Finite Group Databasebeta

← #497988

Subgroups — #497988

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#22816C211yesmaximal, characteristic, complemented
3#22686C211yesmaximal, characteristic, complemented
4#13155C212yesmaximal, complemented
5#13119C212yesmaximal, complemented
6#13040C212yesmaximal
7#12887C211yesmaximal, characteristic, complemented
8#12818C211yesmaximal, characteristic, complemented
9#12817C211yesmaximal, characteristic, complemented
10#12666C211yesmaximal, characteristic, complemented
11#12650C212yesmaximal, complemented
12C2×#509C2211yescharacteristic
13C2×#444C2212yes—
14C4×#135C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16#2906—21no—
17#2357—21no—
18#2177C412yescomplemented
19#2175C412yescomplemented
20C4×SD32C412yescomplemented
21C4×D16C412yescomplemented
22#2159C2211yescharacteristic
23#2150C2211yescharacteristic
24C2×#361C2212yes—
25C2×#360C2212yes—
26#2140C2211yescharacteristic
27#2139C2211yescharacteristic
28C2×#347C2212yes—
29C4×M32C2211yescharacteristic
30#1972C2212yes—
31#1946C2212yes—
32#1925—22no—
33#1866—22no—
34#1848C2211yescharacteristic
35#1846C2211yescharacteristic
36#1806—21no—
37#1804—21no—
38#1776C2212yes—
39#1770C2211yescharacteristic
40#1387C2212yes—
41#1386C2212yes—
42#1383C2212yes—
43#1383C2212yes—
44#1375C2212yes—
45C2×#135C2311yescharacteristic
46C22×D8C2311yescharacteristic
47C2×#131—22no—
48C4×#41—21no—
49C2×C4×D4—21no—
50#509—22no—
51#509—22no—
52C2×SD32C2×C412yes—
53C2×D16C2×C412yes—
54C2×M32C2312yes—
55C2×M32C2311yescharacteristic
56C2×M32C2311yescharacteristic
57#444—24no—
58#444—42no—
59C4×Q16C2×C411yescharacteristic
60C4×Q16C2×C411yescharacteristic
61C4×SD16—21no—
62C4×SD16—21no—
63C4×D8—21no—
64C4×D8—21no—
65C4×D8C2×C412yes—
66C4×D8C2×C412yes—
67C4×D8C2×C411yescharacteristic
68C4×D8C2×C411yescharacteristic
69C2×#108—21no—
70C2×#108C2312yes—
71#427C2311yescharacteristic
72C2×#107—21no—
73C2×#107C2311yescharacteristic
74C2×#106C2312yes—
75#424—22no—
76#424—42no—
77#423—21no—
78#423—41no—
79C2×C4⋊C8—41no—
80C2×C4⋊C8D411yescharacteristic
81C2×#104—22no—
82#416—21no—
83C2×#102—21no—
84C2×#102—22no—
85C2×#97C2312yes—
86C2×C4×C8C2311yescharacteristic
87#390—22no—
88#388—42no—
89#382—21no—
90#381—41no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4D411yescharacteristic
93C4×C22⋊C4—41no—
94#367C2×C412yes—
95#366C2×C412yes—
96C4⋊C16—21no—
97C4⋊C16—21no—
98#361—42no—
99#360—24no—
100#358C2×C412yes—
101#357C2×C412yes—
102#357C2×C412yes—
103#357C2×C412yes—
104#347—42no—
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#4112yes—
116#135#4112yes—
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
126#131—44no—
127#131—82no—
128C2×M16—22no—
129C2×M16—42no—
130C22×C8—42no—
131C22×C8C2×D412yes—
132C22×C8C2×D411yescharacteristic
133C22×C8C2411yescharacteristic
134C4×Q8—21no—
135C4×Q8—21no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144#117—21no—
145#117—41no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C42—41no—
153C2×C42—42no—
154C2×C42C2×D412yes—
155C2×C42C2×D411yescharacteristic
156SD32—24no—
157D16—24no—
158M32—21no—
159M32—21no—
160M32—21no—
161M32—21no—
162M32—22no—
163M32—22no—
164C2×C16—22no—
165C2×C16C22×C412yes—
166C2×C16C22×C412yes—
167#108—22no—
168#108—22no—
169#108—81no—
170#107—21no—
171#107—81no—
172#107C22×C411yescharacteristic
173#107C22×C411yescharacteristic
174#107C22×C412yes—
