Finite Group Databasebeta

← #35228213

Subgroups — #35228213

Order 20864.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104783C211yesmaximal, characteristic, complemented
2C22×#14579700C211yesmaximal, characteristic, complemented
3Q8×C22⋊D163C211yesmaximal, characteristic, complemented
4#23782327C211yesmaximal, characteristic, complemented
5#23782326C211yesmaximal, characteristic, complemented
6#23782318C213yesmaximal, complemented
7#23782317C213yesmaximal, complemented
8#23782311C213yesmaximal, complemented
9#23782306C213yesmaximal, complemented
10#23782245C213yesmaximal, complemented
11#23782240C213yesmaximal, complemented
12#23782174C213yesmaximal, complemented
13C22⋊Q8×D163C213yesmaximal, complemented
14#23782147C211yesmaximal, characteristic, complemented
15D4×Q8×C163C211yesmaximal, characteristic, complemented
16#19104783—28no—
17C2×#14579700—28no—
18C2×#14579700C2213yes—
19C2×#14579700C2213yes—
20C2×#14579700C2211yescharacteristic
21C2×#14579700C2211yescharacteristic
22C2×#14579700C2213yescomplemented
23C2×#14579699C2213yescomplemented
24C2×#14579699C2213yescomplemented
25C2×Q8×D163C2213yescomplemented
26C2×Q8×D163C2211yescharacteristic, complemented
27C22×D652C2213yes—
28C22×C4×D163C2213yes—
29#19104733C2213yescomplemented
30#19104724C2213yescomplemented
31#19104723C2213yescomplemented
32#19104720C2213yescomplemented
33#19104720C2216yescomplemented
34#19104720C2216yescomplemented
35#19104717C2213yescomplemented
36#19104715C2213yescomplemented
37#19104714C2213yes—
38#19104713C2216yescomplemented
39#19104713C2213yescomplemented
40#19104712C2213yes—
41#19104711C2213yescomplemented
42#19104711C2213yescomplemented
43#19104710C2216yescomplemented
44#19104710C2213yescomplemented
45#19104709C2213yescomplemented
46#19104708C2213yescomplemented
47#19104708C2216yescomplemented
48#19104707C2211yescharacteristic
49#19104705C2213yes—
50#19104703C2213yes—
51Q8×Dic163C2211yescharacteristic, complemented
52C4×D652C2213yescomplemented
53C4×D652C2213yescomplemented
54#19104697C2213yescomplemented
55C4×C22⋊D163C2213yes—
56C4×C22⋊D163C2213yescomplemented
57#19104692C2213yescomplemented
58C4⋊C4×D163C2213yescomplemented
59C4⋊C4×D163C2216yescomplemented
60C22⋊C4×D163C2216yes—
61C22×Q8×C163C2211yescharacteristic
62C22×Q8×C163C2211yescharacteristic, complemented
63C4⋊Q8×C163C2213yescomplemented
64C22⋊Q8×C163C2213yes—
65C22⋊Q8×C163C2213yescomplemented
66C4×Q8×C163C2211yescharacteristic, complemented
67C4×D4×C163C2213yes—
68C23×D163C2313yes—
69#14579700—212no—
70#14579700—212no—
71#14579700—212no—
72#14579700—24no—
73#14579700D418yescomplemented
74#14579699—212no—
75#14579699—212no—
76C2×Dic326C2313yes—
77C2×Dic326C2313yes—
78Q8×D163—212no—
79Q8×D163—24no—
80C2×D652—212no—
81C2×D652C2316yes—
82C2×D652C2313yes—
83C2×D652C2313yes—
84C2×D652C2313yes—
85C2×D652C2313yes—
86C2×C22⋊D163C2313yes—
87C2×C22⋊D163C2311yescharacteristic
