Finite Group Databasebeta

← #72260430

Subgroups — #72260430

Order 1024.

598 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#2263C211yesmaximal, characteristic
2#484092C211yesmaximal, characteristic, complemented
3#484073C211yesmaximal, characteristic, complemented
4#151129C212yesmaximal, complemented
5#151117C212yesmaximal, complemented
6#151115C212yesmaximal, complemented
7#151083C212yesmaximal, complemented
8#151055C211yesmaximal, characteristic, complemented
9#150141C211yesmaximal, characteristic, complemented
10#150138C211yesmaximal, characteristic, complemented
11#149509C211yesmaximal, characteristic, complemented
12C2×#2263C411yescharacteristic
13C4×#508C2211yescharacteristic
14C2×C4×D16C2211yescharacteristic
15#18059—21no—
16M16×D8—21no—
17#13654C2211yescharacteristic
18#13654C2211yescharacteristic
19#13652C2211yescharacteristic
20#13652C2211yescharacteristic
21C4×SD64C411yescharacteristic
22C4×SD64C411yescharacteristic
23C4×D32C411yescharacteristic
24C4×D32C411yescharacteristic
25#13640C2211yescharacteristic
26#13634C411yescharacteristic
27#13617C411yescharacteristic
28#13616C411yescharacteristic
29C4×M64C2211yescharacteristic
30#12086C2212yes—
31#12086C2212yescomplemented
32#12073—22no—
33#12071—22no—
34#12045C2212yes—
35#12044C2212yescomplemented
36#12023C2211yescharacteristic
37#12023C2211yescharacteristic
38#11660—21no—
39#11657—21no—
40#11408C2211yescharacteristic
41#7463C2212yescomplemented
42#7463C2212yescomplemented
43#7386C2212yescomplemented
44#7385C2212yes—
45#7385C2212yes—
46#7385C2212yescomplemented
47#7332C2212yescomplemented
48#7330C2212yescomplemented
49C2×#508C2×C411yescharacteristic
50C22×D16C2×C411yescharacteristic
51#2991—41no—
52C4×#135—21no—
53C2×C4×D8—21no—
54D4×M16—41no—
55#2263—24no—
56C2×SD64C2×C411yescharacteristic
57C2×SD64C2×C411yescharacteristic
58C2×D32C2×C411yescharacteristic
59C2×D32C2×C411yescharacteristic
60C2×M64C2×C411yescharacteristic
61C2×M64C2×C411yescharacteristic
62C4×Q32C2×C411yescharacteristic
63C4×Q32C2311yescharacteristic
64C4×SD32—21no—
65C4×SD32—21no—
66C4×D16—21no—
67C4×D16—21no—
68C4×D16C2×C411yescharacteristic
69C4×D16C2×C411yescharacteristic
70C4×D16C2×C411yescharacteristic
71C4×D16C2311yescharacteristic
72C4×D16C2311yescharacteristic
73C4×D16C2311yescharacteristic
74#2162D411yescharacteristic
75#2158C2×C411yescharacteristic
76C2×#366C2×C411yescharacteristic
77#2138—21no—
78C2×#357—21no—
79C2×C4×C16C2311yescharacteristic
80#1589—22no—
81#1589—22no—
82#1588—42no—
83#1581—42no—
84C8×Q16—22no—
85C8×D8—22no—
86#1567—21no—
87#1567—21no—
88#1489—41no—
89#1486—41no—
90C8×M16D411yescharacteristic
91#1424C2×C411yescharacteristic
92#1424C2×C411yescharacteristic
93#1423C2×C411yescharacteristic
94#1423C2×C411yescharacteristic
95C4⋊C32—21no—
96#1416C2×C411yescharacteristic
97#1416C2×C411yescharacteristic
98#1415C2×C411yescharacteristic
99#1415C2×C411yescharacteristic
100#1415C2×C411yescharacteristic
101#1415C2×C411yescharacteristic
102#1415C2×C411yescharacteristic
103#1415C2×C411yescharacteristic
104C22⋊C32—21no—
105#1397C2312yes—
106#1397C2311yescharacteristic
107#1397C2311yescharacteristic
108C4×C32C2×C411yescharacteristic
109C4×C32C2×C411yescharacteristic
110#1372—21no—
111#1372—21no—
112#1372C2312yes—
113#1372C2312yes—
114#1332—22no—
115#1331—22no—
116#1331—22no—
117#1331—22no—
118#1312D412yes—
119#1311C2312yes—
