Finite Group Databasebeta

← #6458913

Subgroups — #6458913

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3#27621C214yesmaximal, complemented
4#27620C214yesmaximal, complemented
5#27619C218yesmaximal, complemented
6#23746C212yesmaximal, complemented
7#23658C212yesmaximal, complemented
8#21033C214yesmaximal, complemented
9#20846C212yesmaximal
10#20702C212yesmaximal
11#20683C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#428C2214yes—
16C2×#407C2212yes—
17#2317—22no—
18#2317C2214yes—
19#2316C2212yes—
20#2316C2212yes—
21#2315—22no—
22#2314—22no—
23#2314—22no—
24#2314C2214yes—
25#2314C2214yes—
26#2296—22no—
27#2296C2212yes—
28#2296C2212yes—
29C2×#388—22no—
30C2×#387—22no—
31#2285—22no—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2212yes—
36#2101C2218yescomplemented
37#2101C2218yescomplemented
38#2084C2218yescomplemented
39#2083C2218yescomplemented
40#1921C2218yescomplemented
41#1920C2218yescomplemented
42#1919C2218yescomplemented
43#1918C2218yescomplemented
44#1854C2218yes—
45#1853C2214yes—
46#1852C2214yes—
47#1825C2218yes—
48#1824C2214yes—
49#1823C2214yes—
50#1818C2214yes—
51#1818C2214yes—
52#1736C2214yes—
53#1736C2214yes—
54#1736C2214yes—
55#1728C2212yes—
56#1728C2212yes—
57C24×C4C2311yescharacteristic
58C22×M16C2314yes—
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—28no—
62C2×#117—28no—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—28no—
66C2×#117D412yes—
67C2×#117D412yes—
68C2×#117C2312yes—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C2×#117C2312yes—
72C2×#117C2312yes—
73C2×#117C2312yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#428—216no—
91C2×#107C2318yes—
92C2×#106C2318yes—
93#407—28no—
94C2×C22⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96#389—24no—
97#389—24no—
98#389C2314yes—
99#389C2314yes—
100#389C2314yes—
101#389C2314yes—
102#388—28no—
103#388—28no—
104#387—28no—
105#387—28no—
106#385—24no—
107#385—24no—
108#385—24no—
109#385—24no—
110#384—24no—
111#384—22no—
112#384C2314yes—
113#382—24no—
114#382—22no—
115#382C2314yes—
116#380—22no—
117#380—22no—
118#380—24no—
119#380—24no—
120#380C2314yes—
121#380C2314yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2318yes—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131#340C2318yes—
132#340C2318yes—
133#340C2318yes—
134#340C2318yes—
135#340C2318yes—
136#340C2318yes—
137#336C2318yes—
138#336C2314yes—
139#336C2314yes—
140C25C2×D411yescharacteristic
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4#4118yes—
154C23×C4C2411yescharacteristic
155C23×C4C2411yescharacteristic
156C23×C4C2411yescharacteristic
157C2×M16—28no—
158C2×M16—28no—
159C2×M16—28no—
160C22×C8C2418yes—
161#117—28no—
162#117—28no—
163#117—28no—
164#117—28no—
165#117—24no—
166#117—28no—
167#117—28no—
168#117—24no—
169#117—24no—
170#117—28no—
171#117—28no—
172#117—44no—
173#117—44no—
174#117#4118yes—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—84no—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2412yes—
198C2×C4⋊C4C2412yes—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—84no—
209C2×C22⋊C4C2×D414yes—
210C2×C22⋊C4C2×D414yes—
211C2×C42—28no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—28no—
219C2×C42—44no—
220C2×C42—84no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2412yes—
224C2×C42C2412yes—
225#107—216no—
226#106—216no—
227C22⋊C8—28no—
228C22⋊C8—28no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237C24—22no—
238C24—22no—
239C24—42no—
240C24—42no—
241C24#12314yes—
242C24C22×D411yescharacteristic
243C24C22×D411yescharacteristic
244C24C22×D411yescharacteristic
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—232no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
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—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4#12314yes—
281C22×C4#12314yes—
282C22×C4#12314yes—
283C22×C4C22×D414yes—
284C22×C4C22×D414yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4114yes—
289C22×C4C2×#4114yes—
290C22×C4C2×#4114yes—
291C22×C4C2×#4114yes—
292C22×C42−1+414yes—
293C22×C4C2511yescharacteristic
294M16—416no—
295C2×C8—28no—
296C2×C8—28no—
297C2×C8—28no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—88no—
315C4⋊C4C2×#4114yes—
316C4⋊C4C2×#4114yes—
317C22⋊C4—28no—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—28no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C42—28no—
326C42—24no—
327C42—24no—
328C42—28no—
329C42—24no—
330C42—24no—
331C42—44no—
332C42—44no—
333C42—88no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—28no—
345C23—48no—
346C23—82no—
347C23—82no—
348C23—84no—
349C23C2×#12312yes—
350C23C2×#12312yes—
351C23C2×#12312yes—
352C23#52512yes—
353C23#53612yes—
354C23C23×D411yescharacteristic
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—88no—
388C2×C4—88no—
389C2×C4—88no—
390C2×C4C2×#12312yes—
391C2×C4C2×#12312yes—
392C2×C4C2×#12312yes—
393C2×C4#52512yes—
394C2×C4#53612yes—
395C2×C4#552116yes—
396C2×C4C22×#4112yes—
397C2×C4C22×#4112yes—
398C2×C4C2×2−1+412yes—
399C8—48no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—22no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—42no—
414C22—88no—
415C22#248414yes—
416C22C22×#12311yescharacteristic
417C22C2×#52511yescharacteristic
418C22C2×#53611yescharacteristic
419C22#358512yes—
420C22#358612yes—
421C4—22no—
422C4—22no—
423C4—44no—
424C4—44no—
425C4—44no—
426C4—44no—
427C4—84no—
428C4—84no—
429C4—84no—
430C4#248414yes—
431C4C2×#55214yes—
432C2—24no—
433C2—42no—
434C2—42no—
435C2C2×#248411yescharacteristic
436C2#6051712yes—
437C2#6056811yescharacteristic
438C2#6056911yescharacteristic
439C2#6087012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27621C2semidirect
4. #27620C2semidirect
5. #27619C2semidirect
6. #23746C2semidirect
7. #23658C2semidirect
8. #21033C2semidirect
36. #2101C22semidirect
37. #2101C22semidirect
38. #2084C22semidirect
39. #2083C22semidirect
40. #1921C22semidirect
41. #1920C22semidirect
42. #1919C22semidirect
43. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#23142
#276212
#276202
#276192
#237462
#236582
#210332
#208462
#207022
#206832