Finite Group Databasebeta

← #41×SD32 #7623985

Subgroups — #41×SD32 #7623985

Order 512.

1437 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q16×#41C211yesmaximal, characteristic, complemented
2D8×#41C211yesmaximal, characteristic
3#33959C211yesmaximal, characteristic, complemented
4Q8×SD32C211yesmaximal, characteristic, complemented
5#33929C211yesmaximal, characteristic, complemented
6#33926C211yesmaximal, characteristic, complemented
7#33902C213yesmaximal, complemented
8#33865C213yesmaximal, complemented
9#33862C213yesmaximal, complemented
10D4×SD32C213yesmaximal, complemented
11#33736C213yesmaximal, complemented
12#33663C213yesmaximal, complemented
13#33661C213yesmaximal, complemented
14C16×#41C211yesmaximal, characteristic, complemented
15C2×C4×SD32C213yesmaximal, complemented
16Q8×#41—21no—
17D4×#41—21no—
18C22×SD32C2213yes—
19#3395C2211yescharacteristic, complemented
20#3391C2211yescharacteristic
21Q8×Q16C2211yescharacteristic, complemented
22Q8×D8C2211yescharacteristic
23#3363—21no—
24#3358—21no—
25#3338C2213yescomplemented
26#3334C2213yes—
27#3299—23no—
28#3294—23no—
29D4×Q16C2213yescomplemented
30D4×D8C2213yes—
31#3170C2213yescomplemented
32#3169C2213yes—
33#3104—23no—
34#3100—23no—
35C8×#41C2211yescharacteristic
36C2×C4×Q16C2213yescomplemented
37C2×C4×D8C2213yes—
38#2254C2213yescomplemented
39#2248C2213yescomplemented
40#2241C2213yescomplemented
41#2240C2213yescomplemented
42#2235C2216yescomplemented
43#2233C2216yescomplemented
44#2226C2213yes—
45#2226C2213yescomplemented
46#2225C2213yescomplemented
47#2225C2213yescomplemented
48#2217C2216yescomplemented
49#2217C2216yescomplemented
50#2208C2213yes—
51#2208C2213yescomplemented
52#2207C2213yescomplemented
53#2207C2213yescomplemented
54#2187C2216yes—
55#2185C2216yescomplemented
56Q8×C16C2211yescharacteristic, complemented
57Q8×C16C2211yescharacteristic, complemented
58C4×SD32C2213yes—
59C4×SD32C2213yes—
60C4×SD32C2213yes—
61C4×SD32C2211yescharacteristic
62C4×SD32C2213yescomplemented
63C4×SD32C2213yescomplemented
64C4×SD32C2216yescomplemented
65C4×SD32C2213yescomplemented
66C4×SD32C2213yescomplemented
67D4×C16C2213yescomplemented
68D4×C16C2213yescomplemented
69#2162C2213yescomplemented
70C2×#367C2213yescomplemented
71C2×#358C2213yescomplemented
72C2×#357C2213yes—
73C2×C4×C16C2213yescomplemented
74C22×#41—41no—
75C22×Q16C2313yes—
76C22×D8C2313yes—
77Q82—21no—
78#557—21no—
79#557—41no—
80#554—23no—
81#554—43no—
82#550—21no—
83D4×Q8—21no—
84D4×Q8—23no—
85#548—41no—
86#547—23no—
87#546—43no—
88D42—23no—
89#533—23no—
90#533—43no—
91#532—23no—
92#525—43no—
93C4×#41—41no—
94C4×#41D411yescharacteristic
95C2×C4×Q8—23no—
96C2×C4×D4—23no—
97C2×SD32—212no—
