Finite Group Databasebeta

← #72234390

Subgroups — #72234390

Order 1024.

552 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#513008C211yesmaximal, characteristic
2#484120C211yesmaximal, characteristic, complemented
3#484093C211yesmaximal, characteristic, complemented
4#151129C211yesmaximal, characteristic, complemented
5#151127C211yesmaximal, characteristic, complemented
6#151117C211yesmaximal, characteristic, complemented
7#151116C211yesmaximal, characteristic, complemented
8#151115C211yesmaximal, characteristic, complemented
9#151115C211yesmaximal, characteristic, complemented
10#151085C211yesmaximal, characteristic, complemented
11#151083C211yesmaximal, characteristic, complemented
12#151069C211yesmaximal, characteristic, complemented
13#150162C211yesmaximal, characteristic, complemented
14#150064C211yesmaximal, characteristic, complemented
15#149509C211yesmaximal, characteristic, complemented
16#33735C2211yescharacteristic
17C2×C4×D16C2211yescharacteristic
18#18069—21no—
19#18054—21no—
20#13714C411yescharacteristic
21#13711C411yescharacteristic
22#13707C411yescharacteristic
23#13707C411yescharacteristic
24#13689C411yescharacteristic
25#13689C411yescharacteristic
26#13664C411yescharacteristic
27#13661C411yescharacteristic
28#13654C2211yescharacteristic
29#13654C2211yescharacteristic
30#13652C2211yescharacteristic
31#13652C2211yescharacteristic
32#13640C2211yescharacteristic
33C4×M64C2211yescharacteristic
34#12086C2211yescharacteristic
35#12086C2211yescharacteristic
36#12086C2211yescharacteristic, complemented
37#12085C2211yescharacteristic, complemented
38#12075—21no—
39#12074—21no—
40#12069—21no—
41#12069—21no—
42#12045C2211yescharacteristic
43#12045C2211yescharacteristic
44#12045C2211yescharacteristic, complemented
45#12044C2211yescharacteristic, complemented
46#12024C2211yescharacteristic
47#12008C2211yescharacteristic
48#11661—21no—
49#11647—21no—
50#11408C2211yescharacteristic
51#7463C2211yescharacteristic, complemented
52#7463C2211yescharacteristic, complemented
53#7463C2211yescharacteristic, complemented
54#7463C2211yescharacteristic, complemented
55#7386C2211yescharacteristic, complemented
56#7386C2211yescharacteristic, complemented
57#7385C2211yescharacteristic
58#7385C2211yescharacteristic
59#7385C2211yescharacteristic
60#7385C2211yescharacteristic
61#7385C2211yescharacteristic, complemented
62#7385C2211yescharacteristic, complemented
63#7332C2211yescharacteristic, complemented
64#7332C2211yescharacteristic, complemented
65#7330C2211yescharacteristic, complemented
66#7330C2211yescharacteristic, complemented
67C22×D16C2×C411yescharacteristic
68#3168—21no—
69#2974—41no—
70#2969—41no—
71C2×C4×D8—21no—
72C2×SD64C2×C411yescharacteristic
73C2×SD64C2×C411yescharacteristic
74C2×D32C2×C411yescharacteristic
75C2×D32C2×C411yescharacteristic
76C2×M64—21no—
77#2253C2×C411yescharacteristic
78#2249C2×C411yescharacteristic
79#2242—21no—
80#2217—21no—
81#2216C2×C411yescharacteristic
82#2215C2×C411yescharacteristic
83C4×Q32C2311yescharacteristic
84C4×SD32—21no—
