Finite Group Databasebeta

← #319121

Subgroups — #319121

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33207C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic
3#27948C211yesmaximal, characteristic
4#25902C211yesmaximal, characteristic
5#25722C211yesmaximal, characteristic
6C2×#1990C211yesmaximal, characteristic
7#10685C212yesmaximal, complemented
8#10318C212yesmaximal, complemented
9#9839C212yesmaximal, complemented
10#9826C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#498C2211yescharacteristic
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#451—22no—
17C22×C4×C8C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2084—22no—
22#2084—22no—
23#2024—22no—
24#2016—22no—
25#1990—22no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1990C2212yes—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1956C2212yes—
34#1941—22no—
35#1872—22no—
36#1853—22no—
37#1853—22no—
38#1844C2212yes—
39#1844C2212yes—
40#1844C2211yescharacteristic
41#1807—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×#322C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C22×Q16—24no—
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⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66#498—42no—
67#494—24no—
68#494—42no—
69#487—24no—
70#487—42no—
71#451—48no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×C22⋊C8D412yes—
79C2×C22⋊C8D412yes—
80C2×C4×C8—42no—
81C2×C4×C8D412yes—
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2312yes—
84C2×C4×C8C2312yes—
85C2×C4×C8C2311yescharacteristic
86#391—21no—
87#391—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#384C2312yes—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102#340—22no—
103#340—22no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336D412yes—
110#336D412yes—
111#322—42no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—44no—
116C22×Q8—41no—
117C22×Q8—44no—
118C22×Q8—82no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×Q16—48no—
125C2×Q16—48no—
126C2×Q16—84no—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8—42no—
130C22×C8C2×D412yes—
131C22×C8C2×D414yes—
132C22×C8C2×D414yes—
133C22×C8C2×D412yes—
134C4⋊Q8—24no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—44no—
143C4⋊Q8—44no—
144C4⋊Q8—42no—
145C4⋊Q8#4114yes—
146C22⋊Q8—44no—
147C22⋊Q8—44no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2411yescharacteristic
184#106—44no—
185#106—44no—
186#106—84no—
187#106—84no—
188#103—48no—
189#103—48no—
190C22⋊C8—24no—
191C22⋊C8SD1618yes—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—42no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203C24#12711yescharacteristic
204C2×Q8—24no—
205C2×Q8—44no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—44no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—816no—
216C2×Q8—82no—
217C2×Q8—82no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4#12711yescharacteristic
238C22×C4#12711yescharacteristic
239C22×C4#12711yescharacteristic
240C22×C4C2×SD1614yes—
241C22×C4#13714yes—
242C22×C4#13714yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246Q16—816no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—48no—
258C2×C8C2×SD1618yes—
259C2×C8#13714yes—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C42—21no—
289C42—21no—
290C42—22no—
291C42—22no—
292C42C2×#4112yes—
293C42H(5,𝔽2)12yes—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23#49212yes—
299C23#49712yes—
300C23#49712yes—
301C23C2×#12711yescharacteristic
302Q8—48no—
303Q8—88no—
304Q8—84no—
305Q8—84no—
306Q8—84no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—44no—
318C2×C4—88no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—164no—
337C2×C4#49212yes—
338C2×C4#49712yes—
339C2×C4#49712yes—
340C2×C4#52511yescharacteristic
341C2×C4#53411yescharacteristic
342C2×C4#54612yes—
343C2×C4#54612yes—
344C2×C4C22×SD1612yes—
345C2×C4C2×#13712yes—
346C2×C4C2×#13712yes—
347C2×C4C2×#13712yes—
348C2×C4#57612yes—
349C2×C4#57712yes—
350C8—24no—
351C8—28no—
352C8—42no—
353C8—44no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22#171614yes—
361C22#253911yescharacteristic
362C22C2×#49211yescharacteristic
363C22C2×#49711yescharacteristic
364C22C2×#49711yescharacteristic
365C22#315211yescharacteristic
366C22#315411yescharacteristic
367C22#315511yescharacteristic
368C4—22no—
369C4—22no—
370C4—22no—
371C4—84no—
372C4—161no—
373C4—161no—
374C4—161no—
375C4—161no—
376C4—162no—
377C4#311312yes—
378C4#319512yes—
379C4#329514yes—
380C4#331614yes—
381C2—24no—
382C2C2×#171611yescharacteristic
383C2#1962211yescharacteristic
384C2#1968311yescharacteristic
385C2#1970511yescharacteristic
386C2#2831311yescharacteristic
387C2#2833211yescharacteristic
388C2#2834711yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. #10685C2semidirect
8. #10318C2semidirect
9. #9839C2semidirect
10. #9826C2semidirect

Maximal subgroups

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

subgroupindex
#332072
#279482
#259022
#257222
C2×#19902
#106852
#103182
#98392
#98262
#81452