Finite Group Databasebeta

← #509527

Subgroups — #509527

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2255C211yesmaximal, characteristic, complemented
2C2×#2237C211yesmaximal, characteristic
3#27792C211yesmaximal, characteristic
4#26066C211yesmaximal, characteristic
5#13531C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic
7#13475C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13382C211yesmaximal, characteristic
10#13366C211yesmaximal, characteristic, complemented
11#13177C211yesmaximal, characteristic, complemented
12#13150C211yesmaximal, characteristic, complemented
13#13121C211yesmaximal, characteristic
14#13112C211yesmaximal, characteristic, complemented
15#12921C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#427C2211yescharacteristic
20#2492—21no—
21#2441—21no—
22#2255—24no—
23#2237—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#2084—21no—
39#2059—21no—
40#2036—21no—
41#2024—21no—
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1928—21no—
47#1854—21no—
48#1845C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×#117—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—22no—
72#500—41no—
73#499—22no—
74#499—41no—
75#467—22no—
76#467—41no—
77#427—22no—
78#427—41no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#400—41no—
97#399—41no—
98#396—41no—
99#393—41no—
100#391—21no—
101#391—21no—
102#391—41no—
103#389—21no—
104#389—21no—
105#389—41no—
106#388—41no—
107#387—41no—
108#384—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4D411yescharacteristic
113#367—22no—
114#367—22no—
115#367—22no—
116#367—41no—
117#366—22no—
118#366D414yes—
119C4⋊C16—22no—
120#358—22no—
121#358—22no—
122#358—22no—
123#358—41no—
124C22⋊C16—22no—
125C22⋊C16—22no—
126#346—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—41no—
136C23×C4C2×D411yescharacteristic
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2411yescharacteristic
153C4⋊Q8—42no—
154C4⋊Q8—81no—
155#125—42no—
156#125—81no—
157C22⋊Q8—42no—
158C22⋊Q8—81no—
159#117—42no—
160#117—81no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
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—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4D811yescharacteristic
181C2×C22⋊C4D811yescharacteristic
182C2×C42—41no—
183C2×C42—41no—
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×C16C2×D412yes—
197C2×C16#4112yes—
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—42no—
211#107—81no—
212#107C2×D412yes—
213#107C2×D412yes—
214#106—42no—
215#106—42no—
216#103—42no—
217#103—42no—
218#103—81no—
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#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95D811yescharacteristic
234#95D811yescharacteristic
235C24C2×D811yescharacteristic
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—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—41no—
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C22×D411yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—41no—
268Q16—41no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8#13712yes—
282C2×C8C22×D411yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C82−1+411yescharacteristic
286C2×C82−1+411yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
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—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—161no—
309C4⋊C4—161no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—161no—
315C22⋊C4—161no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—82no—
321C42—161no—
322C16—22no—
323C16—22no—
324C16—41no—
325C16—41no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23C22×D811yescharacteristic
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—164no—
339C2×C4—21no—
340C2×C4—21no—
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—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
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#51012yes—
382C2×C4#53611yescharacteristic
383C2×C4D4211yescharacteristic
384C2×C4#54711yescharacteristic
385C2×C4C2×#13711yescharacteristic
386C2×C4#57611yescharacteristic
387C2×C4#57811yescharacteristic
388C8—22no—
389C8—41no—
390C8—41no—
391C8#53611yescharacteristic
392C8#54211yescharacteristic
393C8#54811yescharacteristic
394C8#56311yescharacteristic
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#322611yescharacteristic
403C22D4×D811yescharacteristic
404C22#333411yescharacteristic
405C22C2×#51011yescharacteristic
406C22#341411yescharacteristic
407C22#341611yescharacteristic
408C22#341611yescharacteristic
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#267411yescharacteristic
423C4#324311yescharacteristic
424C4#327011yescharacteristic
425C4#335711yescharacteristic
426C2—81no—
427C2#3077311yescharacteristic
428C2#3380311yescharacteristic
429C2#3381011yescharacteristic
430C2#3383311yescharacteristic
431C2#3384411yescharacteristic
432C2#3391111yescharacteristic
433C2#3391711yescharacteristic
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
1. C2×#2255C2semidirect
5. #13531C2semidirect
7. #13475C2semidirect
8. #13452C2semidirect
10. #13366C2semidirect
11. #13177C2semidirect
12. #13150C2semidirect
14. #13112C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22552
C2×#22372
#277922
#260662
#135312
#134942
#134752
#134522
#133822
#133662
#131772
#131502
#131212
#131122
#129212