Finite Group Databasebeta

← #24997164

Subgroups — #24997164

Order 11008.

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

normal subgroups only

This group has 772 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×#14583218C212yesmaximal, complemented
2#19140697C211yesmaximal, characteristic
3C2×#14582634C212yesmaximal, complemented
4#19132864C211yesmaximal, characteristic, complemented
5C43×#1280C211yesmaximal, characteristic, complemented
6C2×#13063299—22no—
7C2×#13063299C2212yes—
8C22×C4⋊Dic43C2211yescharacteristic
9#14583218—24no—
10#14583218—24no—
11C2×#13063257—22no—
12C2×#13063247—22no—
13#14582836—22no—
14#14582809—22no—
15C2×C43⋊#95—22no—
16C2×C43⋊#95C2211yescharacteristic
17#14582634—24no—
18C2×C22⋊C8×C43C2212yes—
19C2×#95×C43C2211yescharacteristic
20C22×D172—22no—
21C22×D172D412yes—
22C22×C22⋊D43—22no—
23C23×Dic43—21no—
24C23×Dic43—21no—
25#13063299—22no—
26#13063299—22no—
27#13063299—22no—
28#13063299—22no—
29#13063299—44no—
30#13063299—44no—
31C2×C4⋊Dic43—22no—
32C2×C4⋊Dic43—22no—
33C2×C4⋊Dic43—22no—
34C2×C4⋊Dic43—21no—
35C2×C4⋊Dic43D412yes—
36C2×C4⋊Dic43D412yescomplemented
37C2×(C2×D43)⋊C4—24no—
38C2×(C2×D43)⋊C4—22no—
39#13063257—28no—
40#13063247—28no—
41C43⋊#95—22no—
42C43⋊#95—44no—
43C23×C4×C43D412yes—
44C23×C4×C43C2311yescharacteristic
45C22×C8×C43—22no—
46C22⋊C8×C43—22no—
47C22⋊C8×C43D414yescomplemented
48#95×C43—22no—
49C24×C43C2×D411yescharacteristic
50C23×D43—22no—
51C22×C4×C43—22no—
52C22×C4×C43—22no—
53C22×C4×C43—42no—
54C22×C4×C43—42no—
55C22×C4×C43—42no—
56C22×C4×C43SD1614yes—
57C22×C4×C43C2×D412yes—
58C22×C4×C43C2×D411yescharacteristic
59C22×C4×C43C2×D412yes—
60C22×C4×C43#4111yescharacteristic
61C2×C22⋊D43—42no—
62C2×C22⋊D43—42no—
63C2×C22⋊D43—44no—
64C2×C22⋊D43—44no—
65C22×Dic43—22no—
66C22×Dic43—22no—
67C22×Dic43—22no—
68C22×Dic43—42no—
69C22×Dic43—41no—
70C22×Dic43—42no—
71C22×Dic43—42no—
72C22×Dic43—41no—
73C22×Dic43—41no—
74C22×Dic43—41no—
75C22×Dic43—42no—
76C2×D172—22no—
77C2×D172—22no—
78C2×D172—44no—
79C2×D172—44no—
80C2×D172—82no—
81C2×D172—82no—
82C2×C8×C43—24no—
83C2×C8×C43—44no—
84(C2×D43)⋊C4—48no—
85(C2×D43)⋊C4—42no—
86(C2×D43)⋊C4—42no—
87C4⋊Dic43—24no—
88C4⋊Dic43—22no—
89C4⋊Dic43—22no—
90C4⋊Dic43—44no—
91C4⋊Dic43—42no—
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—42no—
108C2×C4×C43—41no—
109C2×C4×C43—41no—
110C2×C4×C43—42no—
111C2×C4×C43—42no—
112C2×C4×C43—44no—
113C2×C4×C43—82no—
114C2×C4×C43#12012yes—
115C2×C4×C43#12112yes—
116C22⋊D43—84no—
117C22⋊D43—82no—
118C22⋊D43—82no—
119C2×Dic43—44no—
120C2×Dic43—42no—
121C2×Dic43—44no—
122C2×Dic43—42no—
123C2×Dic43—42no—
124C2×Dic43—44no—
125C2×Dic43—82no—
126C2×Dic43—81no—
127C2×Dic43—82no—
128C2×Dic43—81no—
129C2×Dic43—81no—
130C2×Dic43—81no—
131D172—44no—
132D172—82no—
133D172—88no—
134C8×C43—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—
160Dic43—82no—
161Dic43—82no—
162Dic43—82no—
163C2×#481—432no—
164#2656—431no—
165C2×#327—432no—
166#1445—431no—
167#1280D4311yescharacteristic, complemented
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⋊C4—431no—
180#481—864no—
181#481—864no—
182C2×#106—862no—
183C2×#102—862no—
