Finite Group Databasebeta

← #16776775

Subgroups — #16776775

Order 3328.

10836 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719648C212yesmaximal, complemented
2#13129624C2116yesmaximal, complemented
3#13129617C2116yesmaximal, complemented
4#13129610C2116yesmaximal, complemented
5C2×#11719453C2112yesmaximal, complemented
6C2×C13×#568C211yesmaximal, characteristic, complemented
7C22×C13×#41C2211yescharacteristic
8#11719648—28no—
9#11719648—28no—
10#11719648C22116yescomplemented
11#11719648C22116yescomplemented
12C2×#88257C2218yescomplemented
13C2×#88256C2218yescomplemented
14C2×#41×D13—28no—
15C2×#88254—22no—
16C2×#88251—26no—
17C22×#6127C22112yescomplemented
18C13×#568C22116yescomplemented
19C2×C13×#131C2218yescomplemented
20C22×C13×M16C2216yescomplemented
21#11719458C22164yescomplemented
22#11719457C22164yescomplemented
23#11719456C221128yescomplemented
24#11719453C22116yescomplemented
25#11719453C22196yescomplemented
26#11719453C22148yescomplemented
27C2×#88163C22148yescomplemented
28C2×#88162C22148yescomplemented
29C2×#88158C22148yescomplemented
30C2×C13×#41D418yes—
31C2×C13×#41D414yes—
32C2×C13×#41C2314yes—
33C2×C13×#41C2318yes—
34C22×Q8×C13D411yescharacteristic
35C22×D4×C13D413yes—
36C23×C4×C13C2313yes—
37C22×C22⋊D13—212no—
38#88257—232no—
39#88257C23164yescomplemented
40#88256—232no—
41#88256C23164yescomplemented
42#41×D13—264no—
43#41×D13—264no—
44#88254—216no—
45#88254—216no—
46C2×#6131—28no—
47C2×Q8×D13—28no—
48#88251—248no—
49#88251—248no—
50C2×#6129—224no—
51C2×D4×D13—224no—
52C2×#6127—212no—
53C2×#6127—212no—
54C2×#6127—224no—
55C2×#6127—224no—
56C2×#6127C23148yes—
57C2×#6127C2318yes—
58C2×#6127C23148yescomplemented
59C2×#6127C23124yescomplemented
60C22×D52C23112yescomplemented
61C22×C4×D13—212no—
62C22×Dic26C23112yescomplemented
63C13×#131C23164yescomplemented
64C2×C13×M16C2314yes—
65C2×C13×M16C23148yescomplemented
66C2×C13×M16C23112yescomplemented
67C22×C8×C13C23112yescomplemented
68#88163C23164yescomplemented
69#88163C231192yescomplemented
70#88162C23164yescomplemented
71#88162C231192yescomplemented
72C2×Dic52C23148yescomplemented
73#88158C23164yescomplemented
74#88158C231192yescomplemented
75C2×D104C23148yescomplemented
76C2×#6095C23196yescomplemented
77#63090—131nomaximal, Sylow
78C24×C13C2×D413yes—
79C23×D13—212no—
80C13×#41C2×D4116yes—
81C13×#41C2×D4132yes—
82C13×#41C24116yes—
83C2×Q8×C13C2×D418yes—
84C2×Q8×C13C2×D414yes—
85C2×D4×C13C2×D4124yes—
86C2×D4×C13C2×D4112yes—
87C22×C4×C13C2×D4112yes—
88C22×C4×C13C2×D413yes—
89C22×C4×C13C2×D416yes—
90C22×C4×C13C2416yes—
91C22×C4×C13C2411yescharacteristic
92C22×C4×C13C24112yes—
93C2×C22⋊D13—28no—
94C2×C22⋊D13—248no—
95C2×C22⋊D13—224no—
96C2×C22⋊D13—248no—
97C22×Dic13—212no—
98#6131—264no—
99Q8×D13—264no—
100#6129—2192no—
101D4×D13—2192no—
102#6127—28no—
103#6127—28no—
104#6127—248no—
105#6127—248no—
106#6127—248no—
107#6127—248no—
108#6127—248no—
109#6127C24116yes—
110#6127C24196yes—
111#6127C24116yes—
112C2×D52—212no—
113C2×D52—224no—
114C2×D52C24148yes—
115C2×D52C2418yes—
116C2×C4×D13—28no—
117C2×C4×D13—248no—
118C2×C4×D13—224no—
119C2×C4×D13—248no—
120C2×Dic26—212no—
121C2×Dic26—224no—
122C2×Dic26C24148yes—
123C2×Dic26C2418yes—
124C13×M16C24116yes—
125C13×M16C24148yes—
126C2×C8×C13C24148yes—
127C2×C8×C13C2418yes—
128Dic52C24164yes—
