Finite Group Databasebeta

← #192766

Subgroups — #192766

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic, complemented
3#21982C211yesmaximal, characteristic, complemented
4#20743C211yesmaximal, characteristic
5C2×#1488C211yesmaximal, characteristic
6#8897C212yesmaximal, complemented
7#8895C212yesmaximal, complemented
8#8885C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic, complemented
11#7900C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2355—21no—
17#2355C2212yes—
18#2354—21no—
19#2354C2212yes—
20#2352—22no—
21#2352—22no—
22#2314—22no—
23#2311—21no—
24#2285—21no—
25#2285—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#2274—21no—
30#1948C2212yes—
31#1947—22no—
32#1947C2212yes—
33#1854—22no—
34#1852—22no—
35#1847—21no—
36#1845C2212yes—
37#1836—21no—
38#1834C2212yes—
39#1825—21no—
40#1803C2211yescharacteristic
41#1803C2211yescharacteristic
42#1802C2211yescharacteristic
43#1802C2211yescharacteristic
44#1788—21no—
45#1763—21no—
46#1762—21no—
47#1488C418yescomplemented
48C2×Q8⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×#41D411yescharacteristic
55C4×#41—22no—
56C4×#41—22no—
57C4×#41C2×C414yes—
58C4×#41Q814yes—
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×#117—21no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#432—22no—
79#432C2×C414yes—
80#431—22no—
81#431C2×C414yes—
82C2×#107—24no—
83C2×#107—22no—
84C2×#106—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2312yes—
89C2×#103—21no—
90C2×#103—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8C2312yes—
99#391—22no—
100#390—22no—
101#388—24no—
102#388—24no—
103#387—24no—
104#387—24no—
105#386—24no—
106#384—21no—
107#384—21no—
108#384D412yes—
109#384D411yescharacteristic
110#384D411yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2312yes—
114#382—22no—
115#382—22no—
116#382—21no—
117#382—21no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—21no—
122#380—22no—
123#380—21no—
124#380—21no—
125#380D412yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2312yes—
136C4×C22⋊C4—24no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—24no—
142#340—22no—
143#340—22no—
144Q8⋊C8C2×C418yescomplemented
145D4⋊C8C2×C418yescomplemented
146C2×#41—22no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—41no—
151C2×#41C4⋊C414yes—
152C22×Q8C2×D411yescharacteristic
153C22×D4—21no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
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×D4C4⋊C414yes—
180C4×D4C22×C412yes—
181C4×D4C2×Q812yes—
182#117—22no—
183#117—24no—
184#117—24no—
185#117—42no—
186#117—42no—
187#117—82no—
188#117—82no—
189#117—82no—
190#117—84no—
191#117C4⋊C414yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D414yes—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—41no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—41no—
234C2×C42C22×C412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242#107—82no—
243#107—84no—
244#106—82no—
245C4⋊C8—24no—
246C4⋊C8—22no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8C22×C414yes—
252#103—42no—
253#103—42no—
254#102—42no—
255#102—42no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C22⋊C8—42no—
259#97—24no—
260#97—42no—
261C4×C8—22no—
262C4×C8C22×C414yes—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—41no—
270#95—41no—
271C24—21no—
272C24#12011yescharacteristic
273#41—44no—
274#41—44no—
275#41—44no—
276#41—44no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8C2×C4⋊C412yes—
283C2×D4—24no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4C2×C4⋊C412yes—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4C2×C4⋊C412yes—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4#12312yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—82no—
373C4⋊C4C2×C4⋊C412yes—
374C4⋊C4C2×C4⋊C412yes—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—24no—
379C22⋊C4—44no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—82no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—42no—
388C42—42no—
389C42—82no—
390C42—84no—
391C42—84no—
392C42C2×C4⋊C412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42C22⋊Q812yes—
396C42C22⋊Q812yes—
397C42#12312yes—
398C42C23×C411yescharacteristic
399C42C22×Q811yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23#38512yes—
416C23C2×#12011yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—41no—
420Q8—41no—
421D4—22no—
422D4—22no—
423D4—48no—
424D4—41no—
425D4—41no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—164no—
471C2×C4#38512yes—
472C2×C4#38512yes—
473C2×C4#38512yes—
474C2×C4C22×C4⋊C411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×C22⋊Q811yescharacteristic
478C2×C4C2×C22⋊Q811yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57511yescharacteristic
482C2×C4#57511yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57612yes—
485C2×C4#57812yes—
486C8—44no—
487C8—44no—
488C8—81no—
489C8—81no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22C2×#38511yescharacteristic
505C22#300511yescharacteristic
506C22#300511yescharacteristic
507C22#300811yescharacteristic
508C22#301111yescharacteristic
509C22#301911yescharacteristic
510C22#302011yescharacteristic
511C4—22no—
512C4—21no—
513C4—22no—
514C4—21no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—161no—
523C4—161no—
524C4#291111yescharacteristic
525C4#291111yescharacteristic
526C4#295112yes—
527C4#308312yes—
528C4#309311yescharacteristic
529C4#309311yescharacteristic
530C2—42no—
531C2—82no—
532C2#1881811yescharacteristic
533C2#1882111yescharacteristic
534C2#1883111yescharacteristic
535C2#1883211yescharacteristic
536C2#2084511yescharacteristic
537C2#2086411yescharacteristic
538C2#2088111yescharacteristic
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. #21988C2semidirect
3. #21982C2semidirect
6. #8897C2semidirect
7. #8895C2semidirect
8. #8885C2semidirect
9. #8883C2semidirect
10. #8040C2semidirect
11. #7900C2semidirect
47. #1488C4semidirect
144. Q8⋊C8C2×C4semidirect
145. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341882
#219882
#219822
#207432
C2×#14882
#88972
#88952
#88852
#88832
#80402
#79002