Finite Group Databasebeta

← #510412

Subgroups — #510412

Order 512.

478 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28827C211yesmaximal, characteristic, complemented
2#26253C211yesmaximal, characteristic
3#13530C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic
5#13505C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic
7#13474C211yesmaximal, characteristic
8#13408C211yesmaximal, characteristic
9#13205C211yesmaximal, characteristic
10#13168C211yesmaximal, characteristic, complemented
11#13133C211yesmaximal, characteristic, complemented
12#13131C211yesmaximal, characteristic
13#13105C211yesmaximal, characteristic, complemented
14#12938C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18#2552—21no—
19#2448—21no—
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#2083—21no—
35#2083—21no—
36#2055—21no—
37#2042—21no—
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1947—21no—
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1880—21no—
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×#125—41no—
63C2×#124—21no—
64C2×C4×Q8—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#495—22no—
70#495—41no—
71C4×Q16—22no—
72C4×Q16—41no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×#103—21no—
84C2×#103—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#399—41no—
91#398—41no—
92#398—41no—
93#397—41no—
94#391—21no—
95#390—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—41no—
101#388—41no—
102#388—41no—
103#384—21no—
104#384—21no—
105#384—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108#367—41no—
109#366—22no—
110#366—41no—
111#366—41no—
112C4⋊C16—41no—
113#358—22no—
114#358—22no—
115#358—41no—
116#358—41no—
117#357—22no—
118#357—41no—
119#357—41no—
120#357—41no—
121#346—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C22×D4—21no—
130C2×Q16—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16#4112yes—
136C2×SD16—42no—
137C2×SD16—42no—
138C2×SD16—81no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149#125—84no—
150#124—42no—
151#124—81no—
152C4×Q8—42no—
153C4×Q8—81no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
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×C4⋊C4D811yescharacteristic
171C2×C4⋊C4D811yescharacteristic
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
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—
194C2×C16#4112yes—
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—81no—
209#107—81no—
210C4⋊C8—22no—
211#103—42no—
212#103—81no—
213#102—81no—
214#102—81no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95D811yescharacteristic
228#95D811yescharacteristic
229C24—41no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C22×D411yescharacteristic
256Q16—22no—
257Q16—22no—
258Q16—41no—
259Q16—41no—
260SD16—84no—
261D8—41no—
262D8—41no—
263D8—41no—
264D8—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8#13712yes—
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8H(5,𝔽2)11yescharacteristic
277C2×C8H(5,𝔽2)11yescharacteristic
278C2×C82−1+411yescharacteristic
279C2×C82−1+411yescharacteristic
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
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—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—161no—
301C4⋊C4—161no—
302C4⋊C4—161no—
303C4⋊C4—161no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—161no—
309C22⋊C4—161no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—82no—
317C42—82no—
318C16—22no—
319C16—41no—
320C16—41no—
321C16—41no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23C22×D811yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
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—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
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#50912yes—
376C2×C4#51012yes—
377C2×C4#53611yescharacteristic
378C2×C4#54611yescharacteristic
379C2×C4#55011yescharacteristic
380C2×C4C2×#13711yescharacteristic
381C2×C4#57611yescharacteristic
382C2×C4#57711yescharacteristic
383C8—22no—
384C8—41no—
385C8—41no—
386C8#53911yescharacteristic
387C8#53911yescharacteristic
388C8#55211yescharacteristic
389C8#56211yescharacteristic
390C22—161no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22#322611yescharacteristic
398C22#329411yescharacteristic
399C22#339111yescharacteristic
400C22C2×#50911yescharacteristic
401C22C2×#51011yescharacteristic
402C22#341511yescharacteristic
403C22#341711yescharacteristic
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—82no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4#269411yescharacteristic
418C4#324111yescharacteristic
419C4#331411yescharacteristic
420C4#340011yescharacteristic
421C2—321no—
422C2#3157011yescharacteristic
423C2#3380511yescharacteristic
424C2#3380911yescharacteristic
425C2#3387511yescharacteristic
426C2#3390011yescharacteristic
427C2#3396511yescharacteristic
428C2#3396611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28827C2semidirect
3. #13530C2semidirect
5. #13505C2semidirect
10. #13168C2semidirect
11. #13133C2semidirect
13. #13105C2semidirect
14. #12938C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
#288272
#262532
#135302
#135262
#135052
#134802
#134742
#134082
#132052
#131682
#131332
#131312
#131052
#129382
#128122