Finite Group Databasebeta

← #192661

Subgroups — #192661

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34191C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic, complemented
3#20816C211yesmaximal, characteristic, complemented
4#20743C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8891C212yesmaximal, complemented
7#8890C212yesmaximal, complemented
8#8887C212yesmaximal, complemented
9#8886C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#7898C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2355C2212yes—
17#2352—22no—
18D4×C4⋊C4C2212yes—
19#2346—21no—
20#2343—22no—
21#2339—21no—
22#2320—21no—
23#2318—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2283—21no—
27#2283C2211yescharacteristic
28C2×#382—21no—
29#2275—21no—
30#1948C2212yes—
31#1947C2212yes—
32#1854—22no—
33#1846—21no—
34#1844C2212yes—
35#1835—21no—
36#1834C2212yes—
37#1825—21no—
38#1807—22no—
39#1803C2212yes—
40#1802C2212yes—
41#1788—21no—
42C4×#106—22no—
43#1763—21no—
44#1762—21no—
45#1476C418yescomplemented
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41D411yescharacteristic
55C23×C8—21no—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41C2×C414yes—
59C4×#41Q814yes—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4D411yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C22×C4⋊C4—22no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#431—22no—
77#431C2×C414yescomplemented
78C2×#107—22no—
79C2×#106—24no—
80C2×#106—22no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#103—22no—
85C2×#102—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—22no—
92C2×C4×C8C2×C414yescomplemented
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—22no—
96#389—22no—
97#388—24no—
98#387—24no—
99#387—24no—
100#386—24no—
101#385—24no—
102#384C2312yes—
103#384C2312yes—
104#383—22no—
105#383—21no—
106#382—22no—
107#382—22no—
108#382—22no—
109#382—21no—
110#382—21no—
111#382D411yescharacteristic
112#382D411yescharacteristic
113#381—22no—
114#381—21no—
115#380—21no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380D412yes—
120#380D412yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—21no—
131#376—22no—
132#376—21no—
133C2×#95—22no—
134C2×#95—22no—
135#340—22no—
136#340—22no—
137#340—24no—
138#340—22no—
139#340—22no—
140Q8⋊C8C2×C414yescomplemented
141Q8⋊C8C2×C414yescomplemented
142D4⋊C8C2×C414yes—
143D4⋊C8C2×C414yes—
144C2×#41—22no—
145C2×#41—24no—
146C2×#41—42no—
147C2×#41C4⋊C414yes—
148C22×Q8C2×D411yescharacteristic
149C22×D4—21no—
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C22×C8—21no—
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—44no—
165C4×Q8—22no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8C4⋊C414yes—
169C4×Q8C22×C412yes—
170C4×Q8C2×Q812yes—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—22no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4C4⋊C414yes—
181C4×D4C22×C412yes—
182C4×D4C2×Q812yes—
183#117—22no—
184#117—24no—
185#117—24no—
186#117—81no—
187#117—81no—
188#117—81no—
189#117—81no—
190#117C4⋊C414yes—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D414yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—41no—
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×Q812yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#107—82no—
231#106—48no—
232#106—82no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239#103—44no—
240#102—44no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243#97—42no—
244C4×C8—24no—
245C4×C8—24no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8C22×C412yes—
249C4×C8C22×C412yes—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—21no—
261#95—22no—
262#95—41no—
263#95—41no—
264C24—21no—
265C24#12011yescharacteristic
266#41—28no—
267#41—44no—
268#41—48no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8C2×C4⋊C412yes—
273C2×D4—24no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—44no—
277C2×D4C2×C4⋊C412yes—
278C2×D4#13514yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—48no—
302C22×C4—42no—
303C22×C4—82no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4C2×C4⋊C412yes—
307C22×C4C4×D412yes—
308C22×C4C4×D412yes—
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4C22⋊Q812yes—
313C22×C4C22⋊Q812yes—
314C22×C4#12312yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—48no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—84no—
352C4⋊C4C2×C4⋊C412yes—
353C4⋊C4C2×C4⋊C412yes—
354C22⋊C4—24no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—24no—
358C22⋊C4—48no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C42—22no—
363C42—22no—
364C42—24no—
365C42—24no—
366C42—42no—
367C42—42no—
368C42—82no—
369C42—82no—
370C42—84no—
371C42C2×C4⋊C412yes—
372C42C4×D412yes—
373C42C4×D412yes—
374C42C22⋊Q812yes—
375C42C22⋊Q812yes—
376C42#12312yes—
377C42C23×C411yescharacteristic
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C23—24no—
382C23—22no—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—22no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23#38512yes—
391C23C2×#12011yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—42no—
395D4—24no—
396D4—22no—
397D4—48no—
398D4#42714yes—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—48no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—168no—
439C2×C4#38512yes—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4C22×C4⋊C411yescharacteristic
443C2×C4C2×C4×D411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×#12311yescharacteristic
448C2×C4C2×#13512yes—
449C2×C4#57512yes—
450C2×C4#57711yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57711yescharacteristic
454C8—24no—
455C8—42no—
456C8—42no—
457C8—42no—
458C8—82no—
459C22—22no—
460C22—24no—
461C22—22no—
462C22—22no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22—44no—
469C22C2×#38511yescharacteristic
470C22#300212yes—
471C22#300911yescharacteristic
472C22#301011yescharacteristic
473C22#301812yes—
474C4—22no—
475C4—21no—
476C4—22no—
477C4—21no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—84no—
483C4—82no—
484C4—82no—
485C4—162no—
486C4C2×#42711yescharacteristic
487C4#291111yescharacteristic
488C4#295211yescharacteristic
489C4#295211yescharacteristic
490C4#308411yescharacteristic
491C4#308411yescharacteristic
492C4#309112yes—
493C2—42no—
494C2—44no—
495C2#1865912yes—
496C2#1866012yes—
497C2#2084311yescharacteristic
498C2#2086311yescharacteristic
499C2#2088211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #21985C2semidirect
3. #20816C2semidirect
6. #8891C2semidirect
7. #8890C2semidirect
8. #8887C2semidirect
9. #8886C2semidirect
10. #8046C2semidirect
11. #7898C2semidirect
45. #1476C4semidirect
77. #431C2×C4semidirect
92. C2×C4×C8C2×C4semidirect
140. Q8⋊C8C2×C4semidirect
141. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341912
#219852
#208162
#207432
C2×#14762
#88912
#88902
#88872
#88862
#80462
#78982