175#106—21no—
176#106—22no—
177#106—22no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184#104—44no—
185#103—21no—
186#103—21no—
187#102—22no—
188#102—22no—
189#102—21no—
190#102—21no—
191#102—82no—
192#97—22no—
193#97—22no—
194C4×C8C22×C411yescharacteristic
195C4×C8C22×C411yescharacteristic
196C4×C8#4111yescharacteristic
197C4×C8#4111yescharacteristic
198#95—21no—
199#95—42no—
200#95—41no—
201#95—41no—
202C24—41no—
203#41—22no—
204#41—22no—
205#41—41no—
206#41—41no—
207C2×Q8—21no—
208C2×D4—21no—
209C2×D4—22no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—162no—
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227C22×C4C22×D411yescharacteristic
228Q16—21no—
229Q16#11712yes—
230SD16—22no—
231SD16—22no—
232D8—22no—
233D8—21no—
234D8—22no—
235D8—41no—
236D8—41no—
237D8#11712yes—
238M16—42no—
239M16—42no—
240M16—84no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—42no—
247C2×C8—84no—
248C2×C8C4×D411yescharacteristic
249C2×C8C4×D411yescharacteristic
250C2×C8C23×C411yescharacteristic
251C2×C8C2×#4111yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C2×C8H(5,𝔽2)11yescharacteristic
255C2×C82−1+411yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—81no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—84no—
271C22⋊C4—162no—
272C42—22no—
273C42—22no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—162no—
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42AGL(1,ℤ/8)11yescharacteristic
281C42AGL(1,ℤ/8)11yescharacteristic
282C16—22no—
283C16—22no—
284C16—42no—
285C23—41no—
286C23—42no—
287C23—82no—
288C23—82no—
289C23—81no—
290C23—81no—
291C23C2×AGL(1,ℤ/8)11yescharacteristic
292Q8—22no—
293Q8—41no—
294D4—22no—
295D4—42no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—81no—
301D4—81no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—161no—
319C2×C4—162no—
320C2×C4—162no—
321C2×C4#44211yescharacteristic
322C2×C4#44211yescharacteristic
323C2×C4#44211yescharacteristic
324C2×C4#44211yescharacteristic
325C2×C4C2×C4×D411yescharacteristic
326C2×C4#54611yescharacteristic
327C2×C4#54811yescharacteristic
328C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
329C2×C4#57611yescharacteristic
330C2×C4#57811yescharacteristic
331C8—21no—
332C8—21no—
333C8—42no—
334C8—82no—
335C8C2×#11711yescharacteristic
336C8#52011yescharacteristic
337C8#55211yescharacteristic
338C8#55311yescharacteristic
339C22—21no—
340C22—21no—
341C22—42no—
342C22—81no—
343C22—81no—
344C22—82no—
345C22—82no—
346C22—82no—
347C22—162no—
348C22—162no—
349C22C2×#44211yescharacteristic
350C22#333011yescharacteristic
351C22#336611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—82no—
358C4—81no—
359C4—82no—
360C4—81no—
361C4—162no—
362C4#235211yescharacteristic
363C4#295111yescharacteristic
364C4#332511yescharacteristic
365C4#336711yescharacteristic
366C2—21no—
367C2—21no—
368C2—82no—
369C2—162no—
370C2#2275811yescharacteristic
371C2#3346811yescharacteristic
372C2#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. #22686C2semidirect
4. #13155C2semidirect
5. #13119C2semidirect
7. #12887C2semidirect
8. #12818C2semidirect
9. #12817C2semidirect
10. #12666C2semidirect
11. #12650C2semidirect
18. #2177C4semidirect
19. #2175C4semidirect
20. C4×SD32C4semidirect
21. 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
#226862
#131552
#131192
#130402
#128872
#128182
#128172
#126662
#126502