Finite Group Databasebeta

← #25124006

Subgroups — #25124006

Order 11008.

1506 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1506 conjugacy classes of subgroups — too many to draw. Only its 209 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#543×D43C211yesmaximal, characteristic, complemented
2#19137172C213yesmaximal
3#19135762C213yesmaximal, complemented
4#19135760C213yesmaximal, complemented
5#19135758C211yesmaximal, characteristic, complemented
6#342×D43C213yesmaximal, complemented
7C43×#2030C211yesmaximal, characteristic, complemented
8C23×D172C2211yescharacteristic
9#14583525C2213yes—
10#14583524—26no—
11C22⋊Q8×D43—26no—
12C2×C22⋊C4×D43C2213yes—
13C2×C22⋊C4×D43C2213yes—
14#14582777—26no—
15#14582777—26no—
16#14582762C2213yes—
17#14582754C2213yes—
18#14582754C2213yes—
19#14582666C2213yes—
20#14582665C2216yes—
21C43×#543C2211yescharacteristic
22C43×#381C2213yes—
23C43×#342C2213yes—
24C24×D43C2311yescharacteristic
25C24×D43C2311yescharacteristic
26C2×Q8×D43—22no—
27C22×D172—23no—
28C22×D172—23no—
29C22×D172—26no—
30C22×D172Q814yescomplemented
31C22×C4×D43—23no—
32C22×C4×D43D416yes—
33#13063305—26no—
34#13063304—26no—
35#13063303—212no—
36#13063302—212no—
37#13063301—26no—
38C4⋊C4×D43—26no—
39C2×C22⋊Dic43C2313yes—
40C2×C22⋊Dic43C2313yes—
41C2×(C2×D43)⋊C4—23no—
42C2×(C2×D43)⋊C4—26no—
43C2×(C2×D43)⋊C4—26no—
44C2×(C2×D43)⋊C4D416yes—
45C22⋊C4×D43—26no—
46C22⋊C4×D43—26no—
47C22⋊C4×D43—212no—
48C43⋊#95—26no—
49C23×C4×C43C2311yescharacteristic
50C22⋊Q8×C43—26no—
51C2×C22⋊C4×C43C2313yes—
52C2×C22⋊C4×C43C2313yes—
53#95×C43—26no—
54C24×C43C2411yescharacteristic
55C23×D43—23no—
56C23×D43—23no—
57C23×D43—82no—
58C23×D43—82no—
59C23×D43C2×D413yes—
60C23×D43C2×D413yes—
61C23×D43C2×Q812yes—
62C23×D43C2×Q812yescomplemented
63C23×D43#4116yes—
64C2×Q8×C43—22no—
65C22×C4×C43—23no—
66C22×C4×C43—23no—
67C22×C4×C43—23no—
68C22×C4×C43C2×D416yes—
69C22×C4×C43C2×Q812yescomplemented
70C22×Dic43—23no—
71C22×Dic43C2×D416yes—
72Q8×D43—84no—
73C2×D172—212no—
74C2×D172—212no—
75C2×D172—43no—
76C2×D172—41no—
77C2×D172—46no—
78C2×D172—46no—
79C2×C4×D43—26no—
80C2×C4×D43—26no—
81C2×C4×D43—46no—
82C2×C4×D43—86no—
83C2×C4×D43—86no—
84C2×Dic86—26no—
85C4⋊C4×C43—26no—
86C22⋊C4×C43—26no—
87C22⋊C4×C43—26no—
88C22⋊Dic43—26no—
89C22⋊Dic43—26no—
90(C2×D43)⋊C4—212no—
91(C2×D43)⋊C4—46no—
92(C2×D43)⋊C4—86no—
93(C2×D43)⋊C4—86no—
94C4⋊Dic43—26no—
95Dic43⋊C4—212no—
96C23×C43—42no—
97C23×C43C22×D413yes—
98C23×C43C22×Q811yescharacteristic
99C23×C43C2×#4113yes—
