Finite Group Databasebeta

← #497404

Subgroups — #497404

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3292C211yesmaximal, characteristic
2C2×#2170C211yesmaximal, characteristic, complemented
3#33381C212yesmaximal, complemented
4#22692C211yesmaximal, characteristic, complemented
5#13261C211yesmaximal, characteristic, complemented
6#13202C211yesmaximal, characteristic, complemented
7#13073C212yesmaximal, complemented
8#12996C212yesmaximal, complemented
9#12821C211yesmaximal, characteristic, complemented
10#12748C212yesmaximal, complemented
11#12641C211yesmaximal, characteristic, complemented
12C2×#576—21no—
13C2×#576C2212yes—
14C2×#504C2212yescomplemented
15#3292—24no—
16#3292C418yescomplemented
17C2×#493C2211yescharacteristic
18C2×#472—21no—
19C2×#472C2212yes—
20C2×#453—22no—
21C2×#443C2211yescharacteristic
22C2×#443C2211yescharacteristic
23#2951—21no—
24#2913C2212yescomplemented
25#2170—24no—
26C2×#364C2211yescharacteristic, complemented
27C2×#361C2212yescomplemented
28C2×#359C2211yescharacteristic
29#2140—22no—
30#2140C2212yescomplemented
31#2139—22no—
32#2139C2212yes—
33#2138—22no—
34#2138C2211yescharacteristic
35C2×#357—22no—
36C2×#357C2212yescomplemented
37C2×#357C2212yescomplemented
38C2×#350C2212yescomplemented
39C2×C4×C16C2211yescharacteristic, complemented
40#1952—21no—
41#1859—22no—
42#1845C2211yescharacteristic, complemented
43#1816C2212yescomplemented
44#1772C2211yescharacteristic, complemented
45#1387C2212yescomplemented
46#1382C2211yescharacteristic, complemented
47#1381C2211yescharacteristic, complemented
48C2×H(5,𝔽2)—22no—
49#576—24no—
50#576—28no—
51#576—24no—
52#576—42no—
53#576—44no—
54#576C2×C418yescomplemented
55C2×AGL(1,ℤ/8)—22no—
56C2×AGL(1,ℤ/8)—22no—
57C2×AGL(1,ℤ/8)—42no—
58C2×AGL(1,ℤ/8)—42no—
59C2×#135—22no—
60C2×#135—41no—
61C2×#135D411yescharacteristic
62C2×#135C2311yescharacteristic
63C22×D8—22no—
64C22×D8—21no—
65C22×D8—41no—
66C22×D8C2312yes—
67C2×#131—21no—
68C2×#131C2312yes—
69C2×#127—21no—
70#520—22no—
71#504—24no—
72#504—42no—
73C2×M32—22no—
74C2×M32C2312yes—
75C22×C16—22no—
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#493—22no—
79#493C2×C414yescomplemented
80#472—24no—
81#472—28no—
82#472C2×C418yescomplemented
83#453—216no—
84#443—22no—
85#443C2×C414yes—
86#443C2×C414yescomplemented
87#443D414yes—
88#440—42no—
89C4×Q16—41no—
90C4×D8—41no—
91C2×#108D411yescharacteristic
92C2×#108C2311yescharacteristic
93C2×#108C2311yescharacteristic
94#428—22no—
95#428—42no—
96#427—22no—
97#427C2311yescharacteristic
98C2×#107—22no—
99C2×#107—21no—
100C2×#107C2312yes—
101C2×#104—21no—
102C2×#104D412yes—
103#417—22no—
104C2×#100—22no—
105#405—21no—
106C2×C4×C8C2311yescharacteristic
107#384—21no—
108#364—22no—
109#361—42no—
110#359—22no—
111#358—22no—
112#358D411yescharacteristic, complemented
113#358D411yescharacteristic, complemented
114#357—22no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#357—42no—
120#357D411yescharacteristic, complemented
121#357D411yescharacteristic, complemented
122#350—24no—
123C4×C16—22no—
124#343—22no—
125#339—21no—
126H(5,𝔽2)—28no—
