Finite Group Databasebeta

← #16016268

Subgroups — #16016268

Order 3328.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718856C211yesmaximal, characteristic
2#13123705C211yesmaximal, characteristic, complemented
3#13116705C212yesmaximal, complemented
4#13116700C212yesmaximal, complemented
5C4×#88069C212yesmaximal, complemented
6C4×#88067C212yesmaximal, complemented
7#13116622C211yesmaximal, characteristic, complemented
8#13116543C211yesmaximal, characteristic
9C2×#11718084C211yesmaximal, characteristic, complemented
10#13116433C211yesmaximal, characteristic, complemented
11C13×#2951C211yesmaximal, characteristic
12C2×C13×#135C2211yescharacteristic
13C13×#520—21no—
14C22×Dic52C2211yescharacteristic
15#11719221—21no—
16C13×#440—22no—
17C4×C13×Q16C2212yes—
18C4×C13×D8C2212yes—
19C2×C13×#107C2211yescharacteristic
20C13×#417—21no—
21C13×#405C2211yescharacteristic
22#11718856C418yescomplemented
23C2×#88069C2212yes—
24C2×#88067C2212yes—
25#11718781—21no—
26#11718767C2211yescharacteristic
27C2×#88052C2211yescharacteristic
28C22×#6090C2211yescharacteristic
29#11718377—22no—
30C4×Dic52C2212yes—
31C4×Dic52C2212yes—
32#11718156C2211yescharacteristic
33#11718156C2211yescharacteristic
34#11718154C2211yescharacteristic
35#11718154C2211yescharacteristic
36#11718084C2212yes—
37#11718084C2212yes—
38#11718082C2212yes—
39#11718081C2212yes—
40#11718053C2212yes—
41C4×#6090C2212yes—
42C2×C13×#41—21no—
43C22×Dic26—21no—
44C13×#135—22no—
45C13×#135C2×C414yes—
46C2×C13×Q16C2311yescharacteristic
47C2×C13×SD16—21no—
48C2×C13×D8C2311yescharacteristic
49C2×C13×M16D411yescharacteristic
50C22×C8×C13C2311yescharacteristic
51C4×Q8×C13—22no—
52C4×D4×C13—22no—
53C4×D4×C13—42no—
54C13×#117—41no—
55C13×#117D411yescharacteristic
56C2×C13×C4⋊C4—21no—
57C2×C13×C4⋊C4—41no—
58#88180—21no—
59C2×Dic52—21no—
60C2×Dic52—22no—
61C2×Dic52C2×C414yescomplemented
62C2×Dic52C2312yes—
63#88141—41no—
64#88141—41no—
65#88131—42no—
66C4×Dic26—22no—
67C4×Dic26—22no—
68C13×#107—21no—
69C13×#107C2312yes—
70C13×#103—22no—
71C13×#102—22no—
72C13×#97D412yes—
73C4×C8×C13C2312yes—
74#88069C2×C418yescomplemented
75#88067C2×C418yescomplemented
76#88058C2311yescharacteristic
77#88058C2311yescharacteristic
78#88057—21no—
79#88056—22no—
80#88056—22no—
81#88052C2×C414yescomplemented
82C2×#6090—22no—
83C2×#6090C2311yescharacteristic
84C2×#6090C2311yescharacteristic
85C2×#6090C2312yes—
86#33455—131nomaximal, Sylow
87C13×#41—24no—
88C13×#41—42no—
89C2×Q8×C13—21no—
90C2×D4×C13—21no—
91C2×D4×C13—41no—
92C22×C4×C13—41no—
93C22×C4×C13—41no—
94C22×C4×C13—42no—
95C22×C4×C13C2×D411yescharacteristic
96C22×Dic13—41no—
97C2×Dic26—24no—
98C2×Dic26—22no—
99C2×Dic26—41no—
100C2×Dic26—42no—
101C13×Q16—21no—
102C13×Q16C22×C412yes—
103C13×SD16—24no—
104C13×D8—21no—
105C13×D8C22×C412yes—
106C13×M16—22no—
107C2×C8×C13C22×C412yes—
108C2×C8×C13C2×D412yes—
109C2×C8×C13C2×D411yescharacteristic
110C2×C8×C13#4112yes—
111C2×C8×C13C2411yescharacteristic