129D104C24164yes—
130#6095C241128yes—
131C2×#585—132no—
132#3586—1316no—
133#3585—1316no—
134#3584—1316no—
135C2×#575—1312no—
136C2×#568D1311yescharacteristic, complemented
137C23×C13—83no—
138C23×C13C22×D4112yes—
139C23×C13C22×D411yescharacteristic
140C23×C13C22×D416yes—
141C22×D13—28no—
142C22×D13—248no—
143C22×D13—824no—
144Q8×C13C22×D4116yes—
145D4×C13C22×D4148yes—
146C2×C4×C13C22×D4112yes—
147C2×C4×C13C22×D4124yes—
148C2×C4×C13C22×D412yes—
149C2×C4×C13C22×D4112yes—
150C2×C4×C13C22×D411yescharacteristic
151C2×C4×C13C22×D416yes—
152C2×C4×C13C2511yescharacteristic
153C2×C4×C13C25112yes—
154C2×C4×C13C2512yes—
155C22⋊D13—216no—
156C22⋊D13—216no—
157C22⋊D13—296no—
158C2×Dic13—28no—
159C2×Dic13—248no—
160D52—28no—
161D52—248no—
162D52C25116yes—
163C4×D13—216no—
164C4×D13—216no—
165C4×D13—296no—
166Dic26—28no—
167Dic26—248no—
168Dic26C25116yes—
169C8×C13C25116yes—
170#585—1316no—
171#585—1316no—
172#585—268no—
173#585—268no—
174C2×2−1+4—138no—
175C2×2−1+4—262no—
176C2×H(5,𝔽2)—138no—
177C2×H(5,𝔽2)—266no—
178C22×#41—1312no—
179C22×#41—268no—
180C22×#41C2×D1311yescharacteristic
181#578—1364no—
182#577—13128no—
183#576—1364no—
184#575—1316no—
185#575—1348no—
186#575—1396no—
187C2×#137—1348no—
188C2×AGL(1,ℤ/8)—1348no—
189C2×#135—1348no—
190#568C2×D13116yescomplemented
191C2×#131C2×D1318yescomplemented
192C22×M16C2×D1316yescomplemented
193C22×C13—412no—
194C22×C13—41no—
195C22×C13—46no—
196C22×C13C23×D412yes—
197C22×C13C23×D4112yes—
198C22×C13C23×D411yescharacteristic
199C2×D13—216no—
200C2×D13—416no—
201C2×D13—848no—
202C4×C13C23×D412yes—
203C4×C13C23×D4112yes—
204C4×C13C23×D411yescharacteristic
205C4×C13C2611yescharacteristic
206Dic13—216no—
2072−1+4—1364no—
2082−1+4—2632no—
2092−1+4—2616no—
2102−1+4—2616no—
211H(5,𝔽2)—1364no—
212H(5,𝔽2)—2632no—
213H(5,𝔽2)—2648no—
214H(5,𝔽2)—2648no—
215C2×#41—1348no—
216C2×#41—1348no—
217C2×#41—138no—
218C2×#41—1324no—
219C2×#41—2664no—
220C2×#41—2664no—
221C2×#41—268no—
222C2×#41—2624no—
223C2×#41—2612no—
224C2×#41—2612no—
225C2×#41—2624no—
226C2×#41—2624no—
227C2×#41D5218yes—
228C2×#41D5214yes—
229C2×#41C22×D1314yes—
230C2×#41C22×D1318yes—
231C22×Q8—1312no—
232C22×Q8—268no—
233C22×Q8D5211yescharacteristic
234C22×D4—1312no—
235C22×D4—2612no—
236C22×D4—2624no—
237C22×D4D5213yes—
238C23×C4—2612no—
239C23×C4C22×D1313yes—
240#137—1364no—
241#137—13192no—
242AGL(1,ℤ/8)—1364no—
243AGL(1,ℤ/8)—13192no—
244#135—1364no—
245#135—13192no—
246C2×Q16—1348no—
247C2×SD16—1396no—
248C2×D8—1348no—
249#131C22×D13164yescomplemented
250C2×M16C22×D1314yes—
251C2×M16C22×D13112yescomplemented
252C2×M16C22×D13148yescomplemented
253C22×C8C22×D13112yescomplemented
254C2×C13—22no—
255C2×C13—212no—
256C2×C13—21no—
257C2×C13C24×D411yescharacteristic
258D13—416no—
259C24—2612no—
260C24C2×D5213yes—
261#41—1316no—
262#41—1396no—
263#41—1316no—
264#41—2664no—
265#41—26192no—
266#41—2648no—
267#41—268no—
268#41—268no—
269#41—2648no—
270#41—2648no—
271#41—2648no—
272#41—2648no—
273#41C2×D52116yes—
274#41C2×D52132yes—
275#41C23×D13116yes—
276C2×Q8—1348no—
277C2×Q8—138no—
278C2×Q8—2664no—
279C2×Q8—2612no—
280C2×Q8—2624no—
281C2×Q8C2×D5218yes—
282C2×Q8C2×D5214yes—
283C2×D4—1348no—
284C2×D4—138no—
285C2×D4—2624no—
286C2×D4—268no—
287C2×D4—2648no—
288C2×D4—2648no—
289C2×D4—26192no—
290C2×D4—2612no—
291C2×D4—2624no—
