Finite Group Databasebeta

← #319040

Subgroups — #319040

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33170C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#27961C211yesmaximal, characteristic, complemented
4#25903C211yesmaximal, characteristic
5#25856C211yesmaximal, characteristic, complemented
6C2×#1990C211yesmaximal, characteristic, complemented
7#10681C212yesmaximal, complemented
8#10310C212yesmaximal, complemented
9#9822C212yesmaximal, complemented
10#9821C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#492C2211yescharacteristic
14C2×#487C2212yes—
15C2×#486C2212yes—
16C2×#465—22no—
17C22×C4×C8C2211yescharacteristic
18#2857C2211yescharacteristic
19#2856C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2100—22no—
22#2071—22no—
23#2043—22no—
24#2023—22no—
25#1990—22no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1968—22no—
31#1957C2212yes—
32#1957C2211yescharacteristic
33#1956C2212yes—
34#1956C2211yescharacteristic
35#1874—22no—
36#1874—22no—
37#1844C2211yescharacteristic
38#1834C2212yes—
39#1834C2212yes—
40#1824—22no—
41#1787—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×C82C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C22×SD16—22no—
49C23×C8D412yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2312yes—
57C2×#127C2311yescharacteristic
58C2×C22⋊Q8—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C42C2311yescharacteristic
68#492—42no—
69#487—24no—
70#487—42no—
71#486—24no—
72#486—42no—
73#465—48no—
74C2×#106—22no—
75C2×#103—24no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#102—24no—
79C2×#102—22no—
80C2×#102—24no—
81C2×C22⋊C8D412yes—
82C2×C22⋊C8D412yes—
83C2×C4×C8—42no—
84C2×C4×C8D412yes—
85C2×C4×C8C2312yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#391—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#389—21no—
94#389—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#384C2312yes—
101#380—21no—
102#380—21no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336D412yes—
110#336D412yes—
111C82—42no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—44no—
115C22×Q8—41no—
116C22×Q8—44no—
117C22×Q8—82no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—41no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C2×SD16—44no—
128C2×SD16—44no—
129C2×SD16—44no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8SD1618yes—
133C22×C8C2×D414yes—
134C22×C8C2×D414yes—
135C22×C8C2×D412yes—
136C22×C8C2×D412yes—
137C4⋊Q8—24no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145C4⋊Q8—42no—
146C4⋊Q8#4114yes—
147#127—22no—
148#127—22no—
149C22⋊Q8—44no—
150#121—44no—
151#121—44no—
152#121—44no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—82no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2411yescharacteristic
188#106—44no—
189#103—48no—
190#103—44no—
191#103—84no—
192#102—48no—
193#102—44no—
194#102—84no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—42no—
204#95—41no—
205#95—41no—
206C24—41no—
207C24#12711yescharacteristic
208C2×Q8—24no—
209C2×Q8—44no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—44no—
217C2×Q8—816no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—82no—
231C2×D4—82no—
232C2×D4—82no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4#12711yescharacteristic
252C22×C4#12711yescharacteristic
253C22×C4#12711yescharacteristic
254C22×C4C2×SD1614yes—
255C22×C4#13514yes—
256C22×C4#13514yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260SD16—88no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—48no—
272C2×C8C2×SD1614yes—
273C2×C8C2×SD1614yes—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C42—21no—
300C42—21no—
301C42—22no—
302C42—22no—
303C42C2×#4112yes—
304C42H(5,𝔽2)12yes—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23#49212yes—
317C23#49512yes—
318C23#49512yes—
319C23C2×#12711yescharacteristic
320Q8—48no—
321Q8—88no—
322Q8—84no—
323Q8—82no—
324Q8—82no—
325D4—82no—
326D4—82no—
327D4—84no—
328D4—84no—
329D4—164no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—44no—
341C2×C4—88no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—164no—
357C2×C4#49212yes—
358C2×C4#49512yes—
359C2×C4#49512yes—
360C2×C4#52511yescharacteristic
361C2×C4#53411yescharacteristic
362C2×C4#54612yes—
363C2×C4#54612yes—
364C2×C4C22×SD1612yes—
365C2×C4C2×#13512yes—
366C2×C4C2×#13512yes—
367C2×C4#57512yes—
368C2×C4#57612yes—
369C2×C4#57812yes—
370C8—24no—
371C8—24no—
372C8—44no—
373C8—44no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22#170814yes—
388C22#253911yescharacteristic
389C22C2×#49211yescharacteristic
390C22C2×#49511yescharacteristic
391C22C2×#49511yescharacteristic
392C22#315311yescharacteristic
393C22#315511yescharacteristic
394C22#315611yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—84no—
399C4—161no—
400C4—161no—
401C4—162no—
402C4—161no—
403C4#310312yes—
404C4#318712yes—
405C4#329714yes—
406C4#330014yes—
407C2—24no—
408C2—321no—
409C2C2×#170811yescharacteristic
410C2#1962711yescharacteristic
411C2#1962811yescharacteristic
412C2#1965511yescharacteristic
413C2#2830511yescharacteristic
414C2#2833111yescharacteristic
415C2#2833511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27961C2semidirect
5. #25856C2semidirect
6. C2×#1990C2semidirect
7. #10681C2semidirect
8. #10310C2semidirect
9. #9822C2semidirect
10. #9821C2semidirect
11. C8×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#331702
#279732
#279612
#259032
#258562
C2×#19902
#106812
#103102
#98222
#98212
C8×C22⋊C82