98C2×SD32C2313yes—
99C2×SD32C2313yes—
100C2×SD32C2313yes—
101C2×SD32C2316yes—
102C2×SD32C2311yescharacteristic
103C22×C16C2313yes—
104C22×C16C2313yes—
105#500C2313yes—
106#500C2313yes—
107#494C2313yes—
108#493C2313yes—
109#487—23no—
110#486—23no—
111#484—26no—
112#480—26no—
113#477—23no—
114#477—23no—
115#474—23no—
116#474—23no—
117#467—23no—
118#467C2316yes—
119#466—23no—
120#466C2316yes—
121#462—23no—
122#462—23no—
123#459—23no—
124#459—23no—
125#451—26no—
126#447—26no—
127C8×Q8D411yescharacteristic
128C8×Q8C2311yescharacteristic
129C4×Q16—23no—
130C4×Q16—23no—
131C4×Q16—23no—
132C4×Q16C2313yes—
133C4×Q16C2313yes—
134C4×Q16C2313yes—
135C4×Q16C2311yescharacteristic
136C4×D8—23no—
137C4×D8—23no—
138C4×D8—23no—
139C4×D8C2313yes—
140C4×D8C2313yes—
141C4×D8C2313yes—
142C4×D8C2311yescharacteristic
143C8×D4D413yes—
144C8×D4C2313yes—
145#431D413yes—
146C2×#107C2313yes—
147C2×#107C2313yes—
148C2×#103—23no—
149C2×#102—23no—
150C2×C4×C8C2313yes—
151#367C2311yescharacteristic
152#367C2313yes—
153#367C2313yes—
154#367C2313yes—
155#367C2316yes—
156C4⋊C16C2313yes—
157C4⋊C16C2313yes—
158#358C2311yescharacteristic
159#358C2313yes—
160#358C2313yes—
161#358C2313yes—
162#358C2316yes—
163#357C2311yescharacteristic
164#357C2313yes—
165#357C2313yes—
166#357C2313yes—
167#357C2316yes—
168C22⋊C16C2316yes—
169C4×C16C2311yescharacteristic
170C4×C16C2313yes—
171C4×C16C2313yes—
172C4×C16C2313yes—
173C2×#41—41no—
174C2×#41—43no—
175C2×#41—43no—
176C2×#41—88no—
177C2×#41D811yescharacteristic
178C22×Q8—23no—
179C22×Q8—41no—
180C22×D4—23no—
181C22×D4—43no—
182C23×C4—43no—
183C2×Q16—26no—
184C2×Q16—23no—
185C2×Q16—23no—
186C2×Q16—43no—
187C2×Q16C2413yes—
188C2×Q16C2411yescharacteristic
189C2×D8—26no—
190C2×D8—23no—
191C2×D8—23no—
192C2×D8—43no—
193C2×D8C2413yes—
194C2×D8C2411yescharacteristic
195C22×C8C2×D413yes—
196C22×C8C2413yes—
197C4⋊Q8—23no—
198C4⋊Q8—23no—
199C4⋊Q8—23no—
200C4⋊Q8—43no—
201C4⋊Q8—43no—
202#127—23no—
203#125—43no—
204#125—43no—
205#125—43no—
206#124—43no—
207#123—46no—
208C22⋊Q8—26no—
209C22⋊Q8—43no—
210C22⋊Q8—43no—
211C22⋊Q8—43no—
212C22⋊Q8—46no—
213C22⋊Q8—46no—
214#121—26no—
215#121—43no—
216#121—43no—
217#121—43no—
218#120—46no—
219C4×Q8—21no—
220C4×Q8—23no—
221C4×Q8—23no—
222C4×Q8—23no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8—43no—
226C4×Q8—43no—
227C4×Q8—43no—
228C4×Q8D811yescharacteristic
229C4×Q8C2×D411yescharacteristic
230C4×D4—21no—
231C4×D4—23no—
232C4×D4—23no—
233C4×D4—23no—
234C4×D4—43no—
235C4×D4—43no—
236C4×D4—43no—
237C4×D4—43no—
238C4×D4—43no—
239C4×D4D813yes—
240C4×D4C2×D413yes—
241#117—43no—
242#117D813yes—
243C2×C4⋊C4—23no—
244C2×C4⋊C4—23no—
245C2×C4⋊C4—43no—
246C2×C22⋊C4—43no—
