Finite Group Databasebeta

← #504334

Subgroups — #504334

Order 512.

531 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28827C211yesmaximal, characteristic, complemented
2#23186C211yesmaximal, characteristic
3#13531C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic
7#13254C211yesmaximal, characteristic
8#13205C211yesmaximal, characteristic
9#13177C211yesmaximal, characteristic, complemented
10#13160C211yesmaximal, characteristic, complemented
11#12936C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic
13#12812C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic
15#12674C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2552—21no—
20#2367—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#346C2211yescharacteristic
37#2083—21no—
38#2083—21no—
39#1983—21no—
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1947—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1880—21no—
46#1877—21no—
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×#125—41no—
63C2×#124—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#495—22no—
71#495Q814yes—
72C4×Q16—22no—
73C4×Q16—41no—
74C4×D8—22no—
75C4×D8—41no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×C4⋊C8D411yescharacteristic
86C2×#103—21no—
87C2×#103—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#398—41no—
94#398—41no—
95#391—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#389—41no—
100#388—41no—
101#388—41no—
102#385—41no—
103#384—21no—
104#384—21no—
105#384—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D411yescharacteristic
109C4×C22⋊C4—41no—
110#367—22no—
111#367—41no—
112#367—41no—
113#366—22no—
114C4⋊C16—22no—
115C4⋊C16—41no—
116C4⋊C16—41no—
117#358—22no—
118#358—41no—
119#358—41no—
120#358—41no—
121#357—22no—
122#357—41no—
123#357—41no—
124#357—41no—
125#346—22no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
132C22×Q8—21no—
133C22×D4—21no—
134C23×C4—41no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16C2×Q812yes—
141C2×SD16—24no—
142C2×SD16—81no—
143C2×SD16—81no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8C2×Q812yes—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2411yescharacteristic
154#125—84no—
155#124—24no—
156#124—42no—
157C4×Q8—42no—
158C4×Q8—81no—
159C4×D4—42no—
160C4×D4—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—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—41no—
176C2×C4⋊C4D811yescharacteristic
177C2×C4⋊C4D811yescharacteristic
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
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—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16#4112yes—
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—21no—
212#107—81no—
213#107—81no—
214#107—81no—
215#107#4112yes—
216C4⋊C8—22no—
217#103—42no—
218#103—81no—
219#102—42no—
220#102—81no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8C2×Q812yes—
226C4×C8#4112yes—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95D811yescharacteristic
234#95D811yescharacteristic
235C24—41no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—161no—
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C22×D411yescharacteristic
265Q16—22no—
266Q16—22no—
267Q16—41no—
268Q16—41no—
269SD16—84no—
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×C8C22⋊Q812yes—
283C2×C8C22×Q811yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C8H(5,𝔽2)11yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
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—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—161no—
310C4⋊C4—161no—
311C4⋊C4—161no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—161no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—82no—
322C42—82no—
323C42—161no—
324C42C22⋊Q812yes—
325C42#13712yes—
326C16—22no—
327C16—41no—
328C16—41no—
329C16—41no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23C22×D811yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342D4—42no—
343D4—42no—
344D4—81no—
345D4—81no—
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—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—42no—
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—164no—
382C2×C4#47412yes—
383C2×C4#47412yes—
384C2×C4#50912yes—
385C2×C4#51012yes—
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4#54611yescharacteristic
388C2×C4#55011yescharacteristic
389C2×C4C2×#13711yescharacteristic
390C2×C4#57611yescharacteristic
391C2×C4#57711yescharacteristic
392C8—22no—
393C8—41no—
394C8—41no—
395C8#52911yescharacteristic
396C8#54011yescharacteristic
397C8#55411yescharacteristic
398C8#55411yescharacteristic
399C22—82no—
400C22—82no—
401C22—82no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22C2×#47411yescharacteristic
407C22#329411yescharacteristic
408C22#339111yescharacteristic
409C22C2×#50911yescharacteristic
410C22C2×#51011yescharacteristic
411C22#341611yescharacteristic
412C22#341611yescharacteristic
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4—82no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4#245411yescharacteristic
427C4#308111yescharacteristic
428C4#333311yescharacteristic
429C4#340011yescharacteristic
430C2—162no—
431C2#2652611yescharacteristic
432C2#3362911yescharacteristic
433C2#3363411yescharacteristic
434C2#3387311yescharacteristic
435C2#3388311yescharacteristic
436C2#3396411yescharacteristic
437C2#3396711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28827C2semidirect
3. #13531C2semidirect
5. #13477C2semidirect
9. #13177C2semidirect
10. #13160C2semidirect
11. #12936C2semidirect
13. #12812C2semidirect
15. #12674C2semidirect

Maximal subgroups

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

subgroupindex
#288272
#231862
#135312
#135272
#134772
#134762
#132542
#132052
#131772
#131602
#129362
#129352
#128122
#128112
#126742