Finite Group Databasebeta

← #72234381

Subgroups — #72234381

Order 1024.

581 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
1C4×#2263C211yesmaximal, characteristic
2#484093C211yesmaximal, characteristic, complemented
3#484074C211yesmaximal, characteristic, complemented
4#151127C212yesmaximal, complemented
5#151116C212yesmaximal, complemented
6#151115C212yesmaximal, complemented
7#151085C212yesmaximal, complemented
8#151056C211yesmaximal, characteristic, complemented
9#150142C211yesmaximal, characteristic, complemented
10#150140C211yesmaximal, characteristic, complemented
11#149509C211yesmaximal, characteristic, complemented
12C2×#2263C411yescharacteristic
13C4×#508C2211yescharacteristic
14C2×C4×D16C2211yescharacteristic
15#18054—21no—
16#18035—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#12085C2212yescomplemented
32#12074—22no—
33#12069—22no—
34#12045C2212yes—
35#12045C2212yescomplemented
36#12024C2211yescharacteristic
37#12024C2211yescharacteristic
38#11661—21no—
39#11659—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#2974—41no—
52#2970—41no—
53C4×#135—21no—
54C2×C4×D8—21no—
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#1591—22no—
81#1589—22no—
82#1586—42no—
83#1582—42no—
84C8×D8—22no—
85C8×D8—22no—
86#1568—21no—
87#1568—21no—
88#1490—41no—
89#1488—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#1372C2312yes—
111#1372C2312yes—
112#1371—21no—
113#1371—21no—
114#1332—22no—
115#1331—22no—
116#1331—22no—
117#1331—22no—
118#1312D412yes—
119#1311C2312yes—
120C2×#135—21no—
121C22×D8—21no—
122C4×#41—41no—
123C2×C4×D4—41no—
124#508—22no—
125#508—41no—
126C2×Q32C22×C411yescharacteristic
127C2×SD32—21no—
128C2×D16—22no—
129C2×D16—21no—
130C2×D16—41no—
131C2×D16C22×C411yescharacteristic
132C2×D16C22×C411yescharacteristic
133C2×D16C22×C411yescharacteristic
134C22×C16C22×C411yescharacteristic
135C22×C16#4111yescharacteristic
136#446—42no—
137C4×Q16—21no—
138C4×Q16—21no—
139C4×SD16—41no—
140C4×SD16—41no—
141C4×D8—21no—
142C4×D8—21no—
143C4×D8—21no—
144C4×D8—21no—
145C4×D8—21no—
146C4×D8—21no—
147C4×D8—41no—
148C4×D8—41no—
149#436—42no—
150#435—82no—
151#435—82no—
152C8×D4—42no—
153C8×D4—42no—
154#432—41no—
155#432—41no—
156#432—81no—
157#431D811yescharacteristic
158#427—21no—
159C2×#107—21no—
160#416—41no—
161C2×#102—41no—
162#407—81no—
163C2×C22⋊C8—81no—
164C4×M16D811yescharacteristic
165C2×#97—81no—
166C2×C4×C8C2×D411yescharacteristic
167SD64—22no—
168SD64—22no—
169D32—22no—
170D32M1614yescomplemented
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#334—41no—
202#334—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—
224C22×C8—82no—
225C22×C8—82no—
226C22×C8C4×D411yescharacteristic
227C22×C8#13511yescharacteristic
228C4×Q8—41no—
229C4×Q8—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—81no—
237C4×D4—81no—
238C4×D4—81no—
239C4×D4—81no—
240#117—41no—
241#117—81no—
242C2×C4⋊C4—41no—
243C2×C22⋊C4—81no—
244C2×C42—81no—
245C2×C42C2×D811yescharacteristic
246Q32—21no—
247Q32—21no—
248SD32—42no—
249D16—21no—
250D16—21no—
251D16—21no—
252D16—21no—
253D16—42no—
254D16C2×M1612yes—
255D16C2×M1612yes—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—22no—
259C2×C16—41no—
260C2×C16C4×D411yescharacteristic
261C2×C16C4×D411yescharacteristic
262C2×C16C23×C411yescharacteristic
263C2×C16C2×#4111yescharacteristic
264#107—21no—
265#107—21no—
266#107—21no—
267#107—21no—
268#107—41no—
269#106—41no—
270C4⋊C8—21no—
271C4⋊C8—42no—
272C4⋊C8—42no—
273C4⋊C8—82no—
274#103—41no—