127H(5,𝔽2)—44no—
128H(5,𝔽2)—44no—
129H(5,𝔽2)—82no—
130C2×#41—21no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41C2×D412yes—
134C22×D4—22no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—81no—
138C22×D4—82no—
139AGL(1,ℤ/8)—28no—
140AGL(1,ℤ/8)—28no—
141AGL(1,ℤ/8)—48no—
142AGL(1,ℤ/8)—48no—
143AGL(1,ℤ/8)—44no—
144AGL(1,ℤ/8)—44no—
145AGL(1,ℤ/8)—416no—
146#135—28no—
147#135—21no—
148#135—21no—
149#135—44no—
150#135—48no—
151#135C22⋊C414yes—
152#135SD1614yes—
153#135C22×C412yes—
154#135#4112yes—
155C2×Q16D811yescharacteristic
156C2×Q16C2×D411yescharacteristic
157C2×SD16—21no—
158C2×SD16—42no—
159C2×SD16—42no—
160C2×SD16D812yes—
161C2×D8—28no—
162C2×D8—24no—
163C2×D8—22no—
164C2×D8—22no—
165C2×D8—22no—
166C2×D8—22no—
167C2×D8—22no—
168C2×D8—44no—
169C2×D8—42no—
170C2×D8—42no—
171C2×D8—48no—
172C2×D8—41no—
173C2×D8—41no—
174C2×D8—42no—
175C2×D8D811yescharacteristic
176C2×D8C22×C414yes—
177C2×D8C2×D411yescharacteristic
178#131—22no—
179#131—22no—
180#131—24no—
181#131—42no—
182#131C22×C414yes—
183#131C2×D414yes—
184C2×M16—22no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—41no—
188C2×M16C2×D412yes—
189C22×C8—22no—
190C22×C8—41no—
191C22×C8C2×D411yescharacteristic
192C22×C8C2411yescharacteristic
193#127—24no—
194#127—81no—
195C4×Q8—42no—
196C4×D4—42no—
197C4×D4—82no—
198#117—21no—
199#117—42no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—81no—
203C2×C42C2×D411yescharacteristic
204M32—22no—
205M32—22no—
206M32—44no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—22no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—44no—
214C2×C16C2×D411yescharacteristic
215C2×C16C2×D411yescharacteristic
216#108—21no—
217#108—21no—
218#108—21no—
219#108C22⋊C414yes—
220#108C22×C412yes—
221#107—22no—
222#107—22no—
223#107—42no—
224#107—41no—
225#107—41no—
226#107C2×D411yescharacteristic
227#107C2×D411yescharacteristic
228#106—21no—
229#106—42no—
230#104—24no—
231#104C22⋊C418yes—
232#103—42no—
233#102—42no—
234#100—28no—
235#97—41no—
236C4×C8—41no—
237C4×C8C22×C412yes—
238C4×C8C2×D412yes—
239C24—42no—
240C24—82no—
241#41—22no—
242#41—22no—
243#41—22no—
244#41—22no—
245#41—48no—
246#41—41no—
247#41—41no—
248#41—48no—
249#41—84no—
250#41—84no—
251#41C2×C22⋊C414yes—
252#41C2×SD1614yes—
253C2×Q8—21no—
254C2×Q8C2×D812yes—
255C2×D4—22no—
256C2×D4—24no—
257C2×D4—21no—
258C2×D4—48no—
259C2×D4—41no—
260C2×D4—48no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—84no—
267C2×D4—82no—
268C2×D4—84no—
269C2×D4—84no—
270C2×D4—82no—
271C2×D4—88no—
272C2×D4—88no—
273C2×D4—162no—
274C2×D4—162no—
275C2×D4C2×D812yes—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C22×D411yescharacteristic
283Q16—21no—
284Q16#10214yes—
285Q16C4×D412yes—
286SD16—22no—
287SD16—22no—
288SD16—48no—
289SD16—48no—
290SD16#10218yes—
291D8—24no—
292D8—24no—
293D8—21no—
294D8—48no—
295D8—48no—
296D8—42no—
297D8—42no—
298D8—44no—
299D8—44no—
300D8#10214yes—
301D8C4×D412yes—
302M16—22no—
303M16—24no—
304M16—24no—
305M16—22no—
306M16—41no—
307M16—44no—
308M16—41no—
309M16—41no—
310M16C2×C22⋊C414yes—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—41no—
