Finite Group Databasebeta

← #19137994

Subgroups — #19137994

Order 5504.

814 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13063306C211yesmaximal, characteristic, complemented
2C2×#13063293C211yesmaximal, characteristic
3C22×(C2×D43)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D43C211yesmaximal, characteristic, complemented
5#14582828C212yesmaximal, complemented
6#14582814C212yesmaximal, complemented
7#14582775C212yesmaximal, complemented
8#14582772C211yesmaximal, characteristic, complemented
9#14582760C211yesmaximal, characteristic, complemented
10#14582758C212yesmaximal, complemented
11C43×#386C211yesmaximal, characteristic, complemented
12C24×D43C2211yescharacteristic
13C22×C22⋊D43C2211yescharacteristic
14C22×C22⋊D43C2211yescharacteristic
15C23×Dic43C2211yescharacteristic
16C22×C4×D43C2212yes—
17#13063306—24no—
18#13063293—24no—
19C2×Dic43⋊C4C2212yescomplemented
20C2×Dic43⋊C4C2212yescomplemented
21C2×C22⋊Dic43C2211yescharacteristic
22C2×C22⋊Dic43C2211yescharacteristic
23C2×C22⋊Dic43C2212yescomplemented
24C2×(C2×D43)⋊C4—24no—
25C2×(C2×D43)⋊C4C2212yes—
26C2×(C2×D43)⋊C4C2212yes—
27C2×(C2×D43)⋊C4C2211yescharacteristic
28C2×(C2×D43)⋊C4C2212yes—
29C2×(C2×D43)⋊C4C2211yescharacteristic
30C22⋊C4×D43—24no—
31C43⋊#95C2212yescomplemented
32C43⋊#95C2212yescomplemented
33C43⋊#95C2212yescomplemented
34C22×D4×C43C2211yescharacteristic
35C23×C4×C43C2211yescharacteristic
36C2×C22⋊C4×C43C2211yescharacteristic
37C2×C22⋊C4×C43C2212yescomplemented
38#95×C43C2212yescomplemented
39C24×C43C2311yescharacteristic
40C24×C43C2311yescharacteristic
41C23×D43—22no—
42C23×D43—22no—
43C23×D43—24no—
44C23×D43—42no—
45C23×D43D414yes—
46C23×D43C2311yescharacteristic
47C23×D43C2311yescharacteristic
48C2×D4×C43—22no—
49C2×D4×C43—22no—
50C2×D4×C43—22no—
51C22×C4×C43—22no—
52C22×C4×C43—42no—
53C22×C4×C43C2311yescharacteristic
54C22×C4×C43C2312yes—
55C22×C4×C43C2312yes—
56C2×C22⋊D43—22no—
57C2×C22⋊D43—22no—
58C2×C22⋊D43—24no—
59C2×C22⋊D43—42no—
60C22×Dic43—22no—
61C22×Dic43—22no—
62C22×Dic43—22no—
63C22×Dic43C2311yescharacteristic
64C22×Dic43C2312yes—
65C22×Dic43C2311yescharacteristic
66C22×Dic43C2312yes—
67C2×C4×D43—24no—
68C2×C4×D43—24no—
69C2×C4×D43—42no—
70C22⋊C4×C43—22no—
71C22⋊C4×C43—42no—
72C22⋊Dic43—24no—
73C22⋊Dic43—22no—
74C22⋊Dic43D414yescomplemented
75(C2×D43)⋊C4—24no—
76(C2×D43)⋊C4—24no—
77(C2×D43)⋊C4—24no—
78(C2×D43)⋊C4—22no—
79(C2×D43)⋊C4—22no—
80Dic43⋊C4—24no—
81Dic43⋊C4—42no—
82C23×C43—22no—
83C23×C43—22no—
84C23×C43—21no—
85C23×C43—21no—
86C23×C43—22no—
87C23×C43—21no—
88C23×C43—22no—
89C23×C43C2×D412yes—
90C23×C43C2×D412yes—
91C23×C43#4112yes—
92C23×C43C2411yescharacteristic
