Finite Group Databasebeta

← #25012591

Subgroups — #25012591

Order 11008.

524 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 93 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#19137305C211yesmaximal, characteristic
2C2×#14582693C212yesmaximal
3#19134581C212yesmaximal
4#19132720C211yesmaximal, characteristic
5C43×#1393C211yesmaximal, characteristic
6C2×#13063312C2211yescharacteristic
7C2×#13063290C411yescharacteristic, complemented
8C2×D4×Dic43C2211yescharacteristic
9C2×#13063240C2212yes—
10#14582805C411yescharacteristic, complemented
11#14582793—22no—
12#14582771—22no—
13#14582768C411yescharacteristic, complemented
14#14582757—22no—
15#14582745C411yescharacteristic, complemented
16#14582693—24no—
17#14582665—22no—
18C2×#99×C43C2212yes—
19C43×#390C2211yescharacteristic
20C22×Dic86C2×C411yescharacteristic
21C23×Dic43D412yes—
22#13063312—22no—
23#13063312D412yes—
24#13063312Q812yes—
25#13063290—42no—
26D4×Dic43—42no—
27C2×C4⋊Dic43C2×C411yescharacteristic
28C2×Dic43⋊C4—21no—
29C2×Dic43⋊C4—22no—
30C2×Dic43⋊C4—42no—
31C2×C22⋊Dic43—22no—
32C2×C22⋊Dic43—22no—
33C2×C22⋊Dic43D412yes—
34C2×C4×Dic43—21no—
35C2×C4×Dic43C2×C411yescharacteristic
36C2×C4×Dic43C2×C411yescharacteristic
37#13063240—22no—
38#13063240—22no—
39C43⋊#95—21no—
40C43⋊#95—22no—
41C43⋊#95—21no—
42C43⋊#95—42no—
43C22×D4×C43C2311yescharacteristic
44C2×C4⋊C4×C43C2×C411yescharacteristic
45C2×C22⋊C4×C43—22no—
46C2×C22⋊C4×C43—22no—
47C2×C42×C43C2×C411yescharacteristic
48#99×C43—22no—
49#99×C43—22no—
50C24×C43C2×D412yes—
51C2×D4×C43—41no—
52C2×D4×C43—41no—
53C2×D4×C43C2×D411yescharacteristic
54C2×D4×C43C2×Q811yescharacteristic
55C2×D4×C43#4112yes—
56C22×C4×C43—21no—
57C22×C4×C43—21no—
58C22×C4×C43—42no—
59C22×C4×C43C22×C411yescharacteristic
60C22×Dic43—22no—
61C22×Dic43—22no—
62C22×Dic43—21no—
63C22×Dic43—21no—
64C22×Dic43—42no—
65C22×Dic43—82no—
66C22×Dic43C22⋊C411yescharacteristic
67C22×Dic43C22⋊C412yes—
68C22×Dic43C22⋊C411yescharacteristic
69C2×Dic86—21no—
70C2×Dic86—41no—
71C2×Dic86—41no—
72C2×Dic86C4⋊C412yescomplemented
73C4⋊C4×C43—41no—
74C22⋊C4×C43—42no—
75C22⋊C4×C43—42no—
76C22⋊C4×C43—82no—
77C42×C43—41no—
78C22⋊Dic43—24no—
79C22⋊Dic43—24no—
80C22⋊Dic43—42no—
81C22⋊Dic43—42no—
82C22⋊Dic43—42no—
83C4⋊Dic43—41no—
84Dic43⋊C4—42no—
85Dic43⋊C4—82no—
86Dic43⋊C4—84no—
87C4×Dic43—21no—
88C4×Dic43—22no—
89C4×Dic43—42no—
90C4×Dic43C4⋊C412yescomplemented
91C23×C43—22no—
92C23×C43—22no—
93C23×C43—42no—
94C23×C43—42no—
95C23×C43C2×C22⋊C411yescharacteristic
96C23×C43#12112yes—
97C23×C43C22⋊Q812yes—
98D4×C43—41no—
99D4×C43—41no—
100D4×C43—42no—
101C2×C4×C43—21no—
102C2×C4×C43—21no—
103C2×C4×C43—42no—
104C2×C4×C43—41no—
105C2×C4×C43—41no—
106C2×C4×C43—41no—
107C2×C4×C43—41no—
108C2×C4×C43—82no—
109C2×C4×C43—82no—
110C2×C4×C43—82no—
111C2×C4×C43C2×C4⋊C411yescharacteristic
112C2×C4×C43#11711yescharacteristic
113C2×Dic43—22no—
114C2×Dic43—22no—
115C2×Dic43—21no—
116C2×Dic43—22no—
117C2×Dic43—41no—
118C2×Dic43—44no—
119C2×Dic43—41no—
120C2×Dic43—41no—
121C2×Dic43—41no—
122C2×Dic43—41no—
123C2×Dic43—42no—
124C2×Dic43—41no—
125C2×Dic43—82no—
126C2×Dic43—82no—
127C2×Dic43—82no—
128C2×Dic43—82no—
129C2×Dic43#9912yes—
130Dic86—42no—
131Dic86—42no—
132#12800—431nomaximal, Sylow
133C22×C43—21no—
134C22×C43—21no—
135C22×C43—22no—
136C22×C43—22no—
137C22×C43—42no—
138C22×C43—42no—
139C22×C43—42no—
140C22×C43—42no—
141C22×C43—82no—
142C22×C43#38011yescharacteristic
143C22×C43C2×#9911yescharacteristic
144C22×C43#41011yescharacteristic
145C4×C43—21no—
146C4×C43—21no—
147C4×C43—81no—
148C4×C43—81no—
149C4×C43—162no—
150Dic43—22no—