315C2×C8—44no—
316C2×C8—41no—
317C2×C8C2×C22⋊C412yes—
318C2×C8C2×D811yescharacteristic
319C2×C8C2×D811yescharacteristic
320C2×C8C2×SD1612yes—
321C2×C8C23×C411yescharacteristic
322C2×C8C22×D411yescharacteristic
323C2×C8C22×D411yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8H(5,𝔽2)11yescharacteristic
326C2×C8H(5,𝔽2)11yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—162no—
333C22⋊C4—41no—
334C22⋊C4—82no—
335C42—21no—
336C42—41no—
337C42—82no—
338C42C2×C22⋊C412yes—
339C16—21no—
340C16—21no—
341C16—42no—
342C16—41no—
343C23—22no—
344C23—41no—
345C23—44no—
346C23—88no—
347C23—82no—
348C23—82no—
349C23—162no—
350C23—162no—
351C23—322no—
352C23—322no—
353C23#57611yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8C2×#10214yes—
358D4—24no—
359D4—24no—
360D4—22no—
361D4—22no—
362D4—42no—
363D4—42no—
364D4—44no—
365D4—44no—
366D4—41no—
367D4—88no—
368D4—88no—
369D4—82no—
370D4—82no—
371D4—168no—
372D4C2×#10214yes—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—162no—
387C2×C4—322no—
388C2×C4C22×C22⋊C411yescharacteristic
389C2×C4D4211yescharacteristic
390C2×C4#54611yescharacteristic
391C2×C4C22×D811yescharacteristic
392C2×C4C22×SD1611yescharacteristic
393C2×C4#57711yescharacteristic
394C8—24no—
395C8—42no—
396C8—81no—
397C8C2×#10212yes—
398C8C2×#10212yes—
399C8C2×C4×D411yescharacteristic
400C8#51811yescharacteristic
401C8D4211yescharacteristic
402C8#54611yescharacteristic
403C22—21no—
404C22—21no—
405C22—24no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—81no—
410C22—81no—
411C22—84no—
412C22—164no—
413C22—162no—
414C22—162no—
415C22—322no—
416C22—328no—
417C22#289711yescharacteristic
418C22D4×D811yescharacteristic
419C22#329511yescharacteristic
420C4—24no—
421C4—24no—
422C4—42no—
423C4—81no—
424C4—81no—
425C4—162no—
426C4D4×C22⋊C411yescharacteristic
427C4C22×#10211yescharacteristic
428C4D4×SD1611yescharacteristic
429C4#329411yescharacteristic
430C2—21no—
431C2—21no—
432C2—44no—
433C2—82no—
434C2—164no—
435C2D4×#10211yescharacteristic
436C2#3385611yescharacteristic
437C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3292C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2170C2semidirect
3. #33381C2semidirect
4. #22692C2semidirect
5. #13261C2semidirect
6. #13202C2semidirect
7. #13073C2semidirect
8. #12996C2semidirect
9. #12821C2semidirect
10. #12748C2semidirect
11. #12641C2semidirect
14. C2×#504C22semidirect
16. #3292C4semidirect
24. #2913C22semidirect
26. C2×#364C22semidirect
27. C2×#361C22semidirect
30. #2140C22semidirect
36. C2×#357C22semidirect
37. C2×#357C22semidirect
38. C2×#350C22semidirect
39. C2×C4×C16C22semidirect
42. #1845C22semidirect
43. #1816C22semidirect
44. #1772C22semidirect
45. #1387C22semidirect
46. #1382C22semidirect
47. #1381C22semidirect
54. #576C2×C4semidirect
79. #493C2×C4semidirect
82. #472C2×C4semidirect
86. #443C2×C4semidirect
112. #358D4semidirect
113. #358D4semidirect
120. #357D4semidirect
121. #357D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32922
C2×#21702
#333812
#226922
#132612
#132022
#130732
#129962
#128212
#127482
#126412