Finite Group Databasebeta

← #226939

Subgroups — #226939

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#454C211yesmaximal, characteristic
2#17852C212yesmaximal, complemented
3#17439C212yesmaximal, complemented
4#9639C211yesmaximal, characteristic, complemented
5#9549C212yesmaximal, complemented
6#9095C212yesmaximal, complemented
7#8957C211yesmaximal, characteristic, complemented
8#8840C212yesmaximal, complemented
9C8×#104C212yesmaximal, complemented
10C4×2−1+4C2211yescharacteristic
11C2×#454C411yescharacteristic
12#2988C2212yes—
13#2972C2211yescharacteristic
14C8×#41C2212yes—
15C4×AGL(1,ℤ/8)C2212yes—
16C4×#124C2211yescharacteristic
17#1888C411yescharacteristic
18#1883C412yes—
19#1860C412yes—
20#1857C411yescharacteristic
21#1831C2211yescharacteristic
22#1793C411yescharacteristic
23C4×#104C2212yes—
24C4×#101C2211yescharacteristic
25C42×C8C2211yescharacteristic
26#1580—22no—
27SD16⋊C8—22no—
28C8×SD16—22no—
29C8×D8—22no—
30#1565C2212yes—
31#1479—22no—
32#1479C2212yes—
33#1477—22no—
34#1477C2212yes—
35#1476—22no—
36#1476C2212yes—
37C2×Q8⋊C8—22no—
38C2×Q8⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40#1460C2211yescharacteristic
41C8×M16C2212yes—
42#1283C2212yes—
43#1278C2212yes—
44#1276C2212yes—
45C2×2−1+4C2×C411yescharacteristic
46C2×AGL(1,ℤ/8)C2×C412yes—
47C2×#124C2×C411yescharacteristic
48#520—22no—
49#520—41no—
50#520C2×C412yes—
51#519—22no—
52#519C2×C411yescharacteristic
53C4×#41—22no—
54C4×#41—41no—
55C4×#41C2312yes—
56C2×C4×Q8—22no—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4C2311yescharacteristic
59#454C818yescomplemented
60C8×Q8—22no—
61C8×Q8—22no—
62#442—22no—
63#440—22no—
64C4×SD16—22no—
65C4×D8—22no—
66C8×D4—22no—
67C8×D4—22no—
68C8×D4—22no—
69C8×D4—22no—
70C8×D4—42no—
71C8×D4—42no—
72#431—22no—
73#431—22no—
74#431C2312yes—
75#431C2312yes—
76#428C2×C412yes—
77C2×C4⋊C8—22no—
78C2×C4⋊C8C2312yes—
79C2×#104C2×C412yes—
80C2×#104C2×C412yes—
81#418C2×C412yes—
82#417C2×C412yes—
83C2×#101C2×C411yescharacteristic
84C2×#101C2×C411yescharacteristic
85C2×C22⋊C8—41no—
86C4×M16C2312yes—
87C2×C4×C8—22no—
88C2×C4×C8—41no—
89C2×C4×C8C2312yes—
90#391C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#388—41no—
93#382C2×C411yescharacteristic
94C4×C22⋊C4—41no—
95C43C2311yescharacteristic
96#343C2×C412yes—
97#339C2×C412yes—
98#339C2×C412yes—
99#336—41no—
100#335—22no—
101#334—22no—
102Q8⋊C8—22no—
103Q8⋊C8—22no—
104Q8⋊C8—22no—
105Q8⋊C8—22no—
106Q8⋊C8—22no—
107Q8⋊C8D412yes—
108Q8⋊C8D412yes—
109D4⋊C8—22no—
110D4⋊C8—22no—
111D4⋊C8—22no—
112D4⋊C8D412yes—
113D4⋊C8D412yes—
114#323—41no—
115#322—22no—
116C82—22no—
1172−1+4—24no—
1182−1+4—41no—
1192−1+4C2×C814yescomplemented
120C2×#41—22no—
121C2×#41—41no—
122C2×#41C22×C412yes—
123C22×Q8—22no—
124C22×Q8C22×C411yescharacteristic
125C22×D4C22×C411yescharacteristic
126C23×C4—21no—
127AGL(1,ℤ/8)—24no—
128AGL(1,ℤ/8)C2×C818yescomplemented
129C2×SD16—22no—
130C2×D8—22no—
131C2×M16C22×C412yes—
132C2×M16C22×C412yes—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138#124—41no—
139#124C2×C814yescomplemented
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—42no—
147C4×Q8—41no—
148C4×Q8C2×D412yes—
149C4×Q8#4112yes—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—41no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—41no—
159C4×D4C2×D412yes—
160C4×D4#4112yes—
161#117—22no—
162#117—22no—
163#117—42no—
164#117—41no—
165#117—42no—
