Finite Group Databasebeta

← #24988313

Subgroups — #24988313

Order 11008.

802 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#14583634C211yesmaximal, characteristic
2#19134591C211yesmaximal, characteristic, complemented
3#19133262C212yesmaximal
4#19133025C212yesmaximal, complemented
5C43×#1461C211yesmaximal, characteristic, complemented
6#14583634—24no—
7#14583634—24no—
8C2×#13063322C2211yescharacteristic
9C2×#13063314—21no—
10C2×#13063313C412yes—
11C2×#13063310—22no—
12C2×C4×Dic86C2211yescharacteristic
13C2×C4×D172C412yes—
14C2×#13063287—21no—
15#14582923C2212yes—
16C2×#13063244—21no—
17#14582668—21no—
18#14582642—24no—
19#14582631—24no—
20C2×C4×Q8×C43C2211yescharacteristic
21C43×#431C2212yes—
22C43×#324—22no—
23C22×Dic86C2×C411yescharacteristic
24C22×D172—21no—
25C22×D172—21no—
26C22×D172C2×C412yes—
27C22×C4×D43—22no—
28C2×#10621877—22no—
29#13063322—22no—
30#13063322D414yes—
31#13063314—28no—
32#13063313—24no—
33#13063313—24no—
34#13063310—216no—
35C4×Dic86—22no—
36C4×Dic86D414yes—
37C4×D172—24no—
38C4×D172—24no—
39#13063287—28no—
40C2×C4⋊Dic43C2×C411yescharacteristic
41C2×Dic43⋊C4—21no—
42C2×(C2×D43)⋊C4—22no—
43C2×(C2×D43)⋊C4—22no—
44C2×(C2×D43)⋊C4—22no—
45C2×C4×Dic43—21no—
46C4⋊(C43⋊C8)—22no—
47C4⋊(C43⋊C8)—22no—
48#13063244—42no—
49C22×Q8×C43D411yescharacteristic
50C4×Q8×C43—22no—
51C4×Q8×C43—22no—
52C2×C4⋊C4×C43D411yescharacteristic
53C2×C4⋊C4×C43D412yes—
54C2×C42×C43C2×C412yes—
55C2×C42×C43C2311yescharacteristic
56C4⋊C8×C43—22no—
57C22⋊C8×C43—22no—
58C4×C8×C43—22no—
59C23×D43—22no—
60C2×Q8×C43—22no—
61C2×Q8×C43—22no—
62C2×Q8×C43—21no—
63C2×Q8×C43—21no—
64C22×C4×C43C22⋊C412yes—
65C22×C4×C43C22⋊C412yes—
66C22×C4×C43C22×C411yescharacteristic
67C22×C4×C43C2×D411yescharacteristic
68C22×C4×C43C2×D411yescharacteristic
69C22×Dic43—21no—
70C22×Dic43—21no—
71C2×D172—22no—
72C2×D172—24no—
73C2×D172—24no—
74C2×D172—42no—
75C2×D172—41no—
76C2×D172—41no—
77C2×D172—44no—
78C2×D172—82no—
79C2×D172C22⋊C414yes—
80C2×C4×D43—28no—
81C2×C4×D43—84no—
82C2×Dic86—21no—
83C2×Dic86—42no—
84C2×Dic86C22⋊C412yes—
85C2×C8×C43—42no—
86C4⋊C4×C43—22no—
87C4⋊C4×C43—24no—
88C42×C43—22no—
89C42×C43—22no—
90C42×C43C2×D412yes—
91C42×C43C2×D412yes—
92(C2×D43)⋊C4—28no—
93(C2×D43)⋊C4—28no—
94(C2×D43)⋊C4—28no—
95C4⋊Dic43—21no—
96C4⋊Dic43C22⋊C412yes—
97Dic43⋊C4—24no—
98C4×Dic43—24no—
99#10621877—42no—
100#10621877—42no—
101C2×C43⋊C8—42no—
102C23×C43C2×C22⋊C411yescharacteristic
103C22×D43—24no—
104C22×D43—44no—
105C22×D43—44no—
106C22×D43—42no—
107C22×D43—42no—
108Q8×C43—42no—
109Q8×C43—42no—
110C2×C4×C43—22no—
111C2×C4×C43—22no—
112C2×C4×C43—22no—
113C2×C4×C43—22no—
114C2×C4×C43—22no—
115C2×C4×C43—41no—
116C2×C4×C43—42no—
117C2×C4×C43C2×C22⋊C411yescharacteristic
118C2×C4×C43C2×C22⋊C411yescharacteristic
119C2×C4×C43#12012yes—
120C2×C4×C43#12012yes—
121C2×C4×C43#13512yes—
122C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C43#13711yescharacteristic
124C2×Dic43—22no—
125C2×Dic43—22no—
126C2×Dic43—42no—
127C2×Dic43—42no—
128D172—44no—
129D172—44no—
130D172—44no—
131D172—44no—
132D172—84no—
133C4×D43—88no—
134Dic86—24no—
135Dic86—42no—
136C8×C43—82no—
137C43⋊C8—82no—
138#8386—431nomaximal, Sylow
139C22×C43—21no—
140C22×C43—21no—
141C22×C43#37911yescharacteristic
142C22×C43#41611yescharacteristic
143C22×C43#41711yescharacteristic
144C2×D43—44no—
145C2×D43—44no—
146C2×D43—44no—
147C2×D43—84no—
148C2×D43—84no—
149C4×C43—22no—
150C4×C43—21no—
151C4×C43—42no—
152C4×C43—42no—
153C4×C43#41312yes—
154C4×C43#44912yes—
155C4×C43#45212yes—
156Dic43—42no—
157Dic43—42no—
158C2×#559—431no—
159#1801—431no—
160#1545—432no—
161#1487—432no—
162#1461D4311yescharacteristic, complemented
