Finite Group Databasebeta

← #19138251

Subgroups — #19138251

Order 5504.

852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C23×Dic86C211yesmaximal, characteristic, complemented
2C2×#13063311C212yesmaximal, complemented
3C2×#13063297C211yesmaximal, characteristic
4#14582812C212yesmaximal, complemented
5#14582811C212yesmaximal, complemented
6#14582803C212yesmaximal, complemented
7#14582779C211yesmaximal, characteristic, complemented
8#14582778C211yesmaximal, characteristic, complemented
9#14582776C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic43C211yesmaximal, characteristic, complemented
11C43×#386C211yesmaximal, characteristic, complemented
12C22×Dic86—24no—
13C22×Dic86—28no—
14C22×Dic86C2212yes—
15C22×Dic86C2212yes—
16C22×C22⋊D43C2211yescharacteristic
17C22×C22⋊D43C2211yescharacteristic
18C23×Dic43C2211yescharacteristic
19C23×Dic43C2211yescharacteristic
20#13063311—28no—
21#13063297—24no—
22C2×Dic43⋊C4C2211yescharacteristic
23C2×C22⋊Dic43C2211yescharacteristic
24C2×C22⋊Dic43C2212yes—
25C2×C22⋊Dic43C2211yescharacteristic
26C2×(C2×D43)⋊C4C2211yescharacteristic
27C2×(C2×D43)⋊C4C2212yes—
28C2×(C2×D43)⋊C4C2212yes—
29C2×C4×Dic43C2212yes—
30C43⋊#95C2212yes—
31C43⋊#95C2212yes—
32C43⋊#95C2212yes—
33C43⋊#95C2212yes—
34C22×D4×C43C2211yescharacteristic
35C23×C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2212yes—
37C2×C22⋊C4×C43C2211yescharacteristic
38#95×C43C2212yes—
39C24×C43C2311yescharacteristic
40C24×C43C2311yescharacteristic
41C23×D43C2311yescharacteristic
42C2×D4×C43—22no—
43C2×D4×C43—22no—
44C2×D4×C43—22no—
45C22×C4×C43—24no—
46C22×C4×C43—22no—
47C22×C4×C43C2311yescharacteristic
48C22×C4×C43C2312yes—
49C22×C4×C43C2312yes—
50C2×C22⋊D43—22no—
51C2×C22⋊D43—24no—
52C2×C22⋊D43—22no—
53C2×C22⋊D43—22no—
54C2×C22⋊D43—22no—
55C22×Dic43—24no—
56C22×Dic43—24no—
57C22×Dic43—22no—
58C22×Dic43D414yes—
59C22×Dic43C2311yescharacteristic
60C22×Dic43C2312yes—
61C22×Dic43C2311yescharacteristic
62C22×Dic43C2312yes—
63C22×Dic43C2311yescharacteristic
64C2×Dic86—232no—
65C2×Dic86—24no—
66C2×Dic86—24no—
67C2×Dic86—24no—
68C2×Dic86—24no—
69C2×Dic86—24no—
70C2×Dic86D418yescomplemented
71C22⋊C4×C43—24no—
72C22⋊C4×C43—22no—
73C22⋊Dic43—22no—
74C22⋊Dic43—24no—
75C22⋊Dic43—22no—
76(C2×D43)⋊C4—24no—
77(C2×D43)⋊C4—22no—
78(C2×D43)⋊C4—24no—
79Dic43⋊C4—22no—
80C4×Dic43—24no—
81C23×C43—22no—
82C23×C43—22no—
83C23×C43—22no—
84C23×C43—21no—
85C23×C43—21no—
86C23×C43—21no—
87C23×C43—22no—
88C23×C43C2×D412yes—
89C23×C43#4112yes—
90C23×C43#4112yes—
91C23×C43C2411yescharacteristic
92C22×D43—22no—
93C22×D43—21no—
94C22×D43—21no—
95C22×D43—21no—
96C22×D43—22no—
97D4×C43—44no—
98C2×C4×C43—28no—
99C2×C4×C43—22no—
100C2×C4×C43—22no—
101C2×C4×C43—22no—
102C2×C4×C43—21no—
103C2×C4×C43—22no—
104C2×C4×C43—21no—
105C2×C4×C43—21no—
106C2×C4×C43—22no—
107C2×C4×C43C2×D414yes—
108C22⋊D43—44no—
109C22⋊D43—44no—
110C2×Dic43—28no—
111C2×Dic43—22no—
112C2×Dic43—22no—
113C2×Dic43—22no—
114C2×Dic43—28no—
115C2×Dic43—22no—
116C2×Dic43—21no—
117C2×Dic43—22no—
118C2×Dic43—21no—
119C2×Dic43—21no—
120C2×Dic43—22no—
121C2×Dic43C2×D412yes—
122C2×Dic43C2×D414yes—
123C2×Dic43C2×D414yes—
124C2×Dic43C2×D412yes—
125Dic86—216no—
126Dic86—216no—
127C22×C43—22no—
128C22×C43—22no—
129C22×C43—24no—
130C22×C43—24no—
131C22×C43—41no—
132C22×C43—41no—
133C22×C43—41no—
134C22×C43—42no—
135C22×C43—42no—
136C22×C43#12414yescomplemented
137C22×C43C22×D411yescharacteristic
138C22×C43C22×D412yes—
139C22×C43C2×#4111yescharacteristic
140C22×C43C2×#4111yescharacteristic
141C22×C432−1+412yes—
142C2×D43—41no—
143C2×D43—41no—
144C2×D43—41no—
145C2×D43—42no—
146C2×D43—42no—
147C4×C43—24no—