184C2×C22⋊C8C2×D4312yes—
185#399—862no—
186#392—862no—
187C2×#95—431no—
188C2×#95—862no—
189C2×#95C2×D4311yescharacteristic
190#327—864no—
191C43#996811yescharacteristic, complemented, Sylow
192C22×D4—432no—
193C22×D4—862no—
194C22×D4—862no—
195C23×C4—861no—
196C23×C4—861no—
197C23×C4D17212yes—
198C23×C4C22×D4311yescharacteristic
199C22×C8—22no—
200#121—862no—
201#121—862no—
202#121—862no—
203#121—862no—
204#121—1724no—
205#121—1724no—
206C2×C4⋊C4—432no—
207C2×C4⋊C4—432no—
208C2×C4⋊C4—862no—
209C2×C4⋊C4—862no—
210C2×C4⋊C4—862no—
211C2×C4⋊C4—861no—
212C2×C22⋊C4—864no—
213C2×C22⋊C4—862no—
214#106—868no—
215#102—868no—
216C22⋊C8—22no—
217C22⋊C8D17214yescomplemented
218#95—22no—
219#95—862no—
220#95—1724no—
221C24—862no—
222C24C2×D17211yescharacteristic
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—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—862no—
239C22×C4—862no—
240C22×C4—862no—
241C22×C4—1722no—
242C22×C4—1721no—
243C22×C4—1722no—
244C22×C4—1722no—
245C22×C4—1721no—
246C22×C4—1721no—
247C22×C4—1721no—
248C22×C4—1722no—
249C22×C4#1062187314yes—
250C22×C4C2×D17212yes—
251C22×C4D4×D4312yes—
252C22×C4D4×D4311yescharacteristic
253C22×C4#1062190211yescharacteristic
254C2×C8—24no—
255C2×C8—44no—
256C4⋊C4—864no—
257C4⋊C4—862no—
258C4⋊C4—862no—
259C4⋊C4—1724no—
260C4⋊C4—1722no—
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#1306329511yescharacteristic
273C23#1306330011yescharacteristic
274C23#1306332211yescharacteristic
275C23C2×#1062187312yes—
276C23#1306335312yes—
277D4—3442no—
278D4—3448no—
279D4—3444no—
280D4—3442no—
281D4—3442no—
282D4—1724no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—3442no—
293C2×C4—3441no—
294C2×C4—3442no—
295C2×C4—3441no—
296C2×C4—3441no—
297C2×C4—3441no—
298C2×C4—1722no—
299C2×C4—1724no—
300C2×C4—1724no—
301C2×C4—1724no—
302C2×C4—1722no—
303C2×C4—1722no—
304C2×C4#1306329512yes—
305C2×C4#1306330012yes—
306C8—44no—
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#1458280911yescharacteristic
321C22#1458305212yes—
322C22#1458306911yescharacteristic
323C22#1458307411yescharacteristic
324C22#1458309012yes—
325C22#1458310012yes—
326C22#1458327211yescharacteristic
327C22#1458328811yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—3442no—
332C4—3442no—
333C4—3442no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—6882no—
338C2#1913361712yes—
339C2#1913363412yes—
340C2#1913563911yescharacteristic
341C2#1913568511yescharacteristic
342C2#1913572711yescharacteristic
subgroupG/H
centerC23#13063295
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC43×#1280C2
socleC23×C43#120
prime radical#19140697C2
hypercenter#1280D43
hypocenterC43#9968
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
1. C2×#14583218C2semidirect
3. C2×#14582634C2semidirect
4. #19132864C2semidirect
5. C43×#1280C2semidirect
36. C2×C4⋊Dic43D4semidirect
47. C22⋊C8×C43D4semidirect
167. #1280D43semidirect
191. C43#9968semidirect
217. C22⋊C8D172semidirect

Maximal subgroups

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

subgroupindex
C2×#145832182
#191406972
C2×#145826342
#191328642
C43×#12802
#996843