Finite Group Databasebeta

← #505443

Subgroups — #505443

Order 512.

432 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27838C211yesmaximal, characteristic
2#22613C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13507C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic
6#13494C211yesmaximal, characteristic
7#13178C211yesmaximal, characteristic, complemented
8#13168C211yesmaximal, characteristic, complemented
9#13121C211yesmaximal, characteristic
10#13110C211yesmaximal, characteristic, complemented
11#13110C211yesmaximal, characteristic, complemented
12#13110C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic
14#12921C211yesmaximal, characteristic
15#12674C211yesmaximal, characteristic, complemented
16C2×#427C2211yescharacteristic
17#2495—21no—
18#2346—21no—
19C2×#367C2211yescharacteristic
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28C2×#346C2211yescharacteristic
29#2101—21no—
30#2077—21no—
31#1942C2211yescharacteristic
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1920—21no—
36#1920—21no—
37#1920—21no—
38#1846C2211yescharacteristic
39#1825—21no—
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
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
59C23×C8C2311yescharacteristic
60C2×#117—21no—
61C22×C16C2311yescharacteristic
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65#427—22no—
66#427—22no—
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#106—21no—
76C2×#106—21no—
77C2×C22⋊C8D411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#400—41no—
81#397—41no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#388—41no—
87#388—41no—
88#388—41no—
89#383—41no—
90#382—21no—
91#382—41no—
92#380—21no—
93#380—21no—
94#380—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C22⋊C4D411yescharacteristic
99#376—41no—
100#367—22no—
101#367—22no—
102#367—22no—
103#367—22no—
104#366—22no—
105#366—22no—
106C4⋊C16—22no—
107C22⋊C16—22no—
108C22⋊C16—22no—
109#346—22no—
110#340—21no—
111#340—21no—
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#117—24no—
134#117—42no—
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—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4D811yescharacteristic
152C2×C22⋊C4D811yescharacteristic
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C16—21no—
160C2×C16—21no—
161C2×C16—21no—
162C2×C16—21no—
163C2×C16—21no—
164C2×C16—21no—
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16#4112yes—
170C2×C16#4112yes—
171#107—21no—
172#107—21no—
173#107—21no—
174#107—21no—
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#4112yes—
186#107#4112yes—
187#106—24no—
188#106—42no—
189C22⋊C8—41no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—21no—
193C4×C8—21no—
194#95—41no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95D811yescharacteristic
207#95D811yescharacteristic
208C24C2×D811yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4C2×D811yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C22×D411yescharacteristic
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8C2×#4111yescharacteristic
243C2×C8C2×#4111yescharacteristic
244C2×C8C2×#4111yescharacteristic
245C2×C8C2×#4111yescharacteristic
246C2×C82−1+411yescharacteristic
247C2×C82−1+411yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
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—161no—
265C4⋊C4—161no—
266C4⋊C4—161no—
267C4⋊C4—161no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—44no—
271C22⋊C4—82no—
272C22⋊C4—161no—
273C22⋊C4—161no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—44no—
279C42—82no—
280C42—82no—
281C42—82no—
282C16—22no—
283C16—22no—
284C16—41no—
285C16—41no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C22×D811yescharacteristic
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
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#52511yescharacteristic
338C2×C4#54711yescharacteristic
339C2×C4#54711yescharacteristic
340C2×C4#57511yescharacteristic
341C2×C4#57811yescharacteristic
342C2×C4#57811yescharacteristic
343C8—41no—
344C8—41no—
345C8—41no—
346C8#52911yescharacteristic
347C8#54111yescharacteristic
348C8#55211yescharacteristic
349C8#55611yescharacteristic
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22#310011yescharacteristic
358C22#333411yescharacteristic
359C22#333411yescharacteristic
360C22#341411yescharacteristic
361C22#341411yescharacteristic
362C22#341611yescharacteristic
363C22#341611yescharacteristic
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—82no—
370C4—82no—
371C4—161no—
372C4—161no—
373C4—161no—
374C4—161no—
375C4—161no—
376C4—161no—
377C4#247811yescharacteristic
378C4#312711yescharacteristic
379C4#334911yescharacteristic
380C4#335211yescharacteristic
381C2—81no—
382C2#2745411yescharacteristic
383C2#3367111yescharacteristic
384C2#3368511yescharacteristic
385C2#3391511yescharacteristic
386C2#3391511yescharacteristic
387C2#3391511yescharacteristic
388C2#3391811yescharacteristic
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. #13516C2semidirect
4. #13507C2semidirect
7. #13178C2semidirect
8. #13168C2semidirect
10. #13110C2semidirect
11. #13110C2semidirect
12. #13110C2semidirect
15. #12674C2semidirect

Maximal subgroups

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

subgroupindex
#278382
#226132
#135162
#135072
#134942
#131782
#131682
#131212
#131102
#129212
#126742