262
#191252
#124782
#124762
#124702
#124682
#124592
#124532
#124412
#124322
C2×#13722
#119492
#119472