Finite Group Databasebeta

← #24997153

Subgroups — #24997153

Order 11008.

1020 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14583050C212yesmaximal
2#19140709C211yesmaximal, characteristic, complemented
3C2×#14582635C212yesmaximal, complemented
4#19132438C211yesmaximal, characteristic, complemented
5C43×#1438C211yesmaximal, characteristic
6C22×#10621883—22no—
7C2×#13063304C2212yes—
8#14583050—24no—
9#14583050—24no—
10C2×#13063246—22no—
11C2×C22⋊(C43⋊C8)C2212yes—
12#14582831—22no—
13#14582820—22no—
14#14582759—22no—
15#14582635—24no—
16C23×D4×C43C2211yescharacteristic
17C22×C22⋊C4×C43—22no—
18C2×#95×C43C2211yescharacteristic
19C2×#95×C43C2211yescharacteristic
20C22×Dic86D412yes—
21C2×#10621883—28no—
22C2×#10621883—48no—
23C22×C43⋊C8—22no—
24#13063304—22no—
25#13063304—22no—
26#13063304—22no—
27#13063304—22no—
28C2×C4⋊Dic43—22no—
29C2×C4⋊Dic43—22no—
30C2×C4⋊Dic43D412yes—
31C2×Dic43⋊C4—22no—
32C2×C22⋊Dic43—22no—
33#13063246—28no—
34C22⋊(C43⋊C8)—22no—
35C22⋊(C43⋊C8)D414yes—
36C43⋊#95—24no—
37C25×C43—21no—
38C22×D4×C43—22no—
39C22×D4×C43—22no—
40C22×D4×C43—21no—
41C22×D4×C43—44no—
42C22×D4×C43D412yes—
43C23×C4×C43—21no—
44C23×C4×C43D412yes—
45C23×C4×C43C2311yescharacteristic
46C2×C22⋊C4×C43—44no—
47C2×C22⋊C4×C43—44no—
48C2×C22⋊C4×C43—44no—
49#95×C43—22no—
50#95×C43—22no—
51C24×C43—24no—
52C24×C43—22no—
53C24×C43—42no—
54C24×C43—42no—
55C24×C43—42no—
56C24×C43—41no—
57C24×C43—41no—
58C24×C43—42no—
59C24×C43—42no—
60C24×C43C2×D411yescharacteristic
61C2×D4×C43—28no—
62C2×D4×C43—24no—
63C2×D4×C43—44no—
64C2×D4×C43—44no—
65C2×D4×C43—44no—
66C2×D4×C43—48no—
67C2×D4×C43—42no—
68C22×C4×C43—22no—
69C22×C4×C43—22no—
70C22×C4×C43—22no—
71C22×C4×C43—42no—
72C22×C4×C43—42no—
73C22×C4×C43—42no—
74C22×C4×C43—42no—
75C22×C4×C43—41no—
76C22×C4×C43—41no—
77C22×C4×C43—42no—
78C22×C4×C43SD1614yescomplemented
79C22×C4×C43C2×D412yes—
80C22×C4×C43C2×D411yescharacteristic
81C22×C4×C43C2×D412yes—
82C22×C4×C43#4111yescharacteristic
83C22×Dic43—22no—
84C22×Dic43—22no—
85C2×Dic86—22no—
86C2×Dic86—22no—
87C2×Dic86—44no—
88C22⋊C4×C43—416no—
89C22⋊Dic43—42no—
90C22⋊Dic43—42no—
91#10621883—416no—
92C4⋊Dic43—22no—
93C4⋊Dic43—22no—
94C4⋊Dic43—82no—
95C4⋊Dic43—82no—
96Dic43⋊C4—42no—
97Dic43⋊C4—42no—
98C2×C43⋊C8—24no—
99C2×C43⋊C8—44no—
100C23×C43—21no—
101C23×C43—21no—
102C23×C43—22no—
103C23×C43—24no—
104C23×C43—44no—
105C23×C43—42no—
106C23×C43—44no—
107C23×C43—44no—
108C23×C43—44no—
109C23×C43—44no—
110C23×C43—44no—
111C23×C43—42no—
112C23×C43—44no—
113C23×C43—44no—
114C23×C43—82no—
115C23×C43—82no—
116C23×C43—82no—
117C23×C43—82no—
118C23×C43—82no—
119C23×C43—82no—
120C23×C43—81no—
121C23×C43—81no—
122C23×C43—81no—
123C23×C43—81no—
124C23×C43#12011yescharacteristic
125C23×C43#12111yescharacteristic
126C23×C43#12711yescharacteristic
127C23×C43C2×SD1612yes—
128C23×C43#13712yes—
129D4×C43—28no—
130D4×C43—44no—
131D4×C43—48no—
132C2×C4×C43—24no—
133C2×C4×C43—24no—
134C2×C4×C43—41no—
135C2×C4×C43—42no—
136C2×C4×C43—41no—
137C2×C4×C43—42no—
138C2×C4×C43—42no—
139C2×C4×C43—44no—
140C2×C4×C43—44no—
141C2×C4×C43—42no—
142C2×C4×C43—82no—
143C2×C4×C43—82no—
144C2×C4×C43—81no—
145C2×C4×C43—81no—
146C2×C4×C43#12012yes—
147C2×C4×C43#12112yes—
148C2×Dic43—42no—
149C2×Dic43—42no—
150C2×Dic43—42no—
151C2×Dic43—42no—
152C2×Dic43—44no—
153Dic86—44no—
154Dic86—82no—
155C43⋊C8—44no—
156#9966—431nomaximal, Sylow
157C22×C43—22no—
158C22×C43—22no—
159C22×C43—22no—
160C22×C43—22no—
161C22×C43—21no—
162C22×C43—21no—
163C22×C43—21no—
164C22×C43—21no—
165C22×C43—48no—
166C22×C43—44no—
167C22×C43—84no—
168C22×C43—84no—
169C22×C43—84no—
170C22×C43—84no—
171C22×C43—84no—
172C22×C43—82no—
173C22×C43—82no—
174C22×C43#39211yescharacteristic
175C22×C43#44812yes—
176C22×C43#45311yescharacteristic
177C22×C43#45411yescharacteristic
178C22×C43B(4,𝔽2)12yes—
179C22×C43#46112yes—
180C22×C43#49211yescharacteristic
