Finite Group Databasebeta

← #495706

Subgroups — #495706

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2950C211yesmaximal, characteristic, complemented
2C2×#2185C211yesmaximal, characteristic
3C2×#2175C211yesmaximal, characteristic, complemented
4C2×C4×SD32C211yesmaximal, characteristic, complemented
5#21524C211yesmaximal, characteristic
6#13013C211yesmaximal, characteristic, complemented
7#13013C211yesmaximal, characteristic, complemented
8#12992C211yesmaximal, characteristic
9#12938C211yesmaximal, characteristic, complemented
10#12936C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12732C211yesmaximal, characteristic, complemented
13#12638C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15#12600C211yesmaximal, characteristic, complemented
16C23×Q16C2211yescharacteristic
17C2×#519—21no—
18C22×SD32C2211yescharacteristic
19C22×SD32C2211yescharacteristic
20C2×#466C2211yescharacteristic
21#2950—24no—
22#2950—24no—
23C2×#441—21no—
24C2×#441—21no—
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×D8C2211yescharacteristic
30C2×#427C2211yescharacteristic
31C2×#419—21no—
32C2×#405C2211yescharacteristic
33#2324—21no—
34#2185C418yescomplemented
35#2175—24no—
36C4×SD32—24no—
37C2×#367C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#358C2211yescharacteristic
43C2×#358C2211yescharacteristic
44C2×#358C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47C2×C22⋊C16C2211yescharacteristic
48C2×C22⋊C16C2211yescharacteristic
49C2×#346C2211yescharacteristic
50C2×C4×C16C2211yescharacteristic
51#1970C2211yescharacteristic
52#1942C2211yescharacteristic
53#1936—21no—
54#1895—21no—
55#1808C2211yescharacteristic
56C4×#107C2211yescharacteristic
57#1762—21no—
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62#1374C2211yescharacteristic
63#1374C2211yescharacteristic
64C23×Q8—21no—
65C22×Q16—22no—
66C22×Q16—22no—
67C22×Q16—22no—
68C22×Q16—22no—
69C22×Q16—21no—
70C22×Q16—22no—
71C22×Q16C2×C414yescomplemented
72C22×Q16C2311yescharacteristic
73C22×Q16C2311yescharacteristic
74C22×D8C2311yescharacteristic
75C22×M16D411yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#121—21no—
78#519—28no—
79#519—44no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8—41no—
85C2×C4×Q8—41no—
86C2×C4×D4—21no—
87C2×#117—21no—
88C2×#117—41no—
89C2×#117—41no—
90C2×#117D411yescharacteristic
91C2×SD32—22no—
92C2×SD32—22no—
93C2×SD32—22no—
94C2×SD32—22no—
95C2×SD32C2×C414yes—
96C2×SD32C2×C414yes—
97C22×C16C2311yescharacteristic
98C22×C16C2311yescharacteristic
99C22×C16C2311yescharacteristic
100C22×C16C2311yescharacteristic
101#466—22no—
102#466C2×C414yes—
103#441—28no—
104#441—28no—
105C4×Q16—22no—
106C4×Q16—22no—
107C4×Q16—22no—
108C4×Q16—22no—
109C4×Q16—22no—
110C4×Q16—22no—
111C4×Q16D414yescomplemented
112C4×Q16D414yescomplemented
113C4×D8—22no—
114C4×D8—22no—
115#427—22no—
116#427—22no—
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#107C2311yescharacteristic
121C2×#107C2311yescharacteristic
122C2×#106—21no—
123#419—28no—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×#102—21no—
131#405—22no—
132#405—22no—
133C2×#97D411yescharacteristic
134C2×#97D411yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#390—21no—
138#390—41no—
139#389—21no—
140#388—41no—
141#386—41no—
142#385—41no—
143#380—21no—
144C4×C4⋊C4—21no—
145#367—22no—
146#366—22no—
147#358—22no—
148#358—22no—
149#358—22no—
150#358—22no—
151#358C2×C414yescomplemented
152#358C2×C414yescomplemented
153#357—22no—
154#357—22no—
155C22⋊C16—22no—
156C22⋊C16C2×C414yescomplemented
157#346—22no—
158C4×C16—22no—
159#340—21no—
160#340—21no—
161#340—21no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×D4—21no—
172C22×D4—41no—
173C23×C4—41no—
174C23×C4—41no—
175C23×C4C2×D411yescharacteristic
176C2×Q16—24no—
177C2×Q16—28no—
178C2×Q16—28no—
179C2×Q16—24no—
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16—24no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×Q16—21no—
186C2×Q16—21no—
187C2×Q16—21no—
188C2×Q16—21no—
189C2×Q16C22×C412yes—
190C2×Q16C22×C412yes—
191C2×Q16C2×D412yes—
192C2×Q16C2×D412yes—
193C2×Q16#4112yes—
194C2×Q16#4112yes—
195C2×D8—21no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8C22×C412yes—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—41no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8C22×C412yes—
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8#4112yes—
213C22×C8C2411yescharacteristic
214#121—24no—
215#121—42no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×Q8—48no—
223C4×Q8—48no—
224C4×Q8—42no—
225C4×Q8—42no—
226C4×D4—42no—
227C4×D4—42no—
228#117—22no—
229#117—22no—
230#117—24no—
231#117—48no—
232#117—48no—
233#117—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4D811yescharacteristic
