Finite Group Databasebeta

← #498067

Subgroups — #498067

Order 512.

527 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#22816C211yesmaximal, characteristic, complemented
3#22803C211yesmaximal, characteristic, complemented
4#13249C212yesmaximal
5#13143C212yesmaximal, complemented
6#13142C212yesmaximal, complemented
7#12904C211yesmaximal, characteristic, complemented
8#12818C211yesmaximal, characteristic, complemented
9#12817C211yesmaximal, characteristic, complemented
10#12677C212yesmaximal, complemented
11#12666C211yesmaximal, characteristic, complemented
12C2×#509C2211yescharacteristic
13C4×#135C2211yescharacteristic
14C2×C4×D8C2211yescharacteristic
15#2925—21no—
16#2357—21no—
17#2177C412yescomplemented
18#2175C412yescomplemented
19C4×SD32C412yescomplemented
20C4×D16C412yescomplemented
21#2159C2211yescharacteristic
22C2×#365C2212yes—
23C2×#365C2212yes—
24#2150C2211yescharacteristic
25C2×#361C2212yes—
26C2×#360C2212yes—
27#2140C2211yescharacteristic
28#2139C2211yescharacteristic
29C2×#347C2212yes—
30C4×M32C2211yescharacteristic
31#1975C2212yes—
32#1972C2212yes—
33#1929—22no—
34#1925—22no—
35#1848C2211yescharacteristic
36#1846C2211yescharacteristic
37#1806—21no—
38#1804—21no—
39#1776C2212yes—
40#1773C2212yes—
41#1770C2211yescharacteristic
42#1387C2212yes—
43#1386C2212yes—
44#1375C2212yes—
45C2×#135C2311yescharacteristic
46C22×D8C2311yescharacteristic
47C4×#41—21no—
48C2×C4×D4—21no—
49#509—22no—
50#509—22no—
51C2×SD32C2×C412yes—
52C2×D16C2×C412yes—
53C2×M32C2312yes—
54C2×M32C2311yescharacteristic
55C2×M32C2311yescharacteristic
56C4×Q16C2×C411yescharacteristic
57C4×Q16C2×C411yescharacteristic
58C4×SD16—21no—
59C4×SD16—21no—
60C4×D8—21no—
61C4×D8—21no—
62C4×D8C2×C412yes—
63C4×D8C2×C412yes—
64C4×D8C2×C411yescharacteristic
65C4×D8C2×C411yescharacteristic
66#433—42no—
67#432—41no—
68#431—41no—
69C2×#108—21no—
70C2×#108C2312yes—
71#427C2311yescharacteristic
72C2×#107—21no—
73C2×#107C2311yescharacteristic
74C2×#106C2312yes—
75#423—21no—
76C2×C4⋊C8D411yescharacteristic
77C2×#104—22no—
78#416—21no—
79C2×#102—21no—
80C2×#102—22no—
81#408—22no—
82#405—22no—
83C2×#97C2312yes—
84C2×C4×C8C2311yescharacteristic
85#390—22no—
86#388—42no—
87#382—21no—
88#381—41no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4D411yescharacteristic
91C4×C22⋊C4—41no—
92#367C2×C412yes—
93#366C2×C412yes—
94#365—24no—
95#365—42no—
96C4⋊C16—21no—
97C4⋊C16—21no—
98#361—42no—
99#360—42no—
100#358C2×C412yes—
101#357C2×C412yes—
102#357C2×C412yes—
103#357C2×C412yes—
104#347—24no—
105#346C2×C412yes—
106C4×C16C2×C412yes—
107#340—22no—
108#339—22no—
109#336D412yes—
110C2×#41—21no—
111C22×D4—21no—
112C23×C4—41no—
113#135—21no—
114#135—21no—
115#135—21no—
116#135—21no—
117C2×Q16C22×C411yescharacteristic
118C2×SD16—21no—
119C2×D8—21no—
120C2×D8—21no—
121C2×D8—21no—
122C2×D8—21no—
123C2×D8—21no—
124C2×D8C22×C412yes—
125C2×D8C22×C411yescharacteristic
126C2×M16—22no—
127C22×C8C2×D412yes—
128C22×C8C2×D411yescharacteristic
129C22×C8C2411yescharacteristic
130C4×Q8—21no—
131C4×Q8—21no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140#117—21no—
141#117—41no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C42—41no—
149C2×C42—42no—
150C2×C42C2×D412yes—
151C2×C42C2×D411yescharacteristic
152SD32—24no—
153D16—24no—
154M32—21no—
155M32—21no—
156M32—22no—
157M32—22no—
158M32#4112yes—
159M32#4112yes—
160C2×C16—22no—
161C2×C16C22×C412yes—
162C2×C16C22×C412yes—
163#108—22no—
164#108—22no—
165#108—81no—
166#107—21no—
167#107—81no—
168#107C22×C411yescharacteristic
169#107C22×C411yescharacteristic
