Finite Group Databasebeta

← #6477790

Subgroups — #6477790

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic
3#33149C212yesmaximal, complemented
4#33147C212yesmaximal, complemented
5#29968C212yesmaximal, complemented
6#29966C212yesmaximal, complemented
7#26007C218yesmaximal, complemented
8#26005C218yesmaximal, complemented
9#21942C212yesmaximal
10#21934C211yesmaximal, characteristic
11#20702C212yesmaximal
12C2×#543—21no—
13C22×#117C2211yescharacteristic
14C2×#407C2212yes—
15#2857C2212yes—
16#2857C2212yes—
17#2659—21no—
18#2659—21no—
19#2549—24no—
20#2440C2214yes—
21#2440C2214yes—
22#2347—24no—
23#2347C2218yes—
24#2315—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#388—21no—
29C2×#383—21no—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#2100C2214yescomplemented
37#2099C2214yescomplemented
38#2088C2218yescomplemented
39#2086C2218yescomplemented
40#2058C2214yescomplemented
41#2057C2214yescomplemented
42#1988C2214yescomplemented
43#1987C2214yescomplemented
44#1948C2214yescomplemented
45#1947C2214yescomplemented
46#1921C2218yescomplemented
47#1919C2218yescomplemented
48#1853C2214yes—
49#1853C2214yes—
50#1852C2214yes—
51#1852C2214yes—
52#1851C2218yes—
53#1825C2218yes—
54#1824C2214yes—
55#1823C2214yes—
56#1809C2218yes—
57#1797C2212yes—
58#1797C2212yes—
59#1736C2214yes—
60#1736C2214yes—
61C24×C4C2311yescharacteristic
62C22×M16C2314yes—
63#543—48no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×#117—22no—
71C2×#117—24no—
72C2×#117Q818yescomplemented
73C2×#117C2314yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×#107C2318yes—
93C2×#106C2318yes—
94C2×C4⋊C8C2314yes—
95C2×C4⋊C8C2314yes—
96#407—28no—
97C2×C22⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99#398—24no—
100#398—24no—
101#395—24no—
102#395—24no—
103#391—22no—
104#391—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389C2314yes—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#388—28no—
114#388—28no—
115#385—24no—
116#385—24no—
117#384C2312yes—
118#384C2312yes—
119#383—28no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380C2314yes—
126#380C2314yes—
127#380C2314yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2314yes—
133C4×C22⋊C4D414yes—
134C4×C22⋊C4C2314yes—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25C2×D411yescharacteristic
152C22×Q8—22no—
153C22×Q8—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×Q814yes—
169C23×C4#4114yes—
170C23×C4C2411yescharacteristic
171C23×C4C2412yes—
172C2×M16—28no—
173C2×M16—44no—
174C2×M16—44no—
175C22×C8C2418yes—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
178C22⋊Q8—416no—
179C22⋊Q8—416no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182#117—216no—
183#117—28no—
184#117C2×Q8116yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C2×Q814yes—
210C2×C4⋊C4C2×Q814yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2412yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2412yes—
216C2×C4⋊C4C2412yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4C2×D414yes—
227C2×C22⋊C4C2×D414yes—
228C2×C42—28no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—82no—
235C2×C42—82no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2412yes—
239C2×C42C2412yes—
240#107—48no—
241#106—48no—
242C4⋊C8—44no—
243C4⋊C8—44no—
244C22⋊C8—28no—
245C22⋊C8—28no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95C2×D414yes—
256#95C2×D414yes—
257C24—22no—
258C24—22no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24C22⋊Q814yes—
264C24C22×D412yes—
265C24C22×D411yescharacteristic
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—82no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C22⋊Q814yes—
317C22×C4C22⋊Q818yes—
318C22×C4AGL(1,ℤ/8)14yes—
319C22×C4#13714yes—
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×Q812yes—
327C22×C4C22×Q814yes—
328C22×C4C2×#4112yes—
329C22×C4C2×#4114yes—
330C22×C4C2511yescharacteristic
331M16—88no—
332C2×C8—28no—
333C2×C8—28no—
334C2×C8—28no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4C22×Q814yes—
358C4⋊C4C22×Q814yes—
359C22⋊C4—216no—
360C22⋊C4—28no—
361C22⋊C4—28no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—84no—
365C42—216no—
366C42—24no—
367C42—24no—
368C42—42no—
369C42—42no—
370C42—84no—
371C42—84no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—28no—
383C23—44no—
384C23—44no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23C2×C22⋊Q814yes—
390C23C2×C22⋊Q812yes—
391C23C2×AGL(1,ℤ/8)12yes—
392C23C2×#13712yes—
393C23C23×D411yescharacteristic
394Q8—164no—
395Q8—164no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—164no—
443C2×C4—164no—
444C2×C4C2×C22⋊Q814yes—
445C2×C4C2×C22⋊Q812yes—
446C2×C4C2×AGL(1,ℤ/8)14yes—
447C2×C4C2×#13714yes—
448C2×C4C23×Q811yescharacteristic
449C2×C4C22×#4111yescharacteristic
450C2×C4#58512yes—
451C2×C4#58512yes—
452C8—48no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—88no—
467C22#307914yes—
468C22C22×C22⋊Q811yescharacteristic
469C22#348511yescharacteristic
470C22#348511yescharacteristic
471C22C22×AGL(1,ℤ/8)11yescharacteristic
472C22C22×#13711yescharacteristic
473C22#358511yescharacteristic
474C22#358611yescharacteristic
475C4—24no—
476C4—24no—
477C4—22no—
478C4—24no—
479C4—42no—
480C4—42no—
481C4—44no—
482C4—44no—
483C4—82no—
484C4—84no—
485C4—82no—
486C4—84no—
487C4#345812yes—
488C4#349112yes—
489C2—24no—
490C2—44no—
491C2#3509011yescharacteristic
492C2C2×#307911yescharacteristic
493C2#6049211yescharacteristic
494C2#6170711yescharacteristic
495C2#6170811yescharacteristic
496C2#6172211yescharacteristic
497C2#6172311yescharacteristic
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
3. #33149C2semidirect
4. #33147C2semidirect
5. #29968C2semidirect
6. #29966C2semidirect
7. #26007C2semidirect
8. #26005C2semidirect
36. #2100C22semidirect
37. #2099C22semidirect
38. #2088C22semidirect
39. #2086C22semidirect
40. #2058C22semidirect
41. #2057C22semidirect
42. #1988C22semidirect
43. #1987C22semidirect
44. #1948C22semidirect
45. #1947C22semidirect
46. #1921C22semidirect
47. #1919C22semidirect
72. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23472
#331492
#331472
#299682
#299662
#260072
#260052
#219422
#219342
#207022