292C2×D4C2×D52124yes—
293C2×D4C2×D52112yes—
294C22×C4—2612no—
295C22×C4—268no—
296C22×C4—2624no—
297C22×C4—2648no—
298C22×C4—2648no—
299C22×C4C2×D52112yes—
300C22×C4C2×D5213yes—
301C22×C4C2×D5216yes—
302C22×C4C23×D1316yes—
303C22×C4C23×D1311yescharacteristic
304C22×C4C23×D13112yes—
305Q16—1364no—
306SD16—13128no—
307D8—1364no—
308M16C23×D13116yes—
309M16C23×D13148yes—
310C2×C8C23×D1318yes—
311C2×C8C23×D13148yes—
312C13#6309011yescharacteristic, complemented, Sylow
313C23—83no—
314C23—268no—
315C23—2648no—
316C23—10424no—
317C23C22×D52112yes—
318C23C22×D5211yescharacteristic
319C23C22×D5216yes—
320Q8—1316no—
321Q8—268no—
322Q8—2648no—
323Q8C22×D52116yes—
324D4—1316no—
325D4—2616no—
326D4—2696no—
327D4—2616no—
328D4—268no—
329D4—2648no—
330D4C22×D52148yes—
331C2×C4—268no—
332C2×C4—2648no—
333C2×C4—2616no—
334C2×C4—2696no—
335C2×C4—2616no—
336C2×C4C22×D52112yes—
337C2×C4C22×D52124yes—
338C2×C4C22×D5211yescharacteristic
339C2×C4C22×D5216yes—
340C2×C4C22×D5212yes—
341C2×C4C22×D52112yes—
342C2×C4C24×D13112yes—
343C2×C4C24×D1312yes—
344C2×C4C24×D1311yescharacteristic
345C8C24×D13116yes—
346C22—41no—
347C22—412no—
348C22—46no—
349C22—2616no—
350C22—5216no—
351C22—10448no—
352C22C23×D5212yes—
353C22C23×D52112yes—
354C22C23×D5211yescharacteristic
355C4—2616no—
356C4C23×D5212yes—
357C4C23×D52112yes—
358C4C23×D5211yescharacteristic
359C4C25×D1311yescharacteristic
360C2—22no—
361C2—21no—
362C2—212no—
363C2—5216no—
364C2C24×D5211yescharacteristic
subgroupG/H
centerC4C23×D52
commutator, cosocleC4×C13C26
FrattiniC4C25×D13
FittingC2×C13×#568C2
socleC2×C13C24×D4
hypercenterC2×#568D13
hypocenterC13#63090
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C4 ◁ C22×#41 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
1. C2×#11719648C2semidirect
2. #13129624C2semidirect
3. #13129617C2semidirect
4. #13129610C2semidirect
5. C2×#11719453C2semidirect
6. C2×C13×#568C2semidirect
10. #11719648C22semidirect
11. #11719648C22semidirect
12. C2×#88257C22semidirect
13. C2×#88256C22semidirect
17. C22×#6127C22semidirect
18. C13×#568C22semidirect
19. C2×C13×#131C22semidirect
20. C22×C13×M16C22semidirect
21. #11719458C22semidirect
22. #11719457C22semidirect
23. #11719456C22semidirect
24. #11719453C22semidirect
25. #11719453C22semidirect
26. #11719453C22semidirect
27. C2×#88163C22semidirect
28. C2×#88162C22semidirect
29. C2×#88158C22semidirect
39. #88257C23semidirect
41. #88256C23semidirect
58. C2×#6127C23semidirect
59. C2×#6127C23semidirect
60. C22×D52C23semidirect
62. C22×Dic26C23semidirect
63. C13×#131C23semidirect
65. C2×C13×M16C23semidirect
66. C2×C13×M16C23semidirect
67. C22×C8×C13C23semidirect
68. #88163C23semidirect
69. #88163C23semidirect
70. #88162C23semidirect
71. #88162C23semidirect
72. C2×Dic52C23semidirect
73. #88158C23semidirect
74. #88158C23semidirect
75. C2×D104C23semidirect
76. C2×#6095C23semidirect
136. C2×#568D13semidirect
190. #568C2×D13semidirect
191. C2×#131C2×D13semidirect
192. C22×M16C2×D13semidirect
249. #131C22×D13semidirect
251. C2×M16C22×D13semidirect
252. C2×M16C22×D13semidirect
253. C22×C8C22×D13semidirect
312. C13#63090semidirect

Maximal subgroups

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

subgroupindex
C2×#117196482
#131296242
#131296172
#131296102
C2×#117194532
C2×C13×#5682
#6309013