Finite Group Databasebeta

← #25256616

Subgroups — #25256616

Order 11008.

5430 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10621902C211yesmaximal, characteristic
2C2×#14583373C212yesmaximal, complemented
3#19137424C2116yesmaximal, complemented
4#19137181C211yesmaximal, characteristic, complemented
5#19137172C214yesmaximal, complemented
6#19136974C212yesmaximal, complemented
7#381×D43C214yesmaximal, complemented
8C43×#2310C211yesmaximal, characteristic, complemented
9C22×#10621902—216no—
10C22×#10621902C4116yescomplemented
11C22×#10621902C2218yes—
12C23×D172C2212yes—
13C23×D172C2211yescharacteristic
14C23×C4×D43C2212yes—
15C23×C4×D43C2211yescharacteristic
16#14583373—216no—
17C2×C4×D172C2218yescomplemented
18C2×#13063287C2218yescomplemented
19C2×#13063284C2212yes—
20C2×C22⋊C4×D43C2218yes—
21#14582806C2218yescomplemented
22#14582777C22132yescomplemented
23#14582762C2214yes—
24#14582761C22116yescomplemented
25#14582760C2214yes—
26#14582756C2218yescomplemented
27#14582754C2214yes—
28C4×(C2×D43)⋊C4C2218yescomplemented
29#95×D43C22116yescomplemented
30C23×Q8×C43C2211yescharacteristic
31C2×#117×C43C2212yes—
32C43×#391C2218yescomplemented
33C43×#381C2214yes—
34C24×D43C2311yescharacteristic
35C24×D43C2312yes—
36C2×#10621902—2128no—
37C2×#10621902—232no—
38C2×#10621902—232no—
39C2×#10621902C2×C4164yescomplemented
40C22×D172—216no—
41C22×D172C2×C418yes—
42C22×D172C2×C4116yescomplemented
43C22×D172D4116yes—
44C22×D172C2318yes—
45C22×D172C2314yes—
46C23×Dic43C2311yescharacteristic
47C22×C4×D43—216no—
48C22×C4×D43—28no—
49C22×C4×D43C2×C418yescomplemented
50C22×C4×D43C2×C4116yescomplemented
51C22×C4×D43C2318yes—
52C22×C4×D43C2314yes—
53C22×C4×D43C23116yes—
54C4×D172—232no—
55#13063287—232no—
56#13063284—28no—
57C2×C4⋊Dic43C2314yes—
58C2×C22⋊Dic43C2314yes—
59C2×(C2×D43)⋊C4C23116yes—
60C2×(C2×D43)⋊C4C23116yes—
61C22⋊C4×D43—232no—
62C2×C4×Dic43C2314yes—
63C43⋊#95C23116yes—
64C43⋊#95C23116yes—
65C22×Q8×C43—28no—
66C22×Q8×C43C2×C418yescomplemented
67C22×Q8×C43C2314yes—
68C23×C4×C43C2312yes—
69C23×C4×C43C2311yescharacteristic
70#117×C43—28no—
71C2×C4⋊C4×C43C2314yes—
72C2×C22⋊C4×C43C2314yes—
73C2×C42×C43C2314yes—
74#95×C43C23116yes—
75C24×C43C2411yescharacteristic
76C23×D43—24no—
77C23×D43—48no—
78C23×D43—44no—
79C23×D43C22×C414yes—
80C23×D43C22×C418yes—
81C23×D43C2×D4116yes—
82C23×D43C2412yes—
83C23×D43C2414yes—
84C2×Q8×C43—232no—
85C2×Q8×C43—28no—
86C2×Q8×C43—28no—
87C2×Q8×C43C22×C4116yes—
88C22×C4×C43—28no—
89C22×C4×C43C22×C418yes—
90C22×C4×C43C22×C414yes—
91C22×C4×C43C2×D418yes—
92C22×C4×C43C2418yes—
93C22×C4×C43C2414yes—
94C22×C4×C43C2412yes—
95C22×Dic43—24no—
96C22×Dic43C22×C414yes—
97C22×Dic43C2418yes—
98C22×Dic43C2412yes—
99#10621902—2256no—
100C2×D172—264no—
101C2×D172—232no—
102C2×D172C22⋊C4164yescomplemented
103C2×D172C22×C4132yes—
104C2×D172C22×C4116yes—
105C2×D172C2×D4116yes—
106C2×D172C2×D4116yes—
107C2×C4×D43—232no—
108C2×C4×D43—28no—
109C2×C4×D43—28no—
110C2×C4×D43—264no—
111C2×C4×D43—232no—
112C2×C4×D43—232no—
113C2×C4×D43—432no—
114C2×C4×D43C22×C4116yes—