166#117C22×C412yes—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C22⋊C4—21no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C2411yescharacteristic
183#107—22no—
184#106—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190C4⋊C8C2×D412yes—
191C4⋊C8C2×D412yes—
192#104—24no—
193#104C2×C818yescomplemented
194#103—22no—
195#102—22no—
196#102—22no—
197#102—22no—
198#101—41no—
199#101C2×C814yescomplemented
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—42no—
205C22⋊C8—42no—
206#97—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—42no—
212C4×C8—42no—
213C4×C8—44no—
214C4×C8C2×D412yes—
215C4×C8C2×D412yes—
216#95—41no—
217#95—41no—
218C24—21no—
219#41—22no—
220#41—22no—
221#41—28no—
222#41—44no—
223#41—44no—
224#41—44no—
225#41C22×C814yes—
226#41#13114yes—
227C2×Q8—21no—
228C2×Q8—28no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—44no—
233C2×Q8C4×D412yes—
234C2×Q8C22×C812yes—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4C4×D412yes—
241C2×D4C22×C812yes—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4C23×C411yescharacteristic
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C4H(5,𝔽2)11yescharacteristic
258SD16—28no—
259D8—28no—
260M16—22no—
261M16—22no—
262M16—22no—
263M16C22×C814yes—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—84no—
274C2×C8—82no—
275C2×C8—162no—
276C2×C8—162no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4C4×D412yes—
286C4⋊C4C4×D412yes—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—82no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—21no—
298C42—22no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—41no—
304C42—41no—
305C42C22×C812yes—
306C42C22×D412yes—
307C42C2×#4112yes—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—81no—
313C23—81no—
314C23#51811yescharacteristic
315Q8—22no—
316Q8—24no—
317Q8—44no—
318Q8—42no—
319Q8C8×D414yes—
320D4—24no—
321D4—24no—
322D4—24no—
323D4—22no—
324D4—44no—
325D4—42no—
326D4C8×D414yes—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—48no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—161no—
353C2×C4—162no—
354C2×C4C2×C4×D412yes—
355C2×C4D4211yescharacteristic
356C2×C4#54612yes—
357C2×C4#55111yescharacteristic
358C2×C4C23×C811yescharacteristic
359C2×C4C2×#13112yes—
360C2×C4#56811yescharacteristic
361C8—24no—
362C8—24no—
363C8—42no—
364C8—42no—
365C8—42no—
366C8—42no—
367C8—82no—
368C8—82no—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—82no—
376C22—81no—
377C22—81no—
378C22—161no—
379C22#235811yescharacteristic
380C22#297211yescharacteristic
381C22#297711yescharacteristic
382C4—24no—
383C4—21no—
384C4—22no—
385C4—42no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—81no—
391C4—81no—
392C4C2×C8×D412yes—
393C4#293312yes—
394C4#329012yes—
395C2—21no—
396C2—21no—
397C2—44no—
398C2—82no—
399C2#2284211yescharacteristic
400C2#2289511yescharacteristic
401C2#2289911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC22#2358
prime radicalC2×#454C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17852C2semidirect
3. #17439C2semidirect
4. #9639C2semidirect
5. #9549C2semidirect
6. #9095C2semidirect
7. #8957C2semidirect
8. #8840C2semidirect
9. C8×#104C2semidirect
59. #454C8semidirect
119. 2−1+4C2×C8semidirect
128. AGL(1,ℤ/8)C2×C8semidirect
139. #124C2×C8semidirect
193. #104C2×C8semidirect
199. #101C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#4542
#178522
#174392
#96392
#95492
#90952
#89572
#88402
C8×#1042