Finite Group Databasebeta

← #6498037

Subgroups — #6498037

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C212yesmaximal
2#33315C211yesmaximal, characteristic, complemented
3#33298C211yesmaximal, characteristic, complemented
4#33296C212yesmaximal, complemented
5#29879C214yesmaximal
6#29182C214yesmaximal, complemented
7#29181C214yesmaximal, complemented
8#27883C214yesmaximal
9#27882C214yesmaximal
10#22249C214yesmaximal, complemented
11#20463C211yesmaximal, characteristic
12C2×#499C2214yes—
13C2×#431C2212yes—
14#2864C2214yes—
15#2863—22no—
16#2863—22no—
17#2862C2212yes—
18#2859C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2855—22no—
22#2855—22no—
23#2851C2214yes—
24C4×C4⋊Q8C2212yes—
25C4×C4⋊Q8C2212yes—
26C4×#125C2214yes—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8—22no—
29#2295—22no—
30#2295—22no—
31#2283C2214yes—
32C2×C43C2211yescharacteristic
33#2088C2218yes—
34#2087C2214yes—
35#2086C2214yes—
36#2077C2218yes—
37#2076C2214yes—
38#2075C2214yes—
39#1993C2214yes—
40#1993C2214yes—
41#1993C2214yes—
42#1993C2214yes—
43#1991C2218yes—
44#1921C2218yes—
45#1920C2218yes—
46#1919C2218yes—
47#1918C2218yes—
48#1847C2214yes—
49#1847C2214yes—
50#1836C2214yes—
51#1836C2214yes—
52#1835C2214yes—
53#1767C2214yes—
54#1749C2214yes—
55#1744C2212yes—
56C4×#97C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C2×C4⋊Q8C2312yes—
59C2×C4⋊Q8C2312yes—
60C2×#125C2314yes—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C22×C42D411yescharacteristic
66C22×C42D412yes—
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2312yes—
70#499—216no—
71#431—28no—
72C2×#107C2314yes—
73C2×#107C2314yes—
74C2×#106C2318yes—
75C2×C4⋊C8C2314yes—
76C2×C22⋊C8C2314yes—
77C2×#97C2314yes—
78C2×#97C2314yes—
79C2×C4×C8C2312yes—
80C2×C4×C8C2312yes—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391—22no—
88#391—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389C2314yes—
98#389C2314yes—
99#389C2314yes—
100#389C2314yes—
101#387—24no—
102#387—24no—
103#387—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#385—24no—
110#385—24no—
111#384C2312yes—
112#384C2312yes—
113#383—24no—
114#382C2314yes—
115#382C2314yes—
116#382C2314yes—
117#381—24no—
118#380—22no—
119#380—22no—
120#380C2314yes—
121#380C2314yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2314yes—
127C4×C4⋊C4C2314yes—
128C4×C22⋊C4—24no—
129#376—24no—
130C43D411yescharacteristic
131C43D412yes—
132C43D411yescharacteristic
133C43C2311yescharacteristic
134C43C2312yes—
135C43C2311yescharacteristic
136#340C2314yes—
137#340C2318yes—
138#340C2314yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#336C2318yes—
144C22×Q8—22no—
145C22×Q8—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8C2414yes—
152C22×C8C2414yes—
153C4⋊Q8—28no—
154C4⋊Q8—28no—
155#125—216no—
156C22⋊Q8—48no—
157C22⋊Q8—48no—
158C4×Q8—48no—
159C4×Q8—48no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2414yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C42—24no—
179C2×C42—28no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—44no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2412yes—
200C2×C42C2411yescharacteristic
201C2×C42C2411yescharacteristic
202C2×C42C2412yes—
203#107—28no—
204#107—28no—
205#106—216no—
206C4⋊C8—28no—
207C22⋊C8—28no—
208#97—28no—
209#97—28no—
210C4×C8—24no—
211C4×C8#4118yes—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226C24C22×D411yescharacteristic
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D412yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D412yes—
256C22×C4C22×D412yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C2511yescharacteristic
260C2×C8—28no—
261C2×C8—24no—
262C2×C8—28no—
263C2×C8C2×#4118yes—
264C4⋊C4—28no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C42—22no—
283C42—22no—
284C42—24no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—22no—
290C42—24no—
291C42—41no—
292C42—41no—
293C42—42no—
294C42—41no—
295C42—42no—
296C42—42no—
297C42—41no—
298C42—48no—
299C42—48no—
300C42C2×#4114yes—
301C422−1+418yes—
302C23—21no—
303C23—22no—
304C23—22no—
305C23—21no—
306C23#57512yes—
307C23C23×D411yescharacteristic
308Q8—88no—
309Q8—88no—
310C2×C4—28no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4#53212yes—
346C2×C4#53212yes—
347C2×C4#53614yes—
348C2×C4#53614yes—
349C2×C4#57514yes—
350C2×C4#57512yes—
351C2×C4C22×#4112yes—
352C2×C4C2×2−1+414yes—
353C8—28no—
354C8—44no—
355C22—22no—
356C22—24no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22C2×#53211yescharacteristic
361C22C2×#53612yes—
362C22C2×#57512yes—
363C22C2×#57511yescharacteristic
364C22C2×#57511yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—24no—
369C4—24no—
370C4—24no—
371C4—41no—
372C4—42no—
373C4—41no—
374C4—84no—
375C4—84no—
376C4—84no—
377C4—84no—
378C4#350914yes—
379C2—42no—
380C2#3532211yescharacteristic
381C2#6067611yescharacteristic
382C2#6068811yescharacteristic
383C2#6086612yes—
384C2#6086612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33315C2semidirect
3. #33298C2semidirect
4. #33296C2semidirect
6. #29182C2semidirect
7. #29181C2semidirect
10. #22249C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#333152
#332982
#332962
#298792
#291822
#291812
#278832
#278822
#222492
#204632