151Dic43—42no—
152Dic43—42no—
153Dic43—42no—
154Dic43—41no—
155Dic43—162no—
156#2326—431no—
157C2×#354—432no—
158#1799—432no—
159#1393—431no—
160#1393D4311yescharacteristic
161C2×C43—21no—
162C2×C43—21no—
163C2×C43—42no—
164C2×C43—42no—
165C2×C43#179911yescharacteristic
166C2×C43#212711yescharacteristic
167C2×C43#212911yescharacteristic
168C2×#124—431no—
169C2×C4×Q8—431no—
170C2×C4×D4—431no—
171C2×#99—432no—
172C2×#99C2×D4312yes—
173#391—431no—
174#390C2×D4311yescharacteristic
175#388—862no—
176#385—862no—
177#384—431no—
178#380—862no—
179#376—431no—
180#354—864no—
181#342—862no—
182C43#1280011yescharacteristic, complemented, Sylow
183C22×Q8—431no—
184C22×D4C22×D4311yescharacteristic
185C23×C4—432no—
186#124—432no—
187#124—432no—
188#124—862no—
189C4×Q8—1722no—
190C4×D4—1722no—
191C2×C4⋊C4—431no—
192C2×C4⋊C4—862no—
193C2×C4⋊C4—861no—
194C2×C4⋊C4—1722no—
195C2×C4⋊C4C4×D4311yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—432no—
199C2×C22⋊C4—862no—
200C2×C22⋊C4—862no—
201C2×C42—431no—
202C2×C42—431no—
203C2×C42—861no—
204C2×C42C4×D4311yescharacteristic
205#99—22no—
206#99—22no—
207#99—862no—
208#99—862no—
209#95—861no—
210#95—861no—
211#95—862no—
212#95—1722no—
213C24D4×D4312yes—
214C2×Q8—432no—
215C2×Q8—861no—
216C2×Q8—1721no—
217C2×Q8—1721no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1062189812yes—
221C2×D4D4×D4311yescharacteristic
222C2×D4Q8×D4311yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—431no—
227C22×C4—431no—
228C22×C4—432no—
229C22×C4—862no—
230C22×C4—862no—
231C22×C4—861no—
232C22×C4—861no—
233C22×C4—3442no—
234C22×C4—1722no—
235C22×C4C2×C4×D4311yescharacteristic
236C4⋊C4—41no—
237C4⋊C4—3442no—
238C4⋊C4—3444no—
239C4⋊C4—1721no—
240C4⋊C4—1722no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—864no—
245C22⋊C4—864no—
246C22⋊C4—1722no—
247C22⋊C4—1722no—
248C22⋊C4—1722no—
249C42—41no—
250C42—432no—
251C42—862no—
252C42—861no—
253C42—1722no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23C22⋊C4×D4311yescharacteristic
259C23#1306329812yes—
260C23#1306330312yes—
261Q8—1722no—
262Q8—1722no—
263D4—42no—
264D4—41no—
265D4—41no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—432no—
277C2×C4—861no—
278C2×C4—862no—
279C2×C4—862no—
280C2×C4—862no—
281C2×C4—3442no—
282C2×C4—3442no—
283C2×C4—3442no—
284C2×C4—3442no—
285C2×C4—1721no—
286C2×C4—1724no—
287C2×C4—1721no—
288C2×C4—1721no—
289C2×C4—1721no—
290C2×C4—1721no—
291C2×C4—1722no—
292C2×C4—1721no—
293C2×C4C4⋊C4×D4311yescharacteristic
294C2×C4#1306328211yescharacteristic
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22#1458275811yescharacteristic
305C22#99×D4311yescharacteristic
306C22#1458287111yescharacteristic
307C4—21no—
308C4—21no—
309C4—81no—
310C4—81no—
311C4—162no—
312C4—862no—
313C4—1722no—
314C4—1722no—
315C4—1721no—
316C4—1722no—
317C4—6882no—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—42no—
322C2#1913458211yescharacteristic
323C2#1913626111yescharacteristic
324C2#1913626811yescharacteristic
subgroupG/H
centerC22#99×D43
commutatorC22×C4×C43C22×C4
FrattiniC22×D4C22×D43
FittingC43×#1393C2
socleC22×C43C2×#99
prime radical, cosocleC22×D4×C43C23
hypercenter#1393D43
hypocenterC43#12800
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
7. C2×#13063290C4semidirect
10. #14582805C4semidirect
13. #14582768C4semidirect
15. #14582745C4semidirect
72. C2×Dic86C4⋊C4semidirect
90. C4×Dic43C4⋊C4semidirect
182. C43#12800semidirect

Maximal subgroups

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

subgroupindex
#191373052
C2×#145826932
#191345812
#191327202
C43×#13932
#1280043