Finite Group Databasebeta

← #511335

Subgroups — #511335

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28014C211yesmaximal, characteristic
2#27995C211yesmaximal, characteristic, complemented
3#13537C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic
6#13476C211yesmaximal, characteristic
7#13476C211yesmaximal, characteristic
8#13476C211yesmaximal, characteristic
9#13177C211yesmaximal, characteristic, complemented
10#13177C211yesmaximal, characteristic, complemented
11#13166C211yesmaximal, characteristic, complemented
12#13159C211yesmaximal, characteristic, complemented
13#13147C211yesmaximal, characteristic
14#13131C211yesmaximal, characteristic
15#13111C211yesmaximal, characteristic, complemented
16#2512—21no—
17#2507—21no—
18C2×#367C2211yescharacteristic
19C2×#367C2211yescharacteristic
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C4×C16C2211yescharacteristic
30#2100—21no—
31#2087—21no—
32#2071—21no—
33#2067—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1925—21no—
43#1921—21no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×#126—41no—
61C22×C16C2311yescharacteristic
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C2×#107—21no—
66C2×#107—21no—
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×#102—21no—
76C2×#102—21no—
77C2×#102—21no—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#398—41no—
82#398—41no—
83#397—41no—
84#396—41no—
85#390—21no—
86#390—21no—
87#390—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—41no—
94#388—41no—
95#387—41no—
96#384—21no—
97#384—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D411yescharacteristic
101#367—22no—
102#367—22no—
103#367—41no—
104#367—41no—
105C4⋊C16—22no—
106C4⋊C16—22no—
107C4⋊C16—41no—
108#357—22no—
109#357—22no—
110#357—41no—
111#357—41no—
112C4×C16—41no—
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—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
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×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C16—21no—
164C2×C16—21no—
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#4112yes—
175#107—21no—
176#107—21no—
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—42no—
189#107—81no—
190#107#4112yes—
191C4⋊C8—41no—
192#102—42no—
193#102—81no—
194#102—81no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95D811yescharacteristic
209#95D811yescharacteristic
210C24—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C22×D411yescharacteristic
231D8—41no—
232D8—41no—
233D8—41no—
234D8—41no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8C2×#4111yescharacteristic
245C2×C8C2×#4111yescharacteristic
246C2×C8H(5,𝔽2)11yescharacteristic
247C2×C82−1+411yescharacteristic
248C2×C82−1+411yescharacteristic
249C2×C82−1+411yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—41no—
252C4⋊C4—41no—
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—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—161no—
269C4⋊C4—161no—
270C4⋊C4—161no—
271C4⋊C4—161no—
272C4⋊C4—161no—
273C4⋊C4—161no—
274C4⋊C4SD3214yes—
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—161no—
289C16—22no—
290C16—41no—
291C16—41no—
292C16—41no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23C22×D811yescharacteristic
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
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—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
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×C4C2×SD3212yes—
342C2×C4C2×SD3212yes—
343C2×C4C2×SD3212yes—
344C2×C4#53711yescharacteristic
345C2×C4#54711yescharacteristic
346C2×C4#54811yescharacteristic
347C2×C4#57511yescharacteristic
348C2×C4#57711yescharacteristic
349C2×C4#57811yescharacteristic
350C8—41no—
351C8—41no—
352C8—41no—
353C8#54111yescharacteristic
354C8#54111yescharacteristic
355C8#55311yescharacteristic
356C8#56311yescharacteristic
357C22—161no—
358C22—161no—
359C22—161no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22#324511yescharacteristic
365C22#333411yescharacteristic
366C22#335811yescharacteristic
367C22C22×SD3211yescharacteristic
368C22#341411yescharacteristic
369C22#341511yescharacteristic
370C22#341611yescharacteristic
371C4—22no—
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#272511yescharacteristic
385C4#325811yescharacteristic
386C4#335111yescharacteristic
387C4#337711yescharacteristic
388C2—321no—
389C2#3247211yescharacteristic
390C2#3381211yescharacteristic
391C2#3382311yescharacteristic
392C2#3390211yescharacteristic
393C2#3392211yescharacteristic
394C2#3392911yescharacteristic
395C2#3394311yescharacteristic
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. #27995C2semidirect
3. #13537C2semidirect
4. #13516C2semidirect
9. #13177C2semidirect
10. #13177C2semidirect
11. #13166C2semidirect
12. #13159C2semidirect
15. #13111C2semidirect

Maximal subgroups

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

subgroupindex
#280142
#279952
#135372
#135162
#134802
#134762
#131772
#131662
#131592
#131472
#131312
#131112