247C2×C42—43no—
248C2×C42C2×D413yes—
249SD32—26no—
250SD32—26no—
251SD32—26no—
252SD32—46no—
253SD32#4114yesdirect product factor
254C2×C16—26no—
255C2×C16—43no—
256C2×C16C2411yescharacteristic
257C2×C16C2413yes—
258C2×C16C2413yes—
259C2×C16C2411yescharacteristic
260#107—23no—
261#107—23no—
262#107—23no—
263#107—23no—
264#107C2411yescharacteristic
265#107C2413yes—
266#107C2411yescharacteristic
267#107C2413yes—
268C4⋊C8C2×D413yes—
269C4⋊C8C2×D413yes—
270#103—21no—
271#103—23no—
272#103—23no—
273#103—23no—
274#103—26no—
275#102—21no—
276#102—23no—
277#102—23no—
278#102—23no—
279#102—26no—
280C22⋊C8C2×D416yes—
281C4×C8C2×D413yes—
282C4×C8C2×D413yes—
283C4×C8C2411yescharacteristic
284C4×C8C2413yes—
285C24—43no—
286#41—84no—
287#41—812no—
288#41—812no—
289#41SD3218yesdirect product factor
290C2×Q8—21no—
291C2×Q8—23no—
292C2×Q8—46no—
293C2×Q8—43no—
294C2×Q8—43no—
295C2×Q8—43no—
296C2×Q8—83no—
297C2×Q8—84no—
298C2×Q8C2×D811yescharacteristic
299C2×D4—21no—
300C2×D4—23no—
301C2×D4—46no—
302C2×D4—46no—
303C2×D4—43no—
304C2×D4—43no—
305C2×D4—43no—
306C2×D4—83no—
307C2×D4—812no—
308C2×D4C2×D813yes—
309C22×C4—43no—
310C22×C4—43no—
311C22×C4—43no—
312C22×C4—43no—
313C22×C4—43no—
314C22×C4—41no—
315C22×C4—43no—
316C22×C4—812no—
317C22×C4C2×D813yes—
318C22×C4C2×D813yes—
319C22×C4C22×D413yes—
320Q16—23no—
321Q16—21no—
322Q16—23no—
323Q16—46no—
324Q16—83no—
325Q16C2×#4112yescomplemented
326D8—23no—
327D8—21no—
328D8—23no—
329D8—46no—
330D8—83no—
331D8C2×#4112yes—
332C2×C8—23no—
333C2×C8—23no—
334C2×C8—43no—
335C2×C8C22×D411yescharacteristic
336C2×C8C22×D413yes—
337C2×C8C22×D413yes—
338C2×C8C2511yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—23no—
341C4⋊C4—21no—
342C4⋊C4—23no—
343C4⋊C4—43no—
344C4⋊C4—43no—
345C4⋊C4—41no—
346C4⋊C4—43no—
347C4⋊C4—43no—
348C4⋊C4—43no—
349C4⋊C4—43no—
350C4⋊C4—43no—
351C4⋊C4—43no—
352C4⋊C4—46no—
353C4⋊C4—43no—
354C4⋊C4C2×D813yes—
355C4⋊C4C2×D813yes—
356C22⋊C4—46no—
357C22⋊C4—41no—
358C22⋊C4—43no—
359C22⋊C4—43no—
360C22⋊C4—43no—
361C22⋊C4—46no—
362C22⋊C4C2×D816yes—
363C42—43no—
364C42—43no—
365C42—41no—
366C42—43no—
367C42C2×D813yes—
368C42C2×D813yes—
369C42C22×D413yes—
370C42C22×D411yescharacteristic
371C16—23no—
372C16—21no—
373C16—23no—
374C16C2×#4112yescomplemented
375C23—41no—
376C23—43no—
377C23—86no—
378C23—86no—
379C23—163no—
380C23C22×D813yes—
381Q8—22no—
382Q8—41no—
383Q8—43no—
384Q8—43no—
385Q8—86no—
386Q8—86no—
387Q8—163no—
388Q8C2×SD3214yescomplemented
389D4—22no—
390D4—41no—
391D4—43no—
392D4—43no—
393D4—86no—
394D4—812no—
395D4—86no—
396D4—163no—
397D4C2×SD32112yescomplemented
398C2×C4—23no—
399C2×C4—23no—
400C2×C4—43no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—43no—