120C2×#135—21no—
121C22×D8—21no—
122C22×M16—81no—
123C4×#41—41no—
124C2×C4×D4—41no—
125#508—22no—
126#508—41no—
127C2×Q32C22×C411yescharacteristic
128C2×SD32—21no—
129C2×D16—22no—
130C2×D16—21no—
131C2×D16—41no—
132C2×D16C22×C411yescharacteristic
133C2×D16C22×C411yescharacteristic
134C2×D16C22×C411yescharacteristic
135C22×C16C22×C411yescharacteristic
136C22×C16#4111yescharacteristic
137C8×Q8—42no—
138C4×Q16—21no—
139C4×Q16—21no—
140C4×SD16—41no—
141C4×SD16—41no—
142C4×D8—21no—
143C4×D8—21no—
144C4×D8—21no—
145C4×D8—21no—
146C4×D8—21no—
147C4×D8—21no—
148C4×D8—41no—
149C4×D8—41no—
150#436—42no—
151#436—42no—
152#436—82no—
153#435—82no—
154C8×D4—42no—
155#431—81no—
156#431D811yescharacteristic
157#427—21no—
158C2×#107—21no—
159#423—41no—
160#423—41no—
161#423—81no—
162#416—41no—
163C2×#102—41no—
164#407—81no—
165C4×M16D811yescharacteristic
166C2×C4×C8C2×D411yescharacteristic
167SD64—22no—
168SD64M1614yescomplemented
169D32—22no—
170D32—22no—
171M64—22no—
172M64—41no—
173C2×C32—22no—
174C2×C32C22×C411yescharacteristic
175C2×C32C22×C411yescharacteristic
176C2×C32C22×C411yescharacteristic
177C2×C32C22×C411yescharacteristic
178#367—21no—
179#366—21no—
180#366C22×C411yescharacteristic
181#366C22×C411yescharacteristic
182#366C22×C411yescharacteristic
183#366C22×C411yescharacteristic
184C4⋊C16—21no—
185#358—21no—
186#358—21no—
187#357—21no—
188#357—21no—
189#357—21no—
190#357—21no—
191#357—21no—
192#357—21no—
193C22⋊C16—21no—
194C4×C16C22×C411yescharacteristic
195C4×C16C2×D412yes—
196C4×C16C2×D411yescharacteristic
197C4×C16#4111yescharacteristic
198C4×C16C2411yescharacteristic
199#335—22no—
200#335—22no—
201#335—41no—
202#335—41no—
203Q8⋊C8—42no—
204D4⋊C8—42no—
205D4⋊C8—42no—
206D4⋊C8—42no—
207#322D812yes—
208C82C2×D412yes—
209C2×#41—41no—
210C22×D4—41no—
211C23×C4—81no—
212#135—42no—
213#135—81no—
214C2×Q16—21no—
215C2×SD16—41no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8—42no—
220C2×D8—41no—
221C2×D8—81no—
222C2×M16—21no—
223C2×M16—82no—
224C2×M16—82no—
225C2×M16—81no—
226C2×M16—81no—
227C2×M16—164no—
228C22×C8—82no—
229C22×C8C4×D411yescharacteristic
230C22×C8#13511yescharacteristic
231C4×Q8—41no—
232C4×Q8—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—81no—
240C4×D4—81no—
241C4×D4—81no—
242C4×D4—81no—
243#117—41no—
244#117—81no—
245C2×C4⋊C4—41no—
246C2×C22⋊C4—81no—
247C2×C42—81no—
248C2×C42C2×D811yescharacteristic
249Q32—21no—
250Q32C2×M1612yescomplemented
251SD32—42no—
252D16—21no—
253D16—21no—
254D16—21no—
255D16—21no—
256D16—21no—
257D16—42no—
258D16C2×M1612yes—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—22no—
262C2×C16—41no—
263C2×C16C4×D411yescharacteristic
264C2×C16C4×D411yescharacteristic
265C2×C16C23×C411yescharacteristic
266C2×C16C2×#4111yescharacteristic
267#107—21no—
268#107—21no—
269#107—21no—
270#107—21no—
271#107—41no—
272#106—41no—
273C4⋊C8—21no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—82no—
277C4⋊C8—81no—
278C4⋊C8—81no—
279#103—41no—
280#103—41no—
281#102—41no—
282#102—41no—
283#102—41no—
284#102—41no—
285#102—41no—
286#102—41no—
287C22⋊C8—21no—
288C22⋊C8—82no—
289C22⋊C8—82no—
290C22⋊C8—82no—
291#97—21no—
292C4×C8—82no—
293C4×C8C4×D411yescharacteristic
294C4×C8C2×D812yes—
295C4×C8C2×D812yes—
296C4×C8C2×D811yescharacteristic
297C4×C8#13511yescharacteristic
298C4×C8C22×D411yescharacteristic
299C32—21no—
300C32—21no—
301C32—21no—
