Finite Group Databasebeta

← #510535

Subgroups — #510535

Order 512.

453 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27838C211yesmaximal, characteristic
2#26065C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic
7#13492C211yesmaximal, characteristic
8#13451C211yesmaximal, characteristic, complemented
9#13276C211yesmaximal, characteristic, complemented
10#13168C211yesmaximal, characteristic, complemented
11#13121C211yesmaximal, characteristic
12#13111C211yesmaximal, characteristic, complemented
13#12919C211yesmaximal, characteristic
14#12890C211yesmaximal, characteristic, complemented
15#12866C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C22×#107C2211yescharacteristic
18#2495—21no—
19#2440—21no—
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27C2×#346C2211yescharacteristic
28#2101—21no—
29#2083—21no—
30#2077—21no—
31#2057—21no—
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1920—21no—
37#1852—21no—
38#1846C2211yescharacteristic
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×C4⋊Q8—21no—
61C2×C22⋊Q8—41no—
62C22×C4⋊C4—21no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#500—22no—
68#500—41no—
69C2×#107—22no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—41no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×C22⋊C8D411yescharacteristic
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#400—41no—
87#398—41no—
88#397—41no—
89#395—41no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#388—41no—
95#385—41no—
96#384—21no—
97#383—41no—
98#382—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102C4×C4⋊C4—21no—
103C4×C22⋊C4D411yescharacteristic
104#367—22no—
105#366—22no—
106#366—41no—
107#366—41no—
108C4⋊C16—41no—
109C22⋊C16—22no—
110C22⋊C16—41no—
111#346—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
122C22×Q8—41no—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2411yescharacteristic
135C4⋊Q8—42no—
136C4⋊Q8—81no—
137C22⋊Q8—84no—
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—21no—
146C2×C4⋊C4—42no—
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×C4⋊C4—81no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4D811yescharacteristic
160C2×C22⋊C4D811yescharacteristic
161C2×C42—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
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—21no—
175C2×C16#4112yes—
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—21no—
189#107—21no—
190#107—21no—
191#107—42no—
192#107—42no—
193#107—81no—
194#107#4112yes—
195#106—42no—
196C22⋊C8—22no—
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#95—41no—
209#95—41no—
210#95—41no—
211#95D811yescharacteristic
212#95D811yescharacteristic
213C24C2×D811yescharacteristic
214C2×Q8—82no—
215C2×Q8—82no—
216C2×Q8—161no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—161no—
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C22×D411yescharacteristic
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8#13712yes—
253C2×C8C2×#4111yescharacteristic
254C2×C8C2×#4111yescharacteristic
255C2×C8H(5,𝔽2)11yescharacteristic
256C2×C82−1+411yescharacteristic
257C2×C82−1+411yescharacteristic
258C2×C82−1+411yescharacteristic
259C4⋊C4—22no—
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—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—161no—
281C4⋊C4—161no—
282C4⋊C4—161no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—161no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—82no—
293C16—22no—
294C16—41no—
295C16—41no—
296C16—41no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23#50912yes—
304C23C22×D811yescharacteristic
305Q8—164no—
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—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
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—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—164no—
349C2×C4#51012yes—
350C2×C4#53611yescharacteristic
351C2×C4#54611yescharacteristic
352C2×C4#54711yescharacteristic
353C2×C4C2×#13711yescharacteristic
354C2×C4#57711yescharacteristic
355C2×C4#57811yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C8#54111yescharacteristic
360C8#54211yescharacteristic
361C8#55211yescharacteristic
362C8#56311yescharacteristic
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22#322611yescharacteristic
371C22#329411yescharacteristic
372C22#333411yescharacteristic
373C22C2×#50911yescharacteristic
374C22C2×#51011yescharacteristic
375C22#341511yescharacteristic
376C22#341711yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—82no—
383C4—161no—
384C4—161no—
385C4—161no—
386C4—161no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4#269811yescharacteristic
391C4#324211yescharacteristic
392C4#332811yescharacteristic
393C4#334411yescharacteristic
394C2—42no—
395C2#3168611yescharacteristic
396C2#3380411yescharacteristic
397C2#3380911yescharacteristic
398C2#3387911yescharacteristic
399C2#3388511yescharacteristic
400C2#3391011yescharacteristic
401C2#3392411yescharacteristic
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. #13539C2semidirect
4. #13515C2semidirect
5. #13505C2semidirect
8. #13451C2semidirect
9. #13276C2semidirect
10. #13168C2semidirect
12. #13111C2semidirect
14. #12890C2semidirect

Maximal subgroups

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

subgroupindex
#278382
#260652
#135392
#135152
#135052
#134942
#134922
#134512
#132762
#131682
#131212
#131112
#129192
#128902
#128662