Finite Group Databasebeta

← #510522

Subgroups — #510522

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27840C211yesmaximal, characteristic
2#26066C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic
7#13492C211yesmaximal, characteristic
8#13453C211yesmaximal, characteristic, complemented
9#13277C211yesmaximal, characteristic, complemented
10#13165C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic
12#13109C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic
14#12888C211yesmaximal, characteristic, complemented
15#12868C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×#427C2211yescharacteristic
18#2497—21no—
19#2441—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27C2×C4×C16C2211yescharacteristic
28#2100—21no—
29#2084—21no—
30#2076—21no—
31#2059—21no—
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1921—21no—
37#1854—21no—
38#1845C2211yescharacteristic
39#1845C2211yescharacteristic
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C8×C22⋊C4C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C23×C8C2311yescharacteristic
60C2×#125—21no—
61C2×#117—21no—
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66#499—22no—
67#499—41no—
68#427—22no—
69#427—41no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×C22⋊C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#400—41no—
86#399—41no—
87#396—41no—
88#396—41no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—41no—
94#388—41no—
95#387—41no—
96#387—41no—
97#384—21no—
98#384—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4D411yescharacteristic
104#367—41no—
105#367—41no—
106#366—22no—
107#366—22no—
108C4⋊C16—41no—
109C22⋊C16—22no—
110C22⋊C16—41no—
111C4×C16—22no—
112#340—21no—
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
122C23×C4C2×D411yescharacteristic
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2411yescharacteristic
133#125—42no—
134#125—81no—
135#117—42no—
136#117—81no—
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—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4D811yescharacteristic
160C2×C22⋊C4D811yescharacteristic
161C2×C42—41no—
162C2×C42—41no—
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—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—42no—
193#107—81no—
194#107#4112yes—
195#106—42no—
196#106—42no—
197C22⋊C8—22no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202#95—41no—
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
212C24C2×D811yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C22×D411yescharacteristic
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8#13712yes—
246C2×C8C2×#4111yescharacteristic
247C2×C8C2×#4111yescharacteristic
248C2×C8H(5,𝔽2)11yescharacteristic
249C2×C82−1+411yescharacteristic
250C2×C82−1+411yescharacteristic
251C2×C82−1+411yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
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—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—161no—
274C4⋊C4—161no—
275C4⋊C4—161no—
276C4⋊C4—161no—
277C4⋊C4—161no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—82no—
281C22⋊C4—161no—
282C22⋊C4—161no—
283C22⋊C4—161no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—82no—
289C42—161no—
290C16—22no—
291C16—41no—
292C16—41no—
293C16—41no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23C22×D811yescharacteristic
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
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—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
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#50812yes—
344C2×C4#51012yes—
345C2×C4#53611yescharacteristic
346C2×C4#54611yescharacteristic
347C2×C4#54711yescharacteristic
348C2×C4C2×#13711yescharacteristic
349C2×C4#57711yescharacteristic
350C2×C4#57811yescharacteristic
351C8—22no—
352C8—41no—
353C8—41no—
354C8#54111yescharacteristic
355C8#54211yescharacteristic
356C8#55211yescharacteristic
357C8#56311yescharacteristic
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22#322611yescharacteristic
366C22#329411yescharacteristic
367C22#333411yescharacteristic
368C22C2×#50811yescharacteristic
369C22C2×#51011yescharacteristic
370C22#341511yescharacteristic
371C22#341611yescharacteristic
372C4—22no—
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—82no—
378C4—161no—
379C4—161no—
380C4—161no—
381C4—161no—
382C4—161no—
383C4—161no—
384C4—161no—
385C4#269811yescharacteristic
386C4#324211yescharacteristic
387C4#332811yescharacteristic
388C4#334411yescharacteristic
389C2—81no—
390C2#3168611yescharacteristic
391C2#3379711yescharacteristic
392C2#3380711yescharacteristic
393C2#3387011yescharacteristic
394C2#3390011yescharacteristic
395C2#3390611yescharacteristic
396C2#3391111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
3. #13538C2semidirect
4. #13516C2semidirect
5. #13506C2semidirect
8. #13453C2semidirect
9. #13277C2semidirect
10. #13165C2semidirect
12. #13109C2semidirect
14. #12888C2semidirect

Maximal subgroups

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

subgroupindex
#278402
#260662
#135382
#135162
#135062
#134942
#134922
#134532
#132772
#131652
#131202
#131092
#129212
#128882
#128682