Finite Group Databasebeta

← #6480123

Subgroups — #6480123

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C212yesmaximal
2#33144C214yesmaximal
3#30296C211yesmaximal, characteristic
4#30250C211yesmaximal, characteristic
5#30199C212yesmaximal
6#29575C214yesmaximal, complemented
7#29553C214yesmaximal, complemented
8#29520C214yesmaximal, complemented
9#29501C214yesmaximal, complemented
10#22189C214yesmaximal
11#20364C211yesmaximal, characteristic
12C2×#470C2214yes—
13C2×C4×M16C2212yes—
14#2864—22no—
15#2863—22no—
16#2863C2214yes—
17#2862C2212yes—
18#2859C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2855—22no—
22#2855C2214yes—
23#2851—22no—
24C4×C4⋊Q8C2212yes—
25C4×C4⋊Q8C2212yes—
26C4×#125—22no—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8C2214yes—
29#2295—22no—
30#2295C2214yes—
31#2283—22no—
32C2×C43C2211yescharacteristic
33#2099C2214yes—
34#2096C2214yes—
35#2084C2214yes—
36#2072C2214yes—
37#2057C2214yes—
38#2043C2214yes—
39#2024C2214yes—
40#2016C2214yes—
41#1971C2214yes—
42#1969C2214yes—
43#1943C2214yes—
44#1941C2214yes—
45#1935C2218yes—
46#1933C2218yes—
47#1928C2218yes—
48#1919C2218yes—
49#1894C2218yes—
50#1878C2218yes—
51#1865C2218yes—
52#1809C2214yes—
53#1745C2214yes—
54#1744C2214yes—
55#1728C2211yescharacteristic
56#1728C2212yes—
57#1728C2211yescharacteristic
58C22×Q16C2314yes—
59C22×M16C2312yes—
60C22×M16C2312yes—
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2312yes—
63C2×#125—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2314yes—
66C2×C4×Q8—22no—
67C2×C4×Q8C2314yes—
68C22×C42D412yes—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#470—216no—
74C2×#107C2314yes—
75C2×#106C2314yes—
76C2×#106C2314yes—
77C2×#103C2318yes—
78C2×#103C2318yes—
79C2×#103C2318yes—
80C4×M16—28no—
81C2×#97C2314yes—
82C2×C4×C8C2314yes—
83#391—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391C2314yes—
88#391C2314yes—
89#391C2314yes—
90#391C2314yes—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389C2314yes—
100#389C2314yes—
101#389C2314yes—
102#389C2314yes—
103#387—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#385—24no—
112#385—24no—
113#384C2312yes—
114#384C2312yes—
115#383—24no—
116#382—22no—
117#382—22no—
118#382—22no—
119#381—24no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380C2314yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4C2314yes—
130C4×C22⋊C4—24no—
131#376—24no—
132C43D412yes—
133C43D412yes—
134C43C2312yes—
135C43C2312yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#336C2314yes—
140#336C2314yes—
141#336C2318yes—
142C22×Q8—22no—
143C22×Q8C2414yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C2×Q16—28no—
150C2×Q16—44no—
151C2×Q16—44no—
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—42no—
156C2×M16—42no—
157C2×M16#4118yes—
158C22×C8C2414yes—
159C22×C8C2414yes—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162#125—48no—
163C22⋊Q8—216no—
164C22⋊Q8—48no—
165C4×Q8—44no—
166C4×Q8—44no—
167C4×Q8—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2414yes—
182C2×C4⋊C4C2414yes—
183C2×C4⋊C4C2414yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C42—24no—
187C2×C42—28no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—44no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—42no—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2412yes—
207C2×C42C2412yes—
208C2×C42C2411yescharacteristic
209C2×C42C2411yescharacteristic
210#107—44no—
211#106—28no—
212#106—44no—
213#103—216no—
214#103—48no—
215#103—48no—
216#97—28no—
217C4×C8—28no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232C24C22×D411yescharacteristic
233C2×Q8—28no—
234C2×Q8—24no—
235C2×Q8—28no—
236C2×Q8—24no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C22×D412yes—
258C22×C4C22×D412yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D412yes—
261C22×C4C22×D412yes—
262C22×C4C22×D412yes—
263C22×C4C22×D412yes—
264C22×C4C2×#4114yes—
265C22×C4C2511yescharacteristic
266Q16—88no—
267M16—48no—
268M16—48no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8C2×#4118yes—
275C4⋊C4—28no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—48no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—44no—
291C4⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—24no—
305C42—41no—
306C42—42no—
307C42—41no—
308C42—42no—
309C42—42no—
310C42—48no—
311C42—48no—
312C42#13718yes—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23#53212yes—
318C23C23×D411yescharacteristic
319Q8—48no—
320Q8—48no—
321Q8—88no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4#53212yes—
357C2×C4C2×#13714yes—
358C2×C4C2×#13714yes—
359C2×C4C2×#13714yes—
360C2×C4#57514yes—
361C2×C4C22×#4112yes—
362C2×C4#58511yescharacteristic
363C2×C4#58512yes—
364C2×C4#58511yescharacteristic
365C8—28no—
366C8—44no—
367C22—24no—
368C22—22no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22C2×#53211yescharacteristic
373C22#344911yescharacteristic
374C22#344911yescharacteristic
375C22C22×#13712yes—
376C22C2×#57512yes—
377C4—24no—
378C4—24no—
379C4—24no—
380C4—24no—
381C4—22no—
382C4—42no—
383C4—42no—
384C4—84no—
385C4—84no—
386C4—84no—
387C4—84no—
388C4#349112yes—
389C4#349312yes—
390C2—42no—
391C2#3511911yescharacteristic
392C2#6070112yes—
393C2#6095212yes—
394C2#6096411yescharacteristic
395C2#6097111yescharacteristic
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
6. #29575C2semidirect
7. #29553C2semidirect
8. #29520C2semidirect
9. #29501C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#331442
#302962
#302502
#301992
#295752
#295532
#295202
#295012
#221892
#203642