Finite Group Databasebeta

← #505477

Subgroups — #505477

Order 512.

492 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
1#28013C211yesmaximal, characteristic, complemented
2#22810C211yesmaximal, characteristic, complemented
3#13528C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13448C211yesmaximal, characteristic
7#13239C211yesmaximal, characteristic
8#13178C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#13130C211yesmaximal, characteristic, complemented
12#13105C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic
14#12935C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2513—21no—
19#2357—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2073—21no—
34#2065—21no—
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1942C2211yescharacteristic
41#1926—21no—
42#1926—21no—
43#1926—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1804—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×D16—22no—
63C2×D16—22no—
64C2×D16—41no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C4×D8—22no—
70C4×D8—22no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#106—21no—
78C2×#106—21no—
79C2×C4⋊C8D411yescharacteristic
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#397—41no—
90#397—41no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#389—21no—
97#388—41no—
98#388—41no—
99#388—41no—
100#388—41no—
101#382—21no—
102#382—21no—
103#381—41no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108C4×C22⋊C4—41no—
109#367—22no—
110#367—41no—
111#366—22no—
112#366—41no—
113#366—41no—
114C4⋊C16—22no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#357—41no—
120#357—41no—
121#346—22no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×D4—21no—
129C22×D4—21no—
130C23×C4—41no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8#4112yes—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147C4×D4—24no—
148C4×D4—42no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4D811yescharacteristic
157C2×C4⋊C4D811yescharacteristic
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175D16—44no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16#4112yes—
187C2×C16#4112yes—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107#4112yes—
200#106—81no—
201#106—81no—
202C4⋊C8—41no—
203#102—24no—
204#102—42no—
205#102—81no—
206#102—81no—
207#102—81no—
208#102—81no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95D811yescharacteristic
224#95D811yescharacteristic
225C24—41no—
226C24—41no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—44no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—161no—
247C22×C4—161no—
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C22×D411yescharacteristic
255D8—22no—
256D8—22no—
257D8—41no—
258D8—41no—
259D8—41no—
260D8—41no—
261D8—41no—
262D8—41no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8C2×#4111yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8H(5,𝔽2)11yescharacteristic
277C2×C82−1+411yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—161no—
293C22⋊C4—44no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—161no—
299C22⋊C4—161no—
300C22⋊C4—161no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—161no—
311C42—161no—
312C42—161no—
313C16—22no—
314C16—22no—
315C16—41no—
316C16—41no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23C22×D811yescharacteristic
332D4—42no—
333D4—42no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
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—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
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—164no—
377C2×C4#52511yescharacteristic
378C2×C4#54711yescharacteristic
379C2×C4#55011yescharacteristic
380C2×C4#57511yescharacteristic
381C2×C4#57611yescharacteristic
382C2×C4#57811yescharacteristic
383C8—41no—
384C8—41no—
385C8—41no—
386C8#52911yescharacteristic
387C8#54211yescharacteristic
388C8#55111yescharacteristic
389C8#55211yescharacteristic
390C22—82no—
391C22—82no—
392C22—82no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22#310011yescharacteristic
405C22#333411yescharacteristic
406C22#339111yescharacteristic
407C22#341411yescharacteristic
408C22#341411yescharacteristic
409C22#341511yescharacteristic
410C22#341511yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4#247911yescharacteristic
424C4#312411yescharacteristic
425C4#334711yescharacteristic
426C4#340211yescharacteristic
427C2—162no—
428C2—321no—
429C2#2748611yescharacteristic
430C2#3367111yescharacteristic
431C2#3368211yescharacteristic
432C2#3391411yescharacteristic
433C2#3392211yescharacteristic
434C2#3397011yescharacteristic
435C2#3397011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
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. #28013C2semidirect
2. #22810C2semidirect
3. #13528C2semidirect
4. #13528C2semidirect
5. #13507C2semidirect
8. #13178C2semidirect
9. #13130C2semidirect
10. #13130C2semidirect
11. #13130C2semidirect
12. #13105C2semidirect
14. #12935C2semidirect
15. #12674C2semidirect

Maximal subgroups

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

subgroupindex
#280132
#228102
#135282
#135072
#134482
#132392
#131782
#131302
#131052
#130122
#129352
#126742