Finite Group Databasebeta

← #6458912

Subgroups — #6458912

Order 512.

1786 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×#2317C211yesmaximal, characteristic
2#34048C211yesmaximal, characteristic
3#27622C218yesmaximal, complemented
4#27619C218yesmaximal, complemented
5#23747C212yesmaximal, complemented
6#23657C212yesmaximal, complemented
7#21034C214yesmaximal, complemented
8#20845C212yesmaximal
9#20700C212yesmaximal
10#20683C211yesmaximal, characteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#428C2214yes—
15C2×#407C2212yes—
16#2317—22no—
17#2317—22no—
18#2317C2214yes—
19#2317C2214yes—
20#2314C2214yes—
21#2314C2214yes—
22#2313—21no—
23#2313—21no—
24#2312—21no—
25#2312—21no—
26#2298—22no—
27#2298—22no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#385—21no—
31C2×#385—21no—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34C2×#384—21no—
35C2×#382—21no—
36#2280C2212yes—
37C4×#117C2212yes—
38#2101C2218yescomplemented
39#2100C2214yescomplemented
40#2099C2214yescomplemented
41#2085C2218yescomplemented
42#2084C2214yescomplemented
43#2083C2214yescomplemented
44#1921C2218yescomplemented
45#1920C2218yescomplemented
46#1919C2218yescomplemented
47#1918C2218yescomplemented
48#1854C2218yes—
49#1854C2218yes—
50#1824C2218yes—
51#1823C2218yes—
52#1818C2214yes—
53#1818C2214yes—
54#1736C2214yes—
55#1736C2214yes—
56#1736C2214yes—
57#1728C2212yes—
58#1728C2212yes—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16C2314yes—
62C2×#117—22no—
63C2×#117—28no—
64C2×#117—28no—
65C2×#117D412yes—
66C2×#117D412yes—
67C2×#117C2314yes—
68C2×#117C2312yes—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C2×#117C2312yes—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#428—216no—
89C2×#107C2318yes—
90C2×#106C2318yes—
91#407—28no—
92C2×C22⋊C8C2314yes—
93C2×C22⋊C8C2314yes—
94#389—24no—
95#389—22no—
96#389—22no—
97#389C2314yes—
98#389C2314yes—
99#389C2318yes—
100#388—24no—
101#388—24no—
102#388—24no—
103#388—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#385—24no—
109#385—24no—
110#385—24no—
111#385—24no—
112#384—24no—
113#384—24no—
114#384C2312yes—
115#384C2312yes—
116#382—24no—
117#382—24no—
118#382C2312yes—
119#382C2312yes—
120#381—22no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380C2314yes—
126#380C2314yes—
127#379—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2314yes—
133C4×C4⋊C4C2314yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138#376—22no—
139C2×#95—22no—
140C2×#95—22no—
141C43—22no—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2314yes—
149#336C2314yes—
150#336C2318yes—
151C25C2×D411yescharacteristic
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—44no—
158C23×C4—84no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4118yes—
166C23×C4C2412yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—28no—
171C22×C8C2418yes—
172#117—216no—
173#117—28no—
174#117—28no—
175#117—24no—
176#117—24no—
177#117—24no—
178#117—24no—
179#117—216no—
180#117—28no—
181#117—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—84no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4#4118yes—
207C2×C4⋊C4C2414yes—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2414yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4C2×D412yes—
220C2×C22⋊C4C2×D412yes—
221C2×C22⋊C4C2×D412yes—
222C2×C22⋊C4C2×D412yes—
223C2×C42—28no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2412yes—
235C2×C42C2411yescharacteristic
236C2×C42C2411yescharacteristic
237#107—216no—
238#106—216no—
239C22⋊C8—28no—
240C22⋊C8—28no—
241#95—28no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252C24—22no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24#12314yes—
257C24C22×D412yes—
258C24C22×D411yescharacteristic
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—232no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—88no—
286C22×C4—816no—
287C22×C4#12314yes—
288C22×C4#12318yes—
289C22×C4C22×D412yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4114yes—
298C22×C4C2×#4114yes—
299C22×C4C2×#4114yes—
300C22×C4C2×#4114yes—
301C22×C42−1+414yes—
302C22×C4C2511yescharacteristic
303M16—416no—
304C2×C8—28no—
305C2×C8—28no—
306C2×C8—28no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—48no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—88no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4C2×#4118yes—
327C22⋊C4—216no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C42—216no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—44no—
345C42—44no—
346C42—84no—
347C42—84no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—28no—
355C23—44no—
356C23—44no—
357C23—48no—
358C23—48no—
359C23C2×#12314yes—
360C23C2×#12312yes—
361C23#52512yes—
362C23#53612yes—
363C23C23×D411yescharacteristic
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—216no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—28no—
383C2×C4—416no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—88no—
400C2×C4—816no—
401C2×C4C2×#12314yes—
402C2×C4C2×#12312yes—
403C2×C4#52512yes—
404C2×C4#53612yes—
405C2×C4#552116yes—
406C2×C4C22×#4112yes—
407C2×C4C22×#4112yes—
408C2×C4C2×2−1+412yes—
409C8—48no—
410C22—22no—
411C22—22no—
412C22—24no—
413C22—24no—
414C22—24no—
415C22—24no—
416C22—24no—
417C22—416no—
418C22—44no—
419C22—44no—
420C22#248414yes—
421C22C22×#12311yescharacteristic
422C22C2×#52511yescharacteristic
423C22C2×#53611yescharacteristic
424C22#358512yes—
425C22#358612yes—
426C4—24no—
427C4—42no—
428C4—41no—
429C4—41no—
430C4—42no—
431C4—41no—
432C4—41no—
433C4—48no—
434C4—84no—
435C4—84no—
436C4—84no—
437C4#248414yes—
438C4C2×#55214yes—
439C2—24no—
440C2—44no—
441C2C2×#248411yescharacteristic
442C2#6051811yescharacteristic
443C2#6051911yescharacteristic
444C2#6056512yes—
445C2#6086712yes—
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. #27622C2semidirect
4. #27619C2semidirect
5. #23747C2semidirect
6. #23657C2semidirect
7. #21034C2semidirect
38. #2101C22semidirect
39. #2100C22semidirect
40. #2099C22semidirect
41. #2085C22semidirect
42. #2084C22semidirect
43. #2083C22semidirect
44. #1921C22semidirect
45. #1920C22semidirect
46. #1919C22semidirect
47. #1918C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
#340482
#276222
#276192
#237472
#236572
#210342
#208452
#207002
#206832