88C2×C22⋊D163C2311yescharacteristic
89C2×C22⋊D163C2313yes—
90C22×Dic163C2311yescharacteristic
91C2×C4×D163—212no—
92C2×C4×D163C2313yes—
93C2×C4×D163C2313yes—
94C2×C4×D163C2313yes—
95C2×C4×D163C2316yes—
96C2×C4×D163C2313yes—
97C2×C4×D163C2316yes—
98C2×C4×D163C2313yes—
99C4⋊Dic163C2313yes—
100C4⋊Dic163C2313yes—
101Dic163⋊C4C2313yes—
102Dic163⋊C4C2313yes—
103Dic163⋊C4C2316yes—
104Dic163⋊C4C2311yescharacteristic
105Dic163⋊C4C2313yes—
106C22⋊Dic163C2313yes—
107C22⋊Dic163C2311yescharacteristic
108(C2×D163)⋊C4C2316yes—
109(C2×D163)⋊C4C2316yes—
110(C2×D163)⋊C4C2313yes—
111(C2×D163)⋊C4C2313yes—
112(C2×D163)⋊C4C2313yes—
113(C2×D163)⋊C4C2313yes—
114(C2×D163)⋊C4C2311yescharacteristic
115(C2×D163)⋊C4C2313yes—
116C4×Dic163C2313yes—
117C2×Q8×C163—24no—
118C2×Q8×C163—24no—
119C2×Q8×C163C2313yes—
120C2×Q8×C163C2313yes—
121C2×Q8×C163C2311yescharacteristic
122C2×D4×C163C2311yescharacteristic
123C22×C4×C163C2313yes—
124C22×C4×C163C2313yes—
125C4⋊C4×C163C2313yes—
126C4⋊C4×C163C2313yes—
127C4⋊C4×C163C2316yes—
128C22⋊C4×C163C2313yes—
129C22⋊C4×C163C2313yes—
130C42×C163C2313yes—
131C22×D163—26no—
132C22×D163—26no—
133C22×D163—43no—
134C22×D163C2413yes—
135C22×D163C2411yescharacteristic
136C22×D163C2413yes—
137Dic326—26no—
138Dic326—26no—
139D652—212no—
140D652—26no—
141D652—26no—
142D652—43no—
143D652C2×D4112yes—
144C22⋊D163—22no—
145C22⋊D163—26no—
146C22⋊D163—22no—
147C22⋊D163—43no—
148C2×Dic163—22no—
149C2×Dic163C2411yescharacteristic
150C2×Dic163C2413yes—
151C2×Dic163C2413yes—
152C2×Dic163C2411yescharacteristic
153C2×Dic163C2411yescharacteristic
154C4×D163—26no—
155C4×D163—212no—
156C4×D163—26no—
157C4×D163—26no—
158C4×D163—26no—
159C4×D163—46no—
160C4×D163C2×D4112yes—
161C23×C163C2411yescharacteristic
162C23×C163C2411yescharacteristic
163Q8×C163—26no—
164Q8×C163—26no—
165Q8×C163C2×D414yes—
166D4×C163—22no—
167C2×C4×C163—26no—
168C2×C4×C163—26no—
169C2×C4×C163C2411yescharacteristic
170C2×C4×C163C2413yes—
171C2×C4×C163C2413yes—
172C2×C4×C163C2413yes—
173C2×C4×C163C2413yes—
174C22×C163—21no—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163—21no—
178C22×C163—21no—
179C22×C1632−1+412yescomplemented
180C22×C163C2511yescharacteristic
181C2×D163—23no—
182C2×D163—23no—
183C2×D163—21no—
184C2×D163—23no—
185C2×D163—21no—
186C2×D163—23no—
187C2×D163—23no—
188C2×D163—21no—
189C2×D163—412no—
190C2×D163C22×D416yes—
191C4×C163—23no—
192C4×C163—23no—
193C4×C163—21no—
194C4×C163—23no—
195C4×C163C22×D416yes—
196Dic163—21no—
197Dic163—23no—
198Dic163—21no—
199Dic163—23no—
200Dic163C22×D412yes—
201C2×C163—22no—
202C2×C163—41no—
203C2×C163C23×D411yescharacteristic
204C2×C163C2×2−1+411yescharacteristic
205C2×C163#58511yescharacteristic
206D163—26no—
207D163—41no—
208D163—43no—
209C163#348911yescharacteristic, complemented, Sylow
210#3489—1631nomaximal, Sylow
211C2×2−1+4—1631no—
212C22×#41—1631no—
213D4×Q8—1631no—
214D4×Q8D16311yescharacteristic, complemented