100C22×D43—26no—
101C22×D43—46no—
102C22×D43—46no—
103C22×D43—46no—
104C22×D43—46no—
105C22×D43—41no—
106C22×D43—43no—
107C22×D43—82no—
108C22×D43—82no—
109C22×D43—86no—
110C22×D43—86no—
111C22×D43#12013yes—
112C22×D43#12011yescharacteristic
113C22×D43C22⋊Q816yes—
114C22×D43C22⋊Q816yes—
115C22×D43#12416yes—
116C22×D43#12416yes—
117Q8×C43—82no—
118C2×C4×C43—26no—
119C2×C4×C43—26no—
120C2×C4×C43—41no—
121C2×C4×C43—43no—
122C2×C4×C43—46no—
123C2×C4×C43—46no—
124C2×C4×C43—43no—
125C2×Dic43—26no—
126C2×Dic43—46no—
127C2×Dic43—46no—
128C2×Dic43—43no—
129D172—412no—
130D172—44no—
131C4×D43—412no—
132C4×D43—86no—
133C4×D43—812no—
134Dic86—86no—
135#23468—431nomaximal, Sylow
136C22×C43—26no—
137C22×C43—41no—
138C22×C43—43no—
139C22×C43C2×#12011yescharacteristic
140C22×C43C2×C22⋊Q813yes—
141C22×C43C2×#12413yes—
142C2×D43—22no—
143C2×D43—26no—
144C2×D43—46no—
145C2×D43—46no—
146C2×D43—46no—
147C2×D43—46no—
148C2×D43—46no—
149C2×D43—46no—
150C2×D43—42no—
151C2×D43—86no—
152C2×D43—82no—
153C2×D43#39316yes—
154C2×D43#39312yes—
155C4×C43—42no—
156C4×C43—46no—
157C4×C43—83no—
158Dic43—46no—
159Dic43—83no—
160C2×#543—431no—
161#2308—433no—
162#2030—433no—
163#2030D4311yescharacteristic, complemented
164#2029—433no—
165#2029—431no—
166C2×#342—433no—
167C2×C43—23no—
168C2×C43—23no—
169C2×C43C2×#39311yescharacteristic
170C2×C43#302612yes—
171D43—22no—
172D43—26no—
173D43—82no—
174C23×D4—431no—
175#543—433no—
176#543—866no—
177#543C2×D4311yescharacteristic
178C2×C22⋊Q8—866no—
179C22×C22⋊C4—433no—
180C22×C22⋊C4—433no—
181#386—866no—
182#386—866no—
183#381—433no—
184#381C2×D4313yes—
185#379—433no—
186#379—433no—
187#342—436no—
188#342—433no—
189#342C2×D4313yes—
190C43#2346811yescharacteristic, complemented, Sylow
191C25—431no—
192C25—431no—
193C22×Q8—862no—
194C22×D4—434no—
195C22×D4—863no—
196C22×D4—863no—
197C22×D4—866no—
198C23×C4—436no—
199C23×C4—863no—
200C23×C4C22×D4311yescharacteristic
201C22⋊Q8—26no—
202C22⋊Q8—8612no—
203C22⋊Q8—8612no—
204C22⋊Q8—866no—
205C22⋊Q8—866no—
206C22⋊Q8—866no—
207C2×C4⋊C4—866no—
208C2×C22⋊C4—433no—
209C2×C22⋊C4—436no—
210C2×C22⋊C4—433no—
211C2×C22⋊C4—866no—
212C2×C22⋊C4—866no—
213C2×C22⋊C4—8612no—
214C2×C22⋊C4—866no—
215C2×C22⋊C4—866no—
216C2×C22⋊C4—863no—
217C2×C22⋊C4C22×D4313yes—
218C2×C22⋊C4C22×D4313yes—
219#95—26no—
220#95—866no—
221C24—432no—
222C24—436no—
223C24—433no—
224C24—433no—
225C24—432no—
226C24—863no—
227C24—863no—
228C24—3442no—
229C24—3442no—
230C24C23×D4311yescharacteristic
231C2×Q8—22no—