404C2×C4—43no—
405C2×C4—41no—
406C2×C4—43no—
407C2×C4—86no—
408C2×C4—86no—
409C2×C4—82no—
410C2×C4—86no—
411C2×C4—86no—
412C2×C4—86no—
413C2×C4—163no—
414C2×C4C2×SD32112yescomplemented
415C2×C4C22×D811yescharacteristic
416C2×C4C22×D813yes—
417C2×C4C22×D813yes—
418C2×C4C22×D813yes—
419C2×C4C22×D811yescharacteristic
420C2×C4C23×D411yescharacteristic
421C8—21no—
422C8—23no—
423C8—23no—
424C8C22×#4111yescharacteristic
425C8#58511yescharacteristic
426C22—23no—
427C22—23no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—83no—
432C22—83no—
433C22—83no—
434C22—1612no—
435C22C22×SD3216yes—
436C22C23×D811yescharacteristic
437C4—21no—
438C4—23no—
439C4—23no—
440C4—42no—
441C4—81no—
442C4—83no—
443C4—81no—
444C4—83no—
445C4—83no—
446C4C22×SD3212yes—
447C4C22×SD3216yes—
448C4D4×#4111yescharacteristic
449C4#358411yescharacteristic
450C2—26no—
451C2—82no—
452C2—163no—
453C2D8×#4111yescharacteristic
454C2C23×SD3211yescharacteristic
455C2#6263511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutator, Frattini, cosocleC2×C8C25
socleC22C23×D8
prime radicalD8×#41C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q16×#41C2semidirect
3. #33959C2semidirect
4. Q8×SD32C2semidirect
5. #33929C2semidirect
6. #33926C2semidirect
7. #33902C2semidirect
8. #33865C2semidirect
9. #33862C2semidirect
10. D4×SD32C2semidirect
11. #33736C2semidirect
12. #33663C2semidirect
13. #33661C2semidirect
14. C16×#41C2semidirect
15. C2×C4×SD32C2semidirect
19. #3395C22semidirect
21. Q8×Q16C22semidirect
25. #3338C22semidirect
29. D4×Q16C22semidirect
31. #3170C22semidirect
36. C2×C4×Q16C22semidirect
38. #2254C22semidirect
39. #2248C22semidirect
40. #2241C22semidirect
41. #2240C22semidirect
42. #2235C22semidirect
43. #2233C22semidirect
45. #2226C22semidirect
46. #2225C22semidirect
47. #2225C22semidirect
48. #2217C22semidirect
49. #2217C22semidirect
51. #2208C22semidirect
52. #2207C22semidirect
53. #2207C22semidirect
55. #2185C22semidirect
56. Q8×C16C22semidirect
57. Q8×C16C22semidirect
62. C4×SD32C22semidirect
63. C4×SD32C22semidirect
64. C4×SD32C22semidirect
65. C4×SD32C22semidirect
66. C4×SD32C22semidirect
67. D4×C16C22semidirect
68. D4×C16C22semidirect
69. #2162C22semidirect
70. C2×#367C22semidirect
71. C2×#358C22semidirect
73. C2×C4×C16C22semidirect
253. SD32#41direct (N × H)
289. #41SD32direct (N × H)
325. Q16C2×#41semidirect
374. C16C2×#41semidirect
388. Q8C2×SD32semidirect
397. D4C2×SD32semidirect
414. C2×C4C2×SD32semidirect

Maximal subgroups

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

subgroupindex
Q16×#412
D8×#412
#339592
Q8×SD322
#339292
#339262
#339022
#338652
#338622
D4×SD322
#337362
#336632
#336612
C16×#412
C2×C4×SD322