Finite Group Databasebeta

← #314708

Subgroups — #314708

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27681C211yesmaximal, characteristic
2#27680C211yesmaximal, characteristic
3C8×C22⋊Q8C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#10656C212yesmaximal, complemented
6#10624C212yesmaximal, complemented
7#9616C212yesmaximal, complemented
8#8725C211yesmaximal, characteristic
9#8724C211yesmaximal, characteristic
10#8649C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic
12C2×C8×Q8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C4×C22⋊Q8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17#2100—21no—
18#2099—21no—
19#2096—21no—
20#2095—21no—
21#1984C2211yescharacteristic
22#1948C2212yes—
23#1947C2212yes—
24#1939C2211yescharacteristic
25#1916—21no—
26#1872—21no—
27#1872—21no—
28#1853—21no—
29#1852—21no—
30#1850—21no—
31#1834—21no—
32#1834—21no—
33#1834C2212yes—
34#1834C2212yes—
35#1803C2212yes—
36#1803C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C8×C4⋊C4—21no—
41C8×C4⋊C4C2211yescharacteristic
42C8×C22⋊C4—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#335C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×C8D411yescharacteristic
52C23×C8D411yescharacteristic
53C2×C22⋊Q8D411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C42C2311yescharacteristic
62C8×Q8—24no—
63C2×#107—22no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×#103—22no—
72C2×#103—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C4×C8—24no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83#391D411yescharacteristic
84#389D411yescharacteristic
85#387—21no—
86#385—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384C2312yes—
94#384C2312yes—
95#380D411yescharacteristic
96C4×C4⋊C4—21no—
97C4×C4⋊C4C2311yescharacteristic
98C4×C22⋊C4—21no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#335—24no—
115C22×Q8C2×D411yescharacteristic
116C23×C4—41no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22⋊Q8—22no—
136C22⋊Q8Q1614yes—
137C4×Q8—22no—
138C4×Q8—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42C2411yescharacteristic
175#107—82no—
176#106—82no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—42no—
184C4⋊C8—42no—
185#103—82no—
186#103—82no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—42no—
194C4×C8—42no—
195C4×C8#4114yes—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201C24#12111yescharacteristic
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C2×Q1612yes—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4C2×Q1612yes—
231C22×C4#13512yes—
232C22×C4#13512yes—
233C22×C4#13712yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8#12312yes—
262C2×C8#12312yes—
263C4⋊C4—21no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4C2×Q1612yes—
286C22⋊C4—24no—
287C22⋊C4—42no—
288C42—21no—
289C42—21no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42C2×#4112yes—
297C422−1+412yes—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#46412yes—
303C23#46712yes—
304C23#47112yes—
305C23C2×#12111yescharacteristic
306Q8—24no—
307Q8—42no—
308C2×C4—28no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—164no—
342C2×C4—164no—
343C2×C4#46412yes—
344C2×C4#46712yes—
345C2×C4#47112yes—
346C2×C4C2×#12311yescharacteristic
347C2×C4#53611yescharacteristic
348C2×C4#54712yes—
349C2×C4#55212yes—
350C2×C4C22×Q1611yescharacteristic
351C2×C4C2×#13511yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C2×C4C2×#13711yescharacteristic
354C2×C4#57511yescharacteristic
355C2×C4#57511yescharacteristic
356C2×C4#57612yes—
357C8—22no—
358C8—22no—
359C8—41no—
360C8—41no—
361C8—82no—
362C8—82no—
363C8—81no—
364C8—81no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22#168414yes—
372C22#248611yescharacteristic
373C22C2×#46411yescharacteristic
374C22#304011yescharacteristic
375C22C2×#46711yescharacteristic
376C22#305811yescharacteristic
377C22C2×#47111yescharacteristic
378C22C2×#47111yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—81no—
386C4—81no—
387C4—162no—
388C4—162no—
389C4#309111yescharacteristic
390C4#309511yescharacteristic
391C4#323211yescharacteristic
392C4#323611yescharacteristic
393C4#333812yes—
394C4#333912yes—
395C2—24no—
396C2C2×#168411yescharacteristic
397C2#1921411yescharacteristic
398C2#1927711yescharacteristic
399C2#1936411yescharacteristic
400C2#2765311yescharacteristic
401C2#2766211yescharacteristic
402C2#2767211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10656C2semidirect
6. #10624C2semidirect
7. #9616C2semidirect
10. #8649C2semidirect

Maximal subgroups

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

subgroupindex
#276812
#276802
C8×C22⋊Q82
C2×#18342
#106562
#106242
#96162
#87252
#87242
#86492
#85962