Finite Group Databasebeta

← #6458908

Subgroups — #6458908

Order 512.

1930 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic
3#27618C218yesmaximal, complemented
4#27615C214yesmaximal, complemented
5#27612C214yesmaximal, complemented
6#23747C212yesmaximal, complemented
7#23657C212yesmaximal, complemented
8#21034C214yesmaximal, complemented
9#20845C212yesmaximal
10#20701C212yesmaximal
11#20682C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#428C2214yes—
16C2×#407C2212yes—
17#2296—22no—
18#2296—22no—
19#2296C2214yes—
20#2296C2214yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#385—22no—
24#2285C2212yes—
25#2285C2212yes—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2283—22no—
29#2283—22no—
30#2283C2214yes—
31#2280—22no—
32#2280C2212yes—
33C4×#117—22no—
34C4×#117C2212yes—
35C2×C4×C4⋊C4—22no—
36#2101C2218yescomplemented
37#2100C2214yescomplemented
38#2099C2214yescomplemented
39#2085C2218yescomplemented
40#2084C2214yescomplemented
41#2083C2214yescomplemented
42#1921C2218yescomplemented
43#1920C2218yescomplemented
44#1919C2218yescomplemented
45#1918C2218yescomplemented
46#1854C2218yes—
47#1854C2218yes—
48#1825C2218yes—
49#1825C2218yes—
50#1817C2214yes—
51#1817C2214yes—
52#1736C2214yes—
53#1736C2214yes—
54#1736C2214yes—
55#1728C2212yes—
56#1728C2212yes—
57C24×C4D412yes—
58C24×C4C2311yescharacteristic
59C22×M16C2314yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—28no—
65C2×#117—28no—
66C2×#117C2314yes—
67C2×#117C2314yes—
68C2×#117C2312yes—
69C2×#117C2312yes—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42—28no—
78C22×C42—21no—
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42D412yes—
82C22×C42C2312yes—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#428—216no—
87C2×#107C2318yes—
88C2×#106C2318yes—
89#407—28no—
90C2×C22⋊C8C2318yes—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389C2314yes—
96#389C2314yes—
97#389C2314yes—
98#389C2314yes—
99#385—24no—
100#385—24no—
101#385—24no—
102#385—24no—
103#384C2312yes—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#383—24no—
107#382—22no—
108#382—22no—
109#382C2312yes—
110#382C2312yes—
111#381—24no—
112#381—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380C2314yes—
118#380C2314yes—
119#379—22no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2314yes—
129C4×C4⋊C4C2314yes—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—24no—
133#376—24no—
134#376—22no—
135C2×#95—22no—
136C2×#95—22no—
137C43—22no—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#336C2314yes—
145#336C2314yes—
146#336C2318yes—
147C25C2×D411yescharacteristic
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—28no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4#4118yes—
164C23×C4C2412yes—
165C23×C4C2411yescharacteristic
166C2×M16—28no—
167C2×M16—28no—
168C2×M16—28no—
169C22×C8C2418yes—
170#117—216no—
171#117—216no—
172#117—216no—
173#117—28no—
174#117—28no—
175#117—48no—
176#117—48no—
177#117—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4#4118yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2412yes—
195C2×C4⋊C4C2412yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—44no—
206C2×C42—28no—
207C2×C42—28no—
208C2×C42—28no—
209C2×C42—28no—
210C2×C42—24no—
211C2×C42—28no—
212C2×C42—28no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—44no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2412yes—
229C2×C42C2411yescharacteristic
230C2×C42C2411yescharacteristic
231#107—216no—
232#106—216no—
233C22⋊C8—216no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244C24—22no—
245C24—22no—
246C24—42no—
247C24—42no—
248C24#12314yes—
249C24C22×D412yes—
250C24C22×D411yescharacteristic
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—232no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—432no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4#12314yes—
289C22×C4#12318yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D412yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4118yes—
297C22×C4C2×#4114yes—
298C22×C4C2×#4114yes—
299C22×C42−1+414yes—
300C22×C4C2511yescharacteristic
301M16—416no—
302C2×C8—28no—
303C2×C8—28no—
304C2×C8—28no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—28no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—48no—
313C4⋊C4—48no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4C2×#4118yes—
322C22⋊C4—216no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C42—216no—
329C42—24no—
330C42—22no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—24no—
338C42—22no—
339C42—48no—
340C42—48no—
341C42—44no—
342C42—44no—
343C42—48no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—28no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—48no—
354C23—48no—
355C23—44no—
356C23—44no—
357C23C2×#12314yes—
358C23C2×#12312yes—
359C23#52512yes—
360C23#53612yes—
361C23C23×D411yescharacteristic
362C2×C4—216no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—28no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—216no—
371C2×C4—216no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—28no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—416no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—88no—
398C2×C4C2×#12314yes—
399C2×C4C2×#12312yes—
400C2×C4#52512yes—
401C2×C4#53612yes—
402C2×C4#552116yes—
403C2×C4#57514yes—
404C2×C4#57514yes—
405C2×C4C22×#4112yes—
406C2×C4C22×#4112yes—
407C2×C4C2×2−1+412yes—
408C8—48no—
409C22—24no—
410C22—24no—
411C22—22no—
412C22—22no—
413C22—24no—
414C22—24no—
415C22—24no—
416C22—416no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22#248414yes—
422C22C22×#12311yescharacteristic
423C22C2×#52511yescharacteristic
424C22C2×#53611yescharacteristic
425C22C2×#57512yes—
426C22C2×#57512yes—
427C4—24no—
428C4—24no—
429C4—24no—
430C4—42no—
431C4—42no—
432C4—48no—
433C4—84no—
434C4—84no—
435C4—82no—
436C4—82no—
437C4#248414yes—
438C4C2×#55214yes—
439C2—24no—
440C2—44no—
441C2C2×#248411yescharacteristic
442C2#6051112yes—
443C2#6055912yes—
444C2#6086211yescharacteristic
445C2#6086311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27618C2semidirect
4. #27615C2semidirect
5. #27612C2semidirect
6. #23747C2semidirect
7. #23657C2semidirect
8. #21034C2semidirect
36. #2101C22semidirect
37. #2100C22semidirect
38. #2099C22semidirect
39. #2085C22semidirect
40. #2084C22semidirect
41. #2083C22semidirect
42. #1921C22semidirect
43. #1920C22semidirect
44. #1919C22semidirect
45. #1918C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#340122
#276182
#276152
#276122
#237472
#236572
#210342
#208452
#207012
#206822