Finite Group Databasebeta

← #362448

Subgroups — #362448

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21087C212yesmaximal
2#20763C211yesmaximal, characteristic, complemented
3#20664C211yesmaximal, characteristic
4C2×#1764C211yesmaximal, characteristic
5#11340C212yesmaximal, complemented
6#11078C212yesmaximal
7#10896C212yesmaximal, complemented
8#10841C211yesmaximal, characteristic, complemented
9#10827C211yesmaximal, characteristic, complemented
10#10774C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C2×#428C2212yes—
14C22×#104C2211yescharacteristic
15C2×#417C2212yes—
16C2×#416C2211yescharacteristic
17C2×#405C2212yes—
18C2×#405C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2317C2212yes—
21#2311C2212yes—
22C2×#384C2212yes—
23#1936—22no—
24#1935—22no—
25#1934—22no—
26#1933—22no—
27#1836—22no—
28#1835—21no—
29#1834—21no—
30#1809—22no—
31#1808—21no—
32#1807—21no—
33#1764C418yescomplemented
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38#1736C2212yes—
39#1736C2212yes—
40#1735C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#339C2212yes—
45C2×#339C2211yescharacteristic
46C22×#41C2311yescharacteristic
47C2×#131—22no—
48C2×#131—24no—
49C2×#131—42no—
50C2×#131C2×C414yes—
51C22×M16—21no—
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C23×C8—21no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×#117C2312yes—
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—42no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42C2311yescharacteristic
64#428—24no—
65#428—42no—
66C2×#107—22no—
67C2×#106—22no—
68C2×#104—24no—
69C2×#104C2×C414yes—
70#417—24no—
71#417—42no—
72#416C2×C414yes—
73#416Q814yes—
74C2×#103—22no—
75C2×#103D411yescharacteristic
76C2×#103D411yescharacteristic
77C2×#102—22no—
78C2×#102D411yescharacteristic
79C2×#102D411yescharacteristic
80#405—22no—
81#405—24no—
82#405—42no—
83#405C2×C414yes—
84C2×#97—21no—
85C2×#97—22no—
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88C2×C4×C8—42no—
89C2×C4×C8C2×C414yes—
90#391—22no—
91#390—22no—
92#389—22no—
93#389—22no—
94#386—42no—
95#385—42no—
96#384—22no—
97#384—24no—
98#384—42no—
99#382—22no—
100#380—22no—
101#380—22no—
102#343—24no—
103#343—24no—
104#341—22no—
105#341C2×C414yes—
106#340—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340D411yescharacteristic
117#340D411yescharacteristic
118#340D412yes—
119#339—24no—
120#339C2×C414yes—
121C2×#41—21no—
122C2×#41—21no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41C22×C412yes—
128C2×#41C2×Q812yes—
129C22×Q8C2×D411yescharacteristic
130C22×D4—21no—
131C22×D4C2×D411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C2411yescharacteristic
137#131—28no—
138#131—44no—
139#131—48no—
140C2×M16—21no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—24no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—44no—
152C2×M16C22×C412yes—
153C2×M16C22×C412yes—
154C2×M16#4114yes—
155C2×M16#4114yes—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—24no—
164C22×C8—44no—
165C22×C8C22×C412yes—
166C22×C8C22×C412yes—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×Q812yes—
172C22×C8#4112yes—
173#117—21no—
174#117—21no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117C22×C412yes—
180#117C2×Q812yes—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—84no—
197C2×C4⋊C4—84no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42C22×C412yes—
211C2×C42#4114yes—
212#107—44no—
213#106—44no—
214#104—28no—
215#103—44no—
216#103C4⋊C414yescomplemented
217#103C4⋊C414yescomplemented
218#102—44no—
219#102C4⋊C414yescomplemented
220#102C4⋊C414yescomplemented
221#97—24no—
222#97—44no—
223C4×C8—24no—
224C4×C8—44no—
225C4×C8—44no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—42no—
230C24—21no—
231C24H(5,𝔽2)11yescharacteristic
232#41—22no—
233#41—22no—
234#41—22no—
235#41—22no—
236#41—24no—
237#41—24no—
238#41—42no—
239#41—42no—
240#41—42no—
241#41—42no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8C2×C4⋊C412yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4C2×C4⋊C412yes—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×Q811yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+411yescharacteristic
288M16—22no—
289M16—24no—
290M16—22no—
291M16—22no—
292M16—24no—
293M16—24no—
294M16—48no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16#11714yes—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—48no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8C2×C4⋊C412yes—
318C2×C8C2×C4⋊C412yes—
319C2×C8C4×D412yes—
320C2×C8C4×D412yes—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—816no—
336C4⋊C4—162no—
337C4⋊C4—162no—
338C4⋊C4C2×C4⋊C412yes—
339C4⋊C4C2×C4⋊C412yes—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—162no—
345C22⋊C4—162no—
346C42—22no—
347C42—22no—
348C42—24no—
349C42—22no—
350C42—42no—
351C42—42no—
352C42#11714yes—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23#52011yescharacteristic
362C23#53812yes—
363C23D4211yescharacteristic
364C23#55411yescharacteristic
365C23#55512yes—
366Q8—22no—
367Q8—22no—
368Q8—42no—
369D4—22no—
370D4—22no—
371D4—42no—
372D4—44no—
373D4—42no—
374D4—42no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—162no—
405C2×C4—162no—
406C2×C4C22×C4⋊C411yescharacteristic
407C2×C4C2×#11711yescharacteristic
408C2×C4C2×#11711yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C4×#4111yescharacteristic
411C2×C4#51811yescharacteristic
412C2×C4#52912yes—
413C2×C4#53212yes—
414C2×C4#53212yes—
415C2×C4#53312yes—
416C2×C4#53912yes—
417C2×C4#54612yes—
418C2×C4#54612yes—
419C2×C4#54711yescharacteristic
420C2×C4D4×Q811yescharacteristic
421C2×C4#55112yes—
422C2×C4#55212yes—
423C2×C4#55412yes—
424C8—22no—
425C8—22no—
426C8—24no—
427C8—22no—
428C8—22no—
429C8—42no—
430C8—44no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—82no—
443C22—82no—
444C22#231811yescharacteristic
445C22#234611yescharacteristic
446C22D4×C4⋊C411yescharacteristic
447C22#255512yes—
448C22#257412yes—
449C22#329112yes—
450C22#329211yescharacteristic
451C22#329311yescharacteristic
452C4—28no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—82no—
457C4—82no—
458C4—162no—
459C4—162no—
460C4#228511yescharacteristic
461C4#229311yescharacteristic
462C4#234211yescharacteristic
463C4#234911yescharacteristic
464C4#255312yes—
465C4#255612yes—
466C2—21no—
467C2—22no—
468C2—21no—
469C2—82no—
470C2#1391511yescharacteristic
471C2#2268911yescharacteristic
472C2#2269211yescharacteristic
473C2#2893812yes—
474C2#2893912yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20763C2semidirect
5. #11340C2semidirect
7. #10896C2semidirect
8. #10841C2semidirect
9. #10827C2semidirect
11. #10706C2semidirect
33. #1764C4semidirect
216. #103C4⋊C4semidirect
217. #103C4⋊C4semidirect
219. #102C4⋊C4semidirect
220. #102C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#210872
#207632
#206642
C2×#17642
#113402
#110782
#108962
#108412
#108272
#107742
#107062