Finite Group Databasebeta

← #25001765

Subgroups — #25001765

Order 11008.

842 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582857C211yesmaximal, characteristic
2C2×#14582855C212yesmaximal
3C2×C43⋊#338C211yesmaximal, characteristic
4C2×#14582655C212yesmaximal
5C2×C43×H(3,ℤ/4)C211yesmaximal, characteristic
6C22×#10621877C2212yes—
7C23×C43⋊C8C2211yescharacteristic
8C2×#13063284C2211yescharacteristic
9#14582857—24no—
10#14582855—24no—
11#14582855—24no—
12C2×#13063236C414yescomplemented
13C2×#13063235C412yescomplemented
14C2×C4×C43⋊C8C414yescomplemented
15C2×C8×Dic43C412yescomplemented
16C43⋊#338C414yescomplemented
17#14582655C418yescomplemented
18C22×M16×C43C2211yescharacteristic
19C2×#117×C43C2212yes—
20C43×H(3,ℤ/4)C414yescomplemented
21C2×#10621877—22no—
22C2×#10621877—22no—
23C2×#10621877—22no—
24C2×#10621877—22no—
25C2×#10621877C2×C414yes—
26C2×#10621877C2×C414yes—
27C22×C43⋊C8C2×C414yes—
28C22×C43⋊C8C2×C411yescharacteristic
29C22×C43⋊C8C2×C412yes—
30C22×C43⋊C8C2×C411yescharacteristic
31C22×C43⋊C8C2×C412yes—
32C22×C43⋊C8D412yes—
33C22×C43⋊C8D414yes—
34C22×C43⋊C8Q812yes—
35#13063284—22no—
36#13063284C2×C414yes—
37C2×C4⋊Dic43—21no—
38C2×C22⋊Dic43—21no—
39C2×C4×Dic43C2×C412yes—
40#13063236—28no—
41#13063235—24no—
42C4×C43⋊C8—28no—
43C8×Dic43—24no—
44C23×C4×C43C2311yescharacteristic
45C2×M16×C43—22no—
46C2×M16×C43—22no—
47C2×M16×C43C2×C412yes—
48C2×M16×C43C2×C412yes—
49C22×C8×C43C2×C412yes—
50#117×C43—22no—
51#117×C43—22no—
52#117×C43C2×C414yes—
53#117×C43C2×C414yes—
54C2×C4⋊C4×C43—22no—
55C2×C22⋊C4×C43—22no—
56C2×C42×C43C2×C414yes—
57C24×C43#4111yescharacteristic
58C22×C4×C43—24no—
59C22×C4×C43C22×C412yes—
60C22×C4×C43C22×C411yescharacteristic
61C22×C4×C43C22×C412yes—
62C22×C4×C43C22×C411yescharacteristic
63C22×C4×C43C22×C411yescharacteristic
64C22×C4×C43C2×D411yescharacteristic
65C22×C4×C43C2×D412yes—
66C22×C4×C43C2×Q811yescharacteristic
67C22×C4×C43#4112yes—
68C22×C4×C43#4111yescharacteristic
69C22×Dic43—22no—
70M16×C43—24no—
71M16×C43—24no—
72C2×C8×C43—22no—
73C2×C8×C43—22no—
74C2×C8×C43C4214yes—
75C4⋊C4×C43—24no—
76C4⋊C4×C43—24no—
77C22⋊C4×C43—24no—
78C22⋊C4×C43—24no—
79C42×C43—24no—
80C42×C43C4218yes—
81C22⋊Dic43—24no—
82C4⋊Dic43—24no—
83C4×Dic43—22no—
84C4×Dic43C4214yes—
85#10621877—24no—
86#10621877—24no—
87#10621877—24no—
88#10621877—24no—
89C2×C43⋊C8—24no—
90C2×C43⋊C8—24no—
91C2×C43⋊C8C4218yes—
92C2×C43⋊C8C4211yescharacteristic
93C2×C43⋊C8C4211yescharacteristic
94C2×C43⋊C8C4212yes—
95C2×C43⋊C8C22⋊C412yes—
96C2×C43⋊C8C22⋊C414yes—
97C2×C43⋊C8C22⋊C412yes—
98C2×C43⋊C8C22⋊C414yes—
99C2×C43⋊C8C4⋊C412yes—
100C2×C43⋊C8C4⋊C414yes—
101C2×C43⋊C8C4⋊C412yes—
102C2×C43⋊C8C4⋊C414yes—
103C23×C43—42no—
104C23×C43#11712yes—
105C23×C43#11711yescharacteristic
106C23×C43C4×D412yes—
107C23×C43C4×D411yescharacteristic
