Finite Group Databasebeta

← #192599

Subgroups — #192599

Order 512.

1104 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#34127C211yesmaximal, characteristic
2#21362C211yesmaximal, characteristic, complemented
3#20760C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8895C212yesmaximal, complemented
7#8894C212yesmaximal, complemented
8#8891C212yesmaximal, complemented
9#8890C212yesmaximal, complemented
10#8038C211yesmaximal, characteristic, complemented
11#7890C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2355C2212yes—
17#2352—22no—
18D4×C4⋊C4C2212yes—
19#2347—21no—
20#2343—22no—
21#2339—21no—
22#2295—21no—
23#2290—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2283—21no—
27C42×D4—21no—
28C4×#117C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30#1948C2212yes—
31#1947C2212yes—
32#1854—22no—
33#1835—22no—
34#1834C2212yes—
35#1834C2212yes—
36#1825—21no—
37#1803C2212yes—
38#1802C2212yes—
39#1788—21no—
40C4×#107—21no—
41C4×#106—21no—
42C4×#103—21no—
43C4×#102—21no—
44#1735—22no—
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×#117D412yes—
70C2×#117D411yescharacteristic
71C2×#117D412yes—
72C22×C4⋊C4—22no—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2×C414yescomplemented
80C2×#107—22no—
81C2×#106—22no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#103—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—22no—
92C2×C4×C8—21no—
93C2×C4×C8C2×C414yescomplemented
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—22no—
97#389—22no—
98#388—24no—
99#387—24no—
100#387—24no—
101#386—24no—
102#385—24no—
103#384C2312yes—
104#384C2312yes—
105#383—22no—
106#382—22no—
107#382—21no—
108#382D412yes—
109#381—22no—
110#380—21no—
111#380—21no—
112#380—22no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130#376—22no—
131#376—21no—
132C2×#95—22no—
133C43—21no—
134#340—22no—
135#340—22no—
136#340—24no—
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—24no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—42no—
165C22×C8—42no—
166C4×Q8—22no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8C4⋊C414yes—
170C4×Q8C22×C412yes—
171C4×Q8C2×Q812yes—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—22no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4C4⋊C414yes—
182C4×D4C22×C412yes—
183C4×D4C2×Q812yes—
184#117—22no—
185#117—24no—
186#117—24no—
187#117—24no—
188#117—42no—
189#117—44no—
190#117—44no—
191#117C4⋊C414yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D414yes—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—41no—
226C2×C42—84no—
227C2×C42C22×C412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×Q812yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235#107—44no—
236#106—44no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243#103—44no—
244#102—44no—
245C22⋊C8—24no—
246C22⋊C8—42no—
247C4×C8—24no—
248C4×C8—24no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8C22×C412yes—
253C4×C8C22×C412yes—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—21no—
259#95—22no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—41no—
264#95—41no—
265C24—21no—
266C24#12011yescharacteristic
267#41—28no—
268#41—44no—
269#41—48no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8C2×C4⋊C412yes—
274C2×D4—24no—
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4C2×C4⋊C412yes—
279C2×D4#13514yes—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—44no—
297C22×C4—48no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—48no—
303C22×C4—42no—
304C22×C4—84no—
305C22×C4C2×C4⋊C412yes—
306C22×C4C4×D412yes—
307C22×C4C4×D412yes—
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4C22⋊Q812yes—
312C22×C4C22⋊Q812yes—
313C22×C4#12312yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—48no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4C2×C4⋊C412yes—
351C4⋊C4C2×C4⋊C412yes—
352C4⋊C4#13514yes—
353C22⋊C4—24no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—24no—
357C22⋊C4—44no—
358C22⋊C4—48no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—22no—
362C42—22no—
363C42—24no—
364C42—24no—
365C42—44no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—44no—
371C42—84no—
372C42—82no—
373C42C2×C4⋊C412yes—
374C42C4×D412yes—
375C42C4×D412yes—
376C42C22⋊Q812yes—
377C42C22⋊Q812yes—
378C42#12312yes—
379C42C23×C411yescharacteristic
380C42C22×Q811yescharacteristic
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C23—22no—
384C23—24no—
385C23—22no—
386C23—21no—
387C23—21no—
388C23—22no—
389C23—42no—
390C23—41no—
391C23—41no—
392C23#38512yes—
393C23C2×#12011yescharacteristic
394Q8—22no—
395Q8—22no—
396Q8—42no—
397D4—24no—
398D4—22no—
399D4—48no—
400D4#42714yes—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—816no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#38512yes—
439C2×C4#38512yes—
440C2×C4#38512yes—
441C2×C4#45412yes—
442C2×C4#45512yes—
443C2×C4C22×C4⋊C411yescharacteristic
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C2×C4C2×#13512yes—
450C2×C4C2×#13512yes—
451C2×C4#57512yes—
452C2×C4#57512yes—
453C8—42no—
454C8—42no—
455C8—42no—
456C8—42no—
457C8—82no—
458C22—22no—
459C22—24no—
460C22—22no—
461C22—22no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22C2×#38511yescharacteristic
469C22#300212yes—
470C22#300212yes—
471C22C2×#45511yescharacteristic
472C22C2×#45411yescharacteristic
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—44no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—84no—
482C4—82no—
483C4—162no—
484C4#186014yes—
485C4C2×#42711yescharacteristic
486C4#291111yescharacteristic
487C4C4×#13511yescharacteristic
488C4#294311yescharacteristic
489C4#307612yes—
490C4#309112yes—
491C2—42no—
492C2—44no—
493C2#1848512yes—
494C2#1849412yes—
495C2#2084311yescharacteristic
496C2C2×#186011yescharacteristic
497C2#2087211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×#120
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21362C2semidirect
3. #20760C2semidirect
6. #8895C2semidirect
7. #8894C2semidirect
8. #8891C2semidirect
9. #8890C2semidirect
10. #8038C2semidirect
11. #7890C2semidirect
45. #1476C4semidirect
79. #431C2×C4semidirect
93. 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
#341272
#213622
#207602
#207372
C2×#14762
#88952
#88942
#88912
#88902
#80382
#78902