115C2×C4×D43C22×C4132yes—
116C4⋊C4×C43—28no—
117C22⋊C4×C43—28no—
118C42×C43—28no—
119C22⋊Dic43—28no—
120(C2×D43)⋊C4—232no—
121(C2×D43)⋊C4—416no—
122C4⋊Dic43—28no—
123C4×Dic43—28no—
124C23×C43—24no—
125C23×C43C23×C412yes—
126C23×C43C22×D414yes—
127C23×C43C2511yescharacteristic
128C22×D43—216no—
129C22×D43—28no—
130C22×D43—24no—
131C22×D43—28no—
132C22×D43—416no—
133C22×D43—48no—
134C22×D43—432no—
135C22×D43—416no—
136C22×D43C2×C22⋊C4132yes—
137C22×D43C23×C414yes—
138C22×D43C23×C418yes—
139C22×D43C22×D4116yes—
140Q8×C43—232no—
141C2×C4×C43—216no—
142C2×C4×C43—28no—
143C2×C4×C43—28no—
144C2×C4×C43—216no—
145C2×C4×C43C2×C22⋊C4116yes—
146C2×C4×C43C23×C418yes—
147C2×C4×C43C23×C414yes—
148C2×C4×C43C22×D414yes—
149C2×C4×C43C22×D414yes—
150C2×Dic43—28no—
151C2×Dic43—24no—
152C2×Dic43—28no—
153C2×Dic43—216no—
154C2×Dic43C23×C414yes—
155D172—264no—
156D172C2×C22⋊C4164yes—
157C4×D43—264no—
158C4×D43—232no—
159C4×D43—832no—
160#34172—431nomaximal, Sylow
161C22×C43—24no—
162C22×C43—28no—
163C22×C43C22×C22⋊C414yes—
164C22×C43C24×C411yescharacteristic
165C22×C43C23×D412yes—
166C22×C43#58514yes—
167C2×D43—216no—
168C2×D43—232no—
169C2×D43—24no—
170C2×D43—28no—
171C2×D43—432no—
172C2×D43—416no—
173C2×D43—48no—
174C2×D43C22×C22⋊C4116yes—
175C4×C43—28no—
176C4×C43—48no—
177C4×C43C22×C22⋊C418yes—
178Dic43—24no—
179Dic43—48no—
180C23×#41—431no—
181C2×#518—432no—
182#2339—4316no—
183#2310—431no—
184#2310D4311yescharacteristic, complemented
185#2308—434no—
186#2280—432no—
187C2×#381—434no—
188C2×C43—24no—
189C2×C43C23×C22⋊C411yescharacteristic
190C2×C43#342812yes—
191C2×C43#348514yes—
192D43—216no—
193D43—44no—
194C22×#41—438no—
195C22×#41—4316no—
196C22×#41—8616no—
197C23×Q8C2×D4311yescharacteristic
198C23×D4—431no—
199C23×D4—432no—
200C24×C4—432no—
201C24×C4—431no—
202#518—8616no—
203C2×C4×D4—438no—
204C2×C4×D4—438no—
205C2×#117—432no—
206C2×#117C2×D4312yes—
207C22×C22⋊C4—438no—
208#391—438no—
209#391C2×D4318yescomplemented
210#386—4332no—
211#381—434no—
212#381—434no—
213#381—4316no—
214#381C2×D4314yes—
215#380—438no—
216#379—434no—
217C4×C22⋊C4—438no—
218C2×#95—4316no—
219C43#3417211yescharacteristic, complemented, Sylow
220C25—432no—
221C25—431no—
222C2×#41—4364no—
223C2×#41—86128no—
224C2×#41—8632no—
225C2×#41—8632no—
226C22×Q8—28no—
227C22×Q8C4×D4318yescomplemented
228C22×Q8C22×D4314yes—
229C22×D4—438no—
230C22×D4—4316no—
231C22×D4—4316no—
232C22×D4—434no—
233C22×D4—438no—
234C22×D4—8616no—
235C23×C4—431no—
236C23×C4—4316no—
237C23×C4—438no—
238C23×C4—438no—
239C23×C4—434no—
240C23×C4—4316no—
241C23×C4—8616no—
242C23×C4—868no—
243C23×C4C22×D4312yes—
244C23×C4C22×D4311yescharacteristic
245C4×D4—8632no—
246C4×D4—8632no—
247#117—28no—
248#117—868no—
249C2×C4⋊C4—434no—
250C2×C4⋊C4C22×D4314yes—
251C2×C22⋊C4—434no—
252C2×C22⋊C4—4316no—
253C2×C22⋊C4—4316no—
254C2×C22⋊C4—8632no—
255C2×C22⋊C4C22×D4314yes—
256C2×C42—434no—
257C2×C42C22×D4314yes—
258#95—4316no—
259#95—4316no—
260#95C22×D43116yes—
261C24—434no—
262C24—438no—
263C24—4316no—
264C24—432no—
265C24—434no—
266C24—864no—
267C24—1728no—
