Finite Group Databasebeta

← #329712

Subgroups — #329712

Order 512.

803 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32956C212yesmaximal
2#29876C211yesmaximal, characteristic, complemented
3#21788C211yesmaximal, characteristic
4#10625C212yesmaximal, complemented
5#10481C212yesmaximal, complemented
6#10480C212yesmaximal, complemented
7#10278C212yesmaximal, complemented
8#9136C212yesmaximal, complemented
9#8600C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C2×#500C2211yescharacteristic, complemented
12C2×#476C2212yes—
13C2×#476C2212yes—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2086—21no—
18#2086—21no—
19#2075—21no—
20#2075—21no—
21#1990C2212yescomplemented
22#1955—21no—
23#1955—21no—
24#1954C2212yes—
25#1954C2212yes—
26#1864—22no—
27#1864—22no—
28#1853—22no—
29#1834C2211yescharacteristic, complemented
30#1802C2212yes—
31#1802C2212yes—
32#1797C2212yes—
33#1786—22no—
34#1779—21no—
35#1779—21no—
36C2×#335C2211yescharacteristic, complemented
37C2×#334C2211yescharacteristic, complemented
38C2×#333C2212yescomplemented
39C2×#333C2212yescomplemented
40#1296C2212yescomplemented
41#1296C2212yescomplemented
42#1277C2212yescomplemented
43C23×D4D411yescharacteristic
44C24×C4—21no—
45C2×C4⋊Q8C2312yes—
46C2×C22⋊Q8—22no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4D412yes—
54C2×C4×D4C2312yes—
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C42C2311yescharacteristic
58#500—41no—
59#500—41no—
60#476—42no—
61#476—42no—
62#476—42no—
63#476—42no—
64#432—41no—
65#432—41no—
66#431—41no—
67#431—41no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#106—22no—
71C2×#106—22no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8D411yescharacteristic
88C2×C22⋊C8D411yescharacteristic
89C2×#97—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#391—22no—
93#389—22no—
94#389—22no—
95#384C2312yes—
96#380—22no—
97#380—22no—
98#340—21no—
99#340—21no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#336—22no—
105#335—41no—
106#334—41no—
107#333—42no—
108#333—42no—
109C25—21no—
110C22×Q8—22no—
111C22×D4—24no—
112C22×D4—21no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4—42no—
116C22×D4C2×D412yes—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—82no—
123C23×C4—82no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C22⋊Q8—84no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—42no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—42no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#107—42no—
173#107—42no—
174#106—44no—
175#106—44no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184#102—44no—
185#102—44no—
186#102—44no—
187#102—44no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—44no—
191C22⋊C8—42no—
192C22⋊C8—82no—
193C22⋊C8—82no—
194C22⋊C8—82no—
195C22⋊C8—82no—
196#97—42no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201#95—42no—
202#95—42no—
203C24—22no—
204C24—44no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24—82no—
209C24#12111yescharacteristic
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—44no—
222C2×D4—48no—
223C2×D4—44no—
224C2×D4#13714yes—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—88no—
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—82no—
289C4⋊C4—84no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—84no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42H(5,𝔽2)11yescharacteristic
301C42H(5,𝔽2)11yescharacteristic
302C42H(5,𝔽2)11yescharacteristic
303C42H(5,𝔽2)11yescharacteristic
304C23—21no—
305C23—22no—
306C23—22no—
307C23—21no—
308C23—24no—
309C23—48no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—88no—
317C23—84no—
318C23—88no—
319C23—84no—
320C23#47212yes—
321C23C2×#12111yescharacteristic
322Q8—162no—
323Q8—162no—
324D4—24no—
325D4—24no—
326D4—48no—
327D4—42no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—88no—
350C2×C4—84no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4#53411yescharacteristic
362C2×C4#53411yescharacteristic
363C2×C4#53411yescharacteristic
364C2×C4#53511yescharacteristic
365C2×C4#53811yescharacteristic
366C2×C4#53811yescharacteristic
367C2×C4C2×#13712yes—
368C2×C4#57512yes—
369C2×C4#57712yes—
370C2×C4#57712yes—
371C8—41no—
372C8—41no—
373C8—41no—
374C8—41no—
375C8—82no—
376C8—82no—
377C8—82no—
378C22—24no—
379C22—22no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—816no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22#261711yescharacteristic
392C22#304212yes—
393C22#305612yes—
394C22C2×#47211yescharacteristic
395C22#306711yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—162no—
403C4—162no—
404C4#319511yescharacteristic
405C4#319612yes—
406C4#319711yescharacteristic
407C4#320011yescharacteristic
408C4#320011yescharacteristic
409C4#322212yes—
410C2—42no—
411C2—84no—
412C2#1946512yes—
413C2#1946612yes—
414C2#2972311yescharacteristic
415C2#2974311yescharacteristic
416C2#2974811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
2. #29876C2semidirect
4. #10625C2semidirect
5. #10481C2semidirect
6. #10480C2semidirect
7. #10278C2semidirect
8. #9136C2semidirect
9. #8600C2semidirect
11. C2×#500C22semidirect
21. #1990C22semidirect
29. #1834C22semidirect
36. C2×#335C22semidirect
37. C2×#334C22semidirect
38. C2×#333C22semidirect
39. C2×#333C22semidirect
40. #1296C22semidirect
41. #1296C22semidirect
42. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#329562
#298762
#217882
#106252
#104812
#104802
#102782
#91362
#86002