245C2×C4⋊C4D811yescharacteristic
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4D811yescharacteristic
253C2×C22⋊C4D811yescharacteristic
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262SD32—28no—
263SD32—28no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—21no—
268C2×C16—21no—
269C2×C16—21no—
270C2×C16—21no—
271C2×C16—21no—
272C2×C16—21no—
273C2×C16—21no—
274C2×C16C22×C412yes—
275C2×C16C22×C412yes—
276#107—21no—
277#107—21no—
278#107—21no—
279#107—21no—
280#107—21no—
281#107—21no—
282#107—21no—
283#107C22×C412yes—
284#107C2×D412yes—
285#107C2×D412yes—
286#106—24no—
287#103—24no—
288#103—24no—
289#103—24no—
290#103—24no—
291#102—24no—
292#102—42no—
293#102—42no—
294#97—22no—
295#97—22no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8C2×D412yes—
299C4×C8C2×D412yes—
300#95—41no—
301#95—41no—
302#95—41no—
303C24—41no—
304C24C2×D811yescharacteristic
305C2×Q8—216no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—44no—
324C2×D4—22no—
325C2×D4—44no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—161no—
332C2×D4—161no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—82no—
348C22×C4—161no—
349C22×C4—161no—
350C22×C4C4×D412yes—
351C22×C4C2×D811yescharacteristic
352C22×C4C2×D811yescharacteristic
353C22×C4C2×D811yescharacteristic
354C22×C4C2×D811yescharacteristic
355C22×C4C2×D811yescharacteristic
356C22×C4#13512yes—
357C22×C4C22×D411yescharacteristic
358Q16—216no—
359Q16—22no—
360Q16—22no—
361Q16—22no—
362Q16—22no—
363Q16—22no—
364Q16—22no—
365Q16C4×D414yes—
366Q16C4×D414yes—
367D8—22no—
368D8—22no—
369D8—41no—
370D8—41no—
371M16—44no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8C4×D412yes—
384C2×C8#12012yes—
385C2×C8C23×C411yescharacteristic
386C2×C8C22×D411yescharacteristic
387C2×C8C22×D411yescharacteristic
388C2×C8C2×#4111yescharacteristic
389C2×C8C2×#4111yescharacteristic
390C2×C8C2×#4111yescharacteristic
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—161no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—161no—
414C22⋊C4—161no—
415C22⋊C4—161no—
416C42—21no—
417C42—21no—
418C42—44no—
419C42—44no—
420C42—44no—
421C42—44no—
422C42—44no—
423C42—161no—
424C42#12012yes—
425C42AGL(1,ℤ/8)12yes—
426C16—22no—
427C16—22no—
428C16—41no—
429C16—41no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—42no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23C4×D812yes—
444C23C22×D811yescharacteristic
445Q8—24no—
446Q8—24no—
447Q8—416no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454D4—42no—
455D4—42no—
456D4—81no—
457D4—81no—
458D4—164no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—416no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—164no—
498C2×C4C4×D812yes—
499C2×C4#44712yes—
500C2×C4#44712yes—
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C2×#12011yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4C2×#13511yescharacteristic
505C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
506C2×C4#57511yescharacteristic
507C8—22no—
508C8—22no—
509C8—41no—
510C8C2×C4×D411yescharacteristic
511C8C2×C4×D411yescharacteristic
512C8#52511yescharacteristic
513C8#52511yescharacteristic
514C22—22no—
515C22—22no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—161no—
525C22—161no—
526C22—161no—
527C22—161no—
528C22C2×C4×D811yescharacteristic
529C22C2×#44711yescharacteristic
530C22#310011yescharacteristic
531C22#341611yescharacteristic
532C22#341611yescharacteristic
533C22#341611yescharacteristic
534C22#341711yescharacteristic
535C4—22no—
536C4—22no—
537C4—22no—
538C4—44no—
539C4—44no—
540C4—41no—
541C4—41no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—161no—
546C4—161no—
547C4C4×#12011yescharacteristic
548C4C4×AGL(1,ℤ/8)11yescharacteristic
549C4#300211yescharacteristic
550C4#310511yescharacteristic
551C2—42no—
552C2—162no—
553C2C4×#44711yescharacteristic
554C2#3345511yescharacteristic
555C2#3345711yescharacteristic
556C2#3352411yescharacteristic
557C2#3352711yescharacteristic
558C2#3368111yescharacteristic
559C2#3368611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2950C2semidirect
3. C2×#2175C2semidirect
4. C2×C4×SD32C2semidirect
6. #13013C2semidirect
7. #13013C2semidirect
9. #12938C2semidirect
10. #12936C2semidirect
11. #12921C2semidirect
12. #12732C2semidirect
13. #12638C2semidirect
14. C4×#358C2semidirect
15. #12600C2semidirect
34. #2185C4semidirect
71. C22×Q16C2×C4semidirect
111. C4×Q16D4semidirect
112. C4×Q16D4semidirect
151. #358C2×C4semidirect
152. #358C2×C4semidirect
156. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29502
C2×#21852
C2×#21752
C2×C4×SD322
#215242
#130132
#129922
#129382
#129362
#129212
#127322
#126382
C4×#3582
#126002