108C23×C43C4×Q811yescharacteristic
109C2×C4×C43—28no—
110C2×C4×C43—48no—
111C2×C4×C43C2×C4212yes—
112C2×C4×C43C2×C4211yescharacteristic
113C2×C4×C43C2×C4211yescharacteristic
114C2×C4×C43C2×C4211yescharacteristic
115C2×C4×C43C2×C4212yes—
116C2×C4×C43C2×C22⋊C412yes—
117C2×C4×C43C2×C22⋊C411yescharacteristic
118C2×C4×C43C2×C4⋊C411yescharacteristic
119C2×C4×C43C2×C4⋊C412yes—
120C2×C4×C43#11711yescharacteristic
121C2×C4×C43#11712yes—
122C2×C4×C43C4×D412yes—
123C2×C4×C43C4×D412yes—
124C2×C4×C43C4×D411yescharacteristic
125C2×C4×C43C4×D412yes—
126C2×C4×C43C4×D412yes—
127C2×C4×C43C4×Q812yes—
128C2×C4×C43C4×Q811yescharacteristic
129C2×Dic43—24no—
130C2×Dic43—44no—
131C8×C43—24no—
132C43⋊C8—28no—
133C43⋊C8#9512yes—
134C43⋊C8#9512yes—
135C43⋊C8#9514yes—
136#10699—431nomaximal, Sylow
137C22×C43—22no—
138C22×C43—24no—
139C22×C43—42no—
140C22×C43—41no—
141C22×C43—41no—
142C22×C43#37611yescharacteristic
143C22×C43C4×C22⋊C411yescharacteristic
144C22×C43C4×C22⋊C412yes—
145C22×C43C4×C4⋊C412yes—
146C22×C43C4×C4⋊C411yescharacteristic
147C4×C43—48no—
148C4×C43C4311yescharacteristic
149C4×C43C2×#9511yescharacteristic
150C4×C43C4×C22⋊C412yes—
151C4×C43C4×C22⋊C411yescharacteristic
152C4×C43C4×C4⋊C411yescharacteristic
153C4×C43C4×C4⋊C412yes—
154Dic43—44no—
155C2×#405—431no—
156C2×#405—432no—
157C2×#338—431no—
158C2×H(3,ℤ/4)—432no—
159C2×H(3,ℤ/4)D4311yescharacteristic
160C2×C43—21no—
161C2×C43—22no—
162C2×C43—22no—
163C2×C43—21no—
164C2×C43C4×#9511yescharacteristic
165C2×C43#174712yes—
166C22×M16—432no—
167C22×M16C2×D4311yescharacteristic
168C23×C8—431no—
169C2×#117—431no—
170C2×#117C2×D4312yes—
171#405—864no—
172#405—864no—
173#405—864no—
174C2×#97—432no—
175C2×#97—434no—
176C2×C4×C8—432no—
177C2×C4×C8—434no—
178#338—434no—
179H(3,ℤ/4)—438no—
180H(3,ℤ/4)Dic4314yescomplemented
181C43#1069911yescharacteristic, complemented, Sylow
182C23×C4C22×D4311yescharacteristic
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—434no—
186C2×M16—434no—
187C2×M16—862no—
188C2×M16—862no—
189C2×M16—862no—
190C2×M16—862no—
191C2×M16C4×D4312yes—
192C2×M16C2×Dic4312yes—
193C22×C8—431no—
194C22×C8—432no—
195C22×C8—431no—
196C22×C8—432no—
197C22×C8—432no—
198C22×C8—434no—
199C22×C8—432no—
200C22×C8—434no—
201C22×C8C4×D4312yes—
202#117—22no—
203#117—22no—
204#117—434no—
205#117—862no—
206#117C4×D4314yes—
207#117C2×Dic4314yes—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—861no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—861no—
212C2×C42—432no—
213C2×C42C4×D4314yes—
214#97—868no—
215#97—864no—
216C4×C8—868no—
217C4×C8—864no—
218C24#1062190011yescharacteristic
219C22×C4—24no—
220C22×C4—862no—
221C22×C4C2×C4×D4311yescharacteristic
222C22×C4C2×C4×D4312yes—
223C22×C4C2×C4×D4311yescharacteristic
224C22×C4C2×C4×D4312yes—
225C22×C4D4×D4311yescharacteristic
