Finite Group Databasebeta

← #16785106

Subgroups — #16785106

Order 3328.

598 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13130563C211yesmaximal, characteristic
2#13130012C211yesmaximal, characteristic
3#13129709C214yesmaximal
4#13123183C211yesmaximal, characteristic
5C2×C13×C4⋊Q8C411yescharacteristic, complemented
6C2×#88200—21no—
7C2×C42×D13C2211yescharacteristic
8C2×(C2×#261)⋊C4—21no—
9C2×(C2×#261)⋊C4—21no—
10C2×(C2×#261)⋊C4C2212yes—
11C2×#88111C2212yes—
12C2×#88110—21no—
13#11718891—21no—
14#11718648—21no—
15#11718648C2212yes—
16#11718634—21no—
17#11718634C412yescomplemented
18#11718596C411yescharacteristic, complemented
19#11718290—21no—
20C23×#261—22no—
21C22×Q8×C13—21no—
22C22×C4×D13D411yescharacteristic
23C22×C4×D13D411yescharacteristic
24C22×C4×D13C2311yescharacteristic
25C22×#1057—22no—
26C13×C4⋊Q8—42no—
27C2×C13×C4⋊C4—21no—
28C2×C13×C4⋊C4C2×C412yes—
29C2×C42×C13C2×C411yescharacteristic
30#88200—44no—
31C2×#6103—22no—
32C2×C4⋊Dic13—21no—
33C2×C4⋊Dic13—21no—
34C2×C4⋊Dic13C2×C412yes—
35C2×#6101—22no—
36C2×C4×Dic13C2×C411yescharacteristic
37C2×C4×Dic13D411yescharacteristic
38C2×C4×Dic13D411yescharacteristic
39C42×D13—42no—
40(C2×#261)⋊C4—24no—
41(C2×#261)⋊C4—24no—
42(C2×#261)⋊C4—42no—
43#88111—22no—
44#88111Q814yescomplemented
45#88110—42no—
46#88071—22no—
47#88071—22no—
48#9888—131nomaximal, Sylow
49C23×D13C2×D411yescharacteristic
50C22×#261—24no—
51C22×#261—42no—
52C22×#261—42no—
53C22×#261—42no—
54C22×#261—42no—
55C22×#261—42no—
56C22×#261—42no—
57C2×Q8×C13—42no—
58C2×Q8×C13—42no—
59C2×Q8×C13—42no—
60C22×C4×C13—22no—
61C22×C4×C13C22⋊C411yescharacteristic
62C22×C4×C13C22⋊C411yescharacteristic
63C22×C4×C13C22×C411yescharacteristic
64C22×Dic13—22no—
65C22×Dic13C22⋊C411yescharacteristic
66C22×Dic13C22⋊C411yescharacteristic
67C22×Dic13C2×D411yescharacteristic
68C2×C4×D13—22no—
69C2×C4×D13—22no—
70C2×C4×D13—22no—
71C2×C4×D13—22no—
72C2×C4×D13—41no—
73C2×C4×D13—41no—
74C2×C4×D13—41no—
75C2×C4×D13—41no—
76C2×C4×D13C2×Q812yes—
77C2×C4×D13#4112yes—
78C2×#1057—24no—
79C2×#1057—42no—
80C2×#1057—42no—
81C13×C4⋊C4—42no—
82C13×C4⋊C4—42no—
83C42×C13—22no—
84#6103—44no—
85C4⋊Dic13—42no—
86C4⋊Dic13—42no—
87C4⋊Dic13—42no—
88#6101—44no—
89C4×Dic13—22no—
90C4×Dic13—41no—
91C4×Dic13—41no—
92#2857—131no—
93#1749—131no—
94#1434—131no—
95#1294—134no—
96C23×C13C2×C22⋊C411yescharacteristic
97C22×D13—21no—
98C22×D13—21no—
99C22×D13—21no—
100C22×D13—21no—
101C22×D13#12411yescharacteristic
102C22×D13C4⋊Q811yescharacteristic
103C22×D13AGL(1,ℤ/8)12yes—
104C22×D13#13712yes—
105C2×#261—44no—
106C2×#261—44no—
107C2×#261—44no—
108C2×#261—82no—
109C2×#261—82no—
110C2×#261—82no—
111C2×#261—82no—
112Q8×C13—84no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13#9912yes—
123C2×C4×C13#10412yes—
124C2×C4×C13C4×Q812yes—
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—42no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13C22⋊Q812yes—
137C4×D13—42no—
138C4×D13—42no—
139C4×D13—44no—
140C4×D13—44no—
141#1057—44no—
142C2×C4⋊Q8#26111yescharacteristic, complemented
143C2×C22⋊Q8—261no—
144C22×C42—131no—
145C2×C22⋊C8—132no—
146C2×#97—261no—
147#391—261no—
148#389—132no—
149#389—261no—
150#384—131no—
151#380—132no—
152#380—261no—
153C2×#95—132no—
154C2×#95—261no—
155C2×#95—261no—
156#336—261no—
157C22×C13#39111yescharacteristic
158C22×C13C2×#9911yescharacteristic
159C22×C13#41011yescharacteristic
160C22×C13#41811yescharacteristic
161C22×C13#41911yescharacteristic
162C22×C13C2×#10411yescharacteristic
163C22×C13#42111yescharacteristic
164C2×D13—21no—
165C2×D13—21no—
166C2×D13—21no—
167C2×D13—21no—
168C2×D13—22no—
169C2×D13—22no—
170C2×D13—22no—
171C2×D13—22no—
172C2×D13#48912yes—
173C2×D13#50112yes—