302C32—42no—
303C32C2×M1612yescomplemented
304C24—81no—
305#41—82no—
306#41—161no—
307C2×Q8—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—82no—
312C2×D4—81no—
313C2×D4—81no—
314C2×D4—161no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—162no—
324C22×C4—162no—
325C22×C4C4×D811yescharacteristic
326C22×C4#50911yescharacteristic
327C22×C4#50911yescharacteristic
328Q16—22no—
329Q16—41no—
330SD16—82no—
331D8—22no—
332D8—41no—
333D8—41no—
334D8—41no—
335D8—41no—
336D8—41no—
337D8—82no—
338M16—42no—
339M16—164no—
340M16—162no—
341M16—322no—
342M16—321no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—82no—
349C2×C8—82no—
350C2×C8—164no—
351C2×C8C4×D811yescharacteristic
352C2×C8C4×D811yescharacteristic
353C2×C8C2×C4×D411yescharacteristic
354C2×C8C2×#13511yescharacteristic
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C22⋊C4—81no—
363C22⋊C4—81no—
364C22⋊C4—81no—
365C22⋊C4—81no—
366C22⋊C4—81no—
367C22⋊C4—81no—
368C22⋊C4—81no—
369C42—81no—
370C42—81no—
371C42—81no—
372C42C4×D811yescharacteristic
373C42#50911yescharacteristic
374C42#50911yescharacteristic
375C42C22×D811yescharacteristic
376C16—21no—
377C16—21no—
378C16—21no—
379C16—42no—
380C16C22×M1611yescharacteristic
381C16#56811yescharacteristic
382C23—81no—
383C23—81no—
384C23—81no—
385C23—162no—
386C23—162no—
387C23—321no—
388C23#217711yescharacteristic
389Q8—42no—
390Q8—81no—
391D4—42no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—81no—
397D4—162no—
398D4—162no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—162no—
409C2×C4—161no—
410C2×C4—161no—
411C2×C4—161no—
412C2×C4—161no—
413C2×C4—161no—
414C2×C4—161no—
415C2×C4—162no—
416C2×C4—162no—
417C2×C4—162no—
418C2×C4—162no—
419C2×C4#217711yescharacteristic
420C2×C4#217711yescharacteristic
421C2×C4#217711yescharacteristic
422C2×C4C2×C4×D811yescharacteristic
423C2×C4C2×#50911yescharacteristic
424C2×C4C2×#50911yescharacteristic
425C8—21no—
426C8—21no—
427C8—21no—
428C8—42no—
429C8—42no—
430C8—162no—
431C8—162no—
432C8D4×M1611yescharacteristic
433C8#295711yescharacteristic
434C22—21no—
435C22—21no—
436C22—21no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—324no—
447C22C2×#217711yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—82no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4#1365812yes—
458C4M16×D811yescharacteristic
459C4#3346511yescharacteristic
460C2—41no—
461C2—162no—
462C2—321no—
463C2—321no—
464C2#48407311yescharacteristic
465C2C2×#1365811yescharacteristic
466C2#51277911yescharacteristic
subgroupG/H
center, socleC22C2×#2177
commutatorC2×C16C23×C4
Frattini, cosocleC4×C16C24
prime radicalC2×#2263C4
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. #484092C2semidirect
3. #484073C2semidirect
4. #151129C2semidirect
5. #151117C2semidirect
6. #151115C2semidirect
7. #151083C2semidirect
8. #151055C2semidirect
9. #150141C2semidirect
10. #150138C2semidirect
11. #149509C2semidirect
31. #12086C22semidirect
35. #12044C22semidirect
41. #7463C22semidirect
42. #7463C22semidirect
43. #7386C22semidirect
46. #7385C22semidirect
47. #7332C22semidirect
48. #7330C22semidirect
168. SD64M16semidirect
250. Q32C2×M16semidirect
303. C32C2×M16semidirect

Maximal subgroups

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

subgroupindex
C4×#22632
#4840922
#4840732
#1511292
#1511172
#1511152
#1510832
#1510552
#1501412
#1501382
#1495092