163C2×C43—21no—
164C2×C43—21no—
165C2×C43#178511yescharacteristic
166C2×C43#178811yescharacteristic
167C2×C43#179511yescharacteristic
168D43—84no—
169#559—864no—
170#559—864no—
171C2×#127—431no—
172C2×#124—432no—
173C2×#124—861no—
174C2×#123—862no—
175C2×C4×Q8—431no—
176C2×C4×Q8C2×D4311yescharacteristic
177C2×C4×D4—432no—
178C2×C4×D4—861no—
179#431C2×D4312yes—
180#423—432no—
181C2×#101—861no—
182#343—861no—
183#331—864no—
184Q8⋊C8—864no—
185#324—22no—
186C43#838611yescharacteristic, complemented, Sylow
187C22×Q8—431no—
188C22×Q8D17211yescharacteristic
189C22×D4—432no—
190C22×D4—861no—
191C22×D4—861no—
192C23×C4—862no—
193C2×M16—862no—
194#127—434no—
195#127—862no—
196#124—864no—
197#124—864no—
198#124—868no—
199#123—8616no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—434no—
203C4×Q8—862no—
204C4×D4—864no—
205C4×D4—864no—
206C4×D4—868no—
207C2×C4⋊C4—431no—
208C2×C4⋊C4—861no—
209C2×C4⋊C4D17211yescharacteristic
210C2×C4⋊C4C22⋊D4312yes—
211C2×C22⋊C4—862no—
212C2×C22⋊C4—862no—
213C2×C22⋊C4—862no—
214C2×C42—861no—
215C2×C42C4×D4312yes—
216C2×C42C22×D4311yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—862no—
219C4⋊C8—862no—
220#101—1722no—
221C22⋊C8—22no—
222C4×C8—22no—
223C24—862no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—432no—
229C2×Q8—861no—
230C2×Q8—1722no—
231C2×D4—434no—
232C2×D4—862no—
233C2×D4—864no—
234C2×D4—864no—
235C2×D4—3442no—
236C2×D4—1724no—
237C2×D4—1722no—
238C2×D4—1721no—
239C2×D4—1721no—
240C22×C4—868no—
241C22×C4—861no—
242C22×C4—861no—
243C22×C4—3444no—
244C22×C4(C2×D43)⋊C412yes—
245C22×C4(C2×D43)⋊C412yes—
246C22×C4C2×C4×D4311yescharacteristic
247C22×C4C2×D17211yescharacteristic
248C22×C4C2×C22⋊D4311yescharacteristic
249M16—1722no—
250M16—1722no—
251C2×C8—42no—
252C2×C8—1722no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—432no—
256C4⋊C4—861no—
257C4⋊C4—864no—
258C22⋊C4—868no—
259C22⋊C4—868no—
260C22⋊C4—868no—
261C42—22no—
262C42—22no—
263C42—864no—
264C42D4×D4312yes—
265C42D4×D4312yes—
266C23—864no—
267C23—1724no—
268C23—1724no—
269C23—1722no—
270C23—1722no—
271C23C2×(C2×D43)⋊C411yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—864no—
275Q8—1722no—
276D4—3444no—
277D4—1724no—
278D4—1724no—
279D4—1724no—
280D4—1724no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—862no—
289C2×C4—862no—
290C2×C4—3448no—
291C2×C4—1722no—
292C2×C4—1722no—
293C2×C4C22⋊C4×D4311yescharacteristic
294C2×C4C22⋊C4×D4311yescharacteristic
295C2×C4#1306329312yes—
296C2×C4#1306329512yes—
297C2×C4#1306334712yes—
298C2×C4#1306335311yescharacteristic
299C2×C4#1306335811yescharacteristic
300C8—82no—
301C8—3442no—
302C22—21no—
303C22—21no—
304C22—3444no—
305C22—3444no—
306C22—1724no—
307C22—1724no—
308C22—1724no—
309C22#1458275411yescharacteristic
310C22#1458289611yescharacteristic
311C22#1458290111yescharacteristic
312C4—21no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—1722no—
317C4—1722no—
318C4#1458288212yes—
319C4#1458304612yes—
320C4#1458306312yes—
321C2—21no—
322C2—21no—
323C2—3444no—
324C2#1913452211yescharacteristic
325C2#1913453611yescharacteristic
326C2#1913456311yescharacteristic
subgroupG/H
centerC22#14582754
commutatorC22×C4×C43C22×C4
FrattiniC2×C42C22×D43
FittingC43×#1461C2
socleC22×C43#379
prime radicalC2×#13063322C22
cosocleC2×C42×C43C23
hypercenter#1461D43
hypocenterC43#8386
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #19134591C2semidirect
4. #19133025C2semidirect
5. C43×#1461C2semidirect
162. #1461D43semidirect
186. C43#8386semidirect

Maximal subgroups

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

subgroupindex
C2×#145836342
#191345912
#191332622
#191330252
C43×#14612
#838643