85C4×D16—21no—
86C4×D16—21no—
87C4×D16C2×C411yescharacteristic
88C4×D16C2×C411yescharacteristic
89C4×D16C2311yescharacteristic
90C4×D16C2311yescharacteristic
91C4×D16C2311yescharacteristic
92#2162D411yescharacteristic
93C2×#366C2×C411yescharacteristic
94C2×#357—21no—
95C2×C4×C16C2311yescharacteristic
96#1591—21no—
97#1589—21no—
98#1589—21no—
99#1589—21no—
100#1586—41no—
101#1586—41no—
102#1583—41no—
103#1582—41no—
104C8×Q16—21no—
105C8×D8—21no—
106C8×D8—21no—
107C8×D8—21no—
108#1568—21no—
109#1564—21no—
110#1490—41no—
111#1482—41no—
112C8×M16D411yescharacteristic
113#1424C2×C411yescharacteristic
114#1424C2×C411yescharacteristic
115#1423C2×C411yescharacteristic
116#1423C2×C411yescharacteristic
117C4⋊C32C2×C411yescharacteristic
118C4⋊C32C2×C411yescharacteristic
119#1416C2×C411yescharacteristic
120#1416C2×C411yescharacteristic
121#1415C2×C411yescharacteristic
122#1415C2×C411yescharacteristic
123#1415C2×C411yescharacteristic
124#1415C2×C411yescharacteristic
125#1415C2×C411yescharacteristic
126#1415C2×C411yescharacteristic
127C22⋊C32C2×C411yescharacteristic
128C22⋊C32C2×C411yescharacteristic
129#1397C2311yescharacteristic
130#1397C2311yescharacteristic
131#1397C2311yescharacteristic
132#1397C2311yescharacteristic
133C4×C32—21no—
134#1372C2311yescharacteristic
135#1372C2311yescharacteristic
136#1372C2311yescharacteristic
137#1372C2311yescharacteristic
138#1371—21no—
139#1371—21no—
140#1332—21no—
141#1332—21no—
142#1331—21no—
143#1331—21no—
144#1331—21no—
145#1331—21no—
146#1331—21no—
147#1331—21no—
148#1312D411yescharacteristic
149#1312D411yescharacteristic
150#1311C2311yescharacteristic
151#1311C2311yescharacteristic
152C22×D8—21no—
153#529—41no—
154C2×C4×D4—41no—
155C2×Q32C22×C411yescharacteristic
156C2×SD32—21no—
157C2×D16—22no—
158C2×D16—21no—
159C2×D16—41no—
160C2×D16C22×C411yescharacteristic
161C2×D16C22×C411yescharacteristic
162C2×D16C22×C411yescharacteristic
163C22×C16C22×C411yescharacteristic
164C22×C16#4111yescharacteristic
165#499—21no—
166#495—21no—
167#488—41no—
168#467—21no—
169#466—21no—
170#465—41no—
171#446—41no—
172C8×Q8—41no—
173C4×Q16—21no—
174C4×SD16—41no—
175C4×D8—21no—
176C4×D8—21no—
177C4×D8—21no—
178C4×D8—21no—
179C4×D8—21no—
180C4×D8—41no—
181C4×D8—41no—
182#436—41no—
183#436—41no—
184#436—41no—
185#436—81no—
186#435—81no—
187#435—81no—
188#435—81no—
189C8×D4—41no—
190C8×D4—41no—
191C8×D4—41no—
192#433—81no—
193#432—41no—
194#431—41no—
195#431D811yescharacteristic
196C2×#107—21no—
197C2×#102—41no—
198#407—81no—
199C2×C22⋊C8—81no—
200#405—81no—
201C4×M16D811yescharacteristic
202C2×C4×C8C2×D411yescharacteristic
203SD64—22no—
204SD64—41no—
205D32—22no—
206D32—41no—
207M64—42no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32C22×C411yescharacteristic
211C2×C32C22×C411yescharacteristic
212C2×C32C22×C411yescharacteristic
213C2×C32C22×C411yescharacteristic
214#367—21no—
215#366—21no—
216#366C22×C411yescharacteristic
217#366C22×C411yescharacteristic
218#366C22×C411yescharacteristic
219#366C22×C411yescharacteristic
220C4⋊C16—21no—
221#358—21no—
222#358—21no—
