Finite Group Databasebeta

← #6510748

Subgroups — #6510748

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic
3#29112C212yesmaximal, complemented
4#29106C212yesmaximal, complemented
5#27966C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27724C218yesmaximal, complemented
8#27708C218yesmaximal, complemented
9#21765C212yesmaximal
10#20702C212yesmaximal
11#20524C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#407C2212yes—
15#2851C2212yes—
16#2851C2212yes—
17#2646—22no—
18#2569—24no—
19#2499—24no—
20#2499C2218yes—
21#2442C2214yes—
22#2442C2214yes—
23C2×#397—22no—
24#2315—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#387—22no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2101C2218yescomplemented
33#2100C2218yescomplemented
34#2074C2214yescomplemented
35#2074C2214yescomplemented
36#2026C2214yescomplemented
37#2026C2214yescomplemented
38#1992C2214yescomplemented
39#1991C2214yescomplemented
40#1935C2218yescomplemented
41#1933C2218yescomplemented
42#1894C2218yes—
43#1872C2214yes—
44#1872C2214yes—
45#1836C2214yescomplemented
46#1835C2214yescomplemented
47#1825C2218yes—
48#1824C2214yes—
49#1823C2214yes—
50#1823C2214yes—
51#1787C2214yes—
52#1753C2218yes—
53#1749C2212yes—
54C4×C22⋊C8C2212yes—
55#1736C2214yes—
56#1736C2214yes—
57C23×Q8C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C22×M16C2314yes—
60C2×#125C2312yes—
61C2×#125C2312yes—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8C2314yes—
65C2×#117—22no—
66C2×#117—28no—
67C2×#117C2314yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#103C2318yes—
83C2×#103C2318yes—
84#407—28no—
85C2×C22⋊C8C2314yes—
86C2×C22⋊C8C2314yes—
87C2×#97C2314yes—
88C2×C4×C8C2314yes—
89#400—24no—
90#400—24no—
91#398—24no—
92#398—24no—
93#397—28no—
94#397—28no—
95#391—22no—
96#391—22no—
97#391C2314yes—
98#391C2314yes—
99#389C2314yes—
100#389C2314yes—
101#387—24no—
102#387—24no—
103#387—28no—
104#387—28no—
105#385—24no—
106#385—24no—
107#385—24no—
108#385—24no—
109#384—22no—
110#384C2314yes—
111#382—22no—
112#382C2312yes—
113#382C2312yes—
114#382C2314yes—
115#380—22no—
116#380—22no—
117#380—24no—
118#380—24no—
119#380C2314yes—
120#380C2314yes—
121C4×C22⋊C4D414yes—
122C4×C22⋊C4C2314yes—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D412yes—
128#340C2318yes—
129#340C2318yes—
130#340C2318yes—
131#340C2318yes—
132#340C2318yes—
133#340C2318yes—
134#336C2314yes—
135#336C2314yes—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—24no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8C2412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4#4118yes—
155C23×C4C2411yescharacteristic
156C23×C4C2412yes—
157C2×M16—28no—
158C2×M16—44no—
159C2×M16—44no—
160C22×C8C2418yes—
161#125—44no—
162#125—44no—
163C22⋊Q8—28no—
164C22⋊Q8—28no—
165C22⋊Q8—48no—
166#117—28no—
167#117—28no—
168#117—48no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4C2412yes—
188C2×C4⋊C4C2412yes—
189C2×C4⋊C4C2412yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2414yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—48no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4C2×D414yes—
201C2×C22⋊C4C2×D414yes—
202C2×C42—24no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—44no—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2412yes—
209C2×C42C2412yes—
210#103—48no—
211#103—48no—
212C22⋊C8—28no—
213C22⋊C8—28no—
214#97—44no—
215C4×C8—44no—
216#95—24no—
217#95—24no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95C2×D414yes—
226#95C2×D414yes—
227C24—24no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24#12414yes—
232C24C22×D412yes—
233C24C22×D411yescharacteristic
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—88no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—416no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4#12414yes—
278C22×C4#12418yes—
279C22×C4C22×D414yes—
280C22×C4C22×D414yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D412yes—
283C22×C4C2×#4114yes—
284C22×C4C2×#4114yes—
285C22×C4C2×#4114yes—
286C22×C4C2511yescharacteristic
287M16—88no—
288C2×C8—28no—
289C2×C8—28no—
290C2×C8—28no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—416no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—48no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C42—24no—
314C42—24no—
315C42—42no—
316C42—42no—
317C42—48no—
318C23—22no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—28no—
328C23—44no—
329C23—44no—
330C23—84no—
331C23—84no—
332C23C2×#12414yes—
333C23C2×#12412yes—
334C23#57514yes—
335C23C23×D411yescharacteristic
336Q8—48no—
337Q8—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—164no—
375C2×C4—164no—
376C2×C4C2×#12414yes—
377C2×C4C2×#12412yes—
378C2×C4C22×#4112yes—
379C2×C4#58512yes—
380C2×C4#58512yes—
381C8—48no—
382C22—24no—
383C22—24no—
384C22—22no—
385C22—22no—
386C22—24no—
387C22—24no—
388C22—24no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—88no—
394C22#314214yes—
395C22C22×#12411yescharacteristic
396C22#348511yescharacteristic
397C22#348511yescharacteristic
398C22C2×#57512yes—
399C22#358612yes—
400C4—24no—
401C4—22no—
402C4—44no—
403C4—44no—
404C4—84no—
405C4—84no—
406C4—82no—
407C4—82no—
408C4—82no—
409C4—82no—
410C4#349114yes—
411C2—24no—
412C2—44no—
413C2#3545511yescharacteristic
414C2C2×#314211yescharacteristic
415C2#6062011yescharacteristic
416C2#6171212yes—
417C2#6171612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29112C2semidirect
4. #29106C2semidirect
5. #27966C2semidirect
6. #27964C2semidirect
7. #27724C2semidirect
8. #27708C2semidirect
32. #2101C22semidirect
33. #2100C22semidirect
34. #2074C22semidirect
35. #2074C22semidirect
36. #2026C22semidirect
37. #2026C22semidirect
38. #1992C22semidirect
39. #1991C22semidirect
40. #1935C22semidirect
41. #1933C22semidirect
45. #1836C22semidirect
46. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#24992
#291122
#291062
#279662
#279642
#277242
#277082
#217652
#207022
#205242