Finite Group Databasebeta

← #453850

Subgroups — #453850

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11368C211yesmaximal, characteristic, complemented
2#11321C211yesmaximal, characteristic, complemented
3#11305C211yesmaximal, characteristic, complemented
4#11295C211yesmaximal, characteristic, complemented
5#11194C211yesmaximal, characteristic
6#11177C211yesmaximal, characteristic
7#11151C211yesmaximal, characteristic
8#11137C211yesmaximal, characteristic
9#11114C211yesmaximal, characteristic
10#10956C211yesmaximal, characteristic
11#10929C211yesmaximal, characteristic
12#10902C211yesmaximal, characteristic, complemented
13#10901C211yesmaximal, characteristic, complemented
14#10822C211yesmaximal, characteristic, complemented
15#10818C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#421C2211yescharacteristic
19C22×#104C2211yescharacteristic
20#2896C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#413C2211yescharacteristic
23#2879C2211yescharacteristic
24C2×#408C2211yescharacteristic
25#2873C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#344C2211yescharacteristic
28C2×#344C2211yescharacteristic
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31#1742C2211yescharacteristic
32#1738C2211yescharacteristic
33C2×#341C2211yescharacteristic
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39#1735C2211yescharacteristic
40#1733C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#338C2211yescharacteristic
45C2×H(3,ℤ/4)C2211yescharacteristic
46#1729C2211yescharacteristic
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1728C2211yescharacteristic
50#1728C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#430—22no—
67#430—22no—
68#430—22no—
69#430—41no—
70C2×#108—41no—
71C2×#108—41no—
72#421—22no—
73#421—41no—
74C2×#104—22no—
75C2×#104—22no—
76C2×#104—41no—
77C2×#104—41no—
78#419—21no—
79#418—21no—
80#417—22no—
81#416—22no—
82#416—41no—
83#416—41no—
84C2×#103—21no—
85C2×#102—21no—
86#413—22no—
87#413—41no—
88C2×#101—21no—
89C2×#100—21no—
90#408—22no—
91#408—22no—
92#408—41no—
93#408—41no—
94#407—41no—
95C2×C22⋊C8—41no—
96#405—41no—
97#405—41no—
98C4×M16—41no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8—41no—
102#344—22no—
103#344—22no—
104#344—41no—
105#344—41no—
106#343—22no—
107#341—22no—
108#341—22no—
109#341—22no—
110#341—22no—
111#341—22no—
112#341—41no—
113#341—41no—
114#340—22no—
115#339—22no—
116#339—41no—
117#339—41no—
118#339—41no—
119#338—22no—
120H(3,ℤ/4)—41no—
121#336—41no—
122#336—41no—
123#336—41no—
124#336—41no—
125#336—41no—
126C2×#41—21no—
127C2×#41—21no—
128C2×#41—21no—
129C2×#41—21no—
130C2×#41—21no—
131C2×#41—21no—
132C2×#41—21no—
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
138C22×Q8#4111yescharacteristic
139C22×D4—21no—
140C22×D4#4111yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2411yescharacteristic
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
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#4112yes—
189C2×M16#4112yes—
190C2×M16#4112yes—
191C2×M16#4112yes—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8#4112yes—
213C22×C8#4112yes—
214#117—21no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4#4112yes—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42#4112yes—
241#108—44no—
242#108—44no—
243#104—44no—
244#104—44no—
245#103—42no—
246#103—42no—
247#102—42no—
248#102—42no—
249#101—42no—
250#100—42no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253#97—42no—
254C4×C8—44no—
255C4×C8—42no—
256C24—21no—
257C242−1+411yescharacteristic
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C4H(5,𝔽2)11yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320M16—22no—
321M16—22no—
322M16—22no—
323M16—22no—
324M16—22no—
325M16—22no—
326M16—22no—
327M16—22no—
328M16—41no—
329M16—41no—
330M16—41no—
331M16—41no—
332M16—41no—
333M16—41no—
334M16—41no—
335M16—41no—
336M16—41no—
337M16—41no—
338M16—41no—
339M16—41no—
340M16—41no—
341M16—41no—
342M16—41no—
343M16—41no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C42—22no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—82no—
417C23#53911yescharacteristic
418C23#53911yescharacteristic
419C23#53911yescharacteristic
420C23#55611yescharacteristic
421C23#55611yescharacteristic
422C23#55611yescharacteristic
423C23#56211yescharacteristic
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—81no—
433D4—81no—
434D4—81no—
435D4—81no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4#52911yescharacteristic
455C2×C4#52911yescharacteristic
456C2×C4#52911yescharacteristic
457C2×C4#52911yescharacteristic
458C2×C4#52911yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4#53811yescharacteristic
461C2×C4#53811yescharacteristic
462C2×C4#54011yescharacteristic
463C2×C4#54011yescharacteristic
464C2×C4#54011yescharacteristic
465C2×C4#54011yescharacteristic
466C2×C4#55111yescharacteristic
467C2×C4#55111yescharacteristic
468C2×C4#55111yescharacteristic
469C2×C4#55111yescharacteristic
470C2×C4#55111yescharacteristic
471C2×C4#55111yescharacteristic
472C2×C4#55111yescharacteristic
473C2×C4#55311yescharacteristic
474C2×C4#55311yescharacteristic
475C2×C4#55311yescharacteristic
476C2×C4#55311yescharacteristic
477C2×C4#55311yescharacteristic
478C2×C4#55311yescharacteristic
479C2×C4#55511yescharacteristic
480C2×C4#55911yescharacteristic
481C2×C4H(3,𝔽4)11yescharacteristic
482C8—42no—
483C8—42no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8—42no—
488C8—81no—
489C8—81no—
490C8—81no—
491C8—81no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—82no—
500C22—81no—
501C22—82no—
502C22—161no—
503C22#252311yescharacteristic
504C22#259411yescharacteristic
505C22#259411yescharacteristic
506C22#274011yescharacteristic
507C22#274011yescharacteristic
508C22#278211yescharacteristic
509C22#280811yescharacteristic
510C4—44no—
511C4—81no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4#251611yescharacteristic
516C4#251611yescharacteristic
517C4#252611yescharacteristic
518C4#252611yescharacteristic
519C4#257811yescharacteristic
520C4#257811yescharacteristic
521C4#275311yescharacteristic
522C4#275311yescharacteristic
523C2—21no—
524C2—21no—
525C2—21no—
526C2—21no—
527C2—21no—
528C2—21no—
529C2—161no—
530C2#1615211yescharacteristic
531C2#3270212yes—
subgroupG/H
centerC2×C4H(3,𝔽4)
commutator, Frattini, cosocleC23×C4C24
socleC22#2782
prime radicalC22×#41C23
LCSG ▷ C23×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. #11368C2semidirect
2. #11321C2semidirect
3. #11305C2semidirect
4. #11295C2semidirect
12. #10902C2semidirect
13. #10901C2semidirect
14. #10822C2semidirect
15. #10818C2semidirect

Maximal subgroups

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

subgroupindex
#113682
#113212
#113052
#112952
#111942
#111772
#111512
#111372
#111142
#109562
#109292
#109022
#109012
#108222
#108182