Finite Group Databasebeta

← #203031

Subgroups — #203031

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21815C211yesmaximal, characteristic
2#20546C211yesmaximal, characteristic, complemented
3#17807C211yesmaximal, characteristic, complemented
4#17731C211yesmaximal, characteristic
5C2×#1480C211yesmaximal, characteristic
6#9582C212yesmaximal, complemented
7#9581C212yesmaximal, complemented
8#9578C212yesmaximal, complemented
9#9577C212yesmaximal, complemented
10#8080C211yesmaximal, characteristic, complemented
11#7869C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#1980—22no—
20#1962C2212yes—
21#1962C2212yes—
22#1789—21no—
23#1789—21no—
24#1771—22no—
25#1767C2212yescomplemented
26C4×C4⋊C8C2212yescomplemented
27#1752—21no—
28C8×C22⋊C4—21no—
29#1573—22no—
30#1571C2211yescharacteristic, complemented
31#1571C2211yescharacteristic, complemented
32#1570C2211yescharacteristic, complemented
33#1570C2211yescharacteristic, complemented
34#1565—22no—
35#1557C414yescomplemented
36#1556C412yescomplemented
37#1555C412yescomplemented
38#1540C2212yes—
39#1540C2212yes—
40#1480—22no—
41#1480C2212yes—
42#1480C2212yes—
43C2×D4⋊C8—22no—
44#1301C2212yescomplemented
45#1301C2212yescomplemented
46#1276C2212yescomplemented
47#1276C2212yescomplemented
48C22×#41—21no—
49C23×C8—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532C2×C412yes—
57#532C2312yes—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#127C2×C411yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×#117—21no—
66C22×C42C2311yescharacteristic
67#432—21no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432C2×C412yes—
73#432C2×C412yes—
74#432C2×C412yes—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431C2312yes—
82#431C2311yescharacteristic
83#431C2311yescharacteristic
84#423C2×C412yes—
85#423C2311yescharacteristic
86#423C2311yescharacteristic
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8D412yes—
92C2×C4⋊C8D412yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#104—22no—
97C2×#104—22no—
98C2×#104—22no—
99C2×#104—22no—
100#407—21no—
101#407—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C4×M16—21no—
105C4×M16—21no—
106C2×#97—21no—
107C2×#97—22no—
108C2×C4×C8—21no—
109C2×C4×C8—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—21no—
113C4×C22⋊C4D411yescharacteristic
114C4×C22⋊C4D411yescharacteristic
115#376—21no—
116C43C2312yes—
117#336—22no—
118#336—22no—
119#335—22no—
120#335—22no—
121#334—22no—
122#334—22no—
123#332C2×C414yes—
124#332C2×C414yes—
125#331C2×C414yes—
126#331C2×C414yes—
127D4⋊C8—24no—
128D4⋊C8—24no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—41no—
132C2×#41—41no—
133C2×#41—82no—
134C2×#41—82no—
135C22×Q8—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C2×M16—22no—
145C2×M16—21no—
146C2×M16—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—41no—
153C22×C8—82no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8C22×C412yes—
158#127—21no—
159#127—21no—
160#127—21no—
161#127C22×C412yes—
162#124—22no—
163#124—22no—
164#124—41no—
165#124—41no—
166#121—22no—
167#121—22no—
168#121—22no—
169#121—22no—
170#121—41no—
171#121—41no—
172#121—41no—
173#121—41no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182#117—44no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—41no—
218C4⋊C8C22×C414yes—
219C4⋊C8C22×C412yes—
220C4⋊C8C22×C412yes—
221C4⋊C8C2×D412yes—
222C4⋊C8C2×D412yes—
223#104—82no—
224#104—82no—
225#104—82no—
226#104—82no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236#97—22no—
237#97—22no—
238#97—22no—
239#97—41no—
240#97—82no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—22no—
244C4×C8—41no—
245C4×C8—41no—
246C4×C8—82no—
247#95—22no—
248#95—21no—
249#95—21no—
250C24—41no—
251C24#12311yescharacteristic
252#41—42no—
253#41—42no—
254#41—42no—
255#41—42no—
256#41—82no—
257#41—82no—
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—44no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—84no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4#11711yescharacteristic
288C22×C4#11711yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12411yescharacteristic
299C22×C4#12411yescharacteristic
300C22×C4#12611yescharacteristic
301C22×C4#12611yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305M16—44no—
306M16—42no—
307M16—42no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—82no—
321C2×C8—82no—
322C2×C8—82no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—44no—
339C42—44no—
340C42#11711yescharacteristic
341C42#11711yescharacteristic
342C42C4×D411yescharacteristic
343C42C4×D411yescharacteristic
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42#12111yescharacteristic
347C42#12111yescharacteristic
348C42#12411yescharacteristic
349C42#12411yescharacteristic
350C42#12611yescharacteristic
351C42#12611yescharacteristic
352C42C23×C411yescharacteristic
353C42C22×D411yescharacteristic
354C42C2×#4111yescharacteristic
355C42C2×#4111yescharacteristic
356C23—22no—
357C23—22no—
358C23—41no—
359C23—42no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23#38811yescharacteristic
364C23#38811yescharacteristic
365C23C2×#12311yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—82no—
369D4—42no—
370D4—42no—
371D4—44no—
372D4—82no—
373D4—82no—
374D4—82no—
375D4—82no—
376D4—84no—
377D4—82no—
378D4—82no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#38811yescharacteristic
410C2×C4#38811yescharacteristic
411C2×C4#38811yescharacteristic
412C2×C4#38811yescharacteristic
413C2×C4#38811yescharacteristic
414C2×C4#38811yescharacteristic
415C2×C4C2×#11711yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×#12111yescharacteristic
419C2×C4C2×#12411yescharacteristic
420C2×C4C2×#12611yescharacteristic
421C2×C4#57612yes—
422C2×C4#57712yes—
423C8—44no—
424C8—42no—
425C8—42no—
426C8—42no—
427C8—42no—
428C8—82no—
429C8—81no—
430C8—81no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—44no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—164no—
443C22#193212yes—
444C22C2×#38811yescharacteristic
445C22#309612yes—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—44no—
456C4—82no—
457C4—82no—
458C4#294612yes—
459C4#295512yes—
460C4#304111yescharacteristic
461C4#304311yescharacteristic
462C2—22no—
463C2—22no—
464C2—41no—
465C2—162no—
466C2#2106311yescharacteristic
467C2C2×#193211yescharacteristic
468C2#2110011yescharacteristic
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
2. #20546C2semidirect
3. #17807C2semidirect
6. #9582C2semidirect
7. #9581C2semidirect
8. #9578C2semidirect
9. #9577C2semidirect
10. #8080C2semidirect
11. #7869C2semidirect
25. #1767C22semidirect
26. C4×C4⋊C8C22semidirect
30. #1571C22semidirect
31. #1571C22semidirect
32. #1570C22semidirect
33. #1570C22semidirect
35. #1557C4semidirect
36. #1556C4semidirect
37. #1555C4semidirect
44. #1301C22semidirect
45. #1301C22semidirect
46. #1276C22semidirect
47. #1276C22semidirect

Maximal subgroups

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

subgroupindex
#218152
#205462
#178072
#177312
C2×#14802
#95822
#95812
#95782
#95772
#80802
#78692