112C13×C4⋊C4—22no—
113C13×C4⋊C4—41no—
114C13×C4⋊C4—42no—
115C13×C4⋊C4D811yescharacteristic
116C13×C4⋊C4D811yescharacteristic
117C13×C22⋊C4—42no—
118C13×C22⋊C4D812yes—
119C42×C13—42no—
120C42×C13C2×D412yes—
121C22⋊Dic13—41no—
122C4⋊Dic13—21no—
123C4⋊Dic13—21no—
124#6101—42no—
125#6101—42no—
126C4×Dic13—42no—
127C4×Dic13—42no—
128C4×Dic13—41no—
129Dic52—24no—
130Dic52—22no—
131Dic52C22×C414yes—
132#6090—21no—
133#6090—21no—
134#6090C22×C414yes—
135C2×#510—131no—
136#2951D1311yescharacteristic
137#2950—131no—
138#2176—132no—
139#2175—132no—
140C4×Q32—132no—
141C4×SD32—132no—
142#2159—131no—
143#2138—131no—
144C2×#358—131no—
145#2108—131no—
146C23×C13—41no—
147C23×C13C2×D811yescharacteristic
148Q8×C13—22no—
149Q8×C13—41no—
150D4×C13—22no—
151D4×C13—44no—
152D4×C13—41no—
153C2×C4×C13—41no—
154C2×C4×C13—44no—
155C2×C4×C13—42no—
156C2×C4×C13—162no—
157C2×C4×C13—161no—
158C2×C4×C13C4×D412yes—
159C2×C4×C13C2×D812yes—
160C2×C4×C13C2×D812yes—
161C2×C4×C13C2×D811yescharacteristic
162C2×C4×C13#13512yes—
163C2×C4×C13C22×D411yescharacteristic
164C2×Dic13—42no—
165C2×Dic13—44no—
166C2×Dic13—41no—
167C2×Dic13—41no—
168Dic26—24no—
169Dic26—42no—
170Dic26—44no—
171C8×C13—22no—
172C8×C13C4×D412yes—
173C8×C13C23×C411yescharacteristic
174C8×C13C2×#4111yescharacteristic
175C2×#135C2×D1311yescharacteristic
176C22×Q16—131no—
177#520—21no—
178#519—261no—
179#510—138no—
180C2×Q32—132no—
181C2×SD32—132no—
182C2×M32—131no—
183C22×C16—131no—
184#441—262no—
185#440—22no—
186C4×Q16—132no—
187C4×Q16—132no—
188C4×Q16C2×D1312yes—
189C4×D8C2×D1312yes—
190#427—131no—
191C2×#107C2×D1311yescharacteristic
192#419—261no—
193#417—21no—
194#405C2×D1311yescharacteristic
195#367—132no—
196#366—132no—
197#358—131no—
198#358—131no—
199#358—132no—
200#358—132no—
201#357—131no—
202#357—131no—
203#346—132no—
204C4×C16—132no—
205C22×C13—21no—
206C22×C13—21no—
207C22×C13—42no—
208C22×C13—81no—
209C22×C13—81no—
210C22×C13C4×D812yes—
211C22×C13C22×D811yescharacteristic
212C4×C13—22no—
213C4×C13—22no—
214C4×C13—42no—
215C4×C13—82no—
216C4×C13C4×D812yes—
217C4×C13C2×C4×D411yescharacteristic
218C4×C13C2×#13511yescharacteristic
219Dic13—44no—
220Dic13—81no—
221Dic13—81no—
222C2×#41—21no—
223C22×Q8—261no—
224#135—22no—
225#135C4×D1314yes—
226C2×Q16—134no—
227C2×Q16—132no—
228C2×Q16—262no—
229C2×Q16—261no—
230C2×Q16C22×D1311yescharacteristic
231C2×SD16—21no—
232C2×D8C22×D1311yescharacteristic
233C2×M16C22⋊D1311yescharacteristic
234C22×C8C22×D1311yescharacteristic
235C4×Q8—22no—
236C4×Q8—262no—
237C4×Q8—262no—
238C4×Q8—521no—
239C4×Q8—521no—
240C4×D4—22no—
241C4×D4—42no—
242#117—41no—
243#117—261no—
244#117—522no—
245#117C22⋊D1311yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—41no—
248Q32—138no—
249SD32—138no—
250M32—134no—
251C2×C16—132no—
252C2×C16—131no—
253C2×C16—131no—
254C2×C16—262no—
255#107—21no—
256#107—131no—
257#107—131no—
258#107C22×D1312yes—
259#106—261no—
260#103—22no—
261#103—262no—
262#103—262no—
