Finite Group Databasebeta

← #203052

Subgroups — #203052

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21815C211yesmaximal, characteristic
2#20549C211yesmaximal, characteristic, complemented
3#17816C211yesmaximal, characteristic, complemented
4#17704C211yesmaximal, characteristic
5#17495C211yesmaximal, characteristic
6#9582C212yesmaximal, complemented
7#9581C212yesmaximal, complemented
8#9578C212yesmaximal, complemented
9#9577C212yesmaximal, complemented
10#8078C211yesmaximal, characteristic, complemented
11#7869C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#1980—22no—
20#1962C2212yes—
21#1962C2212yes—
22#1789—22no—
23#1771—22no—
24#1767C2212yescomplemented
25C4×C4⋊C8C2212yescomplemented
26#1753—22no—
27#1572—22no—
28#1571C2212yescomplemented
29#1570C2212yescomplemented
30#1568—22no—
31#1547C414yescomplemented
32#1546C412yescomplemented
33#1545C412yescomplemented
34#1540C2212yes—
35#1540C2212yes—
36#1496—22no—
37#1490—22no—
38#1480C2212yes—
39#1480C2212yes—
40#1301C2212yescomplemented
41#1301C2212yescomplemented
42#1276C2211yescharacteristic, complemented
43#1276C2211yescharacteristic, complemented
44#1276C2211yescharacteristic, complemented
45#1276C2211yescharacteristic, complemented
46C22×#41—21no—
47C22×M16—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532C2×C412yes—
55#532C2312yes—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#124—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×#117—21no—
64C22×C42C2311yescharacteristic
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432D412yes—
72#432D412yes—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431C2×C412yes—
79#431C2311yescharacteristic
80#431C2311yescharacteristic
81#431C2311yescharacteristic
82#431C2311yescharacteristic
83#423—21no—
84#423—21no—
85#423—21no—
86#423C2×C412yes—
87#423C2×C412yes—
88#423C2312yes—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2×C412yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—22no—
96C2×#104—22no—
97#407—21no—
98#407—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C4×M16—21no—
102C4×M16—21no—
103C2×#97—21no—
104C2×#97—22no—
105C2×C4×C8—21no—
106C2×C4×C8—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D412yes—
109C4×C22⋊C4—21no—
110C4×C22⋊C4D412yes—
111#376—21no—
112C43C2311yescharacteristic
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#335—24no—
117#334—24no—
118#332C2×C414yes—
119#332C2×C414yes—
120#331C2×C414yes—
121#331C2×C414yes—
122D4⋊C8—24no—
123D4⋊C8—24no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—41no—
127C2×#41—41no—
128C2×#41—82no—
129C2×#41—82no—
130C22×Q8—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4#4111yescharacteristic
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—41no—
143C2×M16—82no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C22×C412yes—
151#127—21no—
152#127—21no—
153#127—21no—
154#127C22×C412yes—
155#124—22no—
156#124—22no—
157#124—41no—
158#124—41no—
159#121—22no—
160#121—22no—
161#121—22no—
162#121—22no—
163#121—41no—
164#121—41no—
165#121—41no—
166#121—41no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175#117—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42C2411yescharacteristic
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—22no—
209C4⋊C8—41no—
210C4⋊C8C22×C412yes—
211C4⋊C8C22×C412yes—
212C4⋊C8C22×C414yes—
213C4⋊C8C2×D414yes—
214#104—82no—
215#104—82no—
216#104—82no—
217#104—82no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228#97—22no—
229#97—22no—
230#97—41no—
231#97—41no—
232#97—82no—
233C4×C8—22no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—41no—
239C4×C8—82no—
240#95—22no—
241#95—22no—
242C24—41no—
243C24#12311yescharacteristic
244#41—42no—
245#41—42no—
246#41—42no—
247#41—42no—
248#41—82no—
249#41—82no—
250#41—84no—
251#41—84no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—81no—
255C2×Q8—81no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—82no—
265C2×D4—81no—
266C2×D4—81no—
267C2×D4—82no—
268C2×D4—84no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—84no—
278C22×C4—81no—
279C22×C4#11711yescharacteristic
280C22×C4#11711yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12411yescharacteristic
291C22×C4#12411yescharacteristic
292C22×C4#12611yescharacteristic
293C22×C4#12611yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297M16—42no—
298M16—42no—
299M16—82no—
300M16—82no—
301M16—82no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—82no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—44no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C42—24no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—81no—
331C42—81no—
332C42#11711yescharacteristic
333C42#11711yescharacteristic
334C42C4×D411yescharacteristic
335C42C4×D411yescharacteristic
336C42C4×D411yescharacteristic
337C42C4×D411yescharacteristic
338C42#12111yescharacteristic
339C42#12111yescharacteristic
340C42#12411yescharacteristic
341C42#12411yescharacteristic
342C42#12611yescharacteristic
343C42#12611yescharacteristic
344C42C23×C411yescharacteristic
345C42C22×D411yescharacteristic
346C42C2×#4111yescharacteristic
347C42C2×#4111yescharacteristic
348C23—22no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—82no—
353C23—82no—
354C23—82no—
355C23#38811yescharacteristic
356C23#38811yescharacteristic
357C23C2×#12311yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—82no—
361D4—42no—
362D4—42no—
363D4—44no—
364D4—82no—
365D4—82no—
366D4—82no—
367D4—82no—
368D4—84no—
369D4—82no—
370D4—82no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#38811yescharacteristic
402C2×C4#38811yescharacteristic
403C2×C4#38811yescharacteristic
404C2×C4#38811yescharacteristic
405C2×C4#38811yescharacteristic
406C2×C4#38811yescharacteristic
407C2×C4C2×#11711yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C2×#12111yescharacteristic
411C2×C4C2×#12411yescharacteristic
412C2×C4C2×#12611yescharacteristic
413C2×C4#57612yes—
414C2×C4#57712yes—
415C8—44no—
416C8—42no—
417C8—42no—
418C8—44no—
419C8—81no—
420C8—81no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—164no—
435C22C2×#38811yescharacteristic
436C22#309612yes—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—44no—
447C4—82no—
448C4—82no—
449C4#294811yescharacteristic
450C4#294911yescharacteristic
451C4#295612yes—
452C4#304512yes—
453C2—22no—
454C2—41no—
455C2—41no—
456C2—162no—
457C2#2106511yescharacteristic
458C2#2108011yescharacteristic
459C2#2109911yescharacteristic
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. #20549C2semidirect
3. #17816C2semidirect
6. #9582C2semidirect
7. #9581C2semidirect
8. #9578C2semidirect
9. #9577C2semidirect
10. #8078C2semidirect
11. #7869C2semidirect
24. #1767C22semidirect
25. C4×C4⋊C8C22semidirect
28. #1571C22semidirect
29. #1570C22semidirect
31. #1547C4semidirect
32. #1546C4semidirect
33. #1545C4semidirect
40. #1301C22semidirect
41. #1301C22semidirect
42. #1276C22semidirect
43. #1276C22semidirect
44. #1276C22semidirect
45. #1276C22semidirect

Maximal subgroups

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

subgroupindex
#218152
#205492
#178162
#177042
#174952
#95822
#95812
#95782
#95772
#80782
#78692