Finite Group Databasebeta

← #6510742

Subgroups — #6510742

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic
3#29111C212yesmaximal, complemented
4#29106C212yesmaximal, complemented
5#27969C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27723C218yesmaximal, complemented
8#27705C218yesmaximal, complemented
9#21768C212yesmaximal
10#20700C212yesmaximal
11#20524C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#407C2212yes—
15#2857C2212yes—
16#2851C2212yes—
17#2626—22no—
18#2608—22no—
19#2607—22no—
20#2503—22no—
21#2503—22no—
22#2503C2218yes—
23#2441C2214yes—
24#2439C2214yes—
25C2×#400—22no—
26#2329—22no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31#2280C2212yes—
32C2×#381—22no—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2101C2218yescomplemented
35#2101C2218yescomplemented
36#2074C2214yescomplemented
37#2072C2214yescomplemented
38#2026C2214yescomplemented
39#2024C2214yescomplemented
40#1993C2214yescomplemented
41#1991C2214yescomplemented
42#1935C2218yescomplemented
43#1933C2218yescomplemented
44#1894C2218yes—
45#1873C2214yes—
46#1873C2214yes—
47#1836C2214yescomplemented
48#1835C2214yescomplemented
49#1824C2214yes—
50#1824C2218yes—
51#1823C2214yes—
52#1823C2214yes—
53#1787C2214yes—
54#1753C2218yes—
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57#1736C2214yes—
58#1736C2214yes—
59C23×Q8C2311yescharacteristic
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C22×M16C2314yes—
63C2×C4⋊Q8C2312yes—
64C2×#125C2312yes—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8C2314yes—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2314yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#103C2318yes—
84C2×#103C2318yes—
85#407—28no—
86C2×C22⋊C8C2314yes—
87C2×C22⋊C8C2314yes—
88C2×#97C2314yes—
89C2×C4×C8C2314yes—
90#400—28no—
91#400—28no—
92#399—24no—
93#396—24no—
94#396—24no—
95#393—24no—
96#391—22no—
97#391—22no—
98#391—21no—
99#391—21no—
100#391—24no—
101#391C2314yes—
102#391C2314yes—
103#389—22no—
104#389—24no—
105#389C2314yes—
106#389C2314yes—
107#387—24no—
108#387—24no—
109#387—24no—
110#385—24no—
111#384C2312yes—
112#382—22no—
113#382—21no—
114#382—21no—
115#382C2312yes—
116#382C2314yes—
117#382C2314yes—
118#381—22no—
119#381—28no—
120#381—28no—
121#380—22no—
122#380C2314yes—
123#380C2314yes—
124#379—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4C2314yes—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#336C2314yes—
142#336C2314yes—
143C25C2×D411yescharacteristic
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8C2412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—48no—
158C23×C4—44no—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4#4118yes—
164C23×C4C2411yescharacteristic
165C23×C4C2412yes—
166C2×M16—28no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8C2418yes—
170C4⋊Q8—44no—
171#125—44no—
172C22⋊Q8—28no—
173C22⋊Q8—28no—
174C22⋊Q8—48no—
175C22⋊Q8—48no—
176#117—44no—
177#117—44no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4C2412yes—
198C2×C4⋊C4C2412yes—
199C2×C4⋊C4C2412yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4C2×D414yes—
210C2×C22⋊C4C2×D414yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2412yes—
219C2×C42C2412yes—
220#103—48no—
221#103—48no—
222C22⋊C8—28no—
223C22⋊C8—28no—
224#97—44no—
225C4×C8—44no—
226#95—28no—
227#95—24no—
228#95—24no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95C2×D414yes—
238#95C2×D414yes—
239C24—24no—
240C24—41no—
241C24—41no—
242C24—42no—
243C24#12414yes—
244C24C22×D412yes—
245C24C22×D411yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—44no—
252C2×Q8—42no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—88no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—416no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—416no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—48no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—816no—
289C22×C4#12414yes—
290C22×C4#12418yes—
291C22×C4C22×D414yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C2×#4114yes—
297C22×C4C2×#4114yes—
298C22×C4C2×#4114yes—
299C22×C4C2511yescharacteristic
300M16—88no—
301C2×C8—28no—
302C2×C8—28no—
303C2×C8—28no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—28no—
308C4⋊C4—28no—
309C4⋊C4—48no—
310C4⋊C4—44no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—48no—
316C4⋊C4—82no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—82no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—84no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—24no—
328C42—24no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—82no—
333C42—82no—
334C23—22no—
335C23—21no—
336C23—21no—
337C23—24no—
338C23—28no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—44no—
343C23—44no—
344C23—48no—
345C23—48no—
346C23C2×#12414yes—
347C23C2×#12412yes—
348C23#57514yes—
349C23C23×D411yescharacteristic
350Q8—48no—
351Q8—84no—
352Q8—164no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—816no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4C2×#12414yes—
394C2×C4C2×#12412yes—
395C2×C4C22×#4112yes—
396C2×C4#58512yes—
397C2×C4#58512yes—
398C8—48no—
399C22—24no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—416no—
407C22—44no—
408C22—44no—
409C22#314014yes—
410C22C22×#12411yescharacteristic
411C22#348511yescharacteristic
412C22#348511yescharacteristic
413C22C2×#57512yes—
414C22#358612yes—
415C4—24no—
416C4—22no—
417C4—44no—
418C4—42no—
419C4—42no—
420C4—84no—
421C4—84no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4#349114yes—
427C2—24no—
428C2—44no—
429C2#3545511yescharacteristic
430C2C2×#314011yescharacteristic
431C2#6062011yescharacteristic
432C2#6171112yes—
433C2#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. #29111C2semidirect
4. #29106C2semidirect
5. #27969C2semidirect
6. #27964C2semidirect
7. #27723C2semidirect
8. #27705C2semidirect
34. #2101C22semidirect
35. #2101C22semidirect
36. #2074C22semidirect
37. #2072C22semidirect
38. #2026C22semidirect
39. #2024C22semidirect
40. #1993C22semidirect
41. #1991C22semidirect
42. #1935C22semidirect
43. #1933C22semidirect
47. #1836C22semidirect
48. #1835C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#25032
#291112
#291062
#279692
#279642
#277232
#277052
#217682
#207002
#205242