93C22×D43—21no—
94C22×D43—21no—
95C22×D43—22no—
96C22×D43—28no—
97C22×D43—21no—
98C22×D43—22no—
99C22×D43—21no—
100C22×D43—22no—
101C22×D43—28no—
102C22×D43—44no—
103C22×D43—48no—
104C22×D43—48no—
105C22×D43C2×D412yes—
106C22×D43C2×D412yes—
107C22×D43#4114yes—
108D4×C43—44no—
109C2×C4×C43—21no—
110C2×C4×C43—22no—
111C2×C4×C43—21no—
112C2×C4×C43—22no—
113C2×C4×C43—22no—
114C2×C4×C43—21no—
115C2×C4×C43—22no—
116C2×C4×C43—22no—
117C2×C4×C43—22no—
118C2×C4×C43—44no—
119C22⋊D43—44no—
120C22⋊D43—44no—
121C2×Dic43—22no—
122C2×Dic43—22no—
123C2×Dic43—22no—
124C2×Dic43—22no—
125C2×Dic43—21no—
126C2×Dic43—21no—
127C2×Dic43—22no—
128C2×Dic43—22no—
129C2×Dic43—22no—
130C2×Dic43—44no—
131C2×Dic43C2×D414yes—
132C4×D43—48no—
133C22×C43—22no—
134C22×C43—22no—
135C22×C43—24no—
136C22×C43—24no—
137C22×C43—41no—
138C22×C43—41no—
139C22×C43—41no—
140C22×C43—42no—
141C22×C43—42no—
142C22×C43#12114yes—
143C22×C43C22×D411yescharacteristic
144C22×C43C22×D411yescharacteristic
145C22×C43C2×#4112yes—
146C22×C43C2×#4111yescharacteristic
147C22×C43H(5,𝔽2)12yes—
148C2×D43—22no—
149C2×D43—22no—
150C2×D43—24no—
151C2×D43—24no—
152C2×D43—22no—
153C2×D43—22no—
154C2×D43—22no—
155C2×D43—416no—
156C2×D43—42no—
157C2×D43—42no—
158C2×D43#12314yescomplemented
159C4×C43—41no—
160C4×C43—42no—
161C4×C43—42no—
162Dic43—24no—
163Dic43—42no—
164Dic43—41no—
165Dic43—41no—
166#2412—431nomaximal, Sylow
167C2×C43—24no—
168C2×C43—81no—
169C2×C43C2×#12111yescharacteristic
170C2×C43C2×#12311yescharacteristic
171C2×C43#53411yescharacteristic
172C2×C43#54612yes—
173C2×C43#55112yes—
174D43—24no—
175D43—42no—
176C2×#123—431no—
177C2×#120—431no—
178C22×C22⋊C4—431no—
179C22×C22⋊C4—431no—
180#397—432no—
181#394—432no—
182#386—432no—
183#386D4311yescharacteristic, complemented
184#385—431no—
185#381—431no—
186#380—432no—
187C43#241211yescharacteristic, complemented, Sylow
188C25—431no—
189C22×D4—431no—
190C22×D4—431no—
191C22×D4C2×D4311yescharacteristic
192C23×C4—431no—
193C23×C4—432no—
194C23×C4C2×D4311yescharacteristic
195#123—864no—
196#120—864no—
197C2×C4⋊C4—432no—
198C2×C4⋊C4—432no—
199C2×C22⋊C4—431no—
200C2×C22⋊C4—431no—
201C2×C22⋊C4—432no—
202C2×C22⋊C4—432no—
203C2×C22⋊C4—432no—
204C2×C22⋊C4—432no—
205C2×C22⋊C4—431no—
206C2×C22⋊C4—431no—
207C2×C22⋊C4—864no—
208C2×C22⋊C4—864no—
209C2×C22⋊C4C2×D4311yescharacteristic
210C2×C22⋊C4C2×D4312yescomplemented
211#95—432no—
212#95—432no—
213#95—432no—
214#95C2×D4312yescomplemented
215C24—431no—
216C24—431no—
217C24—434no—
218C24—864no—
219C24—862no—
220C24—862no—
221C24—1722no—
222C24C22×D4311yescharacteristic
223C24C22×D4311yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—862no—
