Finite Group Databasebeta

← #497983

Subgroups — #497983

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#22803C211yesmaximal, characteristic, complemented
3#22672C211yesmaximal, characteristic, complemented
4#13153C212yesmaximal, complemented
5#13118C212yesmaximal, complemented
6#13037C212yesmaximal
7#12887C211yesmaximal, characteristic, complemented
8#12818C211yesmaximal, characteristic, complemented
9#12817C211yesmaximal, characteristic, complemented
10#12666C211yesmaximal, characteristic, complemented
11#12647C212yesmaximal, complemented
12C2×#509C2211yescharacteristic
13C2×#442C2212yes—
14C4×#135C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16#2925—21no—
17#2349—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#1975C2212yes—
31#1943C2212yes—
32#1929—22no—
33#1862—22no—
34#1848C2211yescharacteristic
35#1846C2211yescharacteristic
36#1806—21no—
37#1804—21no—
38#1773C2212yes—
39#1770C2211yescharacteristic
40#1387C2212yes—
41#1386C2212yes—
42#1383C2212yes—
43#1383C2212yes—
44#1375C2212yes—
45C2×#135C2311yescharacteristic
46C22×D8C2311yescharacteristic
47C4×#41—21no—
48C2×C4×D4—22no—
49C2×C4×D4—21no—
50#509—22no—
51#509—22no—
52C2×SD32C2×C412yes—
53C2×D16C2×C412yes—
54C2×M32C2312yes—
55C2×M32C2311yescharacteristic
56C2×M32C2311yescharacteristic
57#442—24no—
58#442—42no—
59C4×Q16C2×C411yescharacteristic, complemented
60C4×Q16C2×C411yescharacteristic, complemented
61C4×SD16—21no—
62C4×SD16—21no—
63C4×D8—21no—
64C4×D8—21no—
65C4×D8C2×C412yes—
66C4×D8C2×C412yes—
67C4×D8C2×C411yescharacteristic, complemented
68C4×D8C2×C411yescharacteristic, complemented
69#433—42no—
70#432—41no—
71#431—41no—
72C2×#108—21no—
73C2×#108C2312yes—
74#427C2311yescharacteristic
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#106C2312yes—
78#423—21no—
79C2×C4⋊C8D411yescharacteristic
80C2×#104—22no—
81#416—21no—
82C2×#102—21no—
83C2×#102—22no—
84#408—22no—
85#405—22no—
86C2×#97C2312yes—
87C2×C4×C8C2311yescharacteristic
88#389—22no—
89#385—42no—
90#382—21no—
91#380—41no—
92C4×C4⋊C4D411yescharacteristic
93C2×#95—41no—
94#367C2×C412yescomplemented
95#366C2×C412yescomplemented
96C4⋊C16—21no—
97C4⋊C16—21no—
98#361—24no—
99#360—42no—
100#358C2×C412yescomplemented
101#357C2×C412yes—
102#357C2×C412yes—
103#357C2×C412yescomplemented
104#347—42no—
105#346C2×C412yescomplemented
106C4×C16C2×C412yescomplemented
107#340—22no—
108#339—22no—
109#336D412yes—
110C2×#41—21no—
111C22×D4—21no—
112C23×C4—42no—
113C23×C4—41no—
114#135—21no—
115#135—21no—
116#135—21no—
117#135—21no—
118C2×Q16C22×C411yescharacteristic
119C2×SD16—21no—
120C2×D8—21no—
121C2×D8—21no—
122C2×D8—21no—
123C2×D8C22×C411yescharacteristic
124C2×D8C22×C412yes—
125C2×D8#4112yes—
126C2×D8#4112yes—
127C2×M16—22no—
128C22×C8C2×D412yes—
129C22×C8C2×D411yescharacteristic
130C22×C8C2411yescharacteristic
131C4×Q8—21no—
132C4×Q8—21no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—44no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—82no—
143#117—21no—
144#117—41no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—42no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C42—41no—
152C2×C42C2×D412yes—
153C2×C42C2×D411yescharacteristic
154SD32—24no—
155D16—24no—
156M32—21no—
157M32—21no—
158M32—21no—
159M32—21no—
160M32—22no—
161M32—22no—
162C2×C16—22no—
163C2×C16C22×C412yes—
164C2×C16C22×C412yes—
165#108—22no—
166#108—22no—
167#108—81no—
168#107—21no—
169#107—81no—
170#107C22×C411yescharacteristic
171#107C22×C411yescharacteristic
172#107C22×C412yes—
173#106—21no—
174#106—22no—
175#106—22no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180#104—82no—
