Finite Group Databasebeta

← #25072859

Subgroups — #25072859

Order 11008.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 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#19145052C211yesmaximal, characteristic
2#19144717C212yesmaximal
3#19133735C214yesmaximal, complemented
4#19133673C214yesmaximal, complemented
5#19133321C211yesmaximal, characteristic, complemented
6#19133042C212yesmaximal, complemented
7C2×Q82×C43C211yesmaximal, characteristic, complemented
8C2×#13063356—22no—
9#14583449C2211yescharacteristic
10#14583291—22no—
11#14583278C2212yes—
12#14583273C2212yes—
13#14583212—22no—
14#14583194C2214yes—
15#14583099C2214yes—
16#14583038—22no—
17#14582997—24no—
18#14582956C2212yes—
19C4×#10621877C2211yescharacteristic
20#14582648C2214yes—
21Q8⋊(C43⋊C8)C2218yes—
22Q82×C43—22no—
23Q82×C43—24no—
24Q82×C43C2214yes—
25C2×C4⋊Q8×C43—22no—
26C2×C4⋊Q8×C43—22no—
27C2×C4⋊Q8×C43C2211yescharacteristic
28C2×C4×Q8×C43—22no—
29C2×C4×Q8×C43C2212yes—
30C2×#10621898—21no—
31#13063356—48no—
32C2×#10621885—24no—
33C2×#10621884—24no—
34C2×#10621884—24no—
35C2×#10621877—21no—
36#13063325C2311yescharacteristic
37#13063322C2311yescharacteristic
38#13063313—21no—
39#13063299—22no—
40C4×D172—22no—
41#13063258—24no—
42C4⋊(C43⋊C8)C2314yes—
43#13063249—24no—
44#13063245—24no—
45C22⋊(C43⋊C8)—22no—
46#13063236—21no—
47#13063236—22no—
48C4×C43⋊C8C2312yes—
49C22×Q8×C43—22no—
50C22×Q8×C43D412yes—
51C22×Q8×C43D412yes—
52C4⋊Q8×C43—28no—
53C4⋊Q8×C43—28no—
54C4⋊Q8×C43—28no—
55C4⋊Q8×C43—28no—
56C4⋊Q8×C43—22no—
57C4⋊Q8×C43D412yes—
58C4⋊Q8×C43C2312yes—
59C4×Q8×C43—28no—
60C4×Q8×C43—28no—
61C4×Q8×C43—22no—
62C4×Q8×C43—22no—
63C4×Q8×C43D414yes—
64C4×Q8×C43C2314yes—
65C2×C4⋊C4×C43—24no—
66C2×C4⋊C4×C43—22no—
67C2×C4⋊C4×C43—22no—
68C2×C4⋊C4×C43D412yes—
69C2×C42×C43—22no—
70C2×C42×C43—22no—
71C2×C42×C43C2311yescharacteristic
72C2×Q8×C43—28no—
73C2×Q8×C43—28no—
74C2×Q8×C43—22no—
75C2×Q8×C43—22no—
76C2×Q8×C43—44no—
77C2×Q8×C43—84no—
78C2×Q8×C43C2×D414yes—
79C2×Q8×C43C2×D414yes—
80C22×C4×C43—22no—
81C22×C4×C43—22no—
82C22×C4×C43—22no—
83C22×C4×C43C2×D412yes—
84C22×C4×C43C2×D411yescharacteristic
85C2×C22⋊D43—41no—
86#10621898—44no—
87C2×D172—22no—
88C2×D172—21no—
89C2×C4×D43—41no—
90C2×Dic86—21no—
91C4⋊C4×C43—216no—
92C4⋊C4×C43—24no—
93C4⋊C4×C43—24no—
94C4⋊C4×C43—22no—
95C4⋊C4×C43—24no—
96C4⋊C4×C43—24no—
97C4⋊C4×C43C2×D414yes—
98C42×C43—28no—
99C42×C43—24no—
100C42×C43—24no—
101C42×C43C2×D411yescharacteristic
102C42×C43C2×D412yes—
103C42×C43C2411yescharacteristic
104#10621885—48no—
105#10621884—48no—
106#10621884—48no—
107(C2×D43)⋊C4—42no—
108C4⋊Dic43—22no—
109#10621877—24no—
110C2×C43⋊C8—22no—
111C2×C43⋊C8—24no—
112C23×C43#12011yescharacteristic
113C22×D43—41no—
114Q8×C43—28no—
115Q8×C43—24no—
116Q8×C43—44no—
117Q8×C43—82no—
118Q8×C43—88no—
119Q8×C43#13718yescomplemented
120C2×C4×C43—28no—
121C2×C4×C43—24no—
122C2×C4×C43—22no—
123C2×C4×C43—24no—
124C2×C4×C43—24no—
125C2×C4×C43—24no—
126C2×C4×C43—44no—
127C2×C4×C43—44no—
128C2×C4×C43#12012yes—
129C2×C4×C43#12011yescharacteristic
130C2×C4×C43C22×D411yescharacteristic
131C2×C4×C43C22×D412yes—
132C22⋊D43—82no—
133C2×Dic43—41no—
134D172—42no—
135D172—44no—
136C4×D43—82no—
137Dic86—42no—
138C43⋊C8—24no—
139C43⋊C8—44no—
140#18546—431nomaximal, Sylow
141C22×C43—22no—
142C22×C43#45312yes—
143C22×C43C2×#12011yescharacteristic
144C2×D43—82no—
145C2×D43—81no—
146C4×C43—28no—
147C4×C43—22no—
148C4×C43—21no—
149C4×C43—21no—