275#103—41no—
276#102—41no—
277#102—41no—
278#102—41no—
279#102—41no—
280#102—41no—
281#102—41no—
282C22⋊C8—21no—
283C22⋊C8—82no—
284C22⋊C8—82no—
285C22⋊C8—82no—
286C22⋊C8—81no—
287C22⋊C8—81no—
288#97—21no—
289#97—82no—
290#97—81no—
291#97—81no—
292C4×C8C4×D411yescharacteristic
293C4×C8C2×D812yes—
294C4×C8C2×D812yes—
295C4×C8C2×D811yescharacteristic
296C4×C8#13511yescharacteristic
297C4×C8C22×D411yescharacteristic
298C32—21no—
299C32—21no—
300C32—21no—
301C32—42no—
302C32C2×M1612yes—
303C24—81no—
304#41—82no—
305#41—161no—
306C2×Q8—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—82no—
311C2×D4—81no—
312C2×D4—81no—
313C2×D4—161no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—322no—
323C22×C4C4×D811yescharacteristic
324C22×C4#50911yescharacteristic
325C22×C4#50911yescharacteristic
326Q16—41no—
327Q16—41no—
328SD16—82no—
329D8—22no—
330D8—22no—
331D8—41no—
332D8—41no—
333D8—41no—
334D8—41no—
335D8—82no—
336M16—81no—
337M16—322no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—41no—
343C2×C8—82no—
344C2×C8—82no—
345C2×C8—164no—
346C2×C8—322no—
347C2×C8C4×D811yescharacteristic
348C2×C8C4×D811yescharacteristic
349C2×C8C2×C4×D411yescharacteristic
350C2×C8C2×#13511yescharacteristic
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C22⋊C4—81no—
359C22⋊C4—81no—
360C22⋊C4—81no—
361C22⋊C4—81no—
362C22⋊C4—81no—
363C22⋊C4—81no—
364C22⋊C4—81no—
365C42—81no—
366C42—81no—
367C42—81no—
368C42C4×D811yescharacteristic
369C42#50911yescharacteristic
370C42#50911yescharacteristic
371C42C22×D811yescharacteristic
372C16—21no—
373C16—21no—
374C16—21no—
375C16—42no—
376C16C22×M1611yescharacteristic
377C16#56811yescharacteristic
378C23—81no—
379C23—81no—
380C23—81no—
381C23—162no—
382C23—162no—
383C23—321no—
384C23#217711yescharacteristic
385Q8—81no—
386Q8—81no—
387D4—42no—
388D4—42no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—162no—
394D4—162no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—162no—
405C2×C4—162no—
406C2×C4—162no—
407C2×C4—162no—
408C2×C4—321no—
409C2×C4—321no—
410C2×C4—321no—
411C2×C4—321no—
412C2×C4#217711yescharacteristic
413C2×C4#217711yescharacteristic
414C2×C4#217711yescharacteristic
415C2×C4C2×C4×D811yescharacteristic
416C2×C4C2×#50911yescharacteristic
417C2×C4C2×#50911yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—42no—
422C8—42no—
423C8—162no—
424C8—162no—
425C8D4×M1611yescharacteristic
426C8#295711yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—82no—
431C22—82no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—324no—
440C22C2×#217711yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4M16×D811yescharacteristic
452C4#3346511yescharacteristic
453C2—41no—
454C2—162no—
455C2—162no—
456C2—321no—
457C2#48407311yescharacteristic
458C2#51277811yescharacteristic
459C2#51278111yescharacteristic
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. #484093C2semidirect
3. #484074C2semidirect
4. #151127C2semidirect
5. #151116C2semidirect
6. #151115C2semidirect
7. #151085C2semidirect
8. #151056C2semidirect
9. #150142C2semidirect
10. #150140C2semidirect
11. #149509C2semidirect
31. #12085C22semidirect
35. #12045C22semidirect
41. #7463C22semidirect
42. #7463C22semidirect
43. #7386C22semidirect
46. #7385C22semidirect
47. #7332C22semidirect
48. #7330C22semidirect
170. D32M16semidirect

Maximal subgroups

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

subgroupindex
C4×#22632
#4840932
#4840742
#1511272
#1511162
#1511152
#1510852
#1510562
#1501422
#1501402
#1495092