Finite Group Databasebeta

← #24994332

Subgroups — #24994332

Order 11008.

700 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#19137181C211yesmaximal, characteristic
2#19134519C211yesmaximal, characteristic
3#19133128C212yesmaximal
4#19132448C212yesmaximal
5C43×#2434C211yesmaximal, characteristic
6C2×#13063284—21no—
7C2×#13063244—22no—
8#14582863C2212yes—
9#14582806—24no—
10#14582762—21no—
11#14582762C2212yes—
12#14582636—24no—
13C23×Q8×C43C2211yescharacteristic
14C2×C4⋊Q8×C43C412yescomplemented
15C2×C22⋊Q8×C43C2211yescharacteristic
16C43×#395—22no—
17C43×#391C412yescomplemented
18C43×#385—21no—
19C43×#385—21no—
20C4×C22⋊C4×C43C2211yescharacteristic
21C2×#10621877—22no—
22#13063284—44no—
23C2×C4⋊Dic43—22no—
24C2×C22⋊Dic43—22no—
25C2×C4×Dic43—22no—
26#13063244—44no—
27C22⋊(C43⋊C8)—22no—
28C22⋊(C43⋊C8)—22no—
29C43⋊#95—24no—
30C43⋊#95—22no—
31C43⋊#95D414yes—
32C22×Q8×C43—22no—
33C22×Q8×C43—22no—
34C22×Q8×C43—44no—
35C22×Q8×C43C2×C411yescharacteristic
36C22×Q8×C43C2×C412yes—
37C22×Q8×C43D412yes—
38C23×C4×C43D411yescharacteristic
39C23×C4×C43D411yescharacteristic
40C23×C4×C43C2311yescharacteristic
41C4⋊Q8×C43—44no—
42C22⋊Q8×C43—22no—
43C22⋊Q8×C43—22no—
44C2×C4⋊C4×C43—22no—
45C2×C4⋊C4×C43—22no—
46C2×C4⋊C4×C43—21no—
47C2×C4⋊C4×C43C2×C411yescharacteristic
48C2×C22⋊C4×C43—21no—
49C2×C22⋊C4×C43—21no—
50C2×C42×C43C2×C412yes—
51#95×C43—22no—
52#95×C43—22no—
53#95×C43—21no—
54C24×C43C2×D411yescharacteristic
55C2×Q8×C43—24no—
56C2×Q8×C43—24no—
57C2×Q8×C43—24no—
58C2×Q8×C43—24no—
59C2×Q8×C43—22no—
60C2×Q8×C43—42no—
61C2×Q8×C43—42no—
62C2×Q8×C43—44no—
63C2×Q8×C43—44no—
64C2×Q8×C43—48no—
65C2×Q8×C43—42no—
66C22×C4×C43—24no—
67C22×C4×C43—22no—
68C22×C4×C43—21no—
69C22×C4×C43—21no—
70C22×C4×C43—21no—
71C22×C4×C43—21no—
72C22×C4×C43—21no—
73C22×C4×C43—21no—
74C22×C4×C43—42no—
75C22×C4×C43—42no—
76C22×C4×C43C22⋊C412yes—
77C22×C4×C43C22⋊C412yes—
78C22×C4×C43C22×C411yescharacteristic
79C22×C4×C43C2×D411yescharacteristic
80C22×C4×C43C2×D412yes—
81C22×C4×C43#4112yes—
82C22×Dic43—24no—
83C4⋊C4×C43—22no—
84C4⋊C4×C43—44no—
85C4⋊C4×C43—44no—
86C4⋊C4×C43—42no—
87C22⋊C4×C43—24no—
88C22⋊C4×C43—41no—
89C22⋊C4×C43—41no—
90C42×C43—42no—
91C22⋊Dic43—44no—
92C4⋊Dic43—44no—
93C4×Dic43—44no—
94#10621877—44no—
95C2×C43⋊C8—42no—
96C23×C43—22no—
97C23×C43—22no—
98C23×C43—41no—
99C23×C43C2×C22⋊C411yescharacteristic
100C23×C43#12011yescharacteristic
101C23×C43#12311yescharacteristic
102Q8×C43—48no—
103Q8×C43—48no—
104Q8×C43—82no—
105C2×C4×C43—22no—
106C2×C4×C43—22no—
107C2×C4×C43—24no—
108C2×C4×C43—22no—
109C2×C4×C43—22no—
110C2×C4×C43—22no—
111C2×C4×C43—22no—
112C2×C4×C43—24no—
113C2×C4×C43—22no—
114C2×C4×C43—22no—
115C2×C4×C43—22no—
116C2×C4×C43—42no—
117C2×C4×C43—42no—
118C2×C4×C43—42no—
119C2×C4×C43—42no—
120C2×C4×C43—42no—
121C2×C4×C43—42no—
122C2×C4×C43—42no—
123C2×C4×C43—41no—
124C2×C4×C43—41no—
125C2×C4×C43—42no—
126C2×C4×C43#9914yes—
127C2×C4×C43C4×D412yes—
128C2×C4×C43#12112yes—
129C2×Dic43—44no—
130C2×Dic43—44no—
131C2×Dic43—44no—
132C43⋊C8—82no—
133#9462—431nomaximal, Sylow
134C22×C43—21no—
135C22×C43—21no—
136C22×C43—21no—
137C22×C43—21no—
138C22×C43—21no—
139C22×C43—21no—
140C22×C43—21no—
141C22×C43—44no—