181#103—21no—
182#103—21no—
183#102—22no—
184#102—22no—
185#102—21no—
186#102—21no—
187#102—44no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190#97—22no—
191#97—22no—
192#97—42no—
193C4×C8—42no—
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—41no—
204#41—41no—
205#41—41no—
206#41—41no—
207C2×Q8—21no—
208C2×D4—22no—
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—22no—
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—84no—
225C22×C4—84no—
226C22×C4—162no—
227C22×C4—162no—
228C22×C4AGL(1,ℤ/8)11yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4C22×D411yescharacteristic
231Q16—21no—
232Q16—21no—
233SD16—41no—
234SD16—41no—
235D8—21no—
236D8—22no—
237D8—21no—
238D8—22no—
239D8—22no—
240D8#11714yes—
241M16—42no—
242M16—42no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—42no—
249C2×C8C4×D411yescharacteristic
250C2×C8C4×D411yescharacteristic
251C2×C8C23×C411yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C2×C8C2×#4111yescharacteristic
255C2×C8H(5,𝔽2)11yescharacteristic
256C2×C82−1+411yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—81no—
266C4⋊C4—162no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—84no—
273C42—22no—
274C42—22no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42AGL(1,ℤ/8)11yescharacteristic
281C42AGL(1,ℤ/8)11yescharacteristic
282C16—22no—
283C16—22no—
284C16—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—41no—
289C23—82no—
290C23—82no—
291C23C2×AGL(1,ℤ/8)11yescharacteristic
292Q8—41no—
293Q8—41no—
294D4—24no—
295D4—42no—
296D4—42no—
297D4—41no—
298D4—42no—
299D4—41no—
300D4—81no—
301D4—81no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—161no—
319C2×C4—162no—
320C2×C4—162no—
321C2×C4—168no—
322C2×C4#44211yescharacteristic
323C2×C4#44211yescharacteristic
324C2×C4#44211yescharacteristic
325C2×C4#44211yescharacteristic
326C2×C4C2×C4×D411yescharacteristic
327C2×C4#54611yescharacteristic
328C2×C4#54811yescharacteristic
329C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
330C2×C4#57611yescharacteristic
331C2×C4#57811yescharacteristic
332C8—21no—
333C8—21no—
334C8—42no—
335C8—82no—
336C8C2×#11711yescharacteristic
337C8#52011yescharacteristic
338C8#55211yescharacteristic
339C8#55311yescharacteristic
340C22—21no—
341C22—21no—
342C22—44no—
343C22—84no—
344C22—84no—
345C22—82no—
346C22—82no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22C2×#44211yescharacteristic
351C22#332511yescharacteristic
352C22#336711yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—82no—
358C4—81no—
359C4—82no—
360C4—81no—
361C4—81no—
362C4—162no—
363C4#235211yescharacteristic
364C4#295111yescharacteristic
365C4#333011yescharacteristic
366C4#336611yescharacteristic
367C2—21no—
368C2—21no—
369C2—84no—
370C2—161no—
371C2#2275811yescharacteristic
372C2#3346811yescharacteristic
373C2#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. #22803C2semidirect
3. #22672C2semidirect
4. #13153C2semidirect
5. #13118C2semidirect
7. #12887C2semidirect
8. #12818C2semidirect
9. #12817C2semidirect
10. #12666C2semidirect
11. #12647C2semidirect
18. #2177C4semidirect
19. #2175C4semidirect
20. C4×SD32C4semidirect
21. C4×D16C4semidirect
59. C4×Q16C2×C4semidirect
60. C4×Q16C2×C4semidirect
67. C4×D8C2×C4semidirect
68. C4×D8C2×C4semidirect
94. #367C2×C4semidirect
95. #366C2×C4semidirect
100. #358C2×C4semidirect
103. #357C2×C4semidirect
105. #346C2×C4semidirect
106. C4×C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#228032
#226722
#131532
#131182
#130372
#128872
#128182
#128172
#126662
#126472