Finite Group Databasebeta

← #24994544

Subgroups — #24994544

Order 11008.

684 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 103 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#19143624C211yesmaximal, characteristic
2#19137960C211yesmaximal, characteristic
3#19134518C212yesmaximal, complemented
4#19132502C212yesmaximal, complemented
5C43×#1465C211yesmaximal, characteristic
6C23×Dic86C2211yescharacteristic
7C2×#13063304—21no—
8#14582971—24no—
9#14582956—21no—
10C2×#13063243—24no—
11#14582863C2212yes—
12C2×C22⋊(C43⋊C8)—21no—
13#14582819—21no—
14#14582811—21no—
15#14582805—21no—
16#14582779C412yes—
17#14582771C412yes—
18#14582759C2211yescharacteristic
19C2×C4×D4×C43C2211yescharacteristic
20C43×#407C2212yes—
21C43×#324—21no—
22C43×#323—21no—
23C22×Dic86—21no—
24C22×Dic86—22no—
25C22×Dic86—21no—
26C22×Dic86—22no—
27C22×Dic86—44no—
28C23×Dic43C2×C412yes—
29C2×#10621877D414yes—
30C22×C43⋊C8—22no—
31#13063304—44no—
32C2×C4⋊Dic43—21no—
33C2×Dic43⋊C4—22no—
34C2×C22⋊Dic43—22no—
35C2×C22⋊Dic43C2×C412yes—
36C4⋊(C43⋊C8)—22no—
37#13063243—48no—
38C22⋊(C43⋊C8)—22no—
39C22⋊(C43⋊C8)—22no—
40C22⋊(C43⋊C8)—22no—
41#13063236—22no—
42C43⋊#95—22no—
43C43⋊#95—22no—
44C43⋊#95—21no—
45C22×D4×C43C2×C411yescharacteristic
46C23×C4×C43D411yescharacteristic
47C23×C4×C43C2311yescharacteristic
48C2×M16×C43—22no—
49C4×D4×C43—22no—
50C4×D4×C43—22no—
51C2×C4⋊C4×C43D411yescharacteristic
52C2×C22⋊C4×C43C2×C411yescharacteristic
53C2×C22⋊C4×C43D411yescharacteristic
54C2×C42×C43D411yescharacteristic
55C22⋊C8×C43—22no—
56C22⋊C8×C43—22no—
57C24×C43C22⋊C411yescharacteristic
58C24×C43C22×C411yescharacteristic
59C2×D4×C43—22no—
60C2×D4×C43—22no—
61C2×D4×C43—41no—
62C22×C4×C43—21no—
63C22×C4×C43—21no—
64C22×C4×C43—42no—
65C22×C4×C43—42no—
66C22×C4×C43C22⋊C411yescharacteristic
67C22×C4×C43C22⋊C411yescharacteristic
68C22×C4×C43C22⋊C411yescharacteristic
69C22×C4×C43C2×D411yescharacteristic
70C22×C4×C43C2×D411yescharacteristic
71C22×C4×C43C2×D412yes—
72C22×C4×C43#4112yes—
73C22×Dic43—22no—
74C22×Dic43—22no—
75C22×Dic43—24no—
76C22×Dic43—44no—
77C2×Dic86—28no—
78C2×Dic86—24no—
79C2×Dic86—44no—
80C2×Dic86—48no—
81C2×Dic86—44no—
82C2×Dic86—42no—
83C2×Dic86—41no—
84C2×Dic86—42no—
85C2×Dic86—41no—
86M16×C43—44no—
87C2×C8×C43—42no—
88C4⋊C4×C43—22no—
89C22⋊C4×C43—22no—
90C22⋊C4×C43—22no—
91C42×C43—21no—
92C42×C43—21no—
93C22⋊Dic43—42no—
94C22⋊Dic43—44no—
95C4⋊Dic43—42no—
96Dic43⋊C4—44no—
97#10621877—44no—
98C2×C43⋊C8—24no—
99C2×C43⋊C8—82no—
100C23×C43—21no—
101C23×C43—21no—
102C23×C43—21no—
103C23×C43—21no—
104C23×C43—41no—
105C23×C43—41no—
106C23×C43—41no—
107C23×C43—41no—
108C23×C43#10112yescomplemented
109C23×C43C2×C22⋊C411yescharacteristic
110C23×C43C4×D412yes—
111D4×C43—44no—
112C2×C4×C43—24no—
113C2×C4×C43—22no—
114C2×C4×C43—21no—
115C2×C4×C43—21no—
116C2×C4×C43—21no—
117C2×C4×C43—21no—
118C2×C4×C43—21no—
119C2×C4×C43—41no—
120C2×C4×C43—42no—
121C2×C4×C43—42no—
122C2×C4×C43—41no—
123C2×C4×C43—41no—
124C2×C4×C43—81no—
125C2×C4×C43#12011yescharacteristic
126C2×C4×C43#12112yes—
127C2×C4×C43#12311yescharacteristic
128C2×Dic43—24no—
129C2×Dic43—28no—
130C2×Dic43—42no—
131C2×Dic43—44no—
132C2×Dic43—42no—
133C2×Dic43—42no—
134C2×Dic43—42no—
135C2×Dic43—42no—
136Dic86—48no—
137Dic86—44no—
138Dic86—44no—
139C8×C43—82no—
140C43⋊C8—44no—
141#9505—431nomaximal, Sylow
142C22×C43—21no—
143C22×C43—21no—
144C22×C43—22no—
145C22×C43—22no—
146C22×C43—22no—
147C22×C43—22no—
148C22×C43—42no—
149C22×C43—42no—
150C22×C43—42no—
151C22×C43—41no—
152C22×C43—41no—
153C22×C43—81no—
154C22×C43#38611yescharacteristic
155C22×C43C2×#10111yescharacteristic
156C22×C43#41311yescharacteristic
157C4×C43—22no—
158C4×C43—21no—
159C4×C43—41no—