150C4×C43—44no—
151C4×C43—44no—
152C4×C43C2×#13714yes—
153Dic43—81no—
154C2×Q82D4311yescharacteristic, complemented
155#3152—431no—
156#3112—432no—
157#1633—434no—
158#1623—434no—
159#1558—431no—
160#1491—432no—
161C2×C43—22no—
162C2×C43#300412yes—
163C2×C43C2×#45311yescharacteristic
164D43—161no—
165C2×#137—862no—
166Q82—22no—
167Q82—24no—
168Q82C2×D4314yes—
169#532—431no—
170C2×C4⋊Q8—22no—
171C2×C4⋊Q8—22no—
172C2×C4⋊Q8C2×D4311yescharacteristic
173C2×C4×Q8—22no—
174C2×C4×Q8C2×D4312yes—
175#497—862no—
176#494—432no—
177#492—432no—
178#480—862no—
179#477—434no—
180#460—434no—
181#448—862no—
182#440—864no—
183#432—432no—
184C4×M16—431no—
185#333—434no—
186Q8⋊C8—438no—
187C43#1854611yescharacteristic, complemented, Sylow
188C2×#41—861no—
189C22×Q8—22no—
190C22×Q8C22⋊D4312yes—
191C22×Q8C22⋊D4312yes—
192#137—1728no—
193C2×Q16—864no—
194C2×SD16—864no—
195C2×SD16—864no—
196C2×M16—861no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—431no—
203C4⋊Q8C22⋊D4312yes—
204C4⋊Q8C22×D4312yes—
205#127—431no—
206#124—861no—
207#121—862no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊D4314yes—
213C4×Q8C22×D4314yes—
214C4×D4—862no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C22⋊D4312yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22×D4311yescharacteristic
222#107—864no—
223C4⋊C8—434no—
224#103—864no—
225#102—864no—
226C22⋊C8—862no—
227#97—861no—
228#97—862no—
229C4×C8—432no—
230#41—1724no—
231C2×Q8—28no—
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—84no—
237C2×Q8—861no—
238C2×Q8C2×C22⋊D4314yes—
239C2×Q8C2×C22⋊D4314yes—
240C2×D4—862no—
241C2×D4—861no—
242C2×D4—1721no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—1721no—
247C22×C4C2×C22⋊D4312yes—
248C22×C4C2×C22⋊D4311yescharacteristic
249Q16—1728no—
250SD16—1728no—
251SD16—1728no—
252M16—864no—
253C2×C8—862no—
254C2×C8—864no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—862no—
262C4⋊C4C2×C22⋊D4314yes—
263C22⋊C4—1722no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C2×C22⋊D4312yes—
268C42C2×C22⋊D4311yescharacteristic
269C42C23×D4311yescharacteristic
270C23—1721no—
271C23#1306329411yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—1722no—
278Q8#1306335618yescomplemented
279D4—3442no—
280D4—1722no—
281D4—1724no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—3442no—
291C2×C4—1721no—
292C2×C4#1306329412yes—
293C2×C4#1306329411yescharacteristic
294C2×C4C22×C22⋊D4311yescharacteristic
295C2×C4C22×C22⋊D4312yes—
296C8—864no—
297C8—1724no—
298C22—22no—
299C22—3442no—
300C22—3441no—
301C22#1458306712yes—
302C22C2×#1306329411yescharacteristic
303C4—28no—
304C4—22no—
305C4—21no—
306C4—21no—
307C4—44no—
308C4—44no—
309C4—3441no—
310C4C2×#1306335614yes—
311C2—22no—
312C2—6881no—
313C2#1914382912yes—
314C2C2×#1458306711yescharacteristic
subgroupG/H
centerC22C2×#13063294
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC2×Q82×C43C2
socleC22×C43C2×#120
prime radical#14583449C22
hypercenterC2×Q82D43
hypocenterC43#18546
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #19133735C2semidirect
4. #19133673C2semidirect
5. #19133321C2semidirect
6. #19133042C2semidirect
7. C2×Q82×C43C2semidirect
119. Q8×C43#137semidirect
154. C2×Q82D43semidirect
187. C43#18546semidirect
278. Q8#13063356semidirect

Maximal subgroups

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

subgroupindex
#191450522
#191447172
#191337352
#191336732
#191333212
#191330422
C2×Q82×C432
#1854643