Finite Group Databasebeta

← #498399

Subgroups — #498399

Order 512.

526 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic
2#23160C211yesmaximal, characteristic, complemented
3#22810C211yesmaximal, characteristic
4#13281C211yesmaximal, characteristic, complemented
5#13178C211yesmaximal, characteristic, complemented
6#13147C211yesmaximal, characteristic
7#13147C211yesmaximal, characteristic
8#13131C211yesmaximal, characteristic
9#13130C211yesmaximal, characteristic
10#13112C211yesmaximal, characteristic, complemented
11#13110C211yesmaximal, characteristic, complemented
12C4×#367C211yesmaximal, characteristic, complemented
13C4×#366C211yesmaximal, characteristic, complemented
14C16×C4⋊C4C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2370—21no—
19#2357—21no—
20#2228C418yescomplemented
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×C4×C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1989—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1926—21no—
39#1926—21no—
40#1920—21no—
41#1919—21no—
42#1846C2211yescharacteristic
43#1804—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—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#499—22no—
68#499C2×C414yes—
69C4×D8—22no—
70C4×D8C2×C414yes—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#106—21no—
79C2×#106—21no—
80C2×C4⋊C8D411yescharacteristic
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#390—21no—
88#390—21no—
89#389—21no—
90#389—21no—
91#389—41no—
92#388—41no—
93#388—41no—
94#387—41no—
95#387—41no—
96#382—21no—
97#381—41no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—41no—
104C4×C4⋊C4D411yescharacteristic
105C4×C22⋊C4—41no—
106#367—22no—
107#367C2×C414yes—
108#366—22no—
109#366C2×C414yes—
110C4⋊C16C2×C414yes—
111#357—22no—
112#357—22no—
113#357C2×C414yes—
114#357C2×C414yes—
115C4×C16—22no—
116C4×C16—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—
143C2×C4⋊C4—21no—
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—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4D811yescharacteristic
155C2×C4⋊C4D811yescharacteristic
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16C22×C412yes—
179C2×C16C22×C412yes—
180C2×C16#4112yes—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107C22×C412yes—
192#107C22×C412yes—
193#107C22×C412yes—
194#107#4112yes—
195#106—24no—
196#106—81no—
197C4⋊C8—22no—
198#102—24no—
199#102—81no—
200#102—81no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8C22×C412yes—
206C4×C8#4112yes—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95D811yescharacteristic
217#95D811yescharacteristic
218C24—41no—
219C2×D4—22no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—161no—
234C22×C4—161no—
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C22×D411yescharacteristic
242D8—22no—
243D8—22no—
244D8—41no—
245D8—41no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8C4×D412yes—
255C2×C8C23×C411yescharacteristic
256C2×C8C2×#4111yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C82−1+411yescharacteristic
260C2×C82−1+411yescharacteristic
261C4⋊C4—22no—
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—44no—
272C4⋊C4—44no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—161no—
281C22⋊C4—44no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—161no—
285C22⋊C4—161no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—82no—
291C42—161no—
292C42—161no—
293C42—161no—
294C42—161no—
295C42C4×D412yes—
296C42#13512yes—
297C16—22no—
298C16—22no—
299C16—22no—
300C16—41no—
301C23—42no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23C22×D811yescharacteristic
309D4—42no—
310D4—42no—
311D4—81no—
312D4—81no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
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—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
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—164no—
349C2×C4C4×D812yes—
350C2×C4C4×D812yes—
351C2×C4#50812yes—
352C2×C4#50812yes—
353C2×C4C2×C4×D411yescharacteristic
354C2×C4#54711yescharacteristic
355C2×C4#54811yescharacteristic
356C2×C4C2×#13511yescharacteristic
357C2×C4#57711yescharacteristic
358C2×C4#57811yescharacteristic
359C8—22no—
360C8—41no—
361C8—41no—
362C8C4×#4111yescharacteristic
363C8#51911yescharacteristic
364C8#55211yescharacteristic
365C8#55611yescharacteristic
366C22—82no—
367C22—82no—
368C22—82no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22C2×C4×D811yescharacteristic
374C22#333411yescharacteristic
375C22#335811yescharacteristic
376C22C2×#50811yescharacteristic
377C22C2×#50811yescharacteristic
378C22#341511yescharacteristic
379C22#341611yescharacteristic
380C4—22no—
381C4—22no—
382C4—22no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4#236211yescharacteristic
394C4#294711yescharacteristic
395C4#334011yescharacteristic
396C4#336411yescharacteristic
397C2—162no—
398C2#2306011yescharacteristic
399C2C4×#50811yescharacteristic
400C2#3345411yescharacteristic
401C2#3390411yescharacteristic
402C2#3390611yescharacteristic
403C2#3392811yescharacteristic
404C2#3393211yescharacteristic
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. #13281C2semidirect
5. #13178C2semidirect
10. #13112C2semidirect
11. #13110C2semidirect
12. C4×#367C2semidirect
13. C4×#366C2semidirect
14. C16×C4⋊C4C2semidirect
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
#228102
#132812
#131782
#131472
#131312
#131302
#131122
#131102
C4×#3672
C4×#3662
C16×C4⋊C42
C4×#3572