Finite Group Databasebeta

← #25018049

Subgroups — #25018049

Order 11008.

1486 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14582765C211yesmaximal, characteristic
2C2×#14582745C211yesmaximal, characteristic
3#19132829C214yesmaximal
4#19132822C212yesmaximal
5#19132820C212yesmaximal
6C4×C43⋊#95C212yesmaximal
7#95×Dic43C212yesmaximal
8C2×C43×#376C211yesmaximal, characteristic
9C22×C4⋊Dic43C2212yes—
10C22×C4×Dic43C2214yes—
11C22×C4×Dic43C2212yes—
12#14582765C418yescomplemented
13#14582745C418yescomplemented
14C2×C43⋊#95C2214yes—
15C2×C43⋊#95C2214yes—
16C2×C43⋊#95C2214yes—
17C2×C43⋊#95C2212yes—
18C2×C43⋊#95C2211yescharacteristic
19C2×C43⋊#95C2212yes—
20C2×C43⋊#95C2211yescharacteristic
21C2×C43⋊#95C2212yes—
22C22×C42×C43C2212yes—
23C22×C42×C43C2211yescharacteristic
24C43×#376C418yescomplemented
25C2×#95×C43C2211yescharacteristic
26C2×#95×C43C2211yescharacteristic
27C2×#95×C43C2212yes—
28C23×Dic43C2314yes—
29C23×Dic43C2312yes—
30C23×Dic43C2312yes—
31C2×C4⋊Dic43—42no—
32C2×C4⋊Dic43—42no—
33C2×C4⋊Dic43C2×C418yes—
34C2×C4×Dic43—28no—
35C2×C4×Dic43—28no—
36C2×C4×Dic43—28no—
37C2×C4×Dic43—44no—
38C2×C4×Dic43C2×C418yes—
39C43⋊#95—28no—
40C43⋊#95—28no—
41C43⋊#95—44no—
42C43⋊#95C2×C414yes—
43C43⋊#95C2×C418yes—
44C43⋊#95C2×C414yes—
45C43⋊#95C2×C418yes—
46C43⋊#95C2×C418yes—
47C23×C4×C43C2312yes—
48C23×C4×C43C2311yescharacteristic
49C23×C4×C43C2311yescharacteristic
50C23×C4×C43C2311yescharacteristic
51C23×C4×C43C2312yes—
52C2×C42×C43—28no—
53C2×C42×C43C2×C414yes—
54C2×C42×C43C2×C414yes—
55C2×C42×C43C2×C418yes—
56C2×C42×C43C2×C414yes—
57#95×C43C2×C414yes—
58#95×C43C2×C418yes—
59#95×C43C2×C414yes—
60C24×C43C2411yescharacteristic
61C22×C4×C43—24no—
62C22×C4×C43—24no—
63C22×C4×C43—24no—
64C22×C4×C43—24no—
65C22×C4×C43—24no—
66C22×C4×C43—22no—
67C22×C4×C43—21no—
68C22×C4×C43—21no—
69C22×C4×C43—22no—
70C22×C4×C43—21no—
71C22×C4×C43—21no—
72C22×C4×C43—22no—
73C22×C4×C43—22no—
74C22×C4×C43C22×C412yes—
75C22×C4×C43C22×C412yes—
76C22×C4×C43C22×C412yes—
77C22×C4×C43C22×C412yes—
78C22×C4×C43C22×C412yes—
79C22×C4×C43C22×C412yes—
80C22×C4×C43C22×C414yes—
81C22×C4×C43C22×C412yes—
82C22×C4×C43C22×C412yes—
83C22×C4×C43C22×C412yes—
84C22×C4×C43C22×C414yes—
85C22×C4×C43#4118yes—
86C22×Dic43—24no—
87C22×Dic43—24no—
88C22×Dic43—24no—
89C22×Dic43—24no—
90C22×Dic43—24no—
91C22×Dic43—24no—
92C22×Dic43—24no—
93C22×Dic43—22no—
94C22×Dic43—22no—
95C22×Dic43—24no—
96C22×Dic43—24no—
97C22×Dic43—22no—
98C22×Dic43—22no—
99C22×Dic43C22×C414yes—
100C22×Dic43C22×C418yes—
101C22×Dic43C22×C414yes—
102C22×Dic43#4118yes—
103C42×C43—216no—
104C42×C43C42116yescomplemented
105C4⋊Dic43—48no—
106C4×Dic43—216no—
107C4×Dic43—416no—
108C23×C43C23×C411yescharacteristic
109C23×C43C23×C411yescharacteristic
110C23×C43C23×C411yescharacteristic
111C23×C43C2×#4114yes—
112C23×C43C2×#4114yes—
113C23×C43H(5,𝔽2)11yescharacteristic
114C23×C43H(5,𝔽2)11yescharacteristic
115C23×C432−1+411yescharacteristic
116C23×C432−1+411yescharacteristic
