Finite Group Databasebeta

← #6499262

Subgroups — #6499262

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic
3#33067C214yesmaximal, complemented
4#29983C214yesmaximal, complemented
5#29273C214yesmaximal, complemented
6#29255C214yesmaximal, complemented
7#29159C218yesmaximal
8#21959C212yesmaximal
9#21885C212yesmaximal
10C2×#1831C211yesmaximal, characteristic
11C2×#533—21no—
12C2×#533C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14#2868C2212yes—
15#2862—21no—
16#2859—21no—
17#2859C2212yes—
18#2857—22no—
19#2857C2214yes—
20C4×C4⋊Q8—22no—
21C4×C4⋊Q8C2214yes—
22C4×C4⋊Q8C2212yes—
23C4×C4⋊Q8C2212yes—
24C4×#125—22no—
25C4×C22⋊Q8—22no—
26C4×C22⋊Q8C2214yes—
27#2296—21no—
28#2295—21no—
29#2295C2212yes—
30#2285—21no—
31#2285C2212yes—
32#2285C2211yescharacteristic
33C42×Q8—21no—
34C42×Q8C2212yes—
35C4×#117—21no—
36C2×C43C2211yescharacteristic
37#2099C2218yes—
38#2096C2218yes—
39#2058C2218yes—
40#2024C2218yes—
41#1987C2214yes—
42#1981C2214yes—
43#1948C2218yes—
44#1948C2218yes—
45#1919C22116yes—
46#1892C2218yes—
47#1865C22116yes—
48#1831C2212yes—
49#1831C2212yes—
50#1831C2214yes—
51#1807C2218yes—
52#1803C2218yes—
53C2×#336C2214yes—
54C23×C8C2314yes—
55#533—24no—
56#533—48no—
57#533Q818yescomplemented
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2314yes—
63C2×#125—21no—
64C2×#125—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8C2312yes—
68C2×C4×Q8—21no—
69C2×C4×Q8—24no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2312yes—
72C2×C4×Q8C2314yes—
73C2×#117—21no—
74C2×#117—22no—
75C2×#117—21no—
76C22×C42D411yescharacteristic
77C22×C42D412yes—
78C22×C42D411yescharacteristic
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#106C23116yes—
82C2×C4⋊C8—28no—
83C2×C4⋊C8C2314yes—
84C2×C4⋊C8C2314yes—
85C2×#103C23116yes—
86#391—24no—
87#391—24no—
88#391—22no—
89#391C2314yes—
90#389—22no—
91#389—22no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—22no—
96#389C2314yes—
97#389C2318yes—
98#387—24no—
99#387—24no—
100#385—24no—
101#384—22no—
102#384—22no—
103#384C2312yes—
104#384C2314yes—
105#384C2314yes—
106#382—21no—
107#382—22no—
108#382—22no—
109#380—22no—
110#380—24no—
111#380—21no—
112#380C2312yes—
113#380C2314yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2314yes—
125C4×C4⋊C4C2314yes—
126C4×C4⋊C4C2312yes—
127C4×C22⋊C4—24no—
128#376—22no—
129C43—22no—
130C43D412yes—
131C43D411yescharacteristic
132C43D411yescharacteristic
133C43C2312yes—
134C43C2311yescharacteristic
135C43C2311yescharacteristic
136#340C23116yes—
137#340C23116yes—
138#336C2318yes—
139#336C2318yes—
140C22×Q8—21no—
141C22×Q8—24no—
142C22×Q8C2412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—28no—
149C22×C8—48no—
150C22×C8C2418yes—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—48no—
154C4⋊Q8—48no—
155C4⋊Q8—48no—
156C4⋊Q8C2×Q814yes—
157C4⋊Q8C2×Q814yes—
158#125—28no—
159#125—48no—
160C22⋊Q8—28no—
161C22⋊Q8—24no—
162C22⋊Q8—416no—
163C22⋊Q8C2×Q818yes—
164C4×Q8—28no—
165C4×Q8—24no—
166C4×Q8—48no—
167C4×Q8—48no—
168C4×Q8—416no—
169C4×Q8C2×Q818yes—
170#117—28no—
171#117—48no—
172#117—44no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2418yes—
190C2×C4⋊C4C2412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42SD1618yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q814yes—
