Finite Group Databasebeta

← #501588

Subgroups — #501588

Order 512.

533 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×#2164C211yesmaximal, characteristic
3#33216C211yesmaximal, characteristic
4#21525C211yesmaximal, characteristic
5#13494C211yesmaximal, characteristic
6#13493C211yesmaximal, characteristic
7#13382C211yesmaximal, characteristic
8#13382C211yesmaximal, characteristic
9#13277C211yesmaximal, characteristic, complemented
10#13177C211yesmaximal, characteristic, complemented
11#13167C211yesmaximal, characteristic, complemented
12#13166C211yesmaximal, characteristic, complemented
13#13150C211yesmaximal, characteristic, complemented
14#13148C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2629—21no—
21#2323—21no—
22#2229—24no—
23#2164—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#1992—21no—
39#1969C2211yescharacteristic
40#1943—21no—
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1935—21no—
45#1894—21no—
46#1845C2211yescharacteristic
47#1809—21no—
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1763—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×#117D411yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#499—22no—
73#499—22no—
74#467—22no—
75#467—22no—
76C4×Q16—22no—
77C4×Q16—22no—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90#405—22no—
91#405—22no—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#398—41no—
97#398—41no—
98#396—41no—
99#396—41no—
100#391—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—41no—
105#388—41no—
106#388—41no—
107#387—41no—
108#384—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113#367—22no—
114#367—22no—
115#367—22no—
116#366—22no—
117C4⋊C16—22no—
118C4⋊C16D414yes—
119#358—22no—
120#358—22no—
121#358—22no—
122#358—22no—
123C22⋊C16—22no—
124C22⋊C16—22no—
125#346—22no—
126C4×C16—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C23×C4C2×D411yescharacteristic
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16#4112yes—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—41no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2411yescharacteristic
155#125—42no—
156#125—42no—
157C22⋊Q8—24no—
158C22⋊Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4D811yescharacteristic
180C2×C4⋊C4D811yescharacteristic
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4D811yescharacteristic
186C2×C22⋊C4D811yescharacteristic
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16C2×D412yes—
202C2×C16C2×D412yes—
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—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—42no—
217#107#4112yes—
218#106—42no—
219#103—24no—
220#103—42no—
221#103—42no—
222#97—22no—
223#97—22no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8C2×D412yes—
227C4×C8#4112yes—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24C2×D811yescharacteristic
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C22×D411yescharacteristic
264Q16—22no—
265Q16—22no—
266Q16—41no—
267Q16—41no—
268M16—44no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8#12112yes—
281C2×C8#13712yes—
282C2×C8C22×D411yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C82−1+411yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—41no—
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—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—161no—
309C4⋊C4—161no—
310C4⋊C4—161no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—44no—
316C22⋊C4—161no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C42—21no—
320C42—21no—
321C42—82no—
322C42—161no—
323C42#12112yes—
324C42#13512yes—
325C16—22no—
326C16—22no—
327C16—41no—
328C16—41no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23C22×D811yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—81no—
340Q8—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
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#46612yes—
383C2×C4#46612yes—
384C2×C4C2×#12111yescharacteristic
385C2×C4#52511yescharacteristic
386C2×C4#53611yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4#57511yescharacteristic
390C8—22no—
391C8—22no—
392C8—41no—
393C8#54711yescharacteristic
394C8#54811yescharacteristic
395C8#55211yescharacteristic
396C8#55611yescharacteristic
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22C2×#46611yescharacteristic
405C22#310011yescharacteristic
406C22#322611yescharacteristic
407C22#341611yescharacteristic
408C22#341611yescharacteristic
409C22#341611yescharacteristic
410C22#341711yescharacteristic
411C4—22no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—82no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#241511yescharacteristic
425C4#305611yescharacteristic
426C4#310711yescharacteristic
427C4#323311yescharacteristic
428C2—81no—
429C2#2470011yescharacteristic
430C2#3361111yescharacteristic
431C2#3361311yescharacteristic
432C2#3368011yescharacteristic
433C2#3368611yescharacteristic
434C2#3380811yescharacteristic
435C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2229C2semidirect
9. #13277C2semidirect
10. #13177C2semidirect
11. #13167C2semidirect
12. #13166C2semidirect
13. #13150C2semidirect
14. #13148C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×#21642
#332162
#215252
#134942
#134932
#133822
#132772
#131772
#131672
#131662
#131502
#131482
#128122