Finite Group Databasebeta

← #497064

Subgroups — #497064

Order 512.

590 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2242C211yesmaximal, characteristic
2C4×#499C211yesmaximal, characteristic, complemented
3C4×#495C211yesmaximal, characteristic
4#13281C211yesmaximal, characteristic, complemented
5#13204C211yesmaximal, characteristic
6#13133C211yesmaximal, characteristic, complemented
7#13132C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic
9#13130C211yesmaximal, characteristic
10#12938C211yesmaximal, characteristic, complemented
11#12937C211yesmaximal, characteristic
12#12936C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic
14#12907C211yesmaximal, characteristic, complemented
15#12590C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×SD16—21no—
20C2×C4×D8C2211yescharacteristic
21C4×#125—21no—
22C4×#124—21no—
23#2242C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#1991—21no—
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1968—21no—
42#1958—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1941—21no—
46#1846C2211yescharacteristic
47#1846C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C4×#106—21no—
51C42×C8C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×SD16—21no—
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#499—22no—
72#499C2×C414yescomplemented
73#495—22no—
74#495C2×C414yes—
75C4×Q16—41no—
76C4×Q16—41no—
77C4×SD16—44no—
78C4×D8—41no—
79C4×D8—41no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—21no—
100#390—21no—
101#389—21no—
102#389—21no—
103#388—41no—
104#388—41no—
105#387—41no—
106#387—41no—
107#382—21no—
108#382—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—41no—
112C4×C4⋊C4—41no—
113C4×C22⋊C4—41no—
114C4×C22⋊C4—41no—
115C43D411yescharacteristic
116#367—41no—
117#367—41no—
118#366—41no—
119#366—41no—
120#358—41no—
121#358—41no—
122#358C2×C414yescomplemented
123#358C2×C414yescomplemented
124#357—41no—
125#357—41no—
126#357C2×C414yes—
127#357C2×C414yes—
128C4×C16—22no—
129C4×C16C2×C414yescomplemented
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C22×D4—21no—
136C23×C4—41no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16C22×C412yes—
143C2×SD16—24no—
144C2×SD16—42no—
145C2×SD16—42no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8C22×C412yes—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2411yescharacteristic
156#125—24no—
157#125—42no—
158#124—24no—
159#124—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—42no—
163C4×D4—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42D811yescharacteristic
184C2×C42D811yescharacteristic
185C2×C42D811yescharacteristic
186C2×C42D811yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
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×C16C22×C412yes—
201C2×C16C22×C412yes—
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#107C22×C412yes—
213#107C22×C412yes—
214#106—24no—
215#106—42no—
216#106—42no—
217#103—42no—
218#103—42no—
219#102—42no—
220#102—42no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—22no—
228C4×C8C22×C412yes—
229C4×C8#4112yes—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240C24—41no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C22×D411yescharacteristic
270Q16—22no—
271Q16—22no—
272Q16—41no—
273Q16—41no—
274SD16—48no—
275D8—22no—
276D8—22no—
277D8—41no—
278D8—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8C4×D412yes—
286C2×C8AGL(1,ℤ/8)12yes—
287C2×C8#13712yes—
288C2×C8C23×C411yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8H(5,𝔽2)11yescharacteristic
291C2×C8H(5,𝔽2)11yescharacteristic
292C2×C82−1+411yescharacteristic
293C2×C82−1+411yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—22no—
321C42—22no—
322C42—41no—
323C42—41no—
324C42—82no—
325C42—82no—
326C42—82no—
327C42C4×D412yes—
328C42#13512yes—
329C16—22no—
330C16—22no—
331C16—41no—
332C16—41no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23C22×D811yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345D4—42no—
346D4—42no—
347D4—81no—
348D4—81no—
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—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
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—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—164no—
385C2×C4C4×D812yes—
386C2×C4C4×D812yes—
387C2×C4#50812yes—
388C2×C4#50812yes—
389C2×C4C2×C4×D411yescharacteristic
390C2×C4#53511yescharacteristic
391C2×C4#53611yescharacteristic
392C2×C4C2×#13511yescharacteristic
393C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C8—41no—
396C8—41no—
397C8#44014yes—
398C8#52011yescharacteristic
399C8#52011yescharacteristic
400C8#54211yescharacteristic
401C8#54211yescharacteristic
402C22—82no—
403C22—82no—
404C22—82no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22C2×C4×D811yescharacteristic
410C22#320511yescharacteristic
411C22#322611yescharacteristic
412C22C2×#50811yescharacteristic
413C22C2×#50811yescharacteristic
414C22#341411yescharacteristic
415C22#341411yescharacteristic
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#232611yescharacteristic
430C4C2×#44011yescharacteristic
431C4#321011yescharacteristic
432C4#322811yescharacteristic
433C2—162no—
434C2#2222011yescharacteristic
435C2C4×#50811yescharacteristic
436C2#3345111yescharacteristic
437C2#3377711yescharacteristic
438C2#3377811yescharacteristic
439C2#3379811yescharacteristic
440C2#3379911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#499C2semidirect
4. #13281C2semidirect
6. #13133C2semidirect
7. #13132C2semidirect
10. #12938C2semidirect
12. #12936C2semidirect
14. #12907C2semidirect
15. #12590C2semidirect
23. #2242C4semidirect
72. #499C2×C4semidirect
122. #358C2×C4semidirect
123. #358C2×C4semidirect
129. C4×C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22422
C4×#4992
C4×#4952
#132812
#132042
#131332
#131322
#131312
#131302
#129382
#129372
#129362
#129352
#129072
#125902