Finite Group Databasebeta

← #15763461

Subgroups — #15763461

Order 3328.

944 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#11718701C212yesmaximal, complemented
2C2×#11718129C212yesmaximal, complemented
3#13113106C211yesmaximal, characteristic, complemented
4#13112679C211yesmaximal, characteristic, complemented
5C13×#2656C211yesmaximal, characteristic
6C23×Dic26C2211yescharacteristic
7C2×C13×#121—22no—
8C2×C13×#121C2212yes—
9C22×#6105—22no—
10C13×#399—22no—
11C13×#392—22no—
12C2×C13×#95C2211yescharacteristic
13C2×#88071—22no—
14C2×#88071C2211yescharacteristic
15C2×C22⋊#1055C2212yes—
16#11718701—24no—
17#11718701—24no—
18C2×#88050—22no—
19#11718129—24no—
20C22×D4×C13—22no—
21C22×D4×C13—22no—
22C22×D4×C13D412yes—
23C23×C4×C13D412yes—
24C23×C4×C13C2311yescharacteristic
25C23×Dic13—21no—
26C23×Dic13—21no—
27C22×Dic26—22no—
28C22×Dic26—22no—
29C22×Dic26—21no—
30C22×Dic26—44no—
31C22×Dic26D412yescomplemented
32C13×#121—22no—
33C13×#121—22no—
34C13×#121—22no—
35C13×#121—22no—
36C13×#121—44no—
37C13×#121—44no—
38C2×C13×C4⋊C4—22no—
39C2×C13×C4⋊C4D412yes—
40C2×C13×C22⋊C4—24no—
41C2×C13×C22⋊C4—22no—
42C2×#6105—28no—
43C2×#6105—48no—
44C22×#1055—22no—
45C13×#95—22no—
46#88071—22no—
47#88071—44no—
48C22⋊#1055—22no—
49C22⋊#1055D414yescomplemented
50#88050—28no—
51#9964—131nomaximal, Sylow
52C24×C13—22no—
53C24×C13C2×D411yescharacteristic
54C2×D4×C13—22no—
55C2×D4×C13—22no—
56C2×D4×C13—44no—
57C2×D4×C13—44no—
58C2×D4×C13—42no—
59C2×D4×C13—42no—
60C2×D4×C13—44no—
61C2×D4×C13—44no—
62C2×D4×C13—82no—
63C2×D4×C13—82no—
64C22×C4×C13—22no—
65C22×C4×C13—22no—
66C22×C4×C13—22no—
67C22×C4×C13—42no—
68C22×C4×C13—42no—
69C22×C4×C13—42no—
70C22×C4×C13SD1614yescomplemented
71C22×C4×C13C2×D411yescharacteristic
72C22×C4×C13C2×D412yes—
73C22×C4×C13C2×D412yes—
74C22×C4×C13#4111yescharacteristic
75C22×Dic13—24no—
76C22×Dic13—22no—
77C22×Dic13—22no—
78C22×Dic13—42no—
79C22×Dic13—41no—
80C22×Dic13—41no—
81C22×Dic13—42no—
82C22×Dic13—41no—
83C22×Dic13—41no—
84C22×Dic13—42no—
85C2×Dic26—28no—
86C2×Dic26—24no—
87C2×Dic26—44no—
88C2×Dic26—48no—
89C2×Dic26—44no—
90C2×Dic26—44no—
91C2×Dic26—42no—
92C13×C4⋊C4—22no—
93C13×C4⋊C4—22no—
94C13×C4⋊C4—44no—
95C13×C22⋊C4—48no—
96C13×C22⋊C4—42no—
97C13×C22⋊C4—42no—
98#6105—416no—
99C2×#1055—24no—
100C2×#1055—44no—
101C2×#448—132no—
102#2656D1311yescharacteristic
103C2×#327—132no—
104#1446—131no—
105#1280—131no—
106C23×C13—21no—
107C23×C13—21no—
108C23×C13—22no—
109C23×C13—42no—
110C23×C13—44no—
111C23×C13—42no—
112C23×C13—42no—
113C23×C13—44no—
114C23×C13#12011yescharacteristic
115C23×C13#12111yescharacteristic
116C23×C13#12711yescharacteristic
117C23×C13C2×SD1612yes—
118C23×C13AGL(1,ℤ/8)12yes—
119D4×C13—44no—
120D4×C13—82no—
121D4×C13—88no—
122D4×C13—84no—
123D4×C13—82no—
124D4×C13—82no—
125C2×C4×C13—24no—
126C2×C4×C13—24no—
127C2×C4×C13—41no—
128C2×C4×C13—42no—
129C2×C4×C13—41no—
130C2×C4×C13—42no—
131C2×C4×C13—42no—
132C2×C4×C13—44no—
133C2×C4×C13—42no—
134C2×C4×C13—44no—
135C2×C4×C13—82no—
136C2×C4×C13#12012yes—
137C2×C4×C13#12112yes—
138C2×Dic13—24no—
139C2×Dic13—42no—
140C2×Dic13—44no—
141C2×Dic13—42no—
142C2×Dic13—44no—
143C2×Dic13—44no—
144C2×Dic13—42no—
145C2×Dic13—82no—
146C2×Dic13—81no—
147C2×Dic13—81no—
148Dic26—28no—
149Dic26—44no—
150Dic26—48no—
151#1055—44no—
152C23×Q8—131no—
153C22×SD16—262no—
154C2×#121—22no—
155C2×#121C2×D1312yes—
156#448—264no—
157#448—264no—
158C2×#103—262no—
159C2×C22⋊C8—132no—
160#399—22no—
161#392—22no—
162C2×#95—131no—
163C2×#95—262no—
164C2×#95C2×D1311yescharacteristic
165#327—264no—
166C22×C13—22no—
167C22×C13—22no—
168C22×C13—22no—
169C22×C13—22no—
170C22×C13—21no—
171C22×C13—21no—
172C22×C13—21no—
173C22×C13—21no—
174C22×C13—84no—
175C22×C13—84no—
176C22×C13—82no—
177C22×C13—82no—
178C22×C13—82no—
179C22×C13#39211yescharacteristic
180C22×C13#45012yes—
