Finite Group Databasebeta

← #496865

Subgroups — #496865

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2237C211yesmaximal, characteristic
2C4×#467C211yesmaximal, characteristic
3#21151C211yesmaximal, characteristic
4#13168C211yesmaximal, characteristic, complemented
5#13167C211yesmaximal, characteristic, complemented
6#13150C211yesmaximal, characteristic, complemented
7#13150C211yesmaximal, characteristic, complemented
8#13112C211yesmaximal, characteristic, complemented
9#13110C211yesmaximal, characteristic, complemented
10#12994C211yesmaximal, characteristic
11#12938C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic
14#12868C211yesmaximal, characteristic
15#12599C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C4×C22⋊Q8—21no—
22#2296—21no—
23#2237C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#1969C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1941—21no—
43#1941—21no—
44#1933—21no—
45#1892—21no—
46#1808C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#103—21no—
49#1735—21no—
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C23×C8D411yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×#117—21no—
66C2×#117—21no—
67C22×C42D411yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#467—22no—
73#467C2×C414yes—
74C4×Q16—41no—
75C4×Q16—41no—
76#427—22no—
77#427—41no—
78#427—41no—
79#427C2×C414yes—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#106—21no—
88C2×#106—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×C4×C8—41no—
93C2×C4×C8—41no—
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#387—41no—
105#385—41no—
106#385—41no—
107#385—41no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—41no—
111C4×C4⋊C4—41no—
112C4×C4⋊C4—41no—
113C4×C22⋊C4—41no—
114#367—41no—
115#367C2×C414yescomplemented
116#366—41no—
117#366C2×C414yescomplemented
118#358—41no—
119#358—41no—
120#358C2×C414yescomplemented
121#358C2×C414yescomplemented
122C22⋊C16—22no—
123C22⋊C16C2×C414yes—
124#346—41no—
125#346—41no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133C22×Q8—21no—
134C23×C4D811yescharacteristic
135C23×C4D811yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16C22×C412yes—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—41no—
150C22×C8C22×C412yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8#4112yes—
155C22×C8C2411yescharacteristic
156C22⋊Q8—24no—
157C22⋊Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160#117—24no—
161#117—42no—
162#117—42no—
163#117—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—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42D811yescharacteristic
189C2×C42D811yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
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×C16—21no—
201C2×C16—21no—
202C2×C16C22×C412yes—
203C2×C16C22×C412yes—
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#107—21no—
213#107—21no—
214#107—21no—
215#107C22×C412yes—
216#107C22×C412yes—
217#107C22×C412yes—
218#106—24no—
219#106—42no—
220#103—24no—
221#103—42no—
222#103—42no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—41no—
228C4×C8—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24C2×D811yescharacteristic
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4C4×D412yes—
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4#13512yes—
270C22×C4C22×D411yescharacteristic
271Q16—22no—
272Q16—22no—
273Q16—41no—
274Q16—41no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8C4×D412yes—
288C2×C8#13712yes—
289C2×C8C23×C411yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8H(5,𝔽2)11yescharacteristic
292C2×C8H(5,𝔽2)11yescharacteristic
293C2×C82−1+411yescharacteristic
294C2×C82−1+411yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—41no—
325C42—41no—
326C42—44no—
327C42—82no—
328C42—82no—
329C42—82no—
330C16—22no—
331C16—22no—
332C16—41no—
333C16—41no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23C4×D812yes—
341C23C22×D811yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
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—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
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—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4C4×D812yes—
390C2×C4#51012yes—
391C2×C4C2×C4×D411yescharacteristic
392C2×C4#53411yescharacteristic
393C2×C4#53611yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57511yescharacteristic
397C8—22no—
398C8—22no—
399C8—41no—
400C8#52011yescharacteristic
401C8#52011yescharacteristic
402C8#53911yescharacteristic
403C8#53911yescharacteristic
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22C2×C4×D811yescharacteristic
412C22#318611yescharacteristic
413C22#322611yescharacteristic
414C22C2×#51011yescharacteristic
415C22#341411yescharacteristic
416C22#341411yescharacteristic
417C22#341411yescharacteristic
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#232211yescharacteristic
432C4C4×#13711yescharacteristic
433C4#319011yescharacteristic
434C4#323211yescharacteristic
435C2—42no—
436C2#2207511yescharacteristic
437C2#3345111yescharacteristic
438C2C4×#51011yescharacteristic
439C2#3376211yescharacteristic
440C2#3376311yescharacteristic
441C2#3380311yescharacteristic
442C2#3380611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13168C2semidirect
5. #13167C2semidirect
6. #13150C2semidirect
7. #13150C2semidirect
8. #13112C2semidirect
9. #13110C2semidirect
11. #12938C2semidirect
12. #12936C2semidirect
23. #2237C4semidirect
115. #367C2×C4semidirect
117. #366C2×C4semidirect
120. #358C2×C4semidirect
121. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C4×#4672
#211512
#131682
#131672
#131502
#131122
#131102
#129942
#129382
#129362
#129212
#128682
#125992