Finite Group Databasebeta

← #6510793

Subgroups — #6510793

Order 512.

2184 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3195C212yesmaximal
2C2×#3095C212yesmaximal
3#35011C211yesmaximal, characteristic
4C2×#2499C211yesmaximal, characteristic
5#28355C218yesmaximal, complemented
6#28313C212yesmaximal, complemented
7#28304C212yesmaximal, complemented
8#27940C218yesmaximal, complemented
9#27919C212yesmaximal, complemented
10#27905C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15#3195—216no—
16#3095—216no—
17C2×#484C2214yes—
18C2×#485C2218yes—
19C2×#483C2214yes—
20C2×#483C2214yes—
21C2×#483C2214yes—
22C2×#470C2218yes—
23C2×#451C2214yes—
24C2×#451C2214yes—
25C2×#428C2214yes—
26C2×#419C2214yes—
27C2×#407C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2659—22no—
31#2583—24no—
32#2502C2214yes—
33#2499—24no—
34#2499C2218yes—
35#2434C2214yes—
36C2×#397—22no—
37#2315—22no—
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#387—22no—
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44#2097C2218yescomplemented
45#2096C2218yescomplemented
46#2096C2214yescomplemented
47#2095C2214yescomplemented
48#2016C2214yescomplemented
49#2016C2214yescomplemented
50#1971C2218yescomplemented
51#1969C2218yescomplemented
52#1960C2214yescomplemented
53#1956C2214yescomplemented
54#1847C2214yescomplemented
55#1844C2214yescomplemented
56#1753C2218yes—
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59C23×Q8C2312yes—
60C23×Q8C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C22×Q16C2318yes—
63C22×M16C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8C2314yes—
73C2×C22⋊Q8C2314yes—
74C2×C22⋊Q8C2314yes—
75C2×#117—22no—
76C2×#117—28no—
77C2×#117C2314yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#485—232no—
93#484—216no—
94#483—216no—
95#483—216no—
96#483—216no—
97#470—232no—
98#451—216no—
99#451—216no—
100#428—216no—
101C2×#107C2318yes—
102C2×#106C2318yes—
103C2×#106C2318yes—
104#419—216no—
105C2×#103C2318yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2318yes—
109#407D4116yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×#97C2314yes—
113C2×C4×C8C2314yes—
114#398—28no—
115#397—216no—
116#395—28no—
117#391—22no—
118#391—22no—
119#391C2314yes—
120#391C2314yes—
121#391C2314yes—
122#391C2314yes—
123#389—24no—
124#389—24no—
125#387—216no—
126#385—28no—
127#385—28no—
128#384—22no—
129#384C2312yes—
130#384C2312yes—
131#384C2314yes—
132#382—22no—
133#382C2314yes—
134#380—24no—
135#380—24no—
136C4×C22⋊C4D414yes—
137C4×C22⋊C4C2314yes—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95D412yes—
142#336C2314yes—
143#336C2314yes—
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8C2414yes—
155C22×Q8C2412yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D418yes—
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C2×Q16—216no—
172C2×Q16—48no—
173C2×Q16—48no—
174C2×M16—44no—
175C2×M16—44no—
176C2×M16C2×D4116yes—
177C22×C8C2418yes—
178C4⋊Q8—44no—
179C4⋊Q8—44no—
180C22⋊Q8—28no—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—28no—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189#117—28no—
190#117—28no—
191#117—48no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4C2412yes—
213C2×C4⋊C4C2412yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2414yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4C2×D418yes—
225C2×C42—24no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—44no—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2412yes—
232C2×C42C2412yes—
233#107—216no—
234#106—216no—
235#106—216no—
236#103—216no—
237#103—216no—
238#103—216no—
239#103—216no—
240C22⋊C8C2×D4116yes—
241C22⋊C8C2×D4116yes—
242#97—44no—
243C4×C8—44no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95C2×D418yes—
250C24—24no—
251C24—41no—
252C24—41no—
253C24—42no—
254C24#12714yes—
255C24C22×D412yes—
256C24C22×D411yescharacteristic
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—28no—
261C2×Q8—28no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—416no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—88no—
272C2×Q8—88no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—416no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—84no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4#12714yes—
308C22×C4#12718yes—
309C22×C4#13718yes—
310C22×C4C22×D414yes—
311C22×C4C22×D412yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D414yes—
314C22×C4C22×D414yes—
315C22×C4C22×D414yes—
316C22×C4C22×D414yes—
317C22×C4C2511yescharacteristic
318Q16—816no—
319M16—416no—
320C2×C8—28no—
321C2×C8—28no—
322C2×C8C22×D4116yes—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—416no—
336C4⋊C4—84no—
337C4⋊C4—84no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C22⋊C4—216no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—48no—
345C42—24no—
346C42—24no—
347C42—42no—
348C42—42no—
349C42—48no—
350C23—22no—
351C23—21no—
352C23—21no—
353C23—24no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—28no—
358C23—44no—
359C23—44no—
360C23—84no—
361C23—84no—
362C23C2×#12714yes—
363C23C2×#12712yes—
364C23C2×#13714yes—
365C23C23×D411yescharacteristic
366Q8—48no—
367Q8—48no—
368Q8—48no—
369Q8—84no—
370Q8—88no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—88no—
404C2×C4—88no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4C2×#12714yes—
408C2×C4C2×#12712yes—
409C2×C4C2×#13718yes—
410C2×C4C23×D412yes—
411C2×C4#58512yes—
412C2×C4#58512yes—
413C8—216no—
414C22—24no—
415C22—24no—
416C22—22no—
417C22—22no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—44no—
422C22—44no—
423C22—88no—
424C22#315214yes—
425C22C22×#12711yescharacteristic
426C22#348511yescharacteristic
427C22#348511yescharacteristic
428C22C22×#13712yes—
429C22#358612yes—
430C4—28no—
431C4—22no—
432C4—24no—
433C4—44no—
434C4—84no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4—84no—
439C4—82no—
440C4#348514yes—
441C2—24no—
442C2—44no—
443C2#3546311yescharacteristic
444C2C2×#315211yescharacteristic
445C2#6064511yescharacteristic
446C2#6170512yes—
447C2#6172312yes—
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
5. #28355C2semidirect
6. #28313C2semidirect
7. #28304C2semidirect
8. #27940C2semidirect
9. #27919C2semidirect
10. #27905C2semidirect
44. #2097C22semidirect
45. #2096C22semidirect
46. #2096C22semidirect
47. #2095C22semidirect
48. #2016C22semidirect
49. #2016C22semidirect
50. #1971C22semidirect
51. #1969C22semidirect
52. #1960C22semidirect
53. #1956C22semidirect
54. #1847C22semidirect
55. #1844C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31952
C2×#30952
#350112
C2×#24992
#283552
#283132
#283042
#279402
#279192
#279052
#205242