117C2×C4×C43—24no—
118C2×C4×C43—24no—
119C2×C4×C43—24no—
120C2×C4×C43—24no—
121C2×C4×C43—24no—
122C2×C4×C43—24no—
123C2×C4×C43—24no—
124C2×C4×C43—22no—
125C2×C4×C43—22no—
126C2×C4×C43—22no—
127C2×C4×C43—22no—
128C2×C4×C43—24no—
129C2×C4×C43—22no—
130C2×C4×C43—22no—
131C2×C4×C43—28no—
132C2×C4×C43—28no—
133C2×C4×C43—24no—
134C2×C4×C43—24no—
135C2×C4×C43—24no—
136C2×C4×C43—44no—
137C2×C4×C43—44no—
138C2×C4×C43C2×C4214yes—
139C2×C4×C43C2×C4214yes—
140C2×C4×C43C2×C4214yes—
141C2×C4×C43#11718yes—
142C2×Dic43—24no—
143C2×Dic43—24no—
144C2×Dic43—28no—
145C2×Dic43—28no—
146C2×Dic43—24no—
147C2×Dic43—28no—
148C2×Dic43—24no—
149C2×Dic43—28no—
150C2×Dic43—28no—
151C2×Dic43—24no—
152C2×Dic43—44no—
153C2×Dic43—44no—
154C2×Dic43—44no—
155C2×Dic43—44no—
156C2×Dic43—44no—
157C2×Dic43—44no—
158C2×Dic43#11718yes—
159#13805—431nomaximal, Sylow
160C22×C43C22×C4211yescharacteristic
161C22×C43C2×#11712yes—
162C22×C43C2×#11712yes—
163C22×C43C4×#4112yes—
164C22×C43C4×#4112yes—
165C22×C43C4×#4112yes—
166C22×C43C4×#4112yes—
167C22×C43#51811yescharacteristic
168C22×C43#51911yescharacteristic
169C22×C43#52011yescharacteristic
170C22×C43#52011yescharacteristic
171C22×C43#52011yescharacteristic
172C22×C43#52011yescharacteristic
173C22×C43#55114yes—
174C22×C43#55214yes—
175C22×C43#55214yes—
176C22×C43#55514yes—
177C4×C43—28no—
178C4×C43—24no—
179C4×C43—24no—
180C4×C43—24no—
181C4×C43—44no—
182Dic43—28no—
183Dic43—44no—
184Dic43—48no—
185C2×#382—431no—
186C2×#376—431no—
187C2×#376D4311yescharacteristic
188#1436—434no—
189#1434—432no—
190#1434—432no—
191C4×#95—432no—
192C4×#95—432no—
193C2×C43C4×#11711yescharacteristic
194C2×C43C4×#11711yescharacteristic
195C2×C43#227411yescharacteristic
196C2×C43#234312yes—
197C2×C43#234312yes—
198C2×C43#234612yes—
199C2×C43#234612yes—
200C2×C43#237112yes—
201C2×C43#237212yes—
202C22×C4⋊C4—432no—
203C22×C42—432no—
204C22×C42—434no—
205C22×C42C2×D4312yes—
206C22×C42C2×D4311yescharacteristic
207#382—438no—
208#376—438no—
209#376Dic4318yescomplemented
210C2×#95—431no—
211C2×#95—432no—
212C2×#95—432no—
213C2×#95—431no—
214C2×#95—432no—
215C2×#95—434no—
216C2×#95—434no—
217C2×#95—434no—
218C2×#95C2×D4311yescharacteristic
219C2×#95C2×D4311yescharacteristic
220C2×#95C2×D4312yes—
221C43#1380511yescharacteristic, complemented, Sylow
222C23×C4—432no—
223C23×C4—434no—
224C23×C4—432no—
225C23×C4C22×D4312yes—
226C23×C4C22×D4311yescharacteristic
227C23×C4C22×D4311yescharacteristic
228C23×C4C22×D4311yescharacteristic
229C23×C4C22×D4312yes—
230C2×C4⋊C4—438no—
231C2×C4⋊C4—1722no—
232C2×C4⋊C4—1722no—
233C2×C42—28no—
234C2×C42—438no—
235C2×C42—868no—
236C2×C42—868no—
237C2×C42—868no—
238C2×C42—1724no—
239C2×C42C4×D4314yes—
240C2×C42C4×D4314yes—
241C2×C42C2×Dic4318yes—
242C2×C42C2×Dic4314yes—
243#95—434no—
244#95—438no—
245#95—434no—
246#95—438no—
247#95—438no—
248#95—868no—
249#95—868no—
250#95—1724no—
251#95C2×Dic4314yes—
252#95C2×Dic4318yes—
253#95C2×Dic4314yes—