148C4×C43—41no—
149C4×C43—42no—
150Dic43—22no—
151Dic43—24no—
152Dic43—41no—
153Dic43—42no—
154Dic43#12014yes—
155#2436—431nomaximal, Sylow
156C2×C43—24no—
157C2×C43—81no—
158C2×C43C2×#12011yescharacteristic
159C2×C43C2×#12411yescharacteristic
160C2×C43#53611yescharacteristic
161C2×C43#54812yes—
162C2×C43#54812yes—
163D43—81no—
164C23×Q8—431no—
165C2×#124—432no—
166C2×#121—431no—
167#393—432no—
168#393—432no—
169#391—432no—
170#386—431no—
171#386—431no—
172#386—431no—
173#386D4311yescharacteristic, complemented
174C4×C22⋊C4—431no—
175C43#243611yescharacteristic, complemented, Sylow
176C22×Q8—432no—
177C22×Q8—432no—
178C22×Q8—864no—
179C22×Q8—868no—
180C22×D4—431no—
181C22×D4—431no—
182C22×D4C2×D4311yescharacteristic
183C23×C4—431no—
184C23×C4—431no—
185C23×C4C2×D4311yescharacteristic
186#124—868no—
187#121—864no—
188C2×C4⋊C4—431no—
189C2×C22⋊C4—431no—
190C2×C22⋊C4—432no—
191C2×C22⋊C4—431no—
192C2×C22⋊C4—431no—
193C2×C22⋊C4—432no—
194C2×C22⋊C4—432no—
195C2×C22⋊C4C2×D4312yes—
196C2×C22⋊C4C2×D4311yescharacteristic
197C2×C42—432no—
198#95—432no—
199#95—432no—
200#95—432no—
201#95—432no—
202#95C2×D4312yes—
203C24—431no—
204C24C22×D4311yescharacteristic
205C24C22×D4311yescharacteristic
206C2×Q8—438no—
207C2×Q8—864no—
208C2×Q8—864no—
209C2×Q8—864no—
210C2×Q8—864no—
211C2×Q8—8632no—
212C2×Q8—864no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—862no—
217C2×D4—864no—
218C2×D4—862no—
219C2×D4—862no—
220C2×D4—862no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—431no—
224C22×C4—434no—
225C22×C4—431no—
226C22×C4—432no—
227C22×C4—431no—
228C22×C4—432no—
229C22×C4—864no—
230C22×C4—864no—
231C22×C4—862no—
232C22×C4C22×D4311yescharacteristic
233C22×C4C22×D4312yes—
234C22×C4C22×D4312yes—
235C4⋊C4—862no—
236C22⋊C4—22no—
237C22⋊C4—24no—
238C22⋊C4—862no—
239C22⋊C4—864no—
240C22⋊C4—862no—
241C22⋊C4—864no—
242C22⋊C4—862no—
243C22⋊C4—864no—
244C42—864no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—862no—
253C23—861no—
254C23—861no—
255C23—861no—
256C23—862no—
257C23#1062189812yes—
258C23D4×D4312yes—
259C23#1062190012yes—
260C23C23×D4311yescharacteristic
261Q8—8616no—
262Q8—8616no—
263D4—44no—
264D4—1724no—
265D4—1724no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—434no—
276C2×C4—432no—
277C2×C4—432no—
278C2×C4—434no—
279C2×C4—868no—
280C2×C4—868no—
281C2×C4—862no—
282C2×C4—862no—
283C2×C4—862no—
284C2×C4—862no—
285C2×C4—861no—
286C2×C4—861no—
287C2×C4—862no—
288C2×C4—861no—
289C2×C4—862no—
290C2×C4D4×D4314yes—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—1722no—
301C22—1722no—
302C22—1721no—
303C22—1721no—
304C22—1721no—
305C22#1306331114yescomplemented
306C22C2×D4×D4312yes—
307C22C2×D4×D4311yescharacteristic
308C22C2×#1062190011yescharacteristic
309C22C2×#1062189811yescharacteristic
310C22#1306337212yes—
311C4—24no—
312C4—42no—
313C4—41no—
314C4—434no—
315C4—862no—
316C4—864no—
317C4—1722no—
318C4—1721no—
319C2—24no—
320C2—81no—
321C2—3441no—
322C2#120×D4311yescharacteristic
323C2C2×#1306331111yescharacteristic
324C2#1458347111yescharacteristic
325C2#1458355512yes—
326C2#1458355812yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#386C2
hypercenter#386D43
hypocenterC43#2436
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×Dic86C2semidirect
2. C2×#13063311C2semidirect
4. #14582812C2semidirect
5. #14582811C2semidirect
6. #14582803C2semidirect
7. #14582779C2semidirect
8. #14582778C2semidirect
9. #14582776C2semidirect
10. C22⋊C4×Dic43C2semidirect
11. C43×#386C2semidirect
70. C2×Dic86D4semidirect
136. C22×C43#124semidirect
173. #386D43semidirect
175. C43#2436semidirect
305. C22#13063311semidirect