226C22×C4D4×D4312yes—
227C22×C4#1062190012yes—
228C22×C4Q8×D4311yescharacteristic
229C22×C4#1062190211yescharacteristic
230C22×C4C22×Dic4311yescharacteristic
231M16—24no—
232M16—24no—
233M16—864no—
234M16—864no—
235M16—864no—
236M16—864no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—431no—
240C2×C8—431no—
241C2×C8—432no—
242C2×C8—434no—
243C2×C8—432no—
244C2×C8—434no—
245C2×C8—432no—
246C2×C8—432no—
247C2×C8—432no—
248C2×C8—434no—
249C2×C8—434no—
250C2×C8—438no—
251C2×C8—864no—
252C2×C8—864no—
253C2×C8C4×Dic4314yes—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—864no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—864no—
260C42—24no—
261C42—434no—
262C42—862no—
263C42C4×Dic4318yes—
264C23—42no—
265C23#1306328112yes—
266C23#1306328311yescharacteristic
267C23(C22⋊D43)⋊C412yes—
268C23D4×Dic4311yescharacteristic
269C23#1306329011yescharacteristic
270C2×C4—28no—
271C2×C4—48no—
272C2×C4—864no—
273C2×C4—1724no—
274C2×C4C42×D4312yes—
275C2×C4C42×D4311yescharacteristic
276C2×C4C42×D4311yescharacteristic
277C2×C4C2×C4×Dic4311yescharacteristic
278C2×C4C2×C4×Dic4312yes—
279C2×C4C22⋊C4×D4312yes—
280C2×C4C22⋊C4×D4311yescharacteristic
281C2×C4C4⋊C4×D4311yescharacteristic
282C2×C4C4⋊C4×D4312yes—
283C2×C4#1306328111yescharacteristic
284C2×C4#1306328312yes—
285C2×C4(C22⋊D43)⋊C412yes—
286C2×C4(C22⋊D43)⋊C412yes—
287C2×C4#1306328711yescharacteristic
288C2×C4#1306328712yes—
289C2×C4D4×Dic4312yes—
290C2×C4#1306329012yes—
291C2×C4Q8×Dic4311yescharacteristic
292C8—24no—
293C8—432no—
294C8—434no—
295C8—432no—
296C8—868no—
297C22—24no—
298C22—22no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22#1458274411yescharacteristic
303C22#1458274611yescharacteristic
304C22C22⋊C4×Dic4312yes—
305C22#1458275012yes—
306C22C4⋊C4×Dic4311yescharacteristic
307C4—48no—
308C4—1724no—
309C4C42×Dic4311yescharacteristic
310C4#95×D4311yescharacteristic
311C4#1458274612yes—
312C4C22⋊C4×Dic4311yescharacteristic
313C4#1458275011yescharacteristic
314C4C4⋊C4×Dic4312yes—
315C2—22no—
316C2—21no—
317C2—22no—
318C2—21no—
319C2#95×Dic4311yescharacteristic
320C2#1913434912yes—
subgroupG/H
centerC2×C4C42×D43
commutatorC4×C43C43
FrattiniC23×C4C22×D43
FittingC2×C43×H(3,ℤ/4)C2
socleC22×C43#376
prime radicalC24×C43#41
cosocleC23×C4×C43C23
hypercenterC2×H(3,ℤ/4)D43
hypocenterC43#10699
LCSG ▷ C4×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
12. C2×#13063236C4semidirect
13. C2×#13063235C4semidirect
14. C2×C4×C43⋊C8C4semidirect
15. C2×C8×Dic43C4semidirect
16. C43⋊#338C4semidirect
17. #14582655C4semidirect
20. C43×H(3,ℤ/4)C4semidirect
180. H(3,ℤ/4)Dic43semidirect
181. C43#10699semidirect

Maximal subgroups

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

subgroupindex
C2×#145828572
C2×#145828552
C2×C43⋊#3382
C2×#145826552
C2×C43×H(3,ℤ/4)2
#1069943