174#261—84no—
175C4×C13—22no—
176C4×C13—22no—
177C4×C13—22no—
178C4×C13—82no—
179Dic13—22no—
180Dic13—41no—
181Dic13—41no—
182Dic13—82no—
183C22×Q8—21no—
184C23×C4—131no—
185C23×C4—131no—
186C23×C4—131no—
187C23×C4—262no—
188C22×C8—262no—
189C4⋊Q8—42no—
190C22⋊Q8—524no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—132no—
193C2×C4⋊C4—262no—
194C2×C4⋊C4—261no—
195C2×C4⋊C4—261no—
196C2×C4⋊C4C2×#26112yes—
197C2×C22⋊C4—262no—
198C2×C42—131no—
199C2×C42—131no—
200C2×C42—131no—
201C2×C42—522no—
202C2×C42C2×#26111yescharacteristic
203C22⋊C8—134no—
204C22⋊C8—262no—
205#97—522no—
206#95—262no—
207#95—262no—
208#95—264no—
209#95—264no—
210#95—522no—
211C2×C13#150611yescharacteristic
212C2×C13#150711yescharacteristic
213C2×C13#151111yescharacteristic
214C2×C13#151411yescharacteristic
215C2×C13#198611yescharacteristic
216C2×C13#199311yescharacteristic
217C2×C13#199611yescharacteristic
218D13—22no—
219D13—22no—
220C24—131no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—22no—
225C22×C4—131no—
226C22×C4—132no—
227C22×C4—132no—
228C22×C4—131no—
229C22×C4—131no—
230C22×C4—262no—
231C22×C4—262no—
232C22×C4—262no—
233C22×C4—262no—
234C22×C4—264no—
235C22×C4—262no—
236C22×C4—521no—
237C22×C4—521no—
238C22×C4—521no—
239C22×C4—521no—
240C22×C4—522no—
241C22×C4—522no—
242C22×C4—522no—
243C22×C4—522no—
244C22×C4—522no—
245C22×C4—522no—
246C22×C4C22⋊#26111yescharacteristic
247C22×C4C22⋊#26111yescharacteristic
248C22×C4C22×#26111yescharacteristic
249C2×C8—264no—
250C2×C8—522no—
251C2×C8—522no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—522no—
255C4⋊C4—522no—
256C4⋊C4—522no—
257C4⋊C4—524no—
258C22⋊C4—524no—
259C42—22no—
260C42—262no—
261C42—521no—
262C42—521no—
263C13#988811yescharacteristic, complemented, Sylow
264C23—132no—
265C23—132no—
266C23—131no—
267C23—131no—
268C23—261no—
269C23—261no—
270C23—261no—
271C23—261no—
272C23C2×C22⋊#26111yescharacteristic
273Q8—84no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—132no—
284C2×C4—261no—
285C2×C4—261no—
286C2×C4—261no—
287C2×C4—261no—
288C2×C4—261no—
289C2×C4—261no—
290C2×C4—261no—
291C2×C4—261no—
292C2×C4—522no—
293C2×C4—522no—
294C2×C4—524no—
295C2×C4—524no—
296C2×C4—524no—
297C2×C4—524no—
298C2×C4—524no—
299C2×C4—522no—
300C2×C4—522no—
301C2×C4—522no—
302C2×C4—1042no—
303C2×C4—1042no—
304C2×C4—1042no—
305C2×C4—1042no—
306C2×C4#8811612yes—
307C2×C4#8811912yes—
308C2×C4Q8×#26112yes—
309C8—524no—
310C22—132no—
311C22—132no—
312C22—261no—
313C22—261no—
314C22—262no—
315C22—262no—
316C22—261no—
317C22—261no—
318C22—262no—
319C22—262no—
320C22#1171913811yescharacteristic
321C22C2×#8811611yescharacteristic
322C22#1171915311yescharacteristic
323C22#1171915511yescharacteristic
324C22#1171915611yescharacteristic
325C22(C2×#261)⋊Q811yescharacteristic
326C22C2×#8811911yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—82no—
331C4—262no—
332C4—521no—
333C4—521no—
334C4—1044no—
335C4—1042no—
336C2—262no—
337C2—262no—
338C2#1313022811yescharacteristic
339C2#1313023211yescharacteristic
340C2#1313024611yescharacteristic
341C2#1313025711yescharacteristic
342C2#1313101211yescharacteristic
343C2#1313102011yescharacteristic
344C2#1313102511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C13×C4⋊Q8C4
socleC23×C13C2×C22⋊C4
prime radicalC23×D13C2×D4
cosocleC22×C4×D13C23
hypercenterC2×C4⋊Q8#261
hypocenterC13#9888
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
5. C2×C13×C4⋊Q8C4semidirect
17. #11718634C4semidirect
18. #11718596C4semidirect
44. #88111Q8semidirect
142. C2×C4⋊Q8#261semidirect
263. C13#9888semidirect

Maximal subgroups

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

subgroupindex
#131305632
#131300122
#131297092
#131231832
#988813