Finite Group Databasebeta

← #385424

Subgroups — #385424

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21178C211yesmaximal, characteristic, complemented
2C2×#1938C211yesmaximal, characteristic
3#21072C211yesmaximal, characteristic
4#20886C211yesmaximal, characteristic
5C2×#1861C211yesmaximal, characteristic
6#20852C211yesmaximal, characteristic
7C2×#1849C211yesmaximal, characteristic, complemented
8#11341C211yesmaximal, characteristic, complemented
9#11332C211yesmaximal, characteristic, complemented
10#11272C211yesmaximal, characteristic, complemented
11#11240C211yesmaximal, characteristic, complemented
12#11108C211yesmaximal, characteristic
13#11080C211yesmaximal, characteristic
14#10894C211yesmaximal, characteristic, complemented
15#10819C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C22×#108C2211yescharacteristic
23#2905C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#421C2211yescharacteristic
28C2×#421C2211yescharacteristic
29C22×#104C2211yescharacteristic
30C2×#416C2211yescharacteristic
31#2879C2211yescharacteristic
32C2×#408C2211yescharacteristic
33#2873C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#1950—24no—
36#1938C418yescomplemented
37#1929—24no—
38#1866—24no—
39#1861—24no—
40#1855—24no—
41#1849—24no—
42C2×#344C2211yescharacteristic
43C2×#344C2211yescharacteristic
44C2×#344C2211yescharacteristic
45C2×#344C2211yescharacteristic
46#1738C2211yescharacteristic
47#1738C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1733C2211yescharacteristic
52#1733C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58C22×#41C2311yescharacteristic
59#568—22no—
60#568—41no—
61#568—41no—
62#568—41no—
63C2×#131—22no—
64C2×#131—41no—
65C22×M16—22no—
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#430—22no—
81#430—41no—
82#430—41no—
83#430—41no—
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—22no—
91C2×#108—22no—
92C2×#108—22no—
93C2×#108—22no—
94C2×#108—41no—
95C2×#108—41no—
96C2×#108—41no—
97C2×#108—41no—
98#424—22no—
99#424—22no—
100#424—41no—
101#424C2×C414yes—
102#423—22no—
103#423—22no—
104#423—41no—
105#423—41no—
106#423D414yes—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—41no—
109C2×C4⋊C8—41no—
110#421—22no—
111#421—41no—
112#421—41no—
113#421—41no—
114C2×#104—22no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—22no—
118C2×#104C2×C414yes—
119#416C2×C414yes—
120#416D414yes—
121C2×#103Q812yes—
122C2×#102Q812yes—
123#408—22no—
124#408—22no—
125#408—41no—
126#408C2×C414yes—
127#407—22no—
128C2×C22⋊C8—22no—
129#405—22no—
130#405—41no—
131C4×M16—22no—
132C2×#97—22no—
133C2×#97—22no—
134C2×#97C2×C414yes—
135#344—22no—
136#344—41no—
137#344—41no—
138#344—41no—
139#341—22no—
140#341—22no—
141#341C2×C414yes—
142#341D414yes—
143#340—22no—
144#339—22no—
145#339—22no—
146#339—22no—
147#339—22no—
148#339C2×C414yes—
149#336—22no—
150#336—22no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41C22×C412yes—
162C2×#41C2×D412yes—
163C22×Q8C2×Q811yescharacteristic
164C22×D4—21no—
165C22×D4C2×Q811yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C2411yescharacteristic
169#131—44no—
170#131—44no—
171#131—44no—
172#131—44no—
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—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—42no—
216C2×M16—44no—
217C2×M16—44no—
218C2×M16C22×C412yes—
219C2×M16C22×C412yes—
220C2×M16C2×D412yes—
221C2×M16C2×D412yes—
222C2×M16C2×D412yes—
223C2×M16C2×D412yes—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—42no—
241C22×C8C22×C412yes—
242C22×C8C22×C412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C22×C8C2×Q812yes—
246C22×C8#4112yes—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117C22×C412yes—
258#117C2×D412yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×Q812yes—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42#4112yes—
