Finite Group Databasebeta

← #384897

Subgroups — #384897

Order 512.

819 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21178C211yesmaximal, characteristic, complemented
2C2×#1868C211yesmaximal, characteristic
3#20675C211yesmaximal, characteristic
4#11350C211yesmaximal, characteristic, complemented
5#11284C211yesmaximal, characteristic, complemented
6#11240C211yesmaximal, characteristic, complemented
7#11236C211yesmaximal, characteristic, complemented
8#11108C211yesmaximal, characteristic
9#11094C211yesmaximal, characteristic
10#10956C211yesmaximal, characteristic
11#10951C211yesmaximal, characteristic
12#10917C211yesmaximal, characteristic, complemented
13#10905C211yesmaximal, characteristic, complemented
14#10831C211yesmaximal, characteristic, complemented
15#10787C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C22×#108C2211yescharacteristic
19#2911C2211yescharacteristic
20#2905C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#421C2211yescharacteristic
24C2×#421C2211yescharacteristic
25C2×#421C2211yescharacteristic
26#2896C2211yescharacteristic
27C2×#410C2211yescharacteristic
28C2×#408C2211yescharacteristic
29#2873C2211yescharacteristic
30#2873C2211yescharacteristic
31#1950—24no—
32#1868C418yescomplemented
33#1814—24no—
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42#1742C2211yescharacteristic
43#1740C2211yescharacteristic
44#1740C2211yescharacteristic
45#1738C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1735C2211yescharacteristic
48#1735C2211yescharacteristic
49#1733C2211yescharacteristic
50#1733C2211yescharacteristic
51#1733C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54C22×#41C2311yescharacteristic
55#568—22no—
56#568—41no—
57#568—41no—
58#568C2×C414yes—
59C2×#131—22no—
60C2×#131—41no—
61C22×M16—41no—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76#430—41no—
77#430—41no—
78C2×#108—22no—
79C2×#108—41no—
80C2×#108—41no—
81C2×#108D414yes—
82#428—41no—
83#428—41no—
84#427—22no—
85#424—22no—
86#424—22no—
87#424—41no—
88#424—41no—
89#424C2×C414yes—
90#424C2×C414yes—
91#423—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—41no—
94#421—22no—
95#421—22no—
96#421—22no—
97#421—41no—
98#421—41no—
99#421—41no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—41no—
103#419Q812yes—
104#418Q812yes—
105#417—41no—
106#416—41no—
107#410—22no—
108#410C2×C414yes—
109C2×#99—22no—
110#408—41no—
111#408C2×C414yes—
112C2×C22⋊C8—41no—
113#405—22no—
114#405—41no—
115#405—41no—
116#405—41no—
117C4×M16—22no—
118C4×M16—41no—
119#344—22no—
120#344—22no—
121#344—41no—
122#344—41no—
123#344—41no—
124#344—41no—
125#344C2×C414yes—
126#344C2×C414yes—
127#343—41no—
128#342—22no—
129#342—22no—
130#341—22no—
131#341—22no—
132#340—41no—
133#340—41no—
134#339—41no—
135#339—41no—
136#339—41no—
137H(3,ℤ/4)—22no—
138H(3,ℤ/4)—41no—
139C2×#41—21no—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41C22×C412yes—
150C2×#41#4112yes—
151C22×Q8C2×Q811yescharacteristic
152C22×D4—21no—
153C22×D4C2×Q811yescharacteristic
154C23×C4—21no—
155C23×C4C2411yescharacteristic
156#131—28no—
157#131—44no—
158#131—44no—
159#131—44no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—42no—
196C2×M16—44no—
197C2×M16—44no—
198C2×M16C22×C412yes—
199C2×M16C22×C412yes—
200C2×M16C22×C412yes—
201C2×M16C2×D412yes—
202C2×M16C2×D412yes—
203C2×M16C2×Q812yes—
204C2×M16#4112yes—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—42no—
218C22×C8C22×C412yes—
219C22×C8C2×D412yes—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117C22×C412yes—
241#117C22×C412yes—
242#117C2×Q812yes—
243#117#4112yes—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4#4112yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262#108—44no—
263#108—44no—
264#107—42no—
265#106—42no—
266C4⋊C8—28no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269#104—44no—
270#104—44no—
271#104—44no—
272#103—42no—
273#102—42no—
274#99—28no—
275C22⋊C8—44no—
276#97—42no—
277#97—42no—
278C4×C8—42no—
279C4×C8—42no—
280C24—21no—
281C24C2×#4111yescharacteristic
282#41—22no—
283#41—22no—
284#41—22no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—22no—
289#41—22no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8C4×Q812yes—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4C4×Q812yes—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—22no—
355M16—22no—
356M16—22no—
357M16—48no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—48no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8C4×D414yes—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4#11714yes—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—81no—
441C23—82no—
442C23C2×#11711yescharacteristic
443C23#54011yescharacteristic
444C23#54211yescharacteristic
445C23#54811yescharacteristic
446C23#55311yescharacteristic
447C23#55311yescharacteristic
448C23#55411yescharacteristic
449Q8—22no—
450Q8—22no—
451Q8—41no—
452Q8—41no—
453D4—22no—
454D4—22no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—41no—
460D4—41no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—161no—
477C2×C4—161no—
478C2×C4—161no—
479C2×C4—161no—
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×C4×Q811yescharacteristic
482C2×C4#51811yescharacteristic
483C2×C4#51811yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4#52011yescharacteristic
486C2×C4#53411yescharacteristic
487C2×C4#53511yescharacteristic
488C2×C4#53811yescharacteristic
489C2×C4#53811yescharacteristic
490C2×C4#53811yescharacteristic
491C2×C4#53911yescharacteristic
492C2×C4#54011yescharacteristic
493C2×C4#54211yescharacteristic
494C2×C4#54311yescharacteristic
495C2×C4#54411yescharacteristic
496C2×C4#54611yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4D4×Q811yescharacteristic
499C2×C4#55211yescharacteristic
500C2×C4#55211yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4#55411yescharacteristic
503C2×C4#55411yescharacteristic
504C2×C4#56011yescharacteristic
505C2×C4#56211yescharacteristic
506C2×C4#56211yescharacteristic
507C2×C4#56311yescharacteristic
508C8—24no—
509C8—42no—
510C8—42no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—81no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—161no—
527C22#231911yescharacteristic
528C22#234511yescharacteristic
529C22#235211yescharacteristic
530C22#271111yescharacteristic
531C22#272911yescharacteristic
532C22#277911yescharacteristic
533C22#278011yescharacteristic
534C4—44no—
535C4—42no—
536C4—82no—
537C4—82no—
538C4—161no—
539C4—161no—
540C4—161no—
541C4#232111yescharacteristic
542C4#232911yescharacteristic
543C4#236011yescharacteristic
544C4#237611yescharacteristic
545C4#268811yescharacteristic
546C4#269211yescharacteristic
547C4#276811yescharacteristic
548C4#277011yescharacteristic
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—82no—
557C2#1439211yescharacteristic
subgroupG/H
centerC4#2688
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14392
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21178C2semidirect
4. #11350C2semidirect
5. #11284C2semidirect
6. #11240C2semidirect
7. #11236C2semidirect
12. #10917C2semidirect
13. #10905C2semidirect
14. #10831C2semidirect
32. #1868C4semidirect

Maximal subgroups

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

subgroupindex
#211782
C2×#18682
#206752
#113502
#112842
#112402
#112362
#111082
#110942
#109562
#109512
#109172
#109052
#108312
#107872