Finite Group Databasebeta

← #24997163

Subgroups — #24997163

Order 11008.

772 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14583217C212yesmaximal, complemented
2#19140697C211yesmaximal, characteristic
3C2×#14582632C212yesmaximal, complemented
4#19132438C211yesmaximal, characteristic, complemented
5C43×#1445C211yesmaximal, characteristic, complemented
6C2×#13063299—22no—
7C2×#13063299C2212yes—
8#14583217—24no—
9#14583217—24no—
10C2×#13063256—22no—
11C2×#13063245—22no—
12C2×C22⋊(C43⋊C8)C2212yes—
13#14582836—22no—
14#14582809—22no—
15#14582632—24no—
16C22×C4⋊C4×C43C2211yescharacteristic
17C2×#95×C43—22no—
18C2×#95×C43C2211yescharacteristic
19C2×#95×C43C2211yescharacteristic
20C22×D172—22no—
21C22×D172D412yes—
22C22×C22⋊D43—22no—
23C22×C43⋊C8—22no—
24#13063299—22no—
25#13063299—22no—
26#13063299—22no—
27#13063299—22no—
28#13063299—44no—
29#13063299—44no—
30C2×C4⋊Dic43—22no—
31C2×C4⋊Dic43D412yes—
32C2×(C2×D43)⋊C4—24no—
33C2×(C2×D43)⋊C4—22no—
34#13063256—28no—
35#13063245—28no—
36C22⋊(C43⋊C8)—22no—
37C22⋊(C43⋊C8)D414yescomplemented
38C23×C4×C43—21no—
39C23×C4×C43—21no—
40C23×C4×C43D412yes—
41C23×C4×C43C2311yescharacteristic
42C2×C4⋊C4×C43—22no—
43C2×C4⋊C4×C43—22no—
44C2×C4⋊C4×C43—21no—
45C2×C4⋊C4×C43D412yescomplemented
46#95×C43—22no—
47#95×C43—22no—
48#95×C43—44no—
49C24×C43C2×D411yescharacteristic
50C23×D43—22no—
51C22×C4×C43—22no—
52C22×C4×C43—22no—
53C22×C4×C43—22no—
54C22×C4×C43—22no—
55C22×C4×C43—42no—
56C22×C4×C43—42no—
57C22×C4×C43—42no—
58C22×C4×C43—42no—
59C22×C4×C43—41no—
60C22×C4×C43—42no—
61C22×C4×C43—42no—
62C22×C4×C43—41no—
63C22×C4×C43—41no—
64C22×C4×C43—41no—
65C22×C4×C43—42no—
66C22×C4×C43SD1614yes—
67C22×C4×C43C2×D412yes—
68C22×C4×C43C2×D411yescharacteristic
69C22×C4×C43C2×D412yes—
70C22×C4×C43#4111yescharacteristic
71C2×C22⋊D43—42no—
72C2×C22⋊D43—42no—
73C2×C22⋊D43—44no—
74C2×C22⋊D43—44no—
75C22×Dic43—22no—
76C2×D172—22no—
77C2×D172—22no—
78C2×D172—44no—
79C2×D172—44no—
80C2×D172—82no—
81C2×D172—82no—
82C4⋊C4×C43—24no—
83C4⋊C4×C43—42no—
84(C2×D43)⋊C4—48no—
85(C2×D43)⋊C4—42no—
86(C2×D43)⋊C4—42no—
87C4⋊Dic43—22no—
88C4⋊Dic43—22no—
89C4⋊Dic43—44no—
90C2×C43⋊C8—24no—
91C2×C43⋊C8—44no—
92C23×C43—21no—
93C23×C43—21no—
94C23×C43—22no—
95C23×C43#12011yescharacteristic
96C23×C43#12111yescharacteristic
97C23×C43#12711yescharacteristic
98C23×C43C2×SD1612yes—
99C23×C43AGL(1,ℤ/8)12yes—
100C22×D43—42no—
101C22×D43—44no—
102C22×D43—42no—
103C22×D43—42no—
104C22×D43—44no—
105C2×C4×C43—24no—
106C2×C4×C43—24no—
107C2×C4×C43—41no—
108C2×C4×C43—42no—
109C2×C4×C43—41no—
110C2×C4×C43—42no—
111C2×C4×C43—42no—
112C2×C4×C43—44no—
113C2×C4×C43—44no—
114C2×C4×C43—42no—
115C2×C4×C43—44no—
116C2×C4×C43—42no—
117C2×C4×C43—82no—
118C2×C4×C43—82no—
119C2×C4×C43—81no—
120C2×C4×C43—82no—
121C2×C4×C43—81no—
122C2×C4×C43—81no—
123C2×C4×C43—81no—
124C2×C4×C43#12012yes—
125C2×C4×C43#12112yes—
126C22⋊D43—84no—
127C22⋊D43—82no—
128C22⋊D43—82no—
129C2×Dic43—42no—
130C2×Dic43—44no—
131D172—44no—
132D172—82no—
133D172—88no—
134C43⋊C8—44no—
135#9968—431nomaximal, Sylow
136C22×C43—22no—
137C22×C43—22no—
138C22×C43—22no—
139C22×C43—22no—
140C22×C43—21no—
141C22×C43—21no—
142C22×C43—21no—
143C22×C43—21no—
144C22×C43#39211yescharacteristic
145C22×C43#45012yes—
146C22×C43#45311yescharacteristic
147C22×C43#45411yescharacteristic
148C22×C43#45812yes—
149C22×C43#46012yes—
150C22×C43#49211yescharacteristic
151C22×C43#49611yescharacteristic
152C2×D43—84no—
153C2×D43—84no—
154C2×D43—82no—
155C2×D43—82no—
156C2×D43—82no—
157C4×C43—22no—
158C4×C43—22no—
159C4×C43—44no—
160C4×C43—82no—