160C4×C43—41no—
161C4×C43—41no—
162C4×C43B(4,𝔽2)12yes—
163Dic43—44no—
164Dic43—82no—
165#2973—431no—
166#2409—431no—
167#1784—432no—
168#1465D4311yescharacteristic
169#1305—432no—
170C2×C43—21no—
171C2×C43—21no—
172C2×C43—42no—
173C2×C43—42no—
174C2×C43#188911yescharacteristic
175C2×C43#190111yescharacteristic
176C2×C43#191511yescharacteristic
177C23×Q8—431no—
178C2×C22⋊Q8—861no—
179C2×C4×D4C2×D4311yescharacteristic
180#435—864no—
181#432—861no—
182C2×#101—864no—
183#407—432no—
184#407C2×D4312yes—
185C2×C22⋊C8—861no—
186#395—861no—
187#393—861no—
188#391—861no—
189#386—432no—
190#385—432no—
191#380—431no—
192#324—21no—
193#323—21no—
194C43#950511yescharacteristic, complemented, Sylow
195C22×Q8—861no—
196C22×Q8—862no—
197C22×Q8—862no—
198C22×Q8—861no—
199C22×Q8—1724no—
200C22×D4C4×D4311yescharacteristic
201C23×C4—432no—
202C23×C4D17211yescharacteristic
203C23×C4C22×D4311yescharacteristic
204C2×M16—22no—
205C2×M16—434no—
206C22×C8—862no—
207C22⋊Q8—1724no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—862no—
211C2×C4⋊C4—861no—
212C2×C4⋊C4C22⋊D4311yescharacteristic
213C2×C22⋊C4—432no—
214C2×C22⋊C4—862no—
215C2×C22⋊C4C4×D4311yescharacteristic
216C2×C22⋊C4C22⋊D4311yescharacteristic
217C2×C42D17211yescharacteristic
218C4⋊C8—862no—
219#101—1728no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—862no—
223C22⋊C8—862no—
224C22⋊C8—862no—
225#97—862no—
226#95—862no—
227#95—862no—
228#95—861no—
229C24(C2×D43)⋊C411yescharacteristic
230C24C2×C4×D4311yescharacteristic
231C2×Q8—868no—
232C2×Q8—864no—
233C2×Q8—1722no—
234C2×Q8—1724no—
235C2×Q8—1728no—
236C2×Q8—1724no—
237C2×Q8—1721no—
238C2×Q8—1722no—
239C2×Q8—1721no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—862no—
248C22×C4—864no—
249C22×C4—862no—
250C22×C4—1724no—
251C22×C4(C2×D43)⋊C411yescharacteristic
252C22×C4(C2×D43)⋊C411yescharacteristic
253C22×C4(C2×D43)⋊C411yescharacteristic
254C22×C4C2×D17211yescharacteristic
255C22×C4D4×D4312yes—
256C22×C4#1062190012yes—
257C22×C4C2×C22⋊D4311yescharacteristic
258M16—44no—
259M16—1724no—
260C2×C8—42no—
261C2×C8—864no—
262C2×C8—3442no—
263C4⋊C4—22no—
264C4⋊C4—1722no—
265C4⋊C4—1724no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—1724no—
269C22⋊C4—1722no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#1306324312yescomplemented
281C23C2×(C2×D43)⋊C411yescharacteristic
282C23(C22⋊D43)⋊C412yes—
283Q8—1728no—
284Q8—1724no—
285Q8—1724no—
286D4—44no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—864no—
301C2×C4—868no—
302C2×C4—1722no—
303C2×C4—1722no—
304C2×C4—1722no—
305C2×C4—1722no—
306C2×C4—1724no—
307C2×C4—1722no—
308C2×C4#1306329511yescharacteristic
309C2×C4#1306329712yes—
310C2×C4#1306330911yescharacteristic
311C8—82no—
312C8—1724no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1458277611yescharacteristic
326C22C2×#1306324311yescharacteristic
327C22#1458288111yescharacteristic
328C4—22no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—3442no—
334C4—1724no—
335C4#1458308712yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2#1913500111yescharacteristic
341C2#1913506111yescharacteristic
342C2#1913514511yescharacteristic
subgroupG/H
centerC22#14582776
commutatorC24×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1465C2
socleC22×C43#386
prime radicalC22×D4×C43C2×C4
cosocleC23×C4×C43C23
hypercenter#1465D43
hypocenterC43#9505
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #19134518C2semidirect
4. #19132502C2semidirect
108. C23×C43#101semidirect
194. C43#9505semidirect
280. C23#13063243semidirect

Maximal subgroups

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

subgroupindex
#191436242
#191379602
#191345182
#191325022
C43×#14652
#950543