Finite Group Databasebeta

← #193232

Subgroups — #193232

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic
6#8926C214yesmaximal, complemented
7#8921C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×#107—21no—
15C22×C4⋊C8C2211yescharacteristic
16#2367—22no—
17#2366C2212yes—
18#2365C2212yes—
19#2354—21no—
20#2352—22no—
21D4×C4⋊C4—21no—
22#2325—22no—
23#2311—21no—
24C4×#121—22no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990C2212yes—
29#1957C2212yes—
30#1948C2214yes—
31#1947—22no—
32#1947C2214yes—
33#1874—22no—
34#1853—22no—
35#1844—21no—
36#1834—21no—
37#1802—22no—
38C2×#340—21no—
39#1540C418yescomplemented
40C2×#332C2212yes—
41C2×#331C2212yes—
42#1279C2214yes—
43#1279C2214yes—
44C22×#41—21no—
45C23×C8—21no—
46#532—22no—
47#532—22no—
48#532C2×C414yes—
49#532Q814yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×#127C2311yescharacteristic
52C2×#124—21no—
53C2×#121—21no—
54C2×#121D412yes—
55#520—44no—
56#520—44no—
57C2×C4×Q8—22no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4D412yes—
63C2×#117—22no—
64C2×#117—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75#431—22no—
76#431C2×C414yescomplemented
77C2×#107—24no—
78C2×#107—44no—
79C2×#106—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2×C414yescomplemented
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86C2×#102—24no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#391—22no—
92#390—22no—
93#390—22no—
94#389—22no—
95#388—24no—
96#388—24no—
97#387—24no—
98#386—24no—
99#386—24no—
100#385—24no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384C2312yes—
107#382—21no—
108#380—22no—
109#380—22no—
110#380—21no—
111#380D412yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2312yes—
118C4×C4⋊C4C2312yes—
119C4×C22⋊C4—24no—
120C2×#95—21no—
121#340—24no—
122#340—24no—
123#340—24no—
124#332C2×C418yescomplemented
125#331C2×C418yescomplemented
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—21no—
141C23×C4—41no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C22×C8—24no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—42no—
150C22×C8—42no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C22×C412yes—
154C4⋊Q8C2×Q812yes—
155#127—21no—
156#127—21no—
157#127C22×C412yes—
158#127C2×Q812yes—
159#124—24no—
160#124—24no—
161#121—24no—
162#121—24no—
163#121—24no—
164#121C4⋊C418yes—
165C4×Q8—44no—
166C4×Q8—44no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—44no—
171C4×D4—416no—
172C4×D4—44no—
173C4×D4—84no—
174C4×D4C4⋊C418yes—
175#117—48no—
176#117—42no—
177#117—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4SD1614yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—82no—
217C2×C42C22×C412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×Q812yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225#107—48no—
226#106—84no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8C22×C414yes—
232C4⋊C8C22×C414yes—
233#102—84no—
234C22⋊C8—24no—
235C22⋊C8—42no—
236C4×C8—24no—
237C4×C8—22no—
238C4×C8—22no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245C24—22no—
246C24#12011yescharacteristic
247#41—48no—
248#41—48no—
249#41—48no—
250#41—48no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×D4—28no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4C2×C4⋊C414yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—28no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—48no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C4⋊C412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4#12312yes—
308C22×C4C2×SD1612yes—
309C22×C4#13512yes—
310C22×C4AGL(1,ℤ/8)12yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—48no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4C2×C4⋊C414yes—
348C4⋊C4C2×SD1614yes—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—48no—
352C22⋊C4—44no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—48no—
356C42—42no—
357C42—42no—
358C42—44no—
359C42—82no—
360C42C2×C4⋊C412yes—
361C42C4×D412yes—
362C42C4×D412yes—
363C42C22⋊Q812yes—
364C42C22⋊Q812yes—
365C42#12312yes—
366C42C23×C411yescharacteristic
367C42C22×Q811yescharacteristic
368C42C2×#4111yescharacteristic
369C42C2×#4111yescharacteristic
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—24no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23#38512yes—
382C23#44912yes—
383C23#45012yes—
384C23C2×#12011yescharacteristic
385Q8—42no—
386Q8—42no—
387Q8—42no—
388Q8—42no—
389D4—44no—
390D4—416no—
391D4—42no—
392D4—44no—
393D4—42no—
394D4—42no—
395D4—44no—
396D4—42no—
397D4—82no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—416no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—168no—
430C2×C4—164no—
431C2×C4#38512yes—
432C2×C4#38512yes—
433C2×C4#38512yes—
434C2×C4C4×SD1614yes—
435C2×C4#44214yes—
436C2×C4#44912yes—
437C2×C4#45012yes—
438C2×C4#47614yes—
439C2×C4#47814yes—
440C2×C4C22×C4⋊C411yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4C2×C4×D411yescharacteristic
443C2×C4C2×C22⋊Q811yescharacteristic
444C2×C4C2×C22⋊Q811yescharacteristic
445C2×C4C2×#12311yescharacteristic
446C2×C4C22×SD1611yescharacteristic
447C2×C4C2×#13511yescharacteristic
448C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
449C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
450C2×C4#57612yes—
451C2×C4#57812yes—
452C8—44no—
453C8—44no—
454C8—82no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22#186414yes—
469C22C2×#38511yescharacteristic
470C22C2×#45011yescharacteristic
471C22C2×#44911yescharacteristic
472C22#300811yescharacteristic
473C22#301111yescharacteristic
474C22#302111yescharacteristic
475C22#302211yescharacteristic
476C4—21no—
477C4—21no—
478C4—22no—
479C4—22no—
480C4—44no—
481C4—44no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—162no—
486C4#186414yes—
487C4C2×C4×SD1611yescharacteristic
488C4C2×#44211yescharacteristic
489C4#295112yes—
490C4C2×#47611yescharacteristic
491C4C2×#47811yescharacteristic
492C4#308312yes—
493C2—24no—
494C2—84no—
495C2#1890311yescharacteristic
496C2#1890611yescharacteristic
497C2#1891311yescharacteristic
498C2#1891611yescharacteristic
499C2#2085811yescharacteristic
500C2C2×#186411yescharacteristic
501C2#2088111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21981C2semidirect
4. C2×#1947C2semidirect
6. #8926C2semidirect
7. #8921C2semidirect
8. #7893C2semidirect
9. #7892C2semidirect
39. #1540C4semidirect
76. #431C2×C4semidirect
83. C2×C4⋊C8C2×C4semidirect
124. #332C2×C4semidirect
125. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219812
#218592
C2×#19472
C2×#15402
#89262
#89212
#78932
#78922