223#357—21no—
224#357—21no—
225#357—21no—
226#357—21no—
227#357—21no—
228#357—21no—
229C22⋊C16—21no—
230C4×C16C22×C411yescharacteristic
231C4×C16C2×D411yescharacteristic
232C4×C16C2×D411yescharacteristic
233C4×C16C2×D411yescharacteristic
234C4×C16#4111yescharacteristic
235C4×C16C2411yescharacteristic
236#335—21no—
237#335—21no—
238#335—21no—
239#335—21no—
240#334—41no—
241#334—41no—
242Q8⋊C8—41no—
243Q8⋊C8—41no—
244D4⋊C8—41no—
245D4⋊C8—41no—
246D4⋊C8—41no—
247D4⋊C8—41no—
248D4⋊C8—41no—
249D4⋊C8—41no—
250#322D811yescharacteristic
251#322D811yescharacteristic
252C82C2×D411yescharacteristic
253C82C2×D411yescharacteristic
254C22×D4—41no—
255C23×C4—81no—
256C2×Q16—21no—
257C2×SD16—41no—
258C2×D8—21no—
259C2×D8—21no—
260C2×D8—21no—
261C2×D8—42no—
262C2×D8—41no—
263C2×D8—81no—
264C2×M16—21no—
265C2×M16—81no—
266C2×M16—81no—
267C2×M16—81no—
268C22×C8—81no—
269C22×C8—81no—
270C22×C8—81no—
271C22×C8C4×D411yescharacteristic
272C22×C8#13511yescharacteristic
273#126—81no—
274#125—41no—
275#124—41no—
276#123—81no—
277C22⋊Q8—41no—
278#121—41no—
279C4×Q8—41no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4—41no—
285C4×D4—81no—
286C4×D4—81no—
287C4×D4—81no—
288#117—81no—
289C2×C4⋊C4—41no—
290C2×C22⋊C4—81no—
291C2×C42C2×D811yescharacteristic
292Q32—21no—
293Q32—21no—
294SD32—81no—
295D16—21no—
296D16—21no—
297D16—21no—
298D16—21no—
299D16—21no—
300D16—42no—
301D16#13112yes—
302C2×C16—21no—
303C2×C16—21no—
304C2×C16—21no—
305C2×C16—21no—
306C2×C16—41no—
307C2×C16C4×D411yescharacteristic
308C2×C16C4×D411yescharacteristic
309C2×C16C23×C411yescharacteristic
310C2×C16C2×#4111yescharacteristic
311#107—21no—
312#107—21no—
313#107—21no—
314#107—21no—
315#107—41no—
316#106—41no—
317C4⋊C8—21no—
318C4⋊C8—41no—
319C4⋊C8—41no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8—81no—
323C4⋊C8—81no—
324#103—41no—
325#103—41no—
326#102—41no—
327#102—41no—
328#102—41no—
329#102—41no—
330#102—41no—
331#102—41no—
332C22⋊C8—21no—
333C22⋊C8—81no—
334C22⋊C8—81no—
335C22⋊C8—81no—
336C22⋊C8—81no—
337C22⋊C8—81no—
338C22⋊C8—81no—
339C22⋊C8—81no—
340C22⋊C8—81no—
341#97—21no—
342#97—81no—
343#97—81no—
344C4×C8—81no—
345C4×C8—81no—
346C4×C8C4×D411yescharacteristic
347C4×C8C2×D811yescharacteristic
348C4×C8C2×D811yescharacteristic
349C4×C8C2×D811yescharacteristic
350C4×C8C2×D811yescharacteristic
351C4×C8C2×D811yescharacteristic
352C4×C8#13511yescharacteristic
353C4×C8C22×D411yescharacteristic
354C32—22no—
355C32—21no—
356C32—21no—
357C32—21no—
358C32—21no—
359C32—41no—
360C24—81no—
361C2×Q8—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—82no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—161no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—321no—
376C22×C4—321no—
377C22×C4C4×D811yescharacteristic
378C22×C4#50911yescharacteristic
379C22×C4#50911yescharacteristic
380Q16—41no—
381Q16—41no—
382SD16—161no—
383D8—22no—
384D8—41no—
385D8—41no—
386D8—41no—
387D8—41no—
388D8—41no—
389D8—82no—
390M16—81no—
391M16—162no—
392M16—321no—
393M16—321no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—41no—
