Finite Group Databasebeta

← #510153

Subgroups — #510153

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26065C211yesmaximal, characteristic, complemented
2#25034C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13514C211yesmaximal, characteristic, complemented
5#13505C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13380C211yesmaximal, characteristic
8#13363C211yesmaximal, characteristic, complemented
9#13282C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic, complemented
12#12991C211yesmaximal, characteristic
13#12935C211yesmaximal, characteristic, complemented
14#12866C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C22×#107C2211yescharacteristic
20#2440—21no—
21#2422—21no—
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#2083—21no—
33#2071—21no—
34#2057—21no—
35#2033—21no—
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1862—21no—
41#1852—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
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
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#123—41no—
63C2×C22⋊Q8—41no—
64C2×#121—21no—
65C2×C4×D4—21no—
66C22×C4⋊C4—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—22no—
72#500—41no—
73#466—22no—
74#466—41no—
75C4×D8—22no—
76C4×D8—41no—
77C2×#107—22no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—41no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#398—41no—
97#397—41no—
98#395—41no—
99#394—41no—
100#390—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#385—41no—
105#385—41no—
106#384—21no—
107#384—21no—
108#380—21no—
109#380—21no—
110C4×C22⋊C4D411yescharacteristic
111#366—41no—
112#366—41no—
113C4⋊C16—41no—
114#357—22no—
115#357—22no—
116#357—41no—
117#357—41no—
118C22⋊C16—41no—
119C22⋊C16—41no—
120C4×C16—41no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—41no—
129C22×D4—21no—
130C22×D4—41no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8#4112yes—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2411yescharacteristic
150C4⋊Q8—42no—
151C4⋊Q8—81no—
152#123—84no—
153C22⋊Q8—84no—
154#121—42no—
155#121—81no—
156C4×D4—42no—
157C4×D4—81no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—81no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4D811yescharacteristic
183C2×C22⋊C4D811yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—42no—
212#107—42no—
213#107—81no—
214#107—81no—
215#107#4112yes—
216#102—42no—
217#102—42no—
218#102—42no—
219C22⋊C8—22no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95D811yescharacteristic
230#95D811yescharacteristic
231C24—41no—
232C24C2×D811yescharacteristic
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—161no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—82no—
243C2×D4—82no—
244C2×D4—161no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—161no—
263C22×C4—161no—
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C22×D411yescharacteristic
270D8—22no—
271D8—22no—
272D8—41no—
273D8—41no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8#13712yes—
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8H(5,𝔽2)11yescharacteristic
290C2×C8H(5,𝔽2)11yescharacteristic
291C2×C82−1+411yescharacteristic
292C2×C82−1+411yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C22⋊C4—22no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4D1614yes—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C16—41no—
329C16—41no—
330C16—41no—
331C16—41no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—42no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23C2×D1612yes—
346C23C22×D811yescharacteristic
347Q8—164no—
348D4—42no—
349D4—42no—
350D4—81no—
351D4—81no—
352D4—164no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—164no—
391C2×C4—164no—
392C2×C4C2×D1612yes—
393C2×C4C2×D1612yes—
394C2×C4#51012yes—
395C2×C4#53611yescharacteristic
396C2×C4#54611yescharacteristic
397C2×C4#54611yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57711yescharacteristic
400C2×C4#57711yescharacteristic
401C8—22no—
402C8—41no—
403C8—41no—
404C8#53911yescharacteristic
405C8#54211yescharacteristic
406C8#55211yescharacteristic
407C8#56211yescharacteristic
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22#322611yescharacteristic
423C22#329411yescharacteristic
424C22#329411yescharacteristic
425C22C22×D1611yescharacteristic
426C22C2×#51011yescharacteristic
427C22#341611yescharacteristic
428C22#341611yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—82no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4#268711yescharacteristic
442C4#324411yescharacteristic
443C4#332011yescharacteristic
444C4#333111yescharacteristic
445C2—42no—
446C2—162no—
447C2#3135011yescharacteristic
448C2#3379211yescharacteristic
449C2#3381011yescharacteristic
450C2#3386111yescharacteristic
451C2#3386111yescharacteristic
452C2#3388711yescharacteristic
453C2#3389811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
1. #26065C2semidirect
3. #13537C2semidirect
4. #13514C2semidirect
5. #13505C2semidirect
6. #13492C2semidirect
8. #13363C2semidirect
9. #13282C2semidirect
10. #13147C2semidirect
11. #13120C2semidirect
13. #12935C2semidirect
14. #12866C2semidirect
15. #12811C2semidirect

Maximal subgroups

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

subgroupindex
#260652
#250342
#135372
#135142
#135052
#134922
#133802
#133632
#132822
#131472
#131202
#129912
#129352
#128662
#128112