Finite Group Databasebeta

← #203124

Subgroups — #203124

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic, complemented
2#21815C211yesmaximal, characteristic
3C2×#1540C211yesmaximal, characteristic
4#17478C211yesmaximal, characteristic
5#17326C211yesmaximal, characteristic, complemented
6#9592C214yesmaximal, complemented
7#9590C214yesmaximal, complemented
8#8092C211yesmaximal, characteristic, complemented
9#7891C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1980C2214yes—
18#1962—22no—
19#1950—22no—
20#1950C2214yescomplemented
21#1810—22no—
22#1789—22no—
23#1540C412yescomplemented
24#1540C412yescomplemented
25#1540C2212yes—
26#1540C2212yes—
27C2×#332C411yescharacteristic, complemented
28C2×#332C411yescharacteristic, complemented
29C2×#331C411yescharacteristic, complemented
30C2×#331C411yescharacteristic, complemented
31#1496—22no—
32#1492C2214yes—
33#1478—21no—
34#1478—21no—
35#1454—22no—
36#1452—21no—
37#1451C2212yescomplemented
38#1450—21no—
39C8×M16C2212yescomplemented
40#1303C2212yescomplemented
41#1303C2212yescomplemented
42#1279C2214yescomplemented
43C22×#41—21no—
44C23×C8—21no—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532C2×C412yes—
52#532C2312yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#124—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×#117—21no—
61C22×C42C2311yescharacteristic
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431C2×C411yescharacteristic
75#431C2×C411yescharacteristic
76#431D411yescharacteristic
77#431D411yescharacteristic
78#431D412yes—
79#431C2311yescharacteristic
80#431C2311yescharacteristic
81C2×#108—24no—
82#423D412yes—
83#423C2312yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C411yescharacteristic
89C2×C4⋊C8C2×C411yescharacteristic
90C2×C4⋊C8C2×C411yescharacteristic
91C2×C4⋊C8C2×C411yescharacteristic
92C2×C4⋊C8C2×C412yes—
93C2×C4⋊C8C2312yes—
94C2×#104—24no—
95C2×#104—24no—
96#407—21no—
97#407D412yes—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C4×M16—21no—
101C4×M16C2312yes—
102C2×C4×C8—21no—
103C2×C4×C8C2312yes—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108#376—21no—
109C43—21no—
110#341—24no—
111#332C2×C412yes—
112#332C2×C412yes—
113#332C2×C412yes—
114#332C2×C412yes—
115#331C2×C412yes—
116#331C2×C412yes—
117#331C2×C412yes—
118#331C2×C412yes—
119D4⋊C8—24no—
120D4⋊C8—24no—
121#322—22no—
122#322—22no—
123#322—22no—
124C82—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—82no—
130C2×#41—82no—
131C22×Q8—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—41no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C23×C4#4111yescharacteristic
140C2×M16—22no—
141C2×M16C2×D414yes—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—82no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C22×C412yes—
154#127—21no—
155#127—21no—
156#127—21no—
157#127C22×C412yes—
158#124—22no—
159#124—22no—
160#124—41no—
161#124—41no—
162#121—22no—
163#121—22no—
164#121—22no—
165#121—22no—
166#121—41no—
167#121—41no—
168#121—41no—
169#121—41no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178#117—24no—
179#117—42no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C42—22no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201#108—84no—
202C4⋊C8—22no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8C22×C412yes—
216C4⋊C8C22×C412yes—
217C4⋊C8C22×C411yescharacteristic
218C4⋊C8C22×C411yescharacteristic
219C4⋊C8C22×C411yescharacteristic
220C4⋊C8C22×C411yescharacteristic
221#104—84no—
222#104—84no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—22no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234#97—22no—
235#97—41no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—22no—
240C4×C8—21no—
241C4×C8—22no—
242C4×C8—41no—
243C4×C8C2×D412yes—
244C4×C8C2×D412yes—
245#95—22no—
246#95—41no—
247C24—41no—
248C24#12411yescharacteristic
249#41—44no—
250#41—44no—
251#41—82no—
252#41—82no—
253#41—84no—
254#41—84no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—81no—
258C2×Q8—81no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—81no—
270C2×D4—81no—
271C2×D4—82no—
272C2×D4—84no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—84no—
282C22×C4—81no—
283C22×C4#11711yescharacteristic
284C22×C4#11711yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12311yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4#12411yescharacteristic
295C22×C4#12411yescharacteristic
296C22×C4#12611yescharacteristic
297C22×C4#12611yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301M16—24no—
302M16—24no—
303M16—42no—
304M16—42no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—81no—
319C2×C8—81no—
320C2×C8—82no—
321C2×C8—82no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C42—44no—
334C42—42no—
335C42—42no—
336C42—82no—
337C42#11711yescharacteristic
338C42#11711yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42C4×D411yescharacteristic
342C42C4×D411yescharacteristic
343C42#12111yescharacteristic
344C42#12111yescharacteristic
345C42#12311yescharacteristic
346C42#12311yescharacteristic
347C42#12611yescharacteristic
348C42#12611yescharacteristic
349C42C23×C411yescharacteristic
350C42C22×D411yescharacteristic
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C23—22no—
354C23—22no—
355C23—41no—
356C23—42no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23#38811yescharacteristic
361C23#38811yescharacteristic
362C23C2×#12411yescharacteristic
363Q8—42no—
364Q8—42no—
365Q8—82no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—82no—
371D4—82no—
372D4—82no—
373D4—84no—
374D4—84no—
375D4—82no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—84no—
403C2×C4#38811yescharacteristic
404C2×C4#38811yescharacteristic
405C2×C4#38811yescharacteristic
406C2×C4#38811yescharacteristic
407C2×C4#38811yescharacteristic
408C2×C4#38811yescharacteristic
409C2×C4#44312yes—
410C2×C4#44412yes—
411C2×C4C2×#11711yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×#12111yescharacteristic
415C2×C4C2×#12311yescharacteristic
416C2×C4C2×#12611yescharacteristic
417C2×C4#57612yes—
418C2×C4#57712yes—
419C8—42no—
420C8—42no—
421C8—42no—
422C8—42no—
423C8—41no—
424C8—41no—
425C8—41no—
426C8—41no—
427C8—44no—
428C8—82no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—44no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—164no—
441C22#192912yes—
442C22C2×#38811yescharacteristic
443C22#313612yes—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—84no—
453C4—82no—
454C4#192912yes—
455C4#294811yescharacteristic
456C4#294911yescharacteristic
457C4C2×#44311yescharacteristic
458C4C2×#44411yescharacteristic
459C4#306212yes—
460C2—22no—
461C2—22no—
462C2—41no—
463C2—162no—
464C2C2×#192911yescharacteristic
465C2#2108011yescharacteristic
466C2#2108911yescharacteristic
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. #21969C2semidirect
5. #17326C2semidirect
6. #9592C2semidirect
7. #9590C2semidirect
8. #8092C2semidirect
9. #7891C2semidirect
20. #1950C22semidirect
23. #1540C4semidirect
24. #1540C4semidirect
27. C2×#332C4semidirect
28. C2×#332C4semidirect
29. C2×#331C4semidirect
30. C2×#331C4semidirect
37. #1451C22semidirect
39. C8×M16C22semidirect
40. #1303C22semidirect
41. #1303C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#219692
#218152
C2×#15402
#174782
#173262
#95922
#95902
#80922
#78912