215#548—1631no—
216#548—1631no—
217#547—1633no—
218#547—1633no—
219#546—1633no—
220#546—1633no—
221#537—1633no—
222#536—1633no—
223#525—1633no—
224C2×C22⋊Q8—1633no—
225#520—1631no—
2262−1+4—3268no—
227C2×#41—3268no—
228C2×#41—1633no—
229C2×#41—1633no—
230C2×#41—1631no—
231C2×#41—1633no—
232C2×#41—1631no—
233C2×#41—1633no—
234C2×#41—1633no—
235C22×Q8—1633no—
236C22×Q8—1631no—
237C22×Q8C2×D16311yescharacteristic
238C22×Q8C2×D16311yescharacteristic, complemented
239C22×D4—1633no—
240C23×C4—1633no—
241C4⋊Q8—1633no—
242C4⋊Q8C2×D16313yescomplemented
243#124—1633no—
244#124—1633no—
245#123—1633no—
246#123—1636no—
247#123—1636no—
248#123—1633no—
249C22⋊Q8—1633no—
250C22⋊Q8—1633no—
251C22⋊Q8—1633no—
252C22⋊Q8—1636no—
253C22⋊Q8—1633no—
254C22⋊Q8—1633no—
255C22⋊Q8—1633no—
256C22⋊Q8C2×D16313yes—
257C22⋊Q8C2×D16313yescomplemented
258#121—1631no—
259#121—1633no—
260#121—1636no—
261#121—1633no—
262#121—1633no—
263#121—1636no—
264#120—1633no—
265#120—1633no—
266C4×Q8—1631no—
267C4×Q8C2×D16311yescharacteristic, complemented
268C4×D4—1633no—
269C4×D4—1633no—
270C4×D4—1633no—
271C4×D4—1633no—
272C4×D4—1633no—
273C4×D4C2×D16313yes—
274#117—1633no—
275C2×C4⋊C4—1633no—
276C2×C4⋊C4—1636no—
277C2×C22⋊C4—1636no—
278C24—1633no—
279#41—32612no—
280#41—32612no—
281#41—3264no—
282#41—32612no—
283#41—32612no—
284#41—32612no—
285#41—1638no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—3264no—
289C2×Q8—32612no—
290C2×Q8—1633no—
291C2×Q8—1633no—
292C2×Q8C22×D16313yes—
293C2×Q8C22×D16313yes—
294C2×Q8C22×D16311yescharacteristic
295C2×D4—32612no—
296C2×D4—1631no—
297C2×D4—1633no—
298C2×D4—1631no—
299C2×D4—1633no—
300C2×D4—1633no—
301C2×D4—1636no—
302C2×D4—1633no—
303C2×D4—1633no—
304C2×D4—1633no—
305C2×D4C22×D16311yescharacteristic
306C22×C4—32612no—
307C22×C4—1631no—
308C22×C4—1636no—
309C22×C4—1633no—
310C22×C4—1633no—
311C22×C4—1633no—
312C22×C4—1633no—
313C22×C4—1636no—
314C22×C4—1633no—
315C22×C4C22×D16313yes—
316C22×C4C22×D16313yes—
317C4⋊C4—1633no—
318C4⋊C4—1633no—
319C4⋊C4—1633no—
320C4⋊C4—1633no—
321C4⋊C4—1636no—
322C4⋊C4—1631no—
323C4⋊C4—1633no—
324C4⋊C4C22×D16313yes—
325C4⋊C4C22×D16316yes—
326C4⋊C4C22×D16313yes—
327C22⋊C4—1633no—
328C22⋊C4—1631no—
329C22⋊C4—1636no—
330C22⋊C4—1636no—
331C22⋊C4—1633no—
332C22⋊C4—1633no—
333C22⋊C4—1631no—
334C22⋊C4—1633no—
335C22⋊C4—1633no—
336C22⋊C4—1633no—
337C22⋊C4C22×D16313yes—
338C22⋊C4C22×D16313yes—
339C42—1633no—
340C42C22×D16313yes—
341C23—3266no—
342C23—3266no—
343C23—6523no—
344C23—1633no—
345C23—1631no—
346C23—1633no—
347C23C23×D16311yescharacteristic
348C23C23×D16311yescharacteristic
349Q8—26no—
350Q8—26no—
351Q8—3266no—
352Q8—3266no—
353Q8D4×D16314yes—
354D4—22no—
355D4—32612no—
356D4—3266no—
357D4—3266no—
358D4—3262no—
359D4—3262no—
360D4—3266no—
361D4—6523no—
362D4—6523no—
363D4—16312no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—3266no—
367C2×C4—32612no—
368C2×C4—3266no—