263#102—22no—
264#97C22⋊D1312yes—
265C4×C8C22×D1312yes—
266C2×C13—22no—
267C2×C13—82no—
268C2×C13C2×C4×D811yescharacteristic
269C2×C13#341611yescharacteristic
270C2×C13#341711yescharacteristic
271#41—24no—
272#41—42no—
273C2×Q8—21no—
274C2×Q8—264no—
275C2×Q8—262no—
276C2×Q8—521no—
277C2×Q8—522no—
278C2×D4—21no—
279C2×D4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—521no—
284C22×C4C2×C22⋊D1311yescharacteristic
285Q16—21no—
286Q16—134no—
287Q16—264no—
288Q16—262no—
289Q16C2×C4×D1312yes—
290SD16—24no—
291D8—21no—
292D8C2×C4×D1312yes—
293M16—22no—
294C2×C8C2×C4×D1312yes—
295C2×C8#612712yes—
296C2×C8C2×C22⋊D1312yes—
297C2×C8C2×C22⋊D1311yescharacteristic
298C2×C8C23×D1311yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—261no—
303C4⋊C4—261no—
304C4⋊C4—522no—
305C4⋊C4—522no—
306C4⋊C4#610411yescharacteristic
307C4⋊C4#610411yescharacteristic
308C22⋊C4—42no—
309C22⋊C4—521no—
310C22⋊C4#610412yes—
311C42—42no—
312C42—521no—
313C42—522no—
314C42—522no—
315C42C2×C22⋊D1312yes—
316C16—134no—
317C16—261no—
318C16—261no—
319C13#3345511yescharacteristic, complemented, Sylow
320C23—41no—
321C23C2×#610411yescharacteristic
322Q8—22no—
323Q8—41no—
324Q8—264no—
325Q8—524no—
326Q8—522no—
327D4—22no—
328D4—44no—
329D4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—162no—
334C2×C4—161no—
335C2×C4—524no—
336C2×C4—522no—
337C2×C4—521no—
338C2×C4—521no—
339C2×C4C4×C22⋊D1312yes—
340C2×C4C2×#610412yes—
341C2×C4C2×#610412yes—
342C2×C4C2×#610411yescharacteristic
343C2×C4#8820412yes—
344C2×C4C22×C22⋊D1311yescharacteristic
345C8—22no—
346C8C4×C22⋊D1312yes—
347C8C22×C4×D1311yescharacteristic
348C8C2×#612711yescharacteristic
349C22—21no—
350C22—21no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22C4×#610412yes—
355C22C22×#610411yescharacteristic
356C4—22no—
357C4—22no—
358C4—42no—
359C4—82no—
360C4—524no—
361C4—1041no—
362C4—1041no—
363C4C4×#610412yes—
364C4C2×C4×C22⋊D1311yescharacteristic
365C4C2×#8820411yescharacteristic
366C2—22no—
367C2—82no—
368C2C2×C4×#610411yescharacteristic
369C2#1312798311yescharacteristic
370C2#1312798711yescharacteristic
subgroupG/H
centerC22C22×#6104
commutatorC8×C13C23×C4
FrattiniC2×C8C23×D13
FittingC13×#2951C2
socleC22×C13C22×D8
prime radicalC2×C13×#135C22
cosocleC2×C8×C13C24
hypercenter#2951D13
hypocenterC13#33455
LCSG ▷ C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. #13123705C2semidirect
3. #13116705C2semidirect
4. #13116700C2semidirect
5. C4×#88069C2semidirect
6. C4×#88067C2semidirect
7. #13116622C2semidirect
9. C2×#11718084C2semidirect
10. #13116433C2semidirect
22. #11718856C4semidirect
61. C2×Dic52C2×C4semidirect
74. #88069C2×C4semidirect
75. #88067C2×C4semidirect
81. #88052C2×C4semidirect
319. C13#33455semidirect

Maximal subgroups

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

subgroupindex
C2×#117188562
#131237052
#131167052
#131167002
C4×#880692
C4×#880672
#131166222
#131165432
C2×#117180842
#131164332
C13×#29512
#3345513