Finite Group Databasebeta

← #505471

Subgroups — #505471

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28014C211yesmaximal, characteristic, complemented
2#22807C211yesmaximal, characteristic, complemented
3#13527C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13448C211yesmaximal, characteristic
7#13239C211yesmaximal, characteristic
8#13177C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13131C211yesmaximal, 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#2512—21no—
19#2354—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2071—21no—
34#2067—21no—
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1942C2211yescharacteristic
41#1925—21no—
42#1925—21no—
43#1925—21no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1802—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×#126—41no—
62C2×C4×D4—21no—
63C2×#117—41no—
64C2×D16—22no—
65C2×D16—22no—
66C2×D16—41no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C4×D8—22no—
72C4×D8—22no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#397—41no—
92#396—41no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#389—21no—
99#388—41no—
100#388—41no—
101#388—41no—
102#384—21no—
103#384—21no—
104#380—41no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D411yescharacteristic
109#367—22no—
110#367—41no—
111#367—41no—
112#367—41no—
113#366—22no—
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
147#126—84no—
148C4×D4—24no—
149C4×D4—42no—
150#117—84no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4D811yescharacteristic
163C2×C4⋊C4D811yescharacteristic
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181D16—44no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16#4112yes—
193C2×C16#4112yes—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—81no—
206#107—81no—
207#107#4112yes—
208C4⋊C8—41no—
209#102—24no—
210#102—42no—
211#102—81no—
212#102—81no—
213#102—81no—
214#102—81no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95D811yescharacteristic
226#95D811yescharacteristic
227C24—41no—
228C24—41no—
229C2×D4—22no—
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—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—44no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C22×D411yescharacteristic
257D8—22no—
258D8—22no—
259D8—41no—
260D8—41no—
261D8—41no—
262D8—41no—
263D8—41no—
264D8—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8H(5,𝔽2)11yescharacteristic
279C2×C82−1+411yescharacteristic
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
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—41no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—161no—
298C4⋊C4—161no—
299C22⋊C4—44no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—161no—
307C22⋊C4—161no—
308C22⋊C4—161no—
309C22⋊C4—161no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—82no—
317C42—82no—
318C42—82no—
319C16—22no—
320C16—22no—
321C16—41no—
322C16—41no—
323C23—42no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23C22×D811yescharacteristic
338D4—42no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
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—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—88no—
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#50912yes—
382C2×C4#50912yes—
383C2×C4#52511yescharacteristic
384C2×C4#54711yescharacteristic
385C2×C4#55011yescharacteristic
386C2×C4#57511yescharacteristic
387C2×C4#57611yescharacteristic
388C2×C4#57811yescharacteristic
389C8—41no—
390C8—41no—
391C8—41no—
392C8#52911yescharacteristic
393C8#54211yescharacteristic
394C8#55111yescharacteristic
395C8#55211yescharacteristic
396C22—82no—
397C22—82no—
398C22—82no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22#310011yescharacteristic
411C22#333411yescharacteristic
412C22#339111yescharacteristic
413C22C2×#50911yescharacteristic
414C22C2×#50911yescharacteristic
415C22#341611yescharacteristic
416C22#341611yescharacteristic
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—82no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#247911yescharacteristic
430C4#312411yescharacteristic
431C4#334711yescharacteristic
432C4#340211yescharacteristic
433C2—162no—
434C2—321no—
435C2#2748611yescharacteristic
436C2#3366911yescharacteristic
437C2#3368511yescharacteristic
438C2#3391211yescharacteristic
439C2#3392111yescharacteristic
440C2#3396411yescharacteristic
441C2#3396411yescharacteristic
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. #28014C2semidirect
2. #22807C2semidirect
3. #13527C2semidirect
4. #13527C2semidirect
5. #13506C2semidirect
8. #13177C2semidirect
9. #13131C2semidirect
10. #13131C2semidirect
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
#280142
#228072
#135272
#135062
#134482
#132392
#131772
#131312
#131302
#131052
#130122
#129352
#126742