Finite Group Databasebeta

← #314760

Subgroups — #314760

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27681C211yesmaximal, characteristic
2#27647C211yesmaximal, characteristic
3#21597C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#10657C212yesmaximal, complemented
6#10626C212yesmaximal, complemented
7#9619C212yesmaximal, complemented
8#8725C211yesmaximal, characteristic
9#8723C211yesmaximal, characteristic
10#8655C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic
12C2×#483—21no—
13C2×#446C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C4×C22⋊Q8C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18#2100—21no—
19#2095—21no—
20#2084—21no—
21#2072—21no—
22#1984C2211yescharacteristic
23#1948C2212yes—
24#1947C2212yes—
25#1939C2211yescharacteristic
26#1916—21no—
27#1872—21no—
28#1852—21no—
29#1850—21no—
30#1834—21no—
31#1834—21no—
32#1834C2212yes—
33#1834C2212yes—
34#1823—21no—
35#1803C2212yes—
36#1803C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1771—21no—
41#1770C2211yescharacteristic
42#1752—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
62#483—44no—
63#446—24no—
64C2×#107—22no—
65C2×#106—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×#103—22no—
73C2×#103—22no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×C4×C8—24no—
80C2×C4×C8D411yescharacteristic
81C2×C4×C8D411yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#391D411yescharacteristic
85#389D411yescharacteristic
86#387—21no—
87#385—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384C2312yes—
95#384C2312yes—
96#380D411yescharacteristic
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2311yescharacteristic
99C4×C22⋊C4—21no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#335—24no—
116C22×Q8C2×D411yescharacteristic
117C23×C4—41no—
118C23×C4—41no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4#4111yescharacteristic
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177#107—82no—
178#106—44no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186C4⋊C8—42no—
187#103—44no—
188#103—82no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192#97—42no—
193#97—42no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8#4114yes—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203C24#12111yescharacteristic
204C2×Q8—22no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#13512yes—
233C22×C4AGL(1,ℤ/8)12yes—
234C22×C4#13712yes—
235C22×C4#13712yes—
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C2×#4111yescharacteristic
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—21no—
248C2×C8—21no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8#12312yes—
264C2×C8#12312yes—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—22no—
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—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—22no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42C2×#4112yes—
299C422−1+412yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23#46412yes—
305C23#46812yes—
306C23#47212yes—
307C23C2×#12111yescharacteristic
308Q8—42no—
309Q8—42no—
310C2×C4—28no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—24no—
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—41no—
336C2×C4—41no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—164no—
344C2×C4—164no—
345C2×C4#46412yes—
346C2×C4#46812yes—
347C2×C4#47312yes—
348C2×C4C2×#12311yescharacteristic
349C2×C4#53611yescharacteristic
350C2×C4#54712yes—
351C2×C4#55212yes—
352C2×C4C2×#13511yescharacteristic
353C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57511yescharacteristic
357C2×C4#57511yescharacteristic
358C2×C4#57712yes—
359C8—22no—
360C8—22no—
361C8—41no—
362C8—41no—
363C8—82no—
364C8—82no—
365C8—81no—
366C8—81no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22#169814yes—
374C22#248611yescharacteristic
375C22C2×#46411yescharacteristic
376C22#304011yescharacteristic
377C22C2×#46811yescharacteristic
378C22#305711yescharacteristic
379C22C2×#47211yescharacteristic
380C22C2×#47311yescharacteristic
381C4—22no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—162no—
390C4—162no—
391C4#309111yescharacteristic
392C4#309511yescharacteristic
393C4#323211yescharacteristic
394C4#323611yescharacteristic
395C4#334412yes—
396C4#334512yes—
397C2—24no—
398C2C2×#169811yescharacteristic
399C2#1921311yescharacteristic
400C2#1930011yescharacteristic
401C2#1936211yescharacteristic
402C2#2765311yescharacteristic
403C2#2766311yescharacteristic
404C2#2767511yescharacteristic
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. #10657C2semidirect
6. #10626C2semidirect
7. #9619C2semidirect
10. #8655C2semidirect

Maximal subgroups

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

subgroupindex
#276812
#276472
#215972
C2×#18342
#106572
#106262
#96192
#87252
#87232
#86552
#85962