161C4×C43—82no—
162Dic43—82no—
163C2×#481—432no—
164#2656—431no—
165C2×#327—432no—
166#1445D4311yescharacteristic, complemented
167#1280—431no—
168C2×C43—21no—
169C2×C43—22no—
170C2×C43—21no—
171C2×C43#161312yes—
172C2×C43#161612yes—
173C2×C43#200611yescharacteristic
174C2×C43#201411yescharacteristic
175C2×C43#202211yescharacteristic
176D43—162no—
177C2×#121—432no—
178C2×#121—862no—
179C22×C4⋊C4C2×D4311yescharacteristic
180#481—864no—
181#481—864no—
182C2×#106—862no—
183C2×#102—862no—
184C2×C22⋊C8—432no—
185#399—862no—
186#392—862no—
187C2×#95—22no—
188C2×#95C2×D4311yescharacteristic
189C2×#95C2×D4311yescharacteristic
190#327—864no—
191C43#996811yescharacteristic, complemented, Sylow
192C22×D4—432no—
193C22×D4—862no—
194C22×D4—862no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C22⋊D4312yes—
198C23×C4C22×D4311yescharacteristic
199C22×C8—862no—
200#121—862no—
201#121—862no—
202#121—862no—
203#121—862no—
204#121—1724no—
205#121—1724no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—432no—
210C2×C4⋊C4—862no—
211C2×C4⋊C4D17212yescomplemented
212C2×C22⋊C4—864no—
213C2×C22⋊C4—862no—
214#106—868no—
215#102—868no—
216C22⋊C8—434no—
217C22⋊C8—862no—
218#95—22no—
219#95—22no—
220#95—44no—
221C24—862no—
222C24C2×C22⋊D4311yescharacteristic
223C2×D4—862no—
224C2×D4—862no—
225C2×D4—3442no—
226C2×D4—3442no—
227C2×D4—1724no—
228C2×D4—1724no—
229C2×D4—1722no—
230C2×D4—1722no—
231C2×D4—1724no—
232C2×D4—1724no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—862no—
249C22×C4#1062188414yes—
250C22×C4C2×D17211yescharacteristic
251C22×C4#1062189811yescharacteristic
252C22×C4D4×D4312yes—
253C22×C4D4×D4312yes—
254C2×C8—864no—
255C2×C8—1724no—
256C4⋊C4—24no—
257C4⋊C4—42no—
258C4⋊C4—862no—
259C4⋊C4—862no—
260C4⋊C4—1724no—
261C22⋊C4—1728no—
262C22⋊C4—1722no—
263C22⋊C4—1722no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—1724no—
268C23—1722no—
269C23—1722no—
270C23—1722no—
271C23—1724no—
272C23#1306329311yescharacteristic
273C23#1306329911yescharacteristic
274C23#1306332111yescharacteristic
275C23C2×#1062188412yes—
276C23#1306335012yes—
277D4—3442no—
278D4—3448no—
279D4—3444no—
280D4—3442no—
281D4—3442no—
282D4—1724no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—1724no—
303C2×C4—1722no—
304C2×C4#1306329512yes—
305C2×C4#1306329812yes—
306C8—1724no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—3442no—
316C22—3444no—
317C22—3444no—
318C22—3442no—
319C22—3442no—
320C22#1458280811yescharacteristic
321C22#1458304912yes—
322C22#1458306611yescharacteristic
323C22#1458307111yescharacteristic
324C22#1458308812yes—
325C22#1458309912yes—
326C22#1458327311yescharacteristic
327C22#1458328911yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—82no—
333C4—3442no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—6882no—
338C2#1913361412yes—
339C2#1913363112yes—
340C2#1913563611yescharacteristic
341C2#1913568211yescharacteristic
342C2#1913572611yescharacteristic
subgroupG/H
centerC23#13063293
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC43×#1445C2
socleC23×C43#120
prime radical#19140697C2
hypercenter#1445D43
hypocenterC43#9968
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#14583217C2semidirect
3. C2×#14582632C2semidirect
4. #19132438C2semidirect
5. C43×#1445C2semidirect
37. C22⋊(C43⋊C8)D4semidirect
45. C2×C4⋊C4×C43D4semidirect
166. #1445D43semidirect
191. C43#9968semidirect
211. C2×C4⋊C4D172semidirect

Maximal subgroups

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

subgroupindex
C2×#145832172
#191406972
C2×#145826322
#191324382
C43×#14452
#996843