Finite Group Databasebeta

← #510164

Subgroups — #510164

Order 512.

499 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26066C211yesmaximal, characteristic, complemented
2#25032C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic
8#13365C211yesmaximal, characteristic, complemented
9#13281C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic, complemented
12#12992C211yesmaximal, characteristic
13#12935C211yesmaximal, characteristic, complemented
14#12868C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#427C2211yescharacteristic
20#2441—21no—
21#2420—21no—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#2084—21no—
33#2073—21no—
34#2059—21no—
35#2034—21no—
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1864—21no—
41#1854—21no—
42#1846C2211yescharacteristic
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C8×C22⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#125—21no—
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×#117—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#499—22no—
70#499—41no—
71#466—22no—
72#466—41no—
73C4×D8—22no—
74C4×D8—41no—
75#427—22no—
76#427—41no—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×C22⋊C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#399—41no—
95#397—41no—
96#396—41no—
97#394—41no—
98#390—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—41no—
103#387—41no—
104#386—41no—
105#384—21no—
106#382—21no—
107#381—41no—
108#380—21no—
109#380—21no—
110C4×C22⋊C4D411yescharacteristic
111#367—41no—
112#366—41no—
113C4⋊C16—41no—
114#357—22no—
115#357—22no—
116#357—41no—
117#357—41no—
118C22⋊C16—41no—
119C22⋊C16—41no—
120C4×C16—41no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×D4—21no—
129C22×D4—41no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8#4112yes—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148#125—42no—
149#125—81no—
150#121—42no—
151#121—81no—
152C4×D4—42no—
153C4×D4—81no—
154#117—42no—
155#117—81no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4D811yescharacteristic
178C2×C22⋊C4D811yescharacteristic
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—81no—
208#107#4112yes—
209#106—42no—
210#106—42no—
211#106—81no—
212#102—42no—
213#102—42no—
214#102—42no—
215C22⋊C8—22no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95D811yescharacteristic
228#95D811yescharacteristic
229C24—41no—
230C24C2×D811yescharacteristic
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—161no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—161no—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C22×D411yescharacteristic
262D8—22no—
263D8—22no—
264D8—41no—
265D8—41no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8#13712yes—
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C2×C8H(5,𝔽2)11yescharacteristic
283C2×C82−1+411yescharacteristic
284C2×C82−1+411yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C22⋊C4—22no—
307C22⋊C4—41no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—161no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C22⋊C4—161no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C16—41no—
321C16—41no—
322C16—41no—
323C16—41no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23C22×D811yescharacteristic
339D4—42no—
340D4—42no—
341D4—81no—
342D4—81no—
343D4—164no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—164no—
384C2×C4#50812yes—
385C2×C4#53611yescharacteristic
386C2×C4#54611yescharacteristic
387C2×C4#54611yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4#57711yescharacteristic
390C2×C4#57711yescharacteristic
391C8—22no—
392C8—41no—
393C8—41no—
394C8#53911yescharacteristic
395C8#54211yescharacteristic
396C8#55211yescharacteristic
397C8#56211yescharacteristic
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22#322611yescharacteristic
413C22#329411yescharacteristic
414C22#329411yescharacteristic
415C22C2×#50811yescharacteristic
416C22#341411yescharacteristic
417C22#341511yescharacteristic
418C22#341611yescharacteristic
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#268711yescharacteristic
432C4#324411yescharacteristic
433C4#332011yescharacteristic
434C4#333111yescharacteristic
435C2—81no—
436C2—162no—
437C2#3135011yescharacteristic
438C2#3379911yescharacteristic
439C2#3380711yescharacteristic
440C2#3386411yescharacteristic
441C2#3387011yescharacteristic
442C2#3387611yescharacteristic
443C2#3389011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26066C2semidirect
3. #13538C2semidirect
4. #13516C2semidirect
5. #13507C2semidirect
6. #13492C2semidirect
8. #13365C2semidirect
9. #13281C2semidirect
10. #13147C2semidirect
11. #13120C2semidirect
13. #12935C2semidirect
14. #12868C2semidirect
15. #12811C2semidirect

Maximal subgroups

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

subgroupindex
#260662
#250322
#135382
#135162
#135072
#134922
#133812
#133652
#132812
#131472
#131202
#129922
#129352
#128682
#128112