Finite Group Databasebeta

← #203126

Subgroups — #203126

Order 512.

773 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic, complemented
2#21815C211yesmaximal, characteristic
3#17717C211yesmaximal, characteristic
4C2×#1492C211yesmaximal, characteristic
5#17323C211yesmaximal, characteristic, complemented
6#9592C214yesmaximal, complemented
7#9590C214yesmaximal, complemented
8#8097C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1980C2214yes—
18#1962—21no—
19#1962—21no—
20#1950C2214yescomplemented
21#1855—22no—
22#1848—21no—
23#1837—21no—
24#1789—22no—
25#1553C414yescomplemented
26#1549C411yescharacteristic, complemented
27#1549C411yescharacteristic, complemented
28#1545C411yescharacteristic, complemented
29#1545C411yescharacteristic, complemented
30#1540C2214yes—
31#1492—21no—
32#1492—21no—
33#1492C2212yes—
34#1492C2212yes—
35C2×D4⋊C8—21no—
36C2×D4⋊C8—21no—
37#1452—21no—
38#1451—21no—
39#1451C2212yescomplemented
40#1450—21no—
41C8×M16—21no—
42C8×M16C2212yescomplemented
43#1303C2214yescomplemented
44#1279C2212yescomplemented
45#1279C2212yescomplemented
46C22×#41—21no—
47C22×M16—21no—
48C23×C8—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532C2×C412yes—
56#532C2312yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#124—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×#117—21no—
65C22×C42C2311yescharacteristic
66#432—21no—
67#432—21no—
68#432—21no—
69#432C2×C412yes—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431C2×C411yescharacteristic
76#431C2×C411yescharacteristic
77#431C2×C412yes—
78#431C2312yes—
79C2×#108—24no—
80#423—21no—
81#423—21no—
82#423—21no—
83#423C2×C411yescharacteristic
84#423C2×C411yescharacteristic
85#423D411yescharacteristic
86#423D411yescharacteristic
87#423C2312yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×#104—24no—
92C2×#104—24no—
93#407—21no—
94#407D412yes—
95C2×C22⋊C8—21no—
96C4×M16—21no—
97C4×M16C2312yes—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8D411yescharacteristic
105C2×C4×C8D411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2312yes—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112#376—21no—
113C43—21no—
114#341—24no—
115#332C2×C414yes—
116#332C2×C414yes—
117#331C2×C414yes—
118#331C2×C414yes—
119D4⋊C8—22no—
120D4⋊C8—22no—
121D4⋊C8—22no—
122D4⋊C8—22no—
123#322—22no—
124#322—22no—
125#322—22no—
126C82—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—82no—
132C2×#41—82no—
133C22×Q8—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—41no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4#4111yescharacteristic
141C23×C4#4111yescharacteristic
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—41no—
146C2×M16—82no—
147C2×M16C2×D412yes—
148C2×M16C2×D412yes—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—44no—
157C22×C8—41no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C22×C412yes—
162#127—21no—
163#127—21no—
164#127—21no—
165#127C22×C412yes—
166#124—22no—
167#124—22no—
168#124—41no—
169#124—41no—
170#121—22no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—41no—
175#121—41no—
176#121—41no—
177#121—41no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186#117—24no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—22no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42C2411yescharacteristic
209#108—84no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8C22×C412yes—
222C4⋊C8C22×C412yes—
223C4⋊C8C22×C412yes—
224C4⋊C8C22×C412yes—
225#104—84no—
226#104—84no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234#97—22no—
235#97—22no—
236C4×C8—22no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8C2×D411yescharacteristic
248C4×C8C2×D411yescharacteristic
249C4×C8C2×D411yescharacteristic
250C4×C8C2×D411yescharacteristic
251#95—22no—
252#95—41no—
253C24—41no—
254C24#12411yescharacteristic
255#41—44no—
256#41—44no—
257#41—84no—
258#41—84no—
259#41—82no—
260#41—82no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—82no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—84no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4#11711yescharacteristic
289C22×C4#11711yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12411yescharacteristic
299C22×C4#12411yescharacteristic
300C22×C4#12411yescharacteristic
301C22×C4#12611yescharacteristic
302C22×C4#12611yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306M16—24no—
307M16—24no—
308M16—42no—
309M16—42no—
310M16—44no—
311M16—82no—
312C2×C8—24no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—82no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C22⋊C4—44no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C42—44no—
344C42—42no—
345C42—42no—
346C42—82no—
347C42#11711yescharacteristic
348C42#11711yescharacteristic
349C42C4×D411yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42C4×D411yescharacteristic
353C42#12111yescharacteristic
354C42#12111yescharacteristic
355C42#12311yescharacteristic
356C42#12311yescharacteristic
357C42#12611yescharacteristic
358C42#12611yescharacteristic
359C42C23×C411yescharacteristic
360C42C22×D411yescharacteristic
361C42C2×#4111yescharacteristic
362C42C2×#4111yescharacteristic
363C23—22no—
364C23—22no—
365C23—41no—
366C23—42no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23#38811yescharacteristic
371C23#38811yescharacteristic
372C23C2×#12411yescharacteristic
373Q8—44no—
374Q8—82no—
375D4—44no—
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380D4—84no—
381D4—84no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#38811yescharacteristic
413C2×C4#38811yescharacteristic
414C2×C4#38811yescharacteristic
415C2×C4#38811yescharacteristic
416C2×C4#38811yescharacteristic
417C2×C4#38811yescharacteristic
418C2×C4#44312yes—
419C2×C4#44412yes—
420C2×C4C2×#11711yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12311yescharacteristic
425C2×C4C2×#12611yescharacteristic
426C2×C4#57612yes—
427C2×C4#57712yes—
428C8—22no—
429C8—22no—
430C8—41no—
431C8—41no—
432C8—42no—
433C8—42no—
434C8—42no—
435C8—42no—
436C8—42no—
437C8—42no—
438C8—82no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—44no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—164no—
451C22#192912yes—
452C22C2×#38811yescharacteristic
453C22#313612yes—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—84no—
462C4—82no—
463C4#192912yes—
464C4#193212yes—
465C4#294612yes—
466C4C2×#44311yescharacteristic
467C4C2×#44411yescharacteristic
468C4#305711yescharacteristic
469C4#305811yescharacteristic
470C2—22no—
471C2—22no—
472C2—41no—
473C2—162no—
474C2C2×#192911yescharacteristic
475C2C2×#193211yescharacteristic
476C2#2109111yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21993C2semidirect
5. #17323C2semidirect
6. #9592C2semidirect
7. #9590C2semidirect
8. #8097C2semidirect
9. #7890C2semidirect
20. #1950C22semidirect
25. #1553C4semidirect
26. #1549C4semidirect
27. #1549C4semidirect
28. #1545C4semidirect
29. #1545C4semidirect
39. #1451C22semidirect
42. C8×M16C22semidirect
43. #1303C22semidirect
44. #1279C22semidirect
45. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#219932
#218152
#177172
C2×#14922
#173232
#95922
#95902
#80972
#78902