369C2×C4—3262no—
370C2×C4—3266no—
371C2×C4—3266no—
372C2×C4—6526no—
373C2×C4—1631no—
374C2×C4—1633no—
375C2×C4—1633no—
376C2×C4—1631no—
377C2×C4—1631no—
378C2×C4—16312no—
379C2×C4C23×D16311yescharacteristic
380C2×C4C23×D16313yes—
381C2×C4C23×D16313yes—
382C2×C4C23×D16313yes—
383C2×C4C23×D16313yes—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—3263no—
390C22—3263no—
391C22—3261no—
392C22—3263no—
393C22—3261no—
394C22—3263no—
395C22—3261no—
396C22—3263no—
397C22—65212no—
398C22—1636no—
399C22#1910478312yescomplemented
400C22C24×D16311yescharacteristic
401C4—23no—
402C4—23no—
403C4—21no—
404C4—23no—
405C4—3261no—
406C4—3263no—
407C4—3261no—
408C4—3263no—
409C4—1632no—
410C4C2×D4×D16316yes—
411C2—22no—
412C2—41no—
413C2—3266no—
414C2—6523no—
415C2—6521no—
416C2C22×D4×D16311yescharacteristic
417C2C2×#1910478311yescharacteristic
418C2#2378250211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
FittingD4×Q8×C163C2
hypercenterD4×Q8D163
hypocenterC163#3489
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19104783C2semidirect
2. C22×#14579700C2semidirect
3. Q8×C22⋊D163C2semidirect
4. #23782327C2semidirect
5. #23782326C2semidirect
6. #23782318C2semidirect
7. #23782317C2semidirect
8. #23782311C2semidirect
9. #23782306C2semidirect
10. #23782245C2semidirect
11. #23782240C2semidirect
12. #23782174C2semidirect
13. C22⋊Q8×D163C2semidirect
14. #23782147C2semidirect
15. D4×Q8×C163C2semidirect
22. C2×#14579700C22semidirect
23. C2×#14579699C22semidirect
24. C2×#14579699C22semidirect
25. C2×Q8×D163C22semidirect
26. C2×Q8×D163C22semidirect
29. #19104733C22semidirect
30. #19104724C22semidirect
31. #19104723C22semidirect
32. #19104720C22semidirect
33. #19104720C22semidirect
34. #19104720C22semidirect
35. #19104717C22semidirect
36. #19104715C22semidirect
38. #19104713C22semidirect
39. #19104713C22semidirect
41. #19104711C22semidirect
42. #19104711C22semidirect
43. #19104710C22semidirect
44. #19104710C22semidirect
45. #19104709C22semidirect
46. #19104708C22semidirect
47. #19104708C22semidirect
51. Q8×Dic163C22semidirect
52. C4×D652C22semidirect
53. C4×D652C22semidirect
54. #19104697C22semidirect
56. C4×C22⋊D163C22semidirect
57. #19104692C22semidirect
58. C4⋊C4×D163C22semidirect
59. C4⋊C4×D163C22semidirect
62. C22×Q8×C163C22semidirect
63. C4⋊Q8×C163C22semidirect
65. C22⋊Q8×C163C22semidirect
66. C4×Q8×C163C22semidirect
73. #14579700D4semidirect
179. C22×C1632−1+4semidirect
209. C163#3489semidirect
214. D4×Q8D163semidirect
238. C22×Q8C2×D163semidirect
242. C4⋊Q8C2×D163semidirect
257. C22⋊Q8C2×D163semidirect
267. C4×Q8C2×D163semidirect
399. C22#19104783semidirect

Maximal subgroups

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

subgroupindex
C2×#191047832
C22×#145797002
Q8×C22⋊D1632
#237823272
#237823262
#237823182
#237823172
#237823112
#237823062
#237822452
#237822402
#237821742
C22⋊Q8×D1632
#237821472
D4×Q8×C1632
#3489163