Finite Group Databasebeta

← #6631367

Subgroups — #6631367

Order 512.

1728 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#32786C211yesmaximal, characteristic, complemented
3#32771C212yesmaximal, complemented
4#32762C211yesmaximal, characteristic, complemented
5#32741C214yesmaximal, complemented
6#32725C214yesmaximal, complemented
7#32711C214yesmaximal
8#23756C214yesmaximal, complemented
9#23699C214yesmaximal, complemented
10#23571C214yesmaximal, complemented
11#20714C211yesmaximal, characteristic
12C2×#533C2212yes—
13C23×C42C2211yescharacteristic
14C2×C4×Q16C2214yescomplemented
15C2×#431C2212yes—
16#2857C2212yes—
17#2857C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2855—22no—
21#2855C2214yes—
22#2851—22no—
23#2851—22no—
24#2848—22no—
25#2848C2214yes—
26C4×C22⋊Q8—22no—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8C2214yes—
29C4×C22⋊Q8C2214yes—
30#2296—22no—
31#2285C2212yes—
32#2280—22no—
33#2099C2214yescomplemented
34#2096C2214yescomplemented
35#2086C2214yescomplemented
36#2084C2214yescomplemented
37#2077C2218yes—
38#2075C2214yes—
39#2072C2214yescomplemented
40#2066C2218yescomplemented
41#2057C2214yescomplemented
42#2056C2214yescomplemented
43#2043C2214yescomplemented
44#2036C2218yes—
45#2035C2218yes—
46#2035C2218yes—
47#2024C2214yescomplemented
48#2016C2214yescomplemented
49#2001C2218yes—
50#2001C2214yes—
51#1817C2211yescharacteristic
52#1817C2211yescharacteristic
53#1797C2214yes—
54#1779C2212yes—
55C4×#106C2214yescomplemented
56C4×#103C2218yescomplemented
57C4×C22⋊C8C2214yes—
58C24×C4D411yescharacteristic
59C24×C4D411yescharacteristic
60C24×C4C2311yescharacteristic
61C22×Q16C2314yes—
62#533—48no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#125—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8C2314yes—
70C2×C22⋊Q8C2314yes—
71C2×C22⋊Q8C2314yes—
72C2×C4×Q8—22no—
73C2×C4×Q8C2314yes—
74C2×#117—22no—
75C2×#117—22no—
76C22×C42—44no—
77C22×C42D411yescharacteristic
78C22×C42D412yes—
79C22×C42D412yes—
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2312yes—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C4×Q16—48no—
86#431—44no—
87C2×#107C2314yes—
88C2×#106C2318yes—
89C2×C4⋊C8C2314yes—
90C2×#103C2318yes—
91C2×#103C2318yes—
92C2×#103C2318yes—
93C2×C22⋊C8C2318yes—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96C2×C4×C8C2314yes—
97#391—22no—
98#391—22no—
99#391C2314yes—
100#391C2314yes—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389C2314yes—
108#389C2314yes—
109#387—24no—
110#387—24no—
111#387—24no—
112#387—24no—
113#385—24no—
114#385—24no—
115#385—24no—
116#385—24no—
117#385—24no—
118#385—24no—
119#384C2312yes—
120#384C2312yes—
121#382—22no—
122#381—24no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380C2314yes—
131#380C2314yes—
132#380C2314yes—
133#379—24no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4C2314yes—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—24no—
140#340C2314yes—
141#340C2318yes—
142#340C2314yes—
143#340C2318yes—
144#336C2314yes—
145#336C2314yes—
146C25C2×D411yescharacteristic
147C22×Q8—22no—
148C22×Q8C2414yes—
149C23×C4—24no—
150C23×C4—28no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C23×C4C2412yes—
168C2×Q16—48no—
169C2×Q16—44no—
170C22×C8C2414yes—
171C22×C8C2414yes—
172C4⋊Q8—44no—
173C4⋊Q8—44no—
174#125—48no—
175C22⋊Q8—44no—
176C22⋊Q8—48no—
177C22⋊Q8—44no—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C4×Q8—216no—
183C4×Q8—48no—
184#117—48no—
185#117—48no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—44no—
217C2×C42—42no—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2412yes—
226C2×C42C2411yescharacteristic
227C2×C42C2411yescharacteristic
228#107—44no—
229#106—48no—
230C4⋊C8—44no—
231#103—48no—
232#103—48no—
233#103—48no—
234C22⋊C8—48no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237C4×C8—28no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248C24—22no—
249C24—24no—
250C24—41no—
251C24—42no—
252C24—41no—
253C24C22×D411yescharacteristic
254C24C22×D412yes—
255C2×Q8—28no—
256C2×Q8—24no—
257C2×Q8—28no—
258C2×Q8—24no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—416no—
297C22×C4—416no—
298C22×C4—416no—
299C22×C4#13514yes—
300C22×C4#13718yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2511yescharacteristic
310Q16—416no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—28no—
314C2×C8—28no—
315C4⋊C4—28no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—24no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C22⋊C4—48no—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—24no—
342C42—22no—
343C42—24no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—48no—
348C42—48no—
349C42H(5,𝔽2)14yes—
350C23—22no—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—21no—
360C23—44no—
361C23—48no—
362C23—44no—
363C23—48no—
364C23—44no—
365C23C2×#13512yes—
366C23C2×#13714yes—
367C23C23×D411yescharacteristic
368Q8—48no—
369Q8—48no—
370Q8—88no—
371C2×C4—28no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—216no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4#53412yes—
411C2×C4#53412yes—
412C2×C4C2×#13514yes—
413C2×C4C2×#13714yes—
414C2×C4C2×#13714yes—
415C2×C4#57512yes—
416C2×C4C2×H(5,𝔽2)12yes—
417C2×C4#58514yes—
418C8—44no—
419C8—44no—
420C22—24no—
421C22—24no—
422C22—24no—
423C22—22no—
424C22—24no—
425C22—416no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—41no—
431C22—42no—
432C22—41no—
433C22C2×#53411yescharacteristic
434C22#344912yes—
435C22C22×#13511yescharacteristic
436C22C22×#13712yes—
437C22C2×#57511yescharacteristic
438C4—22no—
439C4—22no—
440C4—24no—
441C4—24no—
442C4—42no—
443C4—41no—
444C4—41no—
445C4—84no—
446C4—84no—
447C4—84no—
448C4—84no—
449C4#319514yes—
450C4#352914yes—
451C2—42no—
452C2—44no—
453C2#3665211yescharacteristic
454C2#6073411yescharacteristic
455C2C2×#319511yescharacteristic
456C2#6094712yes—
457C2#6095212yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #32786C2semidirect
3. #32771C2semidirect
4. #32762C2semidirect
5. #32741C2semidirect
6. #32725C2semidirect
8. #23756C2semidirect
9. #23699C2semidirect
10. #23571C2semidirect
14. C2×C4×Q16C22semidirect
33. #2099C22semidirect
34. #2096C22semidirect
35. #2086C22semidirect
36. #2084C22semidirect
39. #2072C22semidirect
40. #2066C22semidirect
41. #2057C22semidirect
42. #2056C22semidirect
43. #2043C22semidirect
47. #2024C22semidirect
48. #2016C22semidirect
55. C4×#106C22semidirect
56. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#327862
#327712
#327622
#327412
#327252
#327112
#237562
#236992
#235712
#207142