228C2×D4—862no—
229C2×D4—864no—
230C2×D4—1722no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—432no—
234C22×C4—431no—
235C22×C4—432no—
236C22×C4—431no—
237C22×C4—864no—
238C22×C4—864no—
239C22×C4—862no—
240C22×C4—862no—
241C22×C4—862no—
242C22×C4—1722no—
243C22×C4C22×D4312yes—
244C22×C4C22×D4312yes—
245C22×C4C22×D4311yescharacteristic
246C4⋊C4—864no—
247C4⋊C4—1722no—
248C22⋊C4—22no—
249C22⋊C4—42no—
250C22⋊C4—434no—
251C22⋊C4—864no—
252C22⋊C4—864no—
253C22⋊C4—864no—
254C22⋊C4—862no—
255C22⋊C4—862no—
256C22⋊C4—864no—
257C22⋊C4—862no—
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—432no—
266C23—434no—
267C23—432no—
268C23—868no—
269C23—861no—
270C23—861no—
271C23—862no—
272C23—868no—
273C23—861no—
274C23—862no—
275C23—861no—
276C23—862no—
277C23—1724no—
278C23—1728no—
279C23—1728no—
280C23#1062189812yes—
281C23D4×D4312yes—
282C23D4×D4312yes—
283C23C23×D4311yescharacteristic
284D4—44no—
285D4—1724no—
286D4—1724no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—434no—
298C2×C4—862no—
299C2×C4—862no—
300C2×C4—862no—
301C2×C4—861no—
302C2×C4—861no—
303C2×C4—862no—
304C2×C4—862no—
305C2×C4—862no—
306C2×C4—862no—
307C2×C4—1728no—
308C2×C4—1724no—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—434no—
319C22—864no—
320C22—862no—
321C22—862no—
322C22—864no—
323C22—862no—
324C22—862no—
325C22—862no—
326C22—17216no—
327C22—1722no—
328C22—1722no—
329C22#1306329814yes—
330C22C2×D4×D4311yescharacteristic
331C22C2×D4×D4311yescharacteristic
332C22#41×D4312yes—
333C22C2×#1062189811yescharacteristic
334C22#1306337012yes—
335C4—41no—
336C4—42no—
337C4—42no—
338C4—864no—
339C4—1722no—
340C4—1721no—
341C4—1721no—
342C2—24no—
343C2—81no—
344C2—864no—
345C2—1722no—
346C2C2×#1306329811yescharacteristic
347C2#123×D4311yescharacteristic
348C2#1458345911yescharacteristic
349C2#1458353812yes—
350C2#1458357812yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#386C2
hypercenter#386D43
hypocenterC43#2412
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#13063306C2semidirect
3. C22×(C2×D43)⋊C4C2semidirect
4. C2×C22⋊C4×D43C2semidirect
5. #14582828C2semidirect
6. #14582814C2semidirect
7. #14582775C2semidirect
8. #14582772C2semidirect
9. #14582760C2semidirect
10. #14582758C2semidirect
11. C43×#386C2semidirect
19. C2×Dic43⋊C4C22semidirect
20. C2×Dic43⋊C4C22semidirect
23. C2×C22⋊Dic43C22semidirect
31. C43⋊#95C22semidirect
32. C43⋊#95C22semidirect
33. C43⋊#95C22semidirect
37. C2×C22⋊C4×C43C22semidirect
38. #95×C43C22semidirect
74. C22⋊Dic43D4semidirect
158. C2×D43#123semidirect
183. #386D43semidirect
187. C43#2412semidirect
210. C2×C22⋊C4C2×D43semidirect
214. #95C2×D43semidirect