Finite Group Databasebeta

← #388700

Subgroups — #388700

Order 512.

813 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2080C211yesmaximal, characteristic
2C2×#1937C211yesmaximal, characteristic
3C2×#1858C211yesmaximal, characteristic
4#11286C211yesmaximal, characteristic, complemented
5#11281C211yesmaximal, characteristic, complemented
6#11264C211yesmaximal, characteristic, complemented
7#11237C211yesmaximal, characteristic, complemented
8#11151C211yesmaximal, characteristic
9#10956C211yesmaximal, characteristic
10#10902C211yesmaximal, characteristic, complemented
11#10901C211yesmaximal, characteristic, complemented
12#10882C211yesmaximal, characteristic, complemented
13#10822C211yesmaximal, characteristic, complemented
14#10782C211yesmaximal, characteristic
15#10780C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#421C2211yescharacteristic
22C22×#104C2211yescharacteristic
23#2896C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#413C2211yescharacteristic
26C2×#408C2211yescharacteristic
27#2873C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2080C418yescomplemented
30#1937—24no—
31#1858—24no—
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36#1742C2211yescharacteristic
37#1738C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45#1733C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#338C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51#1729C2211yescharacteristic
52#1728C2211yescharacteristic
53#1728C2211yescharacteristic
54C22×#41C2311yescharacteristic
55#568—22no—
56#568—22no—
57#568—22no—
58#568—41no—
59C2×#131—41no—
60C2×#131—41no—
61C22×M16—41no—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#430—22no—
77#430—22no—
78#430—22no—
79#430—22no—
80C2×#108—41no—
81C2×#108—41no—
82#424—22no—
83#424—41no—
84#424—41no—
85#424C2×C414yes—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88#421—41no—
89#421C2×C414yes—
90C2×#104—22no—
91C2×#104—22no—
92C2×#104—41no—
93C2×#104—41no—
94#419—21no—
95#418—21no—
96#417—22no—
97#416—22no—
98#416—41no—
99#416C2×C414yes—
100C2×#103—21no—
101C2×#102—21no—
102#413—22no—
103#413C2×C414yes—
104C2×#101—21no—
105C2×#100—21no—
106#408—41no—
107#408D414yes—
108C2×C22⋊C8—41no—
109#405—41no—
110#405—41no—
111C4×M16—41no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—41no—
115#344—41no—
116#344—41no—
117#344—41no—
118#344C2×C414yes—
119#343—22no—
120#341—22no—
121#341—22no—
122#341—22no—
123#341—22no—
124#341—41no—
125#341—41no—
126#341C2×C414yes—
127#340—22no—
128#339—22no—
129#339—22no—
130#339—22no—
131#339—41no—
132#338—41no—
133H(3,ℤ/4)C2×C414yes—
134#336—41no—
135#336—41no—
136#336—41no—
137C2×#41—21no—
138C2×#41—21no—
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×#41C22×C412yes—
148C2×#41C2×D412yes—
149C22×Q8#4111yescharacteristic
150C22×D4—21no—
151C22×D4#4111yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2411yescharacteristic
155#131—44no—
156#131—44no—
157#131—44no—
158#131—44no—
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—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—44no—
202C2×M16—44no—
203C2×M16—42no—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16C22×C412yes—
207C2×M16C2×D412yes—
208C2×M16#4112yes—
209C2×M16#4112yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—42no—
231C22×C8C22×C412yes—
232C22×C8C2×D412yes—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117C22×C412yes—
244#117C22×C412yes—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4#4112yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42#4112yes—
260#108—44no—
261#108—44no—
262C4⋊C8—44no—
263C4⋊C8—44no—
264#104—44no—
265#104—44no—
266#103—24no—
267#103—42no—
268#102—24no—
269#102—42no—
270#101—24no—
271#100—24no—
272C22⋊C8—44no—
273#97—42no—
274C4×C8—44no—
275C4×C8—42no—
276C24—21no—
277C242−1+411yescharacteristic
278#41—22no—
279#41—22no—
280#41—22no—
281#41—22no—
282#41—22no—
283#41—22no—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8C4×D412yes—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4C4×D412yes—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C42−1+411yescharacteristic
339C22×C42−1+411yescharacteristic
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—48no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
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—
372M16—41no—
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—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—48no—
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×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C23—22no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—82no—
439C23#52011yescharacteristic
440C23#53611yescharacteristic
441C23#53911yescharacteristic
442C23#53911yescharacteristic
443C23#53911yescharacteristic
444C23#54111yescharacteristic
445C23#54211yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—41no—
449Q8—41no—
450D4—22no—
451D4—22no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—41no—
457D4—41no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C4×#4111yescharacteristic
478C2×C4C4×#4111yescharacteristic
479C2×C4#51811yescharacteristic
480C2×C4#52011yescharacteristic
481C2×C4#52011yescharacteristic
482C2×C4#53411yescharacteristic
483C2×C4#53811yescharacteristic
484C2×C4#53811yescharacteristic
485C2×C4#53911yescharacteristic
486C2×C4#54011yescharacteristic
487C2×C4#54211yescharacteristic
488C2×C4#54611yescharacteristic
489C2×C4#54611yescharacteristic
490C2×C4#54611yescharacteristic
491C2×C4#54711yescharacteristic
492C2×C4#55111yescharacteristic
493C2×C4#55111yescharacteristic
494C2×C4#55211yescharacteristic
495C2×C4#55211yescharacteristic
496C2×C4#55311yescharacteristic
497C2×C4#55311yescharacteristic
498C2×C4#55311yescharacteristic
499C2×C4#55611yescharacteristic
500C2×C4H(3,𝔽4)11yescharacteristic
501C2×C4#56211yescharacteristic
502C2×C4#56211yescharacteristic
503C2×C4#56311yescharacteristic
504C8—42no—
505C8—42no—
506C8—42no—
507C8—42no—
508C8—42no—
509C8—42no—
510C8—42no—
511C8—42no—
512C8—81no—
513C8—81no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—82no—
523C22—82no—
524C22—161no—
525C22#232211yescharacteristic
526C22#233411yescharacteristic
527C22#233511yescharacteristic
528C22#268711yescharacteristic
529C22#268911yescharacteristic
530C22#274711yescharacteristic
531C22#278211yescharacteristic
532C4—44no—
533C4—42no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4#235911yescharacteristic
538C4#236411yescharacteristic
539C4#237311yescharacteristic
540C4#237911yescharacteristic
541C4#268411yescharacteristic
542C4#270111yescharacteristic
543C4#277011yescharacteristic
544C4#278611yescharacteristic
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—82no—
552C2#1452511yescharacteristic
553C2#3270212yes—
subgroupG/H
centerC2×C4H(3,𝔽4)
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2782
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
4. #11286C2semidirect
5. #11281C2semidirect
6. #11264C2semidirect
7. #11237C2semidirect
10. #10902C2semidirect
11. #10901C2semidirect
12. #10882C2semidirect
13. #10822C2semidirect
29. #2080C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20802
C2×#19372
C2×#18582
#112862
#112812
#112642
#112372
#111512
#109562
#109022
#109012
#108822
#108222
#107822
#107802