Finite Group Databasebeta

← #498064

Subgroups — #498064

Order 512.

545 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#509C211yesmaximal, characteristic
2#22813C211yesmaximal, characteristic, complemented
3#22803C211yesmaximal, characteristic, complemented
4#13249C212yesmaximal
5#13143C212yesmaximal, complemented
6#13142C212yesmaximal, complemented
7#12903C211yesmaximal, characteristic, complemented
8#12818C211yesmaximal, characteristic, complemented
9#12816C211yesmaximal, characteristic, complemented
10#12677C212yesmaximal, complemented
11#12666C211yesmaximal, characteristic, complemented
12C2×#509C2211yescharacteristic
13C4×#135C2211yescharacteristic
14C2×C4×D8C2211yescharacteristic
15#2925—21no—
16#2354—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#1926—22no—
35#1848C2211yescharacteristic
36#1844C2211yescharacteristic
37#1806—21no—
38#1802—21no—
39#1776C2212yes—
40#1773C2212yes—
41#1770C2211yescharacteristic
42#1387C2212yes—
43#1386C2212yes—
44#1375C2212yes—
45C2×#135C2311yescharacteristic
46C22×D8C2311yescharacteristic
47C4×#41—21no—
48C2×C4×D4—21no—
49C2×#117—41no—
50#509—22no—
51#509—22no—
52C2×SD32C2×C412yes—
53C2×D16C2×C412yes—
54C2×M32C2312yes—
55C2×M32C2311yescharacteristic
56C2×M32C2311yescharacteristic
57C4×Q16C2×C411yescharacteristic
58C4×Q16C2×C411yescharacteristic
59C4×SD16—21no—
60C4×SD16—21no—
61C4×D8—21no—
62C4×D8—21no—
63C4×D8C2×C412yes—
64C4×D8C2×C412yes—
65C4×D8C2×C411yescharacteristic
66C4×D8C2×C411yescharacteristic
67#433—42no—
68#432—41no—
69#431—41no—
70C2×#108—21no—
71C2×#108C2312yes—
72#427C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#106—21no—
75C2×#106C2312yes—
76#423—21no—
77C2×C4⋊C8D411yescharacteristic
78C2×#104—22no—
79#416—21no—
80C2×#102—21no—
81C2×#102—22no—
82#408—22no—
83#405—22no—
84C2×#97C2312yes—
85C2×C4×C8C2311yescharacteristic
86#390—22no—
87#388—42no—
88#384—21no—
89#380—41no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4D411yescharacteristic
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—
142#117—84no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C42—41no—
152C2×C42—42no—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155SD32—24no—
156D16—24no—
157M32—21no—
158M32—21no—
159M32—22no—
160M32—22no—
161M32#4112yes—
162M32#4112yes—
163C2×C16—22no—
164C2×C16C22×C412yes—
165C2×C16C22×C412yes—
166#108—22no—
167#108—22no—
168#108—81no—
169#107—21no—
170#107C22×C411yescharacteristic
171#107C22×C411yescharacteristic
172#107C22×C412yes—
173#106—21no—
174#106—22no—
175#106—22no—
176#106—81no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181#104—82no—
182#103—21no—
183#103—21no—
184#102—22no—
185#102—22no—
186#102—21no—
187#102—21no—
188#102—82no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191#97—22no—
192#97—22no—
193#97—42no—
194C4×C8—42no—
195C4×C8C22×C411yescharacteristic
196C4×C8C22×C411yescharacteristic
197C4×C8#4111yescharacteristic
198C4×C8#4111yescharacteristic
199#95—21no—
200#95—42no—
201C24—41no—
202#41—41no—
203#41—41no—
204#41—41no—
205#41—41no—
206C2×Q8—21no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227C22×C4C22×D411yescharacteristic
228Q16—21no—
229Q16—21no—
230SD16—41no—
231SD16—41no—
232D8—21no—
233D8—22no—
234D8—22no—
235D8—21no—
236D8—41no—
237D8—41no—
238M16—42no—
239M16—42no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—42no—
246C2×C8C4×D411yescharacteristic
247C2×C8C4×D411yescharacteristic
248C2×C8C23×C411yescharacteristic
249C2×C8C2×#4111yescharacteristic
250C2×C8C2×#4111yescharacteristic
251C2×C8C2×#4111yescharacteristic
252C2×C8H(5,𝔽2)11yescharacteristic
253C2×C82−1+411yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—161no—
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—84no—
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42AGL(1,ℤ/8)11yescharacteristic
281C42AGL(1,ℤ/8)11yescharacteristic
282C16—22no—
283C16—42no—
284C16#11714yescomplemented
285C23—41no—
286C23—42no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23C2×AGL(1,ℤ/8)11yescharacteristic
294Q8—41no—
295Q8—41no—
296D4—42no—
297D4—41no—
298D4—42no—
299D4—41no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—81no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—84no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—161no—
324C2×C4#44211yescharacteristic
325C2×C4#44211yescharacteristic
326C2×C4#44211yescharacteristic
327C2×C4#44211yescharacteristic
328C2×C4C2×C4×D411yescharacteristic
329C2×C4#54711yescharacteristic
330C2×C4#55011yescharacteristic
331C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
332C2×C4#57711yescharacteristic
333C2×C4#57711yescharacteristic
334C8—21no—
335C8—21no—
336C8—42no—
337C8—82no—
338C8C2×#11711yescharacteristic
339C8#52011yescharacteristic
340C8#55511yescharacteristic
341C8#55611yescharacteristic
342C22—21no—
343C22—21no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—81no—
348C22—81no—
349C22—82no—
350C22—161no—
351C22—161no—
352C22—161no—
353C22—161no—
354C22C2×#44211yescharacteristic
355C22#335411yescharacteristic
356C22#339711yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4—82no—
364C4—81no—
365C4—162no—
366C4AGL(1,ℤ/16)12yes—
367C4#235411yescharacteristic
368C4#295211yescharacteristic
369C4#335411yescharacteristic
370C4#339711yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2—162no—
375C2#2281311yescharacteristic
376C2C2×AGL(1,ℤ/16)11yescharacteristic
377C2#3346911yescharacteristic
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. #22813C2semidirect
3. #22803C2semidirect
5. #13143C2semidirect
6. #13142C2semidirect
7. #12903C2semidirect
8. #12818C2semidirect
9. #12816C2semidirect
10. #12677C2semidirect
11. #12666C2semidirect
17. #2177C4semidirect
18. #2175C4semidirect
19. C4×SD32C4semidirect
20. C4×D16C4semidirect
284. C16#117semidirect

Maximal subgroups

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

subgroupindex
C4×#5092
#228132
#228032
#132492
#131432
#131422
#129032
#128182
#128162
#126772
#126662