218C2×C42#4114yes—
219C2×C42C2412yes—
220C2×C42C2412yes—
221C2×C42C2411yescharacteristic
222C2×C42C2411yescharacteristic
223#106—416no—
224C4⋊C8—28no—
225C4⋊C8—28no—
226#103—416no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—28no—
234#95—24no—
235#95—24no—
236C24C22×D411yescharacteristic
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—28no—
240C2×Q8—44no—
241C2×Q8—48no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8C22×Q814yes—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—28no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4C22⋊Q814yes—
257C22×C4C22⋊Q814yes—
258C22×C4C2×SD1618yes—
259C22×C4C2×SD1614yes—
260C22×C4#13714yes—
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D412yes—
266C22×C4C22×D412yes—
267C22×C4C22×D412yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q812yes—
271C22×C4C2×#4112yes—
272C22×C4C2511yescharacteristic
273C2×C8—28no—
274C2×C8—216no—
275C2×C8—88no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—22no—
282C4⋊C4—28no—
283C4⋊C4—28no—
284C4⋊C4—28no—
285C4⋊C4—44no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—416no—
291C4⋊C4—48no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4C22×Q814yes—
296C22⋊C4—28no—
297C22⋊C4—48no—
298C22⋊C4—44no—
299C42—28no—
300C42—22no—
301C42—24no—
302C42—22no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—24no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—48no—
312C42—48no—
313C42—48no—
314C42—44no—
315C42C22⋊Q814yes—
316C42C22⋊Q814yes—
317C42C2×SD1618yes—
318C42C2×SD1614yes—
319C42C2×SD1614yes—
320C42C22×Q812yes—
321C42C22×Q812yes—
322C42C2×#4112yes—
323C42C2×#4112yes—
324C23—22no—
325C23—22no—
326C23C2×C22⋊Q812yes—
327C23C22×SD1612yes—
328C23C2×#13712yes—
329C23C23×D411yescharacteristic
330Q8—28no—
331Q8—48no—
332Q8—44no—
333Q8—816no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—48no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4#475116yes—
362C2×C4C2×C22⋊Q812yes—
363C2×C4C2×C22⋊Q812yes—
364C2×C4C2×C22⋊Q812yes—
365C2×C4C2×C22⋊Q812yes—
366C2×C4C2×C22⋊Q812yes—
367C2×C4C22×SD1614yes—
368C2×C4C22×SD1614yes—
369C2×C4C22×SD1612yes—
370C2×C4C22×SD1612yes—
371C2×C4C2×#13712yes—
372C2×C4C2×#13712yes—
373C2×C4#57514yes—
374C2×C4C23×Q811yescharacteristic
375C2×C4C22×#4111yescharacteristic
376C2×C4#58514yes—
377C8—48no—
378C22—22no—
379C22—24no—
380C22—24no—
381C22—22no—
382C22C2×#47514yes—
383C22C22×C22⋊Q811yescharacteristic
384C22#344912yes—
385C22C23×SD1611yescharacteristic
386C22C22×#13711yescharacteristic
387C22C2×#57512yes—
388C4—22no—
389C4—24no—
390C4—24no—
391C4—24no—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—42no—
396C4—44no—
397C4—88no—
398C4—84no—
399C4C2×#47514yes—
400C4C2×#47514yes—
401C4#350412yes—
402C4#351012yes—
403C2—24no—
404C2#3533411yescharacteristic
405C2C22×#47511yescharacteristic
406C2#6048411yescharacteristic
407C2#6094112yes—
408C2#6096312yes—
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
3. #33067C2semidirect
4. #29983C2semidirect
5. #29273C2semidirect
6. #29255C2semidirect
57. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#5332
#330672
#299832
#292732
#292552
#291592
#219592
#218852
C2×#18312