Finite Group Databasebeta

← #498408

Subgroups — #498408

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic
2#23160C211yesmaximal, characteristic, complemented
3#22807C211yesmaximal, characteristic
4#13284C211yesmaximal, characteristic, complemented
5#13177C211yesmaximal, characteristic, complemented
6#13149C211yesmaximal, characteristic
7#13147C211yesmaximal, characteristic
8#13131C211yesmaximal, characteristic
9#13131C211yesmaximal, characteristic
10#13110C211yesmaximal, characteristic, complemented
11#13109C211yesmaximal, characteristic, complemented
12#12674C211yesmaximal, characteristic, complemented
13C4×#367C211yesmaximal, characteristic, complemented
14#12663C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2370—21no—
19#2354—21no—
20#2228C418yescomplemented
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#346C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1989—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1925—21no—
39#1925—21no—
40#1920—21no—
41#1919—21no—
42#1845C2211yescharacteristic
43#1802—21no—
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C2×#125—21no—
62C2×C4×D4—21no—
63C2×#117—41no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#499—22no—
69#499C2×C414yes—
70C4×D8—22no—
71C4×D8C2×C414yes—
72C2×#107—21no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#106—21no—
81C2×C4⋊C8D411yescharacteristic
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#390—21no—
89#390—21no—
90#389—21no—
91#389—21no—
92#389—41no—
93#388—41no—
94#388—41no—
95#387—41no—
96#387—41no—
97#384—21no—
98#380—41no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—41no—
105C4×C4⋊C4D411yescharacteristic
106#367—22no—
107#367—22no—
108#367C2×C414yes—
109#366C2×C414yes—
110C4⋊C16C2×C414yes—
111#357—22no—
112#357—22no—
113#357C2×C414yes—
114#357C2×C414yes—
115#346—22no—
116#346—22no—
117C4×C16—22no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#336D411yescharacteristic
126#336D411yescharacteristic
127C22×D4—21no—
128C23×C4—41no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8C22×C412yes—
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2411yescharacteristic
139#125—24no—
140#125—42no—
141C4×D4—24no—
142C4×D4—42no—
143#117—84no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4D811yescharacteristic
158C2×C4⋊C4D811yescharacteristic
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16C22×C412yes—
182C2×C16C22×C412yes—
183C2×C16#4112yes—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—81no—
195#107C22×C412yes—
196#107C22×C412yes—
197#107C22×C412yes—
198#107#4112yes—
199#106—24no—
200C4⋊C8—22no—
201#102—24no—
202#102—81no—
203#102—81no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8C22×C412yes—
209C4×C8#4112yes—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95D811yescharacteristic
218#95D811yescharacteristic
219C24—41no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C22×D411yescharacteristic
243D8—22no—
244D8—22no—
245D8—41no—
246D8—41no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8C4×D412yes—
256C2×C8C23×C411yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C82−1+411yescharacteristic
261C2×C82−1+411yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—161no—
283C4⋊C4—161no—
284C22⋊C4—44no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—161no—
288C22⋊C4—161no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—82no—
294C42—82no—
295C42—82no—
296C42—161no—
297C42—161no—
298C42C4×D412yes—
299C42#13512yes—
300C16—22no—
301C16—22no—
302C16—22no—
303C16—41no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23C22×D811yescharacteristic
312D4—42no—
313D4—42no—
314D4—81no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—88no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4C4×D812yes—
354C2×C4C4×D812yes—
355C2×C4#50912yes—
356C2×C4C2×C4×D411yescharacteristic
357C2×C4#54711yescharacteristic
358C2×C4#54811yescharacteristic
359C2×C4C2×#13511yescharacteristic
360C2×C4#57711yescharacteristic
361C2×C4#57811yescharacteristic
362C8—22no—
363C8—41no—
364C8—41no—
365C8C4×#4111yescharacteristic
366C8#51911yescharacteristic
367C8#55211yescharacteristic
368C8#55611yescharacteristic
369C22—82no—
370C22—82no—
371C22—82no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22C2×C4×D811yescharacteristic
377C22#333411yescharacteristic
378C22#335811yescharacteristic
379C22C2×#50911yescharacteristic
380C22#341411yescharacteristic
381C22#341611yescharacteristic
382C22#341611yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—82no—
390C4—82no—
391C4—82no—
392C4—82no—
393C4—161no—
394C4—161no—
395C4—161no—
396C4#236211yescharacteristic
397C4#294711yescharacteristic
398C4#334011yescharacteristic
399C4#336411yescharacteristic
400C2—162no—
401C2#2306011yescharacteristic
402C2C4×#50911yescharacteristic
403C2#3346011yescharacteristic
404C2#3391211yescharacteristic
405C2#3391511yescharacteristic
406C2#3393511yescharacteristic
407C2#3394311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×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
2. #23160C2semidirect
4. #13284C2semidirect
5. #13177C2semidirect
10. #13110C2semidirect
11. #13109C2semidirect
12. #12674C2semidirect
13. C4×#367C2semidirect
14. #12663C2semidirect
20. #2228C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
#231602
#228072
#132842
#131772
#131492
#131472
#131312
#131102
#131092
#126742
C4×#3672
#126632
#126372