254C24C23×D4311yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—434no—
269C22×C4—438no—
270C22×C4—434no—
271C22×C4—438no—
272C22×C4—864no—
273C22×C4—862no—
274C22×C4—862no—
275C22×C4—864no—
276C22×C4—864no—
277C22×C4—864no—
278C22×C4—864no—
279C22×C4—864no—
280C22×C4—864no—
281C22×C4—862no—
282C22×C4—864no—
283C22×C4—864no—
284C22×C4—862no—
285C22×C4C2×C4×D4312yes—
286C22×C4C2×C4×D4312yes—
287C22×C4C2×C4×D4312yes—
288C22×C4C2×C4×D4312yes—
289C22×C4C2×C4×D4312yes—
290C22×C4C2×C4×D4312yes—
291C22×C4#1062189818yes—
292C22×C4C22×Dic4314yes—
293C22×C4C22×Dic4312yes—
294C22×C4C22×Dic4312yes—
295C22×C4C22×Dic4312yes—
296C22×C4C22×Dic4314yes—
297C4⋊C4—1728no—
298C42—216no—
299C42—8616no—
300C42—17216no—
301C42C4×Dic43116yescomplemented
302C23C22×C4×D4311yescharacteristic
303C23C22×C4×D4311yescharacteristic
304C23C23×Dic4311yescharacteristic
305C23#41×D4314yes—
306C23C2×#1062189814yes—
307C23#1306337111yescharacteristic
308C23#1306337111yescharacteristic
309C23#1306337211yescharacteristic
310C23#1306337211yescharacteristic
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—438no—
333C2×C4—868no—
334C2×C4—868no—
335C2×C4—864no—
336C2×C4—864no—
337C2×C4—864no—
338C2×C4—868no—
339C2×C4—868no—
340C2×C4—864no—
341C2×C4—868no—
342C2×C4—864no—
343C2×C4—1724no—
344C2×C4—1724no—
345C2×C4—1724no—
346C2×C4—1724no—
347C2×C4—1724no—
348C2×C4—1724no—
349C2×C4C2×C4×Dic4314yes—
350C2×C4C2×C4×Dic4314yes—
351C2×C4C2×C4×Dic4314yes—
352C2×C4#1306328418yes—
353C22C22×C4×Dic4311yescharacteristic
354C22#117×D4312yes—
355C22C2×#1306328412yes—
356C22#1458336712yes—
357C22C4×#1062189812yes—
358C22C4×#1062189812yes—
359C22#41×Dic4312yes—
360C22#1458337311yescharacteristic
361C22#1458337611yescharacteristic
362C22#1458337911yescharacteristic
363C22#1458337911yescharacteristic
364C22#1458338211yescharacteristic
365C22#1458338211yescharacteristic
366C22#1458357314yes—
367C22#1458358214yes—
368C22#1458358814yes—
369C22#1458361214yes—
370C4—24no—
371C4—24no—
372C4—28no—
373C4—24no—
374C4—44no—
375C4—868no—
376C4—1728no—
377C4—1724no—
378C2C4×#1306328411yescharacteristic
379C2#117×Dic4311yescharacteristic
380C2#1913695111yescharacteristic
381C2#1913745512yes—
382C2#1913745612yes—
383C2#1913748012yes—
384C2#1913748112yes—
385C2#1913770012yes—
386C2#1913770512yes—
subgroupG/H
center, FrattiniC24C23×D43
commutatorC22×C43C22×C42
FittingC2×C43×#376C2
socle, prime radical, cosocleC24×C43C24
hypercenterC2×#376D43
hypocenterC43#13805
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C24 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
12. #14582765C4semidirect
13. #14582745C4semidirect
24. C43×#376C4semidirect
104. C42×C43C42semidirect
209. #376Dic43semidirect
221. C43#13805semidirect
301. C42C4×Dic43semidirect

Maximal subgroups

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

subgroupindex
C2×#145827652
C2×#145827452
#191328292
#191328222
#191328202
C4×C43⋊#952
#95×Dic432
C2×C43×#3762
#1380543