268C24—1724no—
269C24C23×D4311yescharacteristic
270#41—86256no—
271C2×Q8—232no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8C2×C4×D43116yes—
275C2×D4—4364no—
276C2×D4—4316no—
277C2×D4—4332no—
278C2×D4—4316no—
279C2×D4—4316no—
280C2×D4—8664no—
281C2×D4—8632no—
282C22×C4—28no—
283C22×C4—434no—
284C22×C4—432no—
285C22×C4—4332no—
286C22×C4—4316no—
287C22×C4—438no—
288C22×C4—8664no—
289C22×C4—8632no—
290C22×C4—8632no—
291C22×C4—864no—
292C22×C4—8632no—
293C22×C4—868no—
294C22×C4—868no—
295C22×C4—17232no—
296C22×C4C2×C4×D4318yes—
297C22×C4C2×C4×D4314yes—
298C22×C4D4×D4318yes—
299C22×C4C23×D4318yes—
300C22×C4C23×D4314yes—
301C22×C4C23×D4312yes—
302C4⋊C4—28no—
303C4⋊C4—868no—
304C22⋊C4—28no—
305C22⋊C4—8632no—
306C22⋊C4—868no—
307C22⋊C4—17216no—
308C42—28no—
309C42—868no—
310C23—24no—
311C23—438no—
312C23—4316no—
313C23—4332no—
314C23—434no—
315C23—8616no—
316C23—868no—
317C23—864no—
318C23—868no—
319C23—17216no—
320C23—1728no—
321C23—17232no—
322C23—17216no—
323C23C22×C4×D4312yes—
324C23C2×D4×D4314yes—
325C23C24×D4311yescharacteristic
326Q8—232no—
327D4—4364no—
328D4—8664no—
329C2×C4—216no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—216no—
333C2×C4—434no—
334C2×C4—8664no—
335C2×C4—868no—
336C2×C4—864no—
337C2×C4—8632no—
338C2×C4—868no—
339C2×C4—8616no—
340C2×C4—34432no—
341C2×C4C22⋊C4×D43116yes—
342C2×C4C22×C4×D4318yes—
343C2×C4C22×C4×D4314yes—
344C2×C4C2×D4×D4314yes—
345C2×C4C2×D4×D4314yes—
346C22—24no—
347C22—28no—
348C22—4316no—
349C22—8616no—
350C22—8632no—
351C22—868no—
352C22—864no—
353C22—17232no—
354C22—17216no—
355C22—1728no—
356C22C2×C22⋊C4×D4314yes—
357C22C23×C4×D4311yescharacteristic
358C22C22×D4×D4312yes—
359C22#1458373614yes—
360C4—28no—
361C4—48no—
362C4—864no—
363C4—1728no—
364C4C2×C22⋊C4×D4318yes—
365C2—24no—
366C2—8616no—
367C2—1724no—
368C2C22×C22⋊C4×D4311yescharacteristic
369C2#1914779712yes—
370C2#1914823814yes—
subgroupG/H
center, FrattiniC23C24×D43
commutatorC22×C43C24×C4
FittingC43×#2310C2
socle, cosocleC23×C43C25
prime radicalC23×#10621902C2
hypercenter#2310D43
hypocenterC43#34172
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#14583373C2semidirect
3. #19137424C2semidirect
4. #19137181C2semidirect
5. #19137172C2semidirect
6. #19136974C2semidirect
7. #381×D43C2semidirect
8. C43×#2310C2semidirect
10. C22×#10621902C4semidirect
17. C2×C4×D172C22semidirect
18. C2×#13063287C22semidirect
21. #14582806C22semidirect
22. #14582777C22semidirect
24. #14582761C22semidirect
26. #14582756C22semidirect
28. C4×(C2×D43)⋊C4C22semidirect
29. #95×D43C22semidirect
32. C43×#391C22semidirect
39. C2×#10621902C2×C4semidirect
42. C22×D172C2×C4semidirect
49. C22×C4×D43C2×C4semidirect
50. C22×C4×D43C2×C4semidirect
66. C22×Q8×C43C2×C4semidirect
102. C2×D172C22⋊C4semidirect
184. #2310D43semidirect
209. #391C2×D43semidirect
219. C43#34172semidirect
227. C22×Q8C4×D43semidirect

Maximal subgroups

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

subgroupindex
C23×#106219022
C2×#145833732
#191374242
#191371812
#191371722
#191369742
#381×D432
C43×#23102
#3417243