Finite Group Databasebeta

← #510521

Subgroups — #510521

Order 512.

470 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27839C211yesmaximal, characteristic
2#26065C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic
7#13494C211yesmaximal, characteristic
8#13452C211yesmaximal, characteristic, complemented
9#13276C211yesmaximal, characteristic, complemented
10#13166C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic
12#13111C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic
14#12888C211yesmaximal, characteristic, complemented
15#12867C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C22×#107C2211yescharacteristic
18#2496—21no—
19#2440—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27C2×C4×C16C2211yescharacteristic
28#2099—21no—
29#2083—21no—
30#2075—21no—
31#2057—21no—
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1918—21no—
37#1852—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×C4⋊Q8—21no—
61C2×#126—41no—
62C2×C22⋊Q8—41no—
63C22×C4⋊C4—21no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#500—22no—
69#500—41no—
70C2×#107—22no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—41no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×C22⋊C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#400—41no—
88#398—41no—
89#397—41no—
90#395—41no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#387—41no—
96#385—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—22no—
105#367—22no—
106#367—41no—
107#367—41no—
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
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—
137#126—84no—
138C22⋊Q8—84no—
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—21no—
147C2×C4⋊C4—42no—
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—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—81no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4D811yescharacteristic
163C2×C22⋊C4D811yescharacteristic
164C2×C42—41no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
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—21no—
177C2×C16—21no—
178C2×C16#4112yes—
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—21no—
193#107—21no—
194#107—42no—
195#107—42no—
196#107—42no—
197#107—81no—
198#107#4112yes—
199C22⋊C8—22no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95D811yescharacteristic
213#95D811yescharacteristic
214C24C2×D811yescharacteristic
215C2×Q8—82no—
216C2×Q8—82no—
217C2×Q8—161no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—161no—
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C22×D411yescharacteristic
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8#13712yes—
254C2×C8C2×#4111yescharacteristic
255C2×C8C2×#4111yescharacteristic
256C2×C8H(5,𝔽2)11yescharacteristic
257C2×C82−1+411yescharacteristic
258C2×C82−1+411yescharacteristic
259C2×C82−1+411yescharacteristic
260C4⋊C4—22no—
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—41no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—161no—
286C22⋊C4—22no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C22⋊C4SD3214yes—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—82no—
296C16—22no—
297C16—41no—
298C16—41no—
299C16—41no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23C2×SD3212yes—
307C23C22×D811yescharacteristic
308Q8—164no—
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—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
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—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—164no—
350C2×C4C2×SD3212yes—
351C2×C4C2×SD3212yes—
352C2×C4#50912yes—
353C2×C4#53611yescharacteristic
354C2×C4#54611yescharacteristic
355C2×C4#54711yescharacteristic
356C2×C4C2×#13711yescharacteristic
357C2×C4#57711yescharacteristic
358C2×C4#57811yescharacteristic
359C8—22no—
360C8—41no—
361C8—41no—
362C8#54111yescharacteristic
363C8#54211yescharacteristic
364C8#55211yescharacteristic
365C8#56311yescharacteristic
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#322611yescharacteristic
374C22#329411yescharacteristic
375C22#333411yescharacteristic
376C22C22×SD3211yescharacteristic
377C22C2×#50911yescharacteristic
378C22#341611yescharacteristic
379C22#341711yescharacteristic
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4—161no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4#269811yescharacteristic
394C4#324211yescharacteristic
395C4#332811yescharacteristic
396C4#334411yescharacteristic
397C2—42no—
398C2#3168611yescharacteristic
399C2#3379311yescharacteristic
400C2#3380811yescharacteristic
401C2#3386511yescharacteristic
402C2#3389111yescharacteristic
403C2#3390211yescharacteristic
404C2#3391011yescharacteristic
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. #13537C2semidirect
4. #13516C2semidirect
5. #13506C2semidirect
8. #13452C2semidirect
9. #13276C2semidirect
10. #13166C2semidirect
12. #13111C2semidirect
14. #12888C2semidirect

Maximal subgroups

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

subgroupindex
#278392
#260652
#135372
#135162
#135062
#134942
#134522
#132762
#131662
#131202
#131112
#129202
#128882
#128672