181C22×C13#45511yescharacteristic
182C22×C13#45611yescharacteristic
183C22×C13B(4,𝔽2)12yes—
184C22×C13#46012yes—
185C22×C13#49211yescharacteristic
186C22×C13#49611yescharacteristic
187C4×C13—22no—
188C4×C13—22no—
189C4×C13—44no—
190C4×C13—82no—
191Dic13—44no—
192Dic13—82no—
193C22×Q8—132no—
194C22×Q8—261no—
195C22×Q8—262no—
196C22×Q8—262no—
197C22×Q8—524no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4C22⋊D1312yes—
201C23×C4—261no—
202C23×C4—261no—
203C23×C4C22⋊D1312yes—
204C23×C4C22×D1311yescharacteristic
205C2×SD16—268no—
206C2×SD16—528no—
207C22×C8—262no—
208#121—22no—
209#121—22no—
210#121—22no—
211#121—22no—
212#121—44no—
213#121—44no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C22⋊D1312yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218#103—268no—
219C22⋊C8—134no—
220C22⋊C8—262no—
221#95—22no—
222#95—262no—
223#95—524no—
224C2×C13—21no—
225C2×C13—22no—
226C2×C13—21no—
227C2×C13—162no—
228C2×C13#160112yes—
229C2×C13#160412yes—
230C2×C13#200711yescharacteristic
231C2×C13#201411yescharacteristic
232C2×C13#202111yescharacteristic
233C24—22no—
234C24C2×C22⋊D1311yescharacteristic
235C2×Q8—268no—
236C2×Q8—264no—
237C2×Q8—524no—
238C2×Q8—524no—
239C2×Q8—524no—
240C2×Q8—528no—
241C2×Q8—522no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—82no—
251C2×D4—82no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—264no—
259C22×C4—262no—
260C22×C4—262no—
261C22×C4—522no—
262C22×C4—521no—
263C22×C4—521no—
264C22×C4—522no—
265C22×C4—521no—
266C22×C4—522no—
267C22×C4—521no—
268C22×C4#610514yescomplemented
269C22×C4D4×D1311yescharacteristic
270C22×C4D4×D1312yes—
271C22×C4#612911yescharacteristic
272C22×C4C2×C22⋊D1312yes—
273SD16—5216no—
274C2×C8—264no—
275C2×C8—524no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C22⋊C4—48no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C13#996411yescharacteristic, complemented, Sylow
283C23—21no—
284C23—22no—
285C23—21no—
286C23—44no—
287C23—42no—
288C23—42no—
289C23—44no—
290C23—42no—
291C23#8818612yes—
292C23C2×#610512yes—
293C23#8819211yescharacteristic
294C23#8819411yescharacteristic
295C23#8820711yescharacteristic
296Q8—268no—
297Q8—524no—
298Q8—528no—
299D4—44no—
300D4—82no—
301D4—88no—
302D4—82no—
303D4—84no—
304D4—82no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—264no—
317C2×C4—522no—
318C2×C4—524no—
319C2×C4—522no—
320C2×C4—524no—
321C2×C4—524no—
322C2×C4—522no—
323C2×C4—1041no—
324C2×C4—1042no—
325C2×C4—1041no—
326C2×C4#8819112yes—
327C2×C4#8819312yes—
328C8—524no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—21no—
336C22—21no—
337C22—82no—
338C22—82no—
339C22—84no—
340C22—84no—
341C22—82no—
342C22#1171873511yescharacteristic
343C22#1171873612yes—
344C22#1171873711yescharacteristic
345C22#1171887912yes—
346C22#1171888411yescharacteristic
347C22#1171890311yescharacteristic
348C22#1171890512yes—
349C22#1171890711yescharacteristic
350C4—22no—
351C4—22no—
352C4—44no—
353C4—82no—
354C4—524no—
355C4—1042no—
356C2—21no—
357C2—22no—
358C2—21no—
359C2—162no—
360C2#1311378812yes—
361C2#1311380512yes—
362C2#1311588311yescharacteristic
363C2#1311592411yescharacteristic
364C2#1311596011yescharacteristic
subgroupG/H
centerC23#88207
commutator, cosocleC23×C4×C13C23
FrattiniC23×C4C22×D13
Fitting, prime radicalC13×#2656C2
socleC23×C13#120
hypercenter#2656D13
hypocenterC13#9964
LCSG ▷ C23×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
1. C2×#11718701C2semidirect
2. C2×#11718129C2semidirect
3. #13113106C2semidirect
4. #13112679C2semidirect
31. C22×Dic26D4semidirect
49. C22⋊#1055D4semidirect
70. C22×C4×C13SD16semidirect
268. C22×C4#6105semidirect
282. C13#9964semidirect

Maximal subgroups

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

subgroupindex
C2×#117187012
C2×#117181292
#131131062
#131126792
C13×#26562
#996413