Finite Group Databasebeta

← #505053

Subgroups — #505053

Order 512.

457 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28014C211yesmaximal, characteristic
2#23160C211yesmaximal, characteristic, complemented
3#13538C211yesmaximal, characteristic, complemented
4#13476C211yesmaximal, characteristic
5#13453C211yesmaximal, characteristic, complemented
6#13365C211yesmaximal, characteristic
7#13177C211yesmaximal, characteristic, complemented
8#13147C211yesmaximal, characteristic
9#13131C211yesmaximal, characteristic
10#13131C211yesmaximal, characteristic
11#13130C211yesmaximal, characteristic
12#13111C211yesmaximal, characteristic, complemented
13#13109C211yesmaximal, characteristic, complemented
14#13104C211yesmaximal, characteristic, complemented
15C4×#367C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17#2512—21no—
18#2370—21no—
19C2×#367C2211yescharacteristic
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×C4×C16C2211yescharacteristic
29#2071—21no—
30#2067—21no—
31#1989—21no—
32#1970C2211yescharacteristic
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1925—21no—
39#1920—21no—
40#1919—21no—
41#1845C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×#126—41no—
61C2×#125—21no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66#499—22no—
67#499—22no—
68C2×#107—21no—
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#106—21no—
77C2×C4⋊C8D411yescharacteristic
78C2×#102—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#397—41no—
85#396—41no—
86#390—21no—
87#390—21no—
88#390—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—41no—
93#388—41no—
94#387—41no—
95#387—41no—
96#384—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—41no—
102C4×C4⋊C4D411yescharacteristic
103#367—22no—
104#367—22no—
105#367—22no—
106#366—22no—
107C4⋊C16—22no—
108#357—22no—
109#357—22no—
110#357—22no—
111#357—22no—
112C4×C16—22no—
113#340—21no—
114#340—21no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#336D411yescharacteristic
121#336D411yescharacteristic
122C22×D4—21no—
123C2×D8—21no—
124C2×D8—21no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2411yescharacteristic
133#126—84no—
134#125—24no—
135#125—42no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4D811yescharacteristic
151C2×C4⋊C4D811yescharacteristic
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16#4112yes—
177#107—21no—
178#107—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—81no—
189#107#4112yes—
190#107#4112yes—
191#107#4112yes—
192#106—24no—
193C4⋊C8—41no—
194#102—42no—
195#102—81no—
196#102—81no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95D811yescharacteristic
211#95D811yescharacteristic
212C24—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C22×D411yescharacteristic
233D8—41no—
234D8—41no—
235D8—41no—
236D8—41no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8C2×#4111yescharacteristic
247C2×C8C2×#4111yescharacteristic
248C2×C8C2×#4111yescharacteristic
249C2×C8C2×#4111yescharacteristic
250C2×C82−1+411yescharacteristic
251C2×C82−1+411yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—161no—
274C4⋊C4—161no—
275C22⋊C4—82no—
276C22⋊C4—82no—
277C22⋊C4—82no—
278C22⋊C4—82no—
279C22⋊C4—161no—
280C22⋊C4—161no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—82no—
288C42—82no—
289C42—161no—
290C42—161no—
291C16—22no—
292C16—41no—
293C16—41no—
294C16—41no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23C22×D811yescharacteristic
303D4—81no—
304D4—81no—
305D4—81no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4#50812yes—
345C2×C4#50912yes—
346C2×C4#52511yescharacteristic
347C2×C4#54711yescharacteristic
348C2×C4#54811yescharacteristic
349C2×C4#57511yescharacteristic
350C2×C4#57711yescharacteristic
351C2×C4#57811yescharacteristic
352C8—41no—
353C8—41no—
354C8—41no—
355C8#52911yescharacteristic
356C8#54111yescharacteristic
357C8#55611yescharacteristic
358C8#55611yescharacteristic
359C22—161no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—161no—
365C22—161no—
366C22#310011yescharacteristic
367C22#333411yescharacteristic
368C22#335811yescharacteristic
369C22C2×#50811yescharacteristic
370C22C2×#50911yescharacteristic
371C22#341611yescharacteristic
372C22#341611yescharacteristic
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—82no—
380C4—82no—
381C4—82no—
382C4—161no—
383C4—161no—
384C4—161no—
385C4—161no—
386C4#246911yescharacteristic
387C4#312111yescharacteristic
388C4#335011yescharacteristic
389C4#337711yescharacteristic
390C2—321no—
391C2#2712611yescharacteristic
392C2#3366611yescharacteristic
393C2#3368511yescharacteristic
394C2#3390411yescharacteristic
395C2#3391211yescharacteristic
396C2#3392711yescharacteristic
397C2#3393511yescharacteristic
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
2. #23160C2semidirect
3. #13538C2semidirect
5. #13453C2semidirect
7. #13177C2semidirect
12. #13111C2semidirect
13. #13109C2semidirect
14. #13104C2semidirect
15. C4×#367C2semidirect

Maximal subgroups

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

subgroupindex
#280142
#231602
#135382
#134762
#134532
#133652
#131772
#131472
#131312
#131302
#131112
#131092
#131042
C4×#3672