Finite Group Databasebeta

← #72234389

Subgroups — #72234389

Order 1024.

517 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#513084C211yesmaximal, characteristic
2#484130C211yesmaximal, characteristic, complemented
3#484120C211yesmaximal, characteristic, complemented
4#151129C211yesmaximal, characteristic, complemented
5#151128C211yesmaximal, characteristic, complemented
6#151117C211yesmaximal, characteristic, complemented
7#151117C211yesmaximal, characteristic, complemented
8#151116C211yesmaximal, characteristic, complemented
9#151116C211yesmaximal, characteristic, complemented
10#151085C211yesmaximal, characteristic, complemented
11#151084C211yesmaximal, characteristic, complemented
12#151038C211yesmaximal, characteristic, complemented
13#150128C211yesmaximal, characteristic, complemented
14#150123C211yesmaximal, characteristic, complemented
15#149509C211yesmaximal, characteristic, complemented
16#33735C2211yescharacteristic
17#33735C2211yescharacteristic
18#18087—21no—
19#18069—21no—
20#13732C411yescharacteristic
21#13728C411yescharacteristic
22#13721C411yescharacteristic
23#13721C411yescharacteristic
24#13699C411yescharacteristic
25#13699C411yescharacteristic
26#13698C411yescharacteristic
27#13697C411yescharacteristic
28#13654C2211yescharacteristic
29#13653C2211yescharacteristic
30#13652C2211yescharacteristic
31#13652C2211yescharacteristic
32#13640C2211yescharacteristic
33C4×M64C2211yescharacteristic
34#12086C2211yescharacteristic
35#12086C2211yescharacteristic
36#12085C2211yescharacteristic
37#12085C2211yescharacteristic
38#12075—21no—
39#12074—21no—
40#12071—21no—
41#12070—21no—
42#12045C2211yescharacteristic
43#12045C2211yescharacteristic
44#12044C2211yescharacteristic
45#12044C2211yescharacteristic
46#12008C2211yescharacteristic
47C2×#1372C2211yescharacteristic
48#11648—21no—
49#11647—21no—
50#11408C2211yescharacteristic
51#7463C2211yescharacteristic
52#7463C2211yescharacteristic
53#7463C2211yescharacteristic
54#7463C2211yescharacteristic
55#7386C2211yescharacteristic
56#7386C2211yescharacteristic
57#7386C2211yescharacteristic
58#7386C2211yescharacteristic
59#7385C2211yescharacteristic
60#7385C2211yescharacteristic
61#7385C2211yescharacteristic
62#7385C2211yescharacteristic
63#7332C2211yescharacteristic
64#7332C2211yescharacteristic
65#7330C2211yescharacteristic
66#7330C2211yescharacteristic
67#3168—21no—
68#3168—21no—
69#2994—41no—
70#2969—41no—
71C2×Q64C2×C411yescharacteristic
72C2×SD64C2×C411yescharacteristic
73C2×SD64C2×C411yescharacteristic
74C2×D32C2×C411yescharacteristic
75C2×M64C2×C411yescharacteristic
76C2×M64C2×C411yescharacteristic
77#2253C2×C411yescharacteristic
78#2253C2×C411yescharacteristic
79#2249C2×C411yescharacteristic
80#2249C2×C411yescharacteristic
81#2242—21no—
82#2242—21no—
83#2217—21no—
84#2217—21no—
85#2216C2×C411yescharacteristic
86#2216C2×C411yescharacteristic
87#2215C2×C411yescharacteristic
88#2215C2×C411yescharacteristic
89C4×Q32C2311yescharacteristic
90C4×Q32C2311yescharacteristic
91C4×SD32—21no—
92C4×SD32—21no—
93C4×D16C2311yescharacteristic
94C4×D16C2311yescharacteristic
95#2162D411yescharacteristic
96C2×C4×C16C2311yescharacteristic
97#1591—21no—
98#1591—21no—
99#1589—21no—
100#1589—21no—
101#1588—41no—
102#1585—41no—
103#1583—41no—
104#1582—41no—
105C8×Q16—21no—
106C8×Q16—21no—
107C8×D8—21no—
108C8×D8—21no—
109#1564—21no—
110C2×#335—21no—
111#1483—41no—
112#1482—41no—
113C8×M16D411yescharacteristic
114#1424C2×C411yescharacteristic
115#1424C2×C411yescharacteristic
116#1423C2×C411yescharacteristic
117#1423C2×C411yescharacteristic
118C4⋊C32—21no—
119#1416C2×C411yescharacteristic
120#1416C2×C411yescharacteristic
121#1416C2×C411yescharacteristic
122#1416C2×C411yescharacteristic
123#1415C2×C411yescharacteristic
124#1415C2×C411yescharacteristic
