Finite Group Databasebeta

← #6434101

Subgroups — #6434101

Order 512.

1882 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2624C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic, complemented
3#29917C218yesmaximal, complemented
4#29916C218yesmaximal, complemented
5#23747C212yesmaximal, complemented
6#23731C212yesmaximal, complemented
7#23657C212yesmaximal, complemented
8#23628C212yesmaximal, complemented
9#20912C212yesmaximal
10#20846C212yesmaximal, complemented
11#20683C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2212yes—
14C22×#117C2211yescharacteristic
15C2×#428C2214yes—
16C2×#418C2214yes—
17C2×#407C2212yes—
18#2624—24no—
19#2624C2218yes—
20C2×#399—21no—
21C2×#397—21no—
22C2×#394—21no—
23C2×#394—21no—
24#2317C2214yescomplemented
25#2317C2214yescomplemented
26#2316—21no—
27#2316—21no—
28#2315—21no—
29#2315—21no—
30#2296—24no—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35#2285C2212yes—
36#2285C2212yes—
37C2×#380C2211yescharacteristic
38#2101C2218yescomplemented
39#2100C2214yescomplemented
40#2099C2214yescomplemented
41#2094C2214yescomplemented
42#2093C2214yescomplemented
43#2092C2218yescomplemented
44#2085C2218yescomplemented
45#2084C2214yescomplemented
46#2083C2214yescomplemented
47#2073C2218yescomplemented
48#2071C2214yescomplemented
49#2071C2214yescomplemented
50#1875C2218yes—
51#1874C2214yes—
52#1874C2214yes—
53#1854C2218yescomplemented
54#1853C2214yescomplemented
55#1852C2214yescomplemented
56#1818C2218yescomplemented
57#1736C2214yes—
58#1736C2214yes—
59#1728C2212yes—
60#1728C2212yes—
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C22×M16C2314yes—
65C2×#121—22no—
66C2×#121—28no—
67C2×#121C2314yes—
68C2×#117—28no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—48no—
72C2×#117D414yes—
73C2×#117C2314yes—
74C2×#117C2314yes—
75C2×#117C2314yes—
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⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—22no—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#428—48no—
98C2×#107C2318yes—
99C2×#106C2318yes—
100#418—48no—
101C2×#102C2318yes—
102C2×#102C2318yes—
103#407—28no—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106#399—28no—
107#397—28no—
108#394—28no—
109#394—28no—
110#390—22no—
111#390—22no—
112#390C2314yes—
113#390C2314yes—
114#389—24no—
115#389—24no—
116#389—24no—
117#389—24no—
118#389—22no—
119#389—22no—
120#389C2314yes—
121#389C2314yes—
122#385—24no—
123#385—24no—
124#385—24no—
125#385—24no—
126#384C2312yes—
127#384C2312yes—
128#384C2312yes—
129#384C2312yes—
130#382C2314yes—
131#382C2314yes—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380C2314yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
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—
153C25—21no—
154C25C2×D411yescharacteristic
155C22×D4—24no—
156C22×D4—21no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—44no—
161C22×D4—82no—
162C22×D4—82no—
163C22×D4C2412yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4118yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C2×M16—28no—
181C2×M16—28no—
182C2×M16—28no—
183C22×C8C2418yes—
184#121—28no—
185#121—28no—
186#121—48no—
187#117—216no—
188#117—28no—
189#117—28no—
190#117—28no—
191#117—28no—
192#117—44no—
193#117—44no—
194#117—48no—
195#117—48no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2414yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2412yes—
225C2×C4⋊C4C2412yes—
226C2×C4⋊C4C2412yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4C2×D414yes—
245C2×C22⋊C4C2×D414yes—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—48no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2412yes—
254C2×C42C2412yes—
255#107—48no—
256#106—48no—
257#102—48no—
258#102—48no—
259C22⋊C8—28no—
260C22⋊C8—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24#12314yes—
283C24C22×D412yes—
284C24C22×D411yescharacteristic
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—48no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—416no—
293C2×D4—84no—
294C2×D4—84no—
295C2×D4—816no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—232no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4#12314yes—
336C22×C4#12318yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4114yes—
344C22×C4C2×#4114yes—
345C22×C4C2×#4114yes—
346C22×C4H(5,𝔽2)14yes—
347C22×C42−1+414yes—
348C22×C4C2511yescharacteristic
349M16—416no—
350C2×C8—28no—
351C2×C8—28no—
352C2×C8—28no—
353C4⋊C4—28no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C22⋊C4—28no—
378C22⋊C4—28no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—48no—
392C42—48no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—28no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23—84no—
421C23—84no—
422C23—84no—
423C23—84no—
424C23C2×#12314yes—
425C23C2×#12312yes—
426C23#53512yes—
427C23#53612yes—
428C23C23×D411yescharacteristic
429D4—48no—
430D4—816no—
431D4—84no—
432C2×C4—28no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—216no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—88no—
473C2×C4—88no—
474C2×C4C2×#12314yes—
475C2×C4C2×#12312yes—
476C2×C4#53512yes—
477C2×C4#53612yes—
478C2×C4#539116yes—
479C2×C4C22×#4112yes—
480C2×C4C2×H(5,𝔽2)12yes—
481C2×C4C2×2−1+412yes—
482C8—48no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—88no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—168no—
505C22#262314yes—
506C22C22×#12311yescharacteristic
507C22C2×#53511yescharacteristic
508C22C2×#53611yescharacteristic
509C22#358511yescharacteristic
510C22#358511yescharacteristic
511C22#358511yescharacteristic
512C22#358611yescharacteristic
513C4—24no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—82no—
520C4—84no—
521C4—84no—
522C4—82no—
523C4#262314yes—
524C4C2×#53914yes—
525C2—24no—
526C2—44no—
527C2—162no—
528C2C2×#262311yescharacteristic
529C2#6051811yescharacteristic
530C2#6052011yescharacteristic
531C2#6080811yescharacteristic
532C2#6081011yescharacteristic
533C2#6086811yescharacteristic
534C2#6087011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. #34052C2semidirect
3. #29917C2semidirect
4. #29916C2semidirect
5. #23747C2semidirect
6. #23731C2semidirect
7. #23657C2semidirect
8. #23628C2semidirect
10. #20846C2semidirect
11. #20683C2semidirect
24. #2317C22semidirect
25. #2317C22semidirect
38. #2101C22semidirect
39. #2100C22semidirect
40. #2099C22semidirect
41. #2094C22semidirect
42. #2093C22semidirect
43. #2092C22semidirect
44. #2085C22semidirect
45. #2084C22semidirect
46. #2083C22semidirect
47. #2073C22semidirect
48. #2071C22semidirect
49. #2071C22semidirect
53. #1854C22semidirect
54. #1853C22semidirect
55. #1852C22semidirect
56. #1818C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26242
#340522
#299172
#299162
#237472
#237312
#236572
#236282
#209122
#208462
#206832