Finite Group Databasebeta

← #203021

Subgroups — #203021

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21815C211yesmaximal, characteristic
2#20550C211yesmaximal, characteristic, complemented
3#17820C211yesmaximal, characteristic, complemented
4#17717C211yesmaximal, characteristic
5#17485C211yesmaximal, characteristic
6#9582C212yesmaximal, complemented
7#9581C212yesmaximal, complemented
8#9578C212yesmaximal, complemented
9#9577C212yesmaximal, complemented
10#8076C211yesmaximal, characteristic, complemented
11#7869C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#1980—21no—
20#1980—21no—
21#1962C2212yes—
22#1962C2212yes—
23#1789—22no—
24#1770—21no—
25C8×C4⋊C4—21no—
26#1767C2212yescomplemented
27C4×C4⋊C8C2212yescomplemented
28#1753—22no—
29#1574—22no—
30#1571C2212yescomplemented
31#1570C2212yescomplemented
32#1569—22no—
33#1553C414yescomplemented
34#1549C412yescomplemented
35#1545C412yescomplemented
36#1540C2212yes—
37#1540C2212yes—
38#1492—22no—
39#1486—22no—
40#1480C2212yes—
41#1480C2212yes—
42#1301C2211yescharacteristic, complemented
43#1301C2211yescharacteristic, complemented
44#1301C2211yescharacteristic, complemented
45#1301C2211yescharacteristic, complemented
46#1276C2212yescomplemented
47#1276C2212yescomplemented
48C22×#41—21no—
49C22×M16—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#432C2×C412yes—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431C2×C412yes—
80#431C2×C412yes—
81#431C2311yescharacteristic
82#431C2311yescharacteristic
83#431C2312yes—
84#423—21no—
85#423—21no—
86#423C2×C412yes—
87#423D412yes—
88#423D412yes—
89#423C2311yescharacteristic
90#423C2311yescharacteristic
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2312yes—
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—22no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4D412yes—
115#376—21no—
116C43C2312yes—
117#336—22no—
118#336—22no—
119#335—24no—
120#334—24no—
121#332C2×C414yes—
122#332C2×C414yes—
123#331C2×C414yes—
124#331C2×C414yes—
125D4⋊C8—24no—
126D4⋊C8—24no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—41no—
130C2×#41—41no—
131C2×#41—81no—
132C2×#41—81no—
133C2×#41—81no—
134C2×#41—81no—
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—22no—
146C2×M16—22no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—81no—
150C2×M16—81no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—41no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8C22×C412yes—
159#127—21no—
160#127—21no—
161#127—21no—
162#127C22×C412yes—
163#124—22no—
164#124—22no—
165#124—41no—
166#124—41no—
167#121—22no—
168#121—22no—
169#121—22no—
170#121—22no—
171#121—41no—
172#121—41no—
173#121—41no—
174#121—41no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183#117—44no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—22no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—41no—
220C4⋊C8C22×C412yes—
221C4⋊C8C22×C412yes—
222C4⋊C8C22×C414yes—
223C4⋊C8C2×D414yes—
224#104—82no—
225#104—82no—
226#104—82no—
227#104—82no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237#97—22no—
238#97—41no—
239#97—82no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—41no—
246C4×C8—41no—
247C4×C8—82no—
248#95—21no—
249#95—21no—
250#95—22no—
251C24—41no—
252C24#12311yescharacteristic
253#41—42no—
254#41—42no—
255#41—42no—
256#41—42no—
257#41—82no—
258#41—82no—
259#41—82no—
260#41—82no—
261#41—82no—
262#41—82no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—82no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—82no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4#11711yescharacteristic
295C22×C4#11711yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4#12311yescharacteristic
305C22×C4#12411yescharacteristic
306C22×C4#12411yescharacteristic
307C22×C4#12611yescharacteristic
308C22×C4#12611yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312M16—41no—
313M16—41no—
314M16—41no—
315M16—41no—
316M16—82no—
317M16—82no—
318M16—82no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—82no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—44no—
350C42—44no—
351C42#11711yescharacteristic
352C42#11711yescharacteristic
353C42C4×D411yescharacteristic
354C42C4×D411yescharacteristic
355C42C4×D411yescharacteristic
356C42C4×D411yescharacteristic
357C42#12111yescharacteristic
358C42#12111yescharacteristic
359C42#12411yescharacteristic
360C42#12411yescharacteristic
361C42#12611yescharacteristic
362C42#12611yescharacteristic
363C42C23×C411yescharacteristic
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23#38811yescharacteristic
376C23#38811yescharacteristic
377C23C2×#12311yescharacteristic
378Q8—42no—
379Q8—42no—
380Q8—82no—
381D4—42no—
382D4—42no—
383D4—44no—
384D4—82no—
385D4—82no—
386D4—82no—
387D4—84no—
388D4—82no—
389D4—82no—
390D4—82no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#38811yescharacteristic
425C2×C4#38811yescharacteristic
426C2×C4#38811yescharacteristic
427C2×C4#38811yescharacteristic
428C2×C4#38811yescharacteristic
429C2×C4#38811yescharacteristic
430C2×C4C2×#11711yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12411yescharacteristic
435C2×C4C2×#12611yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C8—42no—
441C8—42no—
442C8—44no—
443C8—44no—
444C8—81no—
445C8—81no—
446C8—81no—
447C8—81no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—164no—
462C22C2×#38811yescharacteristic
463C22#309611yescharacteristic
464C22#309611yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—44no—
474C4—82no—
475C4—82no—
476C4#193212yes—
477C4#294611yescharacteristic
478C4#294611yescharacteristic
479C4#295712yes—
480C4#304111yescharacteristic
481C4#304311yescharacteristic
482C2—21no—
483C2—21no—
484C2—41no—
485C2—41no—
486C2—162no—
487C2#2106311yescharacteristic
488C2C2×#193211yescharacteristic
489C2#2109811yescharacteristic
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. #20550C2semidirect
3. #17820C2semidirect
6. #9582C2semidirect
7. #9581C2semidirect
8. #9578C2semidirect
9. #9577C2semidirect
10. #8076C2semidirect
11. #7869C2semidirect
26. #1767C22semidirect
27. C4×C4⋊C8C22semidirect
30. #1571C22semidirect
31. #1570C22semidirect
33. #1553C4semidirect
34. #1549C4semidirect
35. #1545C4semidirect
42. #1301C22semidirect
43. #1301C22semidirect
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
#205502
#178202
#177172
#174852
#95822
#95812
#95782
#95772
#80762
#78692