401C2×C8—81no—
402C2×C8—81no—
403C2×C8—81no—
404C2×C8—81no—
405C2×C8—162no—
406C2×C8—321no—
407C2×C8—321no—
408C2×C8C4×D811yescharacteristic
409C2×C8C4×D811yescharacteristic
410C2×C8C2×C4×D411yescharacteristic
411C2×C8C2×#13511yescharacteristic
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C22⋊C4—81no—
422C22⋊C4—81no—
423C22⋊C4—81no—
424C22⋊C4—81no—
425C22⋊C4—81no—
426C22⋊C4—81no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C42—81no—
431C42C4×D811yescharacteristic
432C42#50911yescharacteristic
433C42#50911yescharacteristic
434C42C22×D811yescharacteristic
435C16—21no—
436C16—21no—
437C16—21no—
438C16—41no—
439C16—41no—
440C16C2×#13111yescharacteristic
441C16#56811yescharacteristic
442C23—81no—
443C23—81no—
444C23—81no—
445C23—162no—
446C23—162no—
447C23—321no—
448C23#217711yescharacteristic
449Q8—81no—
450Q8—81no—
451D4—42no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—162no—
458D4—321no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—161no—
469C2×C4—161no—
470C2×C4—161no—
471C2×C4—161no—
472C2×C4—161no—
473C2×C4—161no—
474C2×C4—321no—
475C2×C4—321no—
476C2×C4—321no—
477C2×C4#217711yescharacteristic
478C2×C4#217711yescharacteristic
479C2×C4#217711yescharacteristic
480C2×C4C2×C4×D811yescharacteristic
481C2×C4C2×#50911yescharacteristic
482C2×C4C2×#50911yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—161no—
491C8—161no—
492C8—161no—
493C8—161no—
494C8#293311yescharacteristic
495C8#295611yescharacteristic
496C22—21no—
497C22—21no—
498C22—21no—
499C22—82no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22—324no—
509C22C2×#217711yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4—161no—
519C4—161no—
520C4#1799411yescharacteristic
521C4#3346411yescharacteristic
522C2—41no—
523C2—162no—
524C2—321no—
525C2—321no—
526C2#48401211yescharacteristic
527C2#51277811yescharacteristic
528C2#51278111yescharacteristic
subgroupG/H
center, socleC22C2×#2177
commutatorC2×C16C23×C4
Frattini, cosocleC4×C16C24
prime radical#13661C4
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ #11408 ◁ G
LECSG ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #484120C2semidirect
3. #484093C2semidirect
4. #151129C2semidirect
5. #151127C2semidirect
6. #151117C2semidirect
7. #151116C2semidirect
8. #151115C2semidirect
9. #151115C2semidirect
10. #151085C2semidirect
11. #151083C2semidirect
12. #151069C2semidirect
13. #150162C2semidirect
14. #150064C2semidirect
15. #149509C2semidirect
36. #12086C22semidirect
37. #12085C22semidirect
44. #12045C22semidirect
45. #12044C22semidirect
51. #7463C22semidirect
52. #7463C22semidirect
53. #7463C22semidirect
54. #7463C22semidirect
55. #7386C22semidirect
56. #7386C22semidirect
61. #7385C22semidirect
62. #7385C22semidirect
63. #7332C22semidirect
64. #7332C22semidirect
65. #7330C22semidirect
66. #7330C22semidirect

Maximal subgroups

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

subgroupindex
#5130082
#4841202
#4840932
#1511292
#1511272
#1511172
#1511162
#1511152
#1510852
#1510832
#1510692
#1501622
#1500642
#1495092