232C2×Q8—866no—
233C2×Q8—3444no—
234C2×D4—8612no—
235C2×D4—8612no—
236C2×D4—1723no—
237C2×D4—1721no—
238C2×D4—1726no—
239C2×D4—1726no—
240C22×C4—23no—
241C22×C4—23no—
242C22×C4—23no—
243C22×C4—436no—
244C22×C4—866no—
245C22×C4—866no—
246C22×C4—863no—
247C22×C4—3446no—
248C22×C4—3446no—
249C22×C4—1726no—
250C22×C4D4×D4316yes—
251C22×C4Q8×D4312yescomplemented
252C4⋊C4—26no—
253C4⋊C4—8612no—
254C4⋊C4—866no—
255C22⋊C4—26no—
256C22⋊C4—26no—
257C22⋊C4—8612no—
258C22⋊C4—866no—
259C22⋊C4—866no—
260C22⋊C4—3446no—
261C22⋊C4—3446no—
262C22⋊C4—1726no—
263C23—42no—
264C23—433no—
265C23—431no—
266C23—436no—
267C23—436no—
268C23—436no—
269C23—436no—
270C23—866no—
271C23—3446no—
272C23—3442no—
273C23—3442no—
274C23—3446no—
275C23—1726no—
276C23—1726no—
277C23—1726no—
278C23—1726no—
279C23—1721no—
280C23—1723no—
281C23C2×D4×D4313yes—
282C23C2×Q8×D4311yescharacteristic
283C23#41×D4313yes—
284Q8—82no—
285Q8—3446no—
286D4—17212no—
287D4—1724no—
288C2×C4—26no—
289C2×C4—26no—
290C2×C4—41no—
291C2×C4—43no—
292C2×C4—46no—
293C2×C4—46no—
294C2×C4—43no—
295C2×C4—866no—
296C2×C4—3446no—
297C2×C4—34412no—
298C2×C4—17212no—
299C2×C4—1726no—
300C2×C4—1723no—
301C2×C4—1726no—
302C22—26no—
303C22—41no—
304C22—43no—
305C22—436no—
306C22—432no—
307C22—866no—
308C22—862no—
309C22—3446no—
310C22—3442no—
311C22—1726no—
312C22—1726no—
313C22—1726no—
314C22—1726no—
315C22—1726no—
316C22—1726no—
317C22—1722no—
318C22#120×D4311yescharacteristic
319C22C22⋊Q8×D4313yes—
320C22#124×D4313yes—
321C4—42no—
322C4—46no—
323C4—83no—
324C4—3443no—
325C4—1726no—
326C2—23no—
327C2—23no—
328C2—866no—
329C2—862no—
330C2—3442no—
331C2#393×D4311yescharacteristic
332C2#1914399312yes—
subgroupG/H
centerC22#120×D43
commutator, cosocleC24×C43C24
FrattiniC24C23×D43
FittingC43×#2030C2
socleC22×C43C2×#120
prime radicalC23×D172C22
hypercenter#2030D43
hypocenterC43#23468
LCSG ▷ C24×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C24 ◁ #2030

Complements

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

NH ≅ G/Ntype
1. #543×D43C2semidirect
3. #19135762C2semidirect
4. #19135760C2semidirect
5. #19135758C2semidirect
6. #342×D43C2semidirect
7. C43×#2030C2semidirect
30. C22×D172Q8semidirect
62. C23×D43C2×Q8semidirect
69. C22×C4×C43C2×Q8semidirect
163. #2030D43semidirect
190. C43#23468semidirect
251. C22×C4Q8×D43semidirect

Maximal subgroups

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

subgroupindex
#543×D432
#191371722
#191357622
#191357602
#191357582
#342×D432
C43×#20302
#2346843