125#1415C2×C411yescharacteristic
126#1415C2×C411yescharacteristic
127C22⋊C32—21no—
128#1397C2311yescharacteristic
129#1397C2311yescharacteristic
130#1397C2311yescharacteristic
131#1397C2311yescharacteristic
132C4×C32C2×C411yescharacteristic
133C4×C32C2×C411yescharacteristic
134#1372—21no—
135#1372C2311yescharacteristic
136#1372C2311yescharacteristic
137#1372C2311yescharacteristic
138#1372C2311yescharacteristic
139#1371—21no—
140#1332—21no—
141#1332—21no—
142#1332—21no—
143#1332—21no—
144#1331—21no—
145#1331—21no—
146#1331—21no—
147#1331—21no—
148#1312D411yescharacteristic
149#1312D411yescharacteristic
150#1311C2311yescharacteristic
151#1311C2311yescharacteristic
152#529—41no—
153#529—41no—
154C2×Q32C22×C411yescharacteristic
155C2×Q32C22×C411yescharacteristic
156C2×SD32—21no—
157C2×SD32—21no—
158C2×D16C22×C411yescharacteristic
159C2×D16C22×C411yescharacteristic
160C22×C16C22×C411yescharacteristic
161C22×C16#4111yescharacteristic
162#499—21no—
163#499—21no—
164#495—21no—
165#495—21no—
166#488—41no—
167#488—41no—
168#467—21no—
169#467—21no—
170#466—21no—
171#466—21no—
172#465—41no—
173#465—41no—
174#446—41no—
175#446—41no—
176C8×Q8—41no—
177C8×Q8—41no—
178C4×Q16—21no—
179C4×Q16—21no—
180C4×SD16—41no—
181C4×SD16—41no—
182C4×D8—21no—
183C4×D8—21no—
184#436—41no—
185#436—41no—
186#436—81no—
187#435—81no—
188#435—81no—
189C8×D4—41no—
190C8×D4—41no—
191C8×D4—81no—
192#433—81no—
193#433—81no—
194#431—41no—
195#431D811yescharacteristic
196#424—81no—
197C2×C4⋊C8—41no—
198#405—81no—
199C4×M16D811yescharacteristic
200C2×C4×C8C2×D411yescharacteristic
201Q64—22no—
202SD64—41no—
203SD64—41no—
204D32—22no—
205M64—22no—
206M64—41no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32C22×C411yescharacteristic
210C2×C32C22×C411yescharacteristic
211C2×C32C22×C411yescharacteristic
212C2×C32C22×C411yescharacteristic
213#367—21no—
214#367—21no—
215#366C22×C411yescharacteristic
216#366C22×C411yescharacteristic
217#366C22×C411yescharacteristic
218#366C22×C411yescharacteristic
219C4⋊C16—21no—
220#358—21no—
221#358—21no—
222#358—21no—
223#358—21no—
224#357—21no—
225#357—21no—
226#357—21no—
227#357—21no—
228C22⋊C16—21no—
229C4×C16C22×C411yescharacteristic
230C4×C16C2×D411yescharacteristic
231C4×C16C2×D411yescharacteristic
232C4×C16C2×D411yescharacteristic
233C4×C16#4111yescharacteristic
234C4×C16C2411yescharacteristic
235#335—21no—
236#335—21no—
237#335—21no—
238#335—21no—
239#335—41no—
240#334—41no—
241Q8⋊C8—41no—
242Q8⋊C8—41no—
243Q8⋊C8—41no—
244Q8⋊C8—41no—
245D4⋊C8—41no—
246D4⋊C8—41no—
247D4⋊C8—41no—
248D4⋊C8—41no—
249#322D811yescharacteristic
250#322D811yescharacteristic
251C82C2×D411yescharacteristic
252C82C2×D411yescharacteristic
253C2×Q16—21no—
254C2×Q16—21no—
255C2×SD16—41no—
256C2×SD16—41no—
257C2×D8—21no—
258C2×D8—21no—
259C2×M16—21no—
260C2×M16—81no—
261C2×M16—81no—
262C22×C8—81no—
263C22×C8—81no—
264C22×C8—81no—
265C22×C8C4×D411yescharacteristic
266C22×C8#13511yescharacteristic
267#126—81no—
268#126—81no—
269#125—41no—
270#125—41no—
271#124—41no—
272#124—41no—
273#123—81no—
274#123—81no—
275C22⋊Q8—41no—
276C22⋊Q8—41no—
277#121—41no—
278#121—41no—
279C4×Q8—41no—
280C4×Q8—41no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—81no—
284C4×D4—81no—
285#117—81no—
286#117—81no—
287C2×C42C2×D811yescharacteristic
288Q32—21no—
289Q32—21no—
290Q32—21no—
291Q32—21no—
292SD32—81no—
293SD32—81no—
294D16—21no—
295D16—21no—
296D16—21no—
297D16—21no—
298C2×C16—21no—
299C2×C16—21no—
300C2×C16—21no—
301C2×C16—21no—
302C2×C16—41no—
303C2×C16C4×D411yescharacteristic
304C2×C16C4×D411yescharacteristic
305C2×C16C23×C411yescharacteristic
306C2×C16C2×#4111yescharacteristic
307#107—21no—
308#107—21no—
309#107—21no—
310#107—21no—
311#106—41no—
312#106—41no—
313C4⋊C8—21no—
314C4⋊C8—41no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317C4⋊C8—41no—