170#107C22×C412yes—
171#106—21no—
172#106—22no—
173#106—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178#104—82no—
179#103—21no—
180#103—21no—
181#102—22no—
182#102—22no—
183#102—21no—
184#102—21no—
185#102—82no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188#97—22no—
189#97—22no—
190#97—42no—
191C4×C8—42no—
192C4×C8C22×C411yescharacteristic
193C4×C8C22×C411yescharacteristic
194C4×C8#4111yescharacteristic
195C4×C8#4111yescharacteristic
196#95—21no—
197#95—42no—
198#95—41no—
199#95—41no—
200C24—41no—
201#41—41no—
202#41—41no—
203#41—41no—
204#41—41no—
205C2×Q8—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—162no—
223C22×C4AGL(1,ℤ/8)11yescharacteristic
224C22×C4AGL(1,ℤ/8)11yescharacteristic
225C22×C4C22×D411yescharacteristic
226Q16—21no—
227Q16—21no—
228SD16—41no—
229SD16—41no—
230D8—21no—
231D8—22no—
232D8—22no—
233D8—21no—
234D8—41no—
235D8—41no—
236M16—42no—
237M16—42no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—42no—
244C2×C8C4×D411yescharacteristic
245C2×C8C4×D411yescharacteristic
246C2×C8C23×C411yescharacteristic
247C2×C8C2×#4111yescharacteristic
248C2×C8C2×#4111yescharacteristic
249C2×C8C2×#4111yescharacteristic
250C2×C8H(5,𝔽2)11yescharacteristic
251C2×C82−1+411yescharacteristic
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—81no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—84no—
267C22⋊C4—162no—
268C42—22no—
269C42—22no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—162no—
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42AGL(1,ℤ/8)11yescharacteristic
277C42AGL(1,ℤ/8)11yescharacteristic
278C16—22no—
279C16—42no—
280C16#11714yes—
281C23—41no—
282C23—42no—
283C23—82no—
284C23—82no—
285C23—81no—
286C23—81no—
287C23C2×AGL(1,ℤ/8)11yescharacteristic
288Q8—41no—
289Q8—41no—
290D4—42no—
291D4—41no—
292D4—42no—
293D4—41no—
294D4—81no—
295D4—81no—
296D4—81no—
297D4—81no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—161no—
315C2×C4—162no—
316C2×C4—162no—
317C2×C4#44211yescharacteristic
318C2×C4#44211yescharacteristic
319C2×C4#44211yescharacteristic
320C2×C4#44211yescharacteristic
321C2×C4C2×C4×D411yescharacteristic
322C2×C4#54711yescharacteristic
323C2×C4#55011yescharacteristic
324C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
325C2×C4#57711yescharacteristic
326C2×C4#57711yescharacteristic
327C8—21no—
328C8—21no—
329C8—42no—
330C8—82no—
331C8C2×#11711yescharacteristic
332C8#52011yescharacteristic
333C8#55511yescharacteristic
334C8#55611yescharacteristic
335C22—21no—
336C22—21no—
337C22—81no—
338C22—82no—
339C22—81no—
340C22—82no—
341C22—82no—
342C22—81no—
343C22—162no—
344C22—162no—
345C22C2×#44211yescharacteristic
346C22#335411yescharacteristic
347C22#339711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4—82no—
356C4—81no—
357C4—162no—
358C4#235411yescharacteristic
359C4#295211yescharacteristic
360C4#335411yescharacteristic
361C4#339711yescharacteristic
362C2—21no—
363C2—21no—
364C2—161no—
365C2—162no—
366C2#2281311yescharacteristic
367C2#3346811yescharacteristic
368C2#3347111yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#509C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22816C2semidirect
3. #22803C2semidirect
5. #13143C2semidirect
6. #13142C2semidirect
7. #12904C2semidirect
8. #12818C2semidirect
9. #12817C2semidirect
10. #12677C2semidirect
11. #12666C2semidirect
17. #2177C4semidirect
18. #2175C4semidirect
19. C4×SD32C4semidirect
20. C4×D16C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#228162
#228032
#132492
#131432
#131422
#129042
#128182
#128172
#126772
#126662