Finite Group Databasebeta

← #226936

Subgroups — #226936

Order 512.

771 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#456C211yesmaximal, characteristic
2#17853C212yesmaximal, complemented
3#17439C212yesmaximal
4#9641C211yesmaximal, characteristic, complemented
5#9549C212yesmaximal, complemented
6#9096C212yesmaximal, complemented
7#8958C211yesmaximal, characteristic, complemented
8#8840C212yesmaximal, complemented
9#8191C212yesmaximal, complemented
10C4×2−1+4C2211yescharacteristic
11C2×#456C411yescharacteristic
12#2988C2212yes—
13#2982C2211yescharacteristic
14C8×#41C2212yes—
15C4×#137C2212yes—
16C4×C4⋊Q8C2211yescharacteristic
17#1886C411yescharacteristic
18#1882C412yes—
19#1859C412yes—
20#1857C411yescharacteristic
21#1831C2211yescharacteristic
22#1793C411yescharacteristic
23C4×#104C2212yes—
24C4×#101C2211yescharacteristic
25#1744C2211yescharacteristic
26#1579—22no—
27SD16⋊C8—22no—
28C8×Q16—22no—
29C8×SD16—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×Q8⋊C8C2212yes—
40#1461C2211yescharacteristic
41C8×M16C2212yes—
42#1283C2212yes—
43#1278C2212yes—
44#1276C2212yes—
45C2×2−1+4C2×C411yescharacteristic
46C2×#137C2×C412yes—
47C2×C4⋊Q8C2×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×Q8C2311yescharacteristic
59#456C818yescomplemented
60C8×Q8—22no—
61C8×Q8—22no—
62C8×Q8—22no—
63C8×Q8—22no—
64#441—22no—
65#440—22no—
66C4×Q16—22no—
67C4×SD16—22no—
68C8×D4—22no—
69C8×D4—22no—
70C8×D4—42no—
71C8×D4—42no—
72#431—22no—
73#431—22no—
74#431—41no—
75#431C2312yes—
76#431C2312yes—
77#428C2×C412yes—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2312yes—
80C2×#104C2×C412yes—
81C2×#104C2×C412yes—
82#419C2×C412yes—
83#417C2×C412yes—
84C2×#101C2×C411yescharacteristic
85C2×#101C2×C411yescharacteristic
86C4×M16C2312yes—
87C2×C4×C8—22no—
88C2×C4×C8C2312yes—
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#389—41no—
92#384C2×C411yescharacteristic
93C4×C4⋊C4—41no—
94C43C2311yescharacteristic
95#343C2×C412yes—
96#339C2×C412yes—
97#339C2×C412yes—
98#336—41no—
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⋊C8—22no—
108Q8⋊C8—22no—
109Q8⋊C8D412yes—
110Q8⋊C8D412yes—
111D4⋊C8—22no—
112D4⋊C8D412yes—
113D4⋊C8D412yes—
114#324—41no—
115#322—22no—
116C82—22no—
1172−1+4—24no—
1182−1+4—41no—
1192−1+4C2×C814yes—
120C2×#41—22no—
121C2×#41—41no—
122C2×#41C22×C412yes—
123C22×Q8—22no—
124C22×Q8C22×C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126#137—24no—
127#137C2×C818yescomplemented
128C2×Q16—22no—
129C2×SD16—22no—
130C2×M16C22×C412yes—
131C2×M16C22×C412yes—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—42no—
135C22×C8—42no—
136C4⋊Q8—41no—
137C4⋊Q8C2×C814yescomplemented
138C4×Q8—22no—
139C4×Q8—22no—
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—41no—
154C4×D4—42no—
155C4×D4—42no—
156C4×D4—41no—
157C4×D4C2×D412yes—
158C4×D4#4112yes—
159#117—22no—
160#117—22no—
161#117—42no—
162#117—41no—
163#117—42no—
164#117C22×C412yes—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—42no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2411yescharacteristic
182#107—22no—
183#106—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190C4⋊C8C2×D412yes—
191C4⋊C8C2×D412yes—
192#104—24no—
193#104C2×C818yescomplemented
194#103—22no—
195#103—22no—
196#103—22no—
197#102—22no—
198#101—41no—
199#101C2×C814yescomplemented
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—42no—
203C22⋊C8—42no—
204#97—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—42no—
210C4×C8—42no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8C2×D412yes—
214C4×C8C2×D412yes—
215#95—41no—
216#95—41no—
217#41—22no—
218#41—22no—
219#41—28no—
220#41—44no—
221#41—44no—
222#41—44no—
223#41C22×C814yes—
224#41#13114yes—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—28no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—44no—
234C2×Q8C4×D412yes—
235C2×Q8C22×C812yes—
236C2×Q8C22×C812yes—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4C4×D412yes—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C23×C411yescharacteristic
252C22×C4H(5,𝔽2)11yescharacteristic
253C22×C4H(5,𝔽2)11yescharacteristic
254Q16—28no—
255SD16—28no—
256M16—22no—
257M16—22no—
258M16—22no—
259M16C22×C814yes—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—82no—
269C2×C8—84no—
270C2×C8—82no—
271C2×C8—162no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4C4×D412yes—
284C4⋊C4C4×D412yes—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—42no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—21no—
293C42—22no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—81no—
302C42C22×C812yes—
303C42C22×D412yes—
304C42C2×#4112yes—
305C23—22no—
306C23#51811yescharacteristic
307Q8—22no—
308Q8—24no—
309Q8—24no—
310Q8—44no—
311Q8—42no—
312Q8—42no—
313Q8C8×D414yes—
314D4—24no—
315D4—24no—
316D4—22no—
317D4—44no—
318D4C8×D414yes—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—48no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—162no—
347C2×C4C2×C4×D412yes—
348C2×C4D4211yescharacteristic
349C2×C4#54612yes—
350C2×C4#55111yescharacteristic
351C2×C4C23×C811yescharacteristic
352C2×C4C2×#13112yes—
353C2×C4#56811yescharacteristic
354C8—24no—
355C8—42no—
356C8—42no—
357C8—42no—
358C8—42no—
359C8—42no—
360C8—82no—
361C8—82no—
362C22—21no—
363C22—21no—
364C22—24no—
365C22—42no—
366C22—42no—
367C22#235811yescharacteristic
368C22#297211yescharacteristic
369C22#297711yescharacteristic
370C4—24no—
371C4—21no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4C2×C8×D412yes—
382C4#293312yes—
383C4#328912yes—
384C2—21no—
385C2—21no—
386C2—44no—
387C2#2284211yescharacteristic
388C2#2289211yescharacteristic
389C2#2289811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC22#2358
prime radicalC2×2−1+4C2×C4
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. #17853C2semidirect
4. #9641C2semidirect
5. #9549C2semidirect
6. #9096C2semidirect
7. #8958C2semidirect
8. #8840C2semidirect
9. #8191C2semidirect
59. #456C8semidirect
127. #137C2×C8semidirect
137. C4⋊Q8C2×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×#4562
#178532
#174392
#96412
#95492
#90962
#89582
#88402
#81912