318C4⋊C8—81no—
319C4⋊C8—81no—
320C4⋊C8—81no—
321C4⋊C8—81no—
322C4⋊C8—81no—
323#103—41no—
324#103—41no—
325#103—41no—
326#103—41no—
327#102—41no—
328#102—41no—
329#102—41no—
330#102—41no—
331C22⋊C8—21no—
332C22⋊C8—81no—
333C22⋊C8—81no—
334C22⋊C8—81no—
335C22⋊C8—81no—
336C22⋊C8—81no—
337#97—21no—
338#97—81no—
339#97—81no—
340C4×C8—81no—
341C4×C8—81no—
342C4×C8—81no—
343C4×C8C4×D411yescharacteristic
344C4×C8C2×D811yescharacteristic
345C4×C8C2×D811yescharacteristic
346C4×C8C2×D811yescharacteristic
347C4×C8C2×D811yescharacteristic
348C4×C8C2×D811yescharacteristic
349C4×C8#13511yescharacteristic
350C4×C8C22×D411yescharacteristic
351C32—21no—
352C32—21no—
353C32—21no—
354C32—41no—
355C32—41no—
356C32#13112yes—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—81no—
362C2×D4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4C4×D811yescharacteristic
366C22×C4#50911yescharacteristic
367C22×C4#50911yescharacteristic
368Q16—41no—
369Q16—41no—
370Q16—41no—
371Q16—41no—
372SD16—161no—
373SD16—161no—
374D8—41no—
375D8—41no—
376D8—41no—
377D8—41no—
378M16—81no—
379M16—162no—
380M16—321no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—41no—
388C2×C8—81no—
389C2×C8—81no—
390C2×C8—81no—
391C2×C8—81no—
392C2×C8—162no—
393C2×C8—321no—
394C2×C8—321no—
395C2×C8C4×D811yescharacteristic
396C2×C8C4×D811yescharacteristic
397C2×C8C2×C4×D411yescharacteristic
398C2×C8C2×#13511yescharacteristic
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—81no—
409C4⋊C4—81no—
410C4⋊C4—81no—
411C22⋊C4—81no—
412C22⋊C4—81no—
413C22⋊C4—81no—
414C22⋊C4—81no—
415C22⋊C4—81no—
416C22⋊C4—81no—
417C22⋊C4—81no—
418C22⋊C4—81no—
419C22⋊C4—81no—
420C22⋊C4—81no—
421C42—81no—
422C42—81no—
423C42C4×D811yescharacteristic
424C42#50911yescharacteristic
425C42#50911yescharacteristic
426C42C22×D811yescharacteristic
427C16—21no—
428C16—21no—
429C16—21no—
430C16—41no—
431C16—41no—
432C16C2×#13111yescharacteristic
433C16#56811yescharacteristic
434C23—81no—
435C23—81no—
436C23#217711yescharacteristic
437Q8—81no—
438Q8—81no—
439Q8—81no—
440Q8—81no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—162no—
446D4—321no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—161no—
458C2×C4—161no—
459C2×C4—161no—
460C2×C4—161no—
461C2×C4#217711yescharacteristic
462C2×C4#217711yescharacteristic
463C2×C4#217711yescharacteristic
464C2×C4C2×C4×D811yescharacteristic
465C2×C4C2×#50911yescharacteristic
466C2×C4C2×#50911yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—161no—
475C8—161no—
476C8—161no—
477C8—161no—
478C8#293511yescharacteristic
479C8#295711yescharacteristic
480C22—21no—
481C22—21no—
482C22—21no—
483C22—161no—
484C22—161no—
485C22—161no—
486C22—161no—
487C22—161no—
488C22—161no—
489C22C2×#217711yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—161no—
496C4—161no—
497C4—161no—
498C4—161no—
499C4—161no—
500C4—161no—
501C4#1806611yescharacteristic
502C4#3346511yescharacteristic
503C2—41no—
504C2—321no—
505C2—321no—
506C2#48411811yescharacteristic
507C2#51277811yescharacteristic
508C2#51277911yescharacteristic
subgroupG/H
center, socleC22C2×#2177
commutatorC2×C16C23×C4
Frattini, cosocleC4×C16C24
prime radical#13697C4
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. #484130C2semidirect
3. #484120C2semidirect
4. #151129C2semidirect
5. #151128C2semidirect
6. #151117C2semidirect
7. #151117C2semidirect
8. #151116C2semidirect
9. #151116C2semidirect
10. #151085C2semidirect
11. #151084C2semidirect
12. #151038C2semidirect
13. #150128C2semidirect
14. #150123C2semidirect
15. #149509C2semidirect

Maximal subgroups

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

subgroupindex
#5130842
#4841302
#4841202
#1511292
#1511282
#1511172
#1511162
#1510852
#1510842
#1510382
#1501282
#1501232
#1495092