142C22×C43#38611yescharacteristic
143C22×C43C2×#9912yes—
144C22×C43#45612yes—
145C4×C43—22no—
146C4×C43—24no—
147C4×C43—22no—
148C4×C43—44no—
149C4×C43—42no—
150C4×C43—81no—
151C4×C43—81no—
152Dic43—84no—
153#2434D4311yescharacteristic
154#2310—431no—
155#1784—431no—
156#1511—432no—
157#1284—432no—
158C2×C43—22no—
159C2×C43—22no—
160C2×C43—41no—
161C2×C43#188611yescharacteristic
162C2×C43#189811yescharacteristic
163C2×C43#191211yescharacteristic
164C23×Q8C2×D4311yescharacteristic
165C2×C4⋊Q8Dic4312yescomplemented
166C2×C22⋊Q8C2×D4311yescharacteristic
167C2×#117—861no—
168C2×#101—862no—
169#407—432no—
170#395—22no—
171#391—864no—
172#391Dic4312yescomplemented
173#385—21no—
174#385—21no—
175#381—432no—
176#381—861no—
177C4×C22⋊C4C2×D4311yescharacteristic
178#328—864no—
179C43#946211yescharacteristic, complemented, Sylow
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8C2×Dic4311yescharacteristic
184C22×Q8C2×Dic4312yes—
185C22×Q8C22⋊D4312yes—
186C23×C4C22⋊D4311yescharacteristic
187C23×C4C22⋊D4311yescharacteristic
188C23×C4C22×D4311yescharacteristic
189C2×M16—862no—
190C4⋊Q8—44no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193#117—1724no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—862no—
198C2×C4⋊C4C2×Dic4311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—862no—
202C2×C42—862no—
203C2×C42C2×Dic4312yes—
204#101—1724no—
205C22⋊C8—862no—
206C22⋊C8—862no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—434no—
211#95—864no—
212#95—862no—
213C24C2×C22⋊D4311yescharacteristic
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—864no—
236C22×C4C22⋊Dic4312yes—
237C22×C4C22⋊Dic4312yes—
238C22×C4D4×D4312yes—
239C22×C4#1062190012yes—
240C22×C4C22×Dic4311yescharacteristic
241C22×C4C2×C22⋊D4311yescharacteristic
242M16—1724no—
243C2×C8—1722no—
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—1724no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—1724no—
253C42—42no—
254C42—1724no—
255C23—22no—
256C23—22no—
257C23—41no—
258C23C2×C22⋊Dic4311yescharacteristic
259C23#1306329311yescharacteristic
260C23#1306330611yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—82no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—1724no—
286C2×C4—1724no—
287C2×C4—1724no—
288C2×C4#1306324014yes—
289C2×C4D4×Dic4312yes—
290C2×C4#1306329712yes—
291C8—3442no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1458277911yescharacteristic
301C22C2×#1306324012yes—
302C22#1458308112yes—
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—3444no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#1913498911yescharacteristic
315C2#1913504911yescharacteristic
316C2#1913513311yescharacteristic
subgroupG/H
centerC22#14582779
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#2434C2
socleC22×C43#386
prime radicalC24×C43C2×D4
cosocleC23×C4×C43C23
hypercenter#2434D43
hypocenterC43#9462
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
14. C2×C4⋊Q8×C43C4semidirect
17. C43×#391C4semidirect
165. C2×C4⋊Q8Dic43semidirect
172. #391Dic43semidirect
179. C43#9462semidirect

Maximal subgroups

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

subgroupindex
#191371812
#191345192
#191331282
#191324482
C43×#24342
#946243