274#108—44no—
275#108—44no—
276#108—44no—
277#108—44no—
278#108—44no—
279#108—44no—
280C4⋊C8—28no—
281C4⋊C8—44no—
282C4⋊C8—44no—
283C4⋊C8—44no—
284#104—28no—
285#104—44no—
286#104—44no—
287#103C4⋊C418yes—
288#102C4⋊C418yes—
289C22⋊C8—28no—
290C22⋊C8—44no—
291#97—28no—
292#97—42no—
293C4×C8—42no—
294C24—21no—
295C242−1+411yescharacteristic
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—22no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8C2×C4⋊C412yes—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4C2×C4⋊C412yes—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—82no—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357C22×C42−1+411yescharacteristic
358M16—22no—
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—48no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—48no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8C2×C4⋊C414yes—
429C2×C8C4×D414yes—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4C2×C4⋊C414yes—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42C4×D414yes—
452C23—22no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—82no—
457C23#52011yescharacteristic
458C23#53611yescharacteristic
459C23#53911yescharacteristic
460C23#54711yescharacteristic
461C23D4×Q811yescharacteristic
462C23#56311yescharacteristic
463C23#56311yescharacteristic
464Q8—22no—
465Q8—22no—
466Q8—41no—
467Q8—41no—
468D4—22no—
469D4—22no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—41no—
475D4—41no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4C22×C4⋊C411yescharacteristic
494C2×C4C2×C4×D411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4#52011yescharacteristic
497C2×C4#52011yescharacteristic
498C2×C4#52011yescharacteristic
499C2×C4#53411yescharacteristic
500C2×C4#53511yescharacteristic
501C2×C4#53511yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#53611yescharacteristic
504C2×C4#53911yescharacteristic
505C2×C4#54211yescharacteristic
506C2×C4#54211yescharacteristic
507C2×C4#54411yescharacteristic
508C2×C4#54411yescharacteristic
509C2×C4D4211yescharacteristic
510C2×C4D4211yescharacteristic
511C2×C4D4211yescharacteristic
512C2×C4#54711yescharacteristic
513C2×C4#54711yescharacteristic
514C2×C4#54811yescharacteristic
515C2×C4D4×Q811yescharacteristic
516C2×C4#55311yescharacteristic
517C2×C4#55411yescharacteristic
518C2×C4#55411yescharacteristic
519C2×C4#56011yescharacteristic
520C2×C4#56311yescharacteristic
521C8—24no—
522C8—24no—
523C8—42no—
524C8—42no—
525C8—42no—
526C8—42no—
527C8—42no—
528C8—42no—
529C8—81no—
530C8—81no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—44no—
540C22—161no—
541C22—161no—
542C22#232211yescharacteristic
543C22D4×C4⋊C411yescharacteristic
544C22#238311yescharacteristic
545C22#267411yescharacteristic
546C22#267711yescharacteristic
547C22#271811yescharacteristic
548C22#272011yescharacteristic
549C4—44no—
550C4—42no—
551C4—44no—
552C4—161no—
553C4—161no—
554C4#231711yescharacteristic
555C4#232611yescharacteristic
556C4D4×C4⋊C411yescharacteristic
557C4#236111yescharacteristic
558C4#266911yescharacteristic
559C4#268011yescharacteristic
560C4#272911yescharacteristic
561C4#279011yescharacteristic
562C2—21no—
563C2—21no—
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—82no—
569C2#1440511yescharacteristic
570C2#2269512yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×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
7. C2×#1849C2semidirect
8. #11341C2semidirect
9. #11332C2semidirect
10. #11272C2semidirect
11. #11240C2semidirect
14. #10894C2semidirect
15. #10819C2semidirect
36. #1938C4semidirect

Maximal subgroups

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

subgroupindex
#211782
C2×#19382
#210722
#208862
C2×#18612
#208522
C2×#18492
#113412
#113322
#112722
#112402
#111082
#110802
#108942
#108192