Finite Group Databasebeta

← #6434102

Subgroups — #6434102

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3#29919C218yesmaximal, complemented
4#29918C218yesmaximal, complemented
5#23747C212yesmaximal, complemented
6#23732C212yesmaximal, complemented
7#23657C212yesmaximal, complemented
8#23629C212yesmaximal, complemented
9#20911C212yesmaximal
10#20846C212yesmaximal
11#20683C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2212yes—
14C22×#117C2211yescharacteristic
15C2×#428C2214yes—
16C2×#419C2214yes—
17C2×#407C2212yes—
18#2625—24no—
19#2625C2218yes—
20C2×#400—21no—
21C2×#398—21no—
22C2×#397—21no—
23C2×#397—21no—
24#2317C2214yes—
25#2317C2214yes—
26#2316—21no—
27#2316—21no—
28#2315—21no—
29#2315—21no—
30#2296—24no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35#2285C2212yes—
36#2285C2212yes—
37C2×#380C2211yescharacteristic
38#2101C2218yescomplemented
39#2100C2214yescomplemented
40#2099C2214yescomplemented
41#2097C2218yescomplemented
42#2096C2214yescomplemented
43#2095C2214yescomplemented
44#2085C2218yescomplemented
45#2084C2214yescomplemented
46#2083C2214yescomplemented
47#2074C2218yescomplemented
48#2072C2214yescomplemented
49#2072C2214yescomplemented
50#1873C2218yes—
51#1872C2214yes—
52#1872C2214yes—
53#1854C2218yes—
54#1853C2214yes—
55#1852C2214yes—
56#1818C2218yes—
57#1736C2214yes—
58#1736C2214yes—
59#1728C2212yes—
60#1728C2212yes—
61C23×Q8C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—28no—
66C2×C22⋊Q8C2314yes—
67C2×#117—28no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—48no—
71C2×#117D414yes—
72C2×#117C2314yes—
73C2×#117C2314yes—
74C2×#117C2314yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C42—22no—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#428—48no—
94C2×#107C2318yes—
95C2×#106C2318yes—
96#419—48no—
97C2×#103C2318yes—
98C2×#103C2318yes—
99#407—28no—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102#400—28no—
103#398—28no—
104#397—28no—
105#397—28no—
106#391—22no—
107#391—22no—
108#391C2314yes—
109#391C2314yes—
110#389—24no—
111#389—24no—
112#389—24no—
113#389—24no—
114#389—22no—
115#389—22no—
116#389C2314yes—
117#389C2314yes—
118#385—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#384C2312yes—
123#384C2312yes—
124#384C2312yes—
125#384C2312yes—
126#382C2314yes—
127#382C2314yes—
128#380—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380C2314yes—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151#336C2314yes—
152#336C2314yes—
153C25C2×D411yescharacteristic
154C22×Q8—21no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8C2412yes—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4118yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C2×M16—28no—
177C2×M16—28no—
178C2×M16—28no—
179C22×C8C2418yes—
180C22⋊Q8—28no—
181C22⋊Q8—28no—
182C22⋊Q8—48no—
183#117—216no—
184#117—28no—
185#117—28no—
186#117—28no—
187#117—28no—
188#117—44no—
189#117—44no—
190#117—48no—
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—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2414yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2412yes—
224C2×C4⋊C4C2412yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4C2×D414yes—
235C2×C22⋊C4C2×D414yes—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—48no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2412yes—
244C2×C42C2412yes—
245#107—48no—
246#106—48no—
247#103—48no—
248#103—48no—
249C22⋊C8—28no—
250C22⋊C8—28no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24#12314yes—
269C24C22×D412yes—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—416no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—232no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4#12314yes—
322C22×C4#12318yes—
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4114yes—
330C22×C4C2×#4114yes—
331C22×C4C2×#4114yes—
332C22×C4H(5,𝔽2)14yes—
333C22×C42−1+414yes—
334C22×C4C2511yescharacteristic
335M16—416no—
336C2×C8—28no—
337C2×C8—28no—
338C2×C8—28no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C22⋊C4—28no—
365C22⋊C4—28no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—84no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—48no—
375C42—48no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—28no—
387C23—44no—
388C23—44no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23C2×#12314yes—
394C23C2×#12312yes—
395C23#53512yes—
396C23#53612yes—
397C23C23×D411yescharacteristic
398Q8—48no—
399Q8—84no—
400C2×C4—28no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—216no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4—84no—
446C2×C4C2×#12314yes—
447C2×C4C2×#12312yes—
448C2×C4#53512yes—
449C2×C4#53612yes—
450C2×C4#539116yes—
451C2×C4C22×#4112yes—
452C2×C4C2×H(5,𝔽2)12yes—
453C2×C4C2×2−1+412yes—
454C8—48no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—88no—
469C22#262314yes—
470C22C22×#12311yescharacteristic
471C22C2×#53511yescharacteristic
472C22C2×#53611yescharacteristic
473C22#358511yescharacteristic
474C22#358611yescharacteristic
475C22#358611yescharacteristic
476C22#358611yescharacteristic
477C4—24no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—82no—
484C4—84no—
485C4—82no—
486C4—84no—
487C4—82no—
488C4#262314yes—
489C4C2×#53914yes—
490C2—24no—
491C2—44no—
492C2C2×#262311yescharacteristic
493C2#6051911yescharacteristic
494C2#6052011yescharacteristic
495C2#6080911yescharacteristic
496C2#6081011yescharacteristic
497C2#6086911yescharacteristic
498C2#6087011yescharacteristic
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. #29919C2semidirect
4. #29918C2semidirect
5. #23747C2semidirect
6. #23732C2semidirect
7. #23657C2semidirect
8. #23629C2semidirect
38. #2101C22semidirect
39. #2100C22semidirect
40. #2099C22semidirect
41. #2097C22semidirect
42. #2096C22semidirect
43. #2095C22semidirect
44. #2085C22semidirect
45. #2084C22semidirect
46. #2083C22semidirect
47. #2074C22semidirect
48. #2072C22semidirect
49. #2072C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
#340522
#299192
#299182
#237472
#237322
#236572
#236292
#209112
#208462
#206832