181C22×C43#49711yescharacteristic
182C4×C43—22no—
183C4×C43—22no—
184C4×C43—44no—
185C4×C43—82no—
186Dic43—82no—
187Dic43—82no—
188C2×#450—432no—
189#2659—431no—
190C2×#328—432no—
191#1438D4311yescharacteristic
192#1280—431no—
193C2×C43—21no—
194C2×C43—22no—
195C2×C43—21no—
196C2×C43—84no—
197C2×C43#159612yes—
198C2×C43#159912yes—
199C2×C43#200611yescharacteristic
200C2×C43#201311yescharacteristic
201C2×C43#202111yescharacteristic
202C23×D4C2×D4311yescharacteristic
203C22×SD16—862no—
204C2×C22⋊Q8—432no—
205C22×C22⋊C4—22no—
206#450—864no—
207#450—864no—
208C2×#102—862no—
209C2×C22⋊C8—432no—
210#398—862no—
211#395—862no—
212#380—862no—
213C2×#95C2×D4311yescharacteristic
214C2×#95C2×D4311yescharacteristic
215#328—864no—
216C43#996611yescharacteristic, complemented, Sylow
217C25—21no—
218C22×Q8—432no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—44no—
223C22×D4D17212yes—
224C23×C4—21no—
225C23×C4C22⋊D4312yes—
226C23×C4C22×D4311yescharacteristic
227C2×SD16—868no—
228C2×SD16—1728no—
229C22×C8—862no—
230C22⋊Q8—862no—
231C22⋊Q8—862no—
232C22⋊Q8—862no—
233C22⋊Q8—862no—
234C2×C4⋊C4—432no—
235C2×C4⋊C4—862no—
236C2×C4⋊C4—862no—
237C2×C4⋊C4—862no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—862no—
242#102—868no—
243C22⋊C8—434no—
244C22⋊C8—862no—
245#95—22no—
246#95—22no—
247#95—864no—
248C24—24no—
249C24—22no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24C2×C22⋊D4311yescharacteristic
258C2×Q8—862no—
259C2×Q8—862no—
260C2×Q8—1724no—
261C2×D4—28no—
262C2×D4—24no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—48no—
266C2×D4—44no—
267C2×D4—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—862no—
279C22×C4—862no—
280C22×C4#1062188314yescomplemented
281C22×C4C2×D17211yescharacteristic
282C22×C4#1062189811yescharacteristic
283C22×C4D4×D4312yes—
284C22×C4D4×D4312yes—
285SD16—17216no—
286C2×C8—864no—
287C2×C8—1724no—
288C4⋊C4—862no—
289C4⋊C4—862no—
290C4⋊C4—3442no—
291C4⋊C4—3442no—
292C4⋊C4—1722no—
293C4⋊C4—1722no—
294C22⋊C4—416no—
295C22⋊C4—1722no—
296C22⋊C4—1722no—
297C23—21no—
298C23—21no—
299C23—22no—
300C23—24no—
301C23—44no—
302C23—44no—
303C23—42no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—82no—
320C23—81no—
321C23#1306329311yescharacteristic
322C23#1306329911yescharacteristic
323C23#1306332111yescharacteristic
324C23C2×#1062188312yes—
325C23#1306335512yes—
326Q8—3442no—
327Q8—1724no—
328D4—28no—
329D4—44no—
330D4—48no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—1722no—
346C2×C4—1722no—
347C2×C4—1722no—
348C2×C4—1722no—
349C2×C4—1724no—
350C2×C4#1306329512yes—
351C2×C4#1306329812yes—
352C8—1724no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—48no—
362C22—44no—
363C22—84no—
364C22—84no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22—82no—
369C22—82no—
370C22#1458280811yescharacteristic
371C22#1458303712yes—
372C22#1458306611yescharacteristic
373C22#1458307111yescharacteristic
374C22#1458308512yes—
375C22#1458310412yes—
376C22#1458327111yescharacteristic
377C22#1458329411yescharacteristic
378C4—22no—
379C4—22no—
380C4—44no—
381C4—82no—
382C4—3442no—
383C4—3442no—
384C2—21no—
385C2—22no—
386C2—21no—
387C2—84no—
388C2#1913351712yes—
389C2#1913353412yes—
390C2#1913563611yescharacteristic
391C2#1913567711yescharacteristic
392C2#1913572111yescharacteristic
subgroupG/H
centerC23#13063293
commutator, cosocleC23×C4×C43C23
FrattiniC23×C4C22×D43
FittingC43×#1438C2
socleC23×C43#120
prime radicalC23×D4×C43C22
hypercenter#1438D43
hypocenterC43#9966
LCSG ▷ C23×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. #19140709C2semidirect
3. C2×#14582635C2semidirect
4. #19132438C2semidirect
78. C22×C4×C43SD16semidirect
216. C43#9966semidirect
280. C22×C4#10621883semidirect

Maximal subgroups

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

subgroupindex
C2×#145830502
#191407092
C2×#145826352
#191324382
C43×#14382
#996643