Finite Group Databasebeta

← #24994374

Subgroups — #24994374

Order 11008.

716 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 115 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#19134529C212yesmaximal, complemented
4#19132462C212yesmaximal, complemented
5C43×#1799C211yesmaximal, characteristic
6C23×Dic86C2211yescharacteristic
7C2×#13063304—21no—
8#14582971—24no—
9#14582956—21no—
10C2×#13063248—24no—
11#14582863—21no—
12C2×C22⋊(C43⋊C8)C2212yes—
13#14582819C412yescomplemented
14#14582811C412yescomplemented
15#14582805—21no—
16#14582779—21no—
17#14582771—21no—
18#14582759C2211yescharacteristic
19C2×C4×D4×C43C2211yescharacteristic
20C2×#99×C43—21no—
21C43×#380C2212yes—
22C43×#342—21no—
23C22×Dic86—22no—
24C22×Dic86—21no—
25C22×Dic86—22no—
26C22×Dic86—44no—
27C22×Dic86C2×C412yes—
28C23×Dic43—21no—
29C2×#10621877—22no—
30C22×C43⋊C8D414yes—
31#13063304—44no—
32C2×C4⋊Dic43—21no—
33C2×Dic43⋊C4—22no—
34C2×C22⋊Dic43—22no—
35C2×C22⋊Dic43—21no—
36C4⋊(C43⋊C8)—22no—
37#13063248—48no—
38C22⋊(C43⋊C8)—22no—
39C22⋊(C43⋊C8)—22no—
40C22⋊(C43⋊C8)—22no—
41#13063236—22no—
42C43⋊#95—22no—
43C43⋊#95—22no—
44C43⋊#95C2×C412yes—
45C22×D4×C43D411yescharacteristic
46C23×C4×C43D411yescharacteristic
47C23×C4×C43C2311yescharacteristic
48C4×D4×C43—22no—
49C4×D4×C43—22no—
50C2×C4⋊C4×C43—22no—
51C2×C4⋊C4×C43C2×C411yescharacteristic
52C2×C22⋊C4×C43—22no—
53C2×C22⋊C4×C43C2×C411yescharacteristic
54C2×C22⋊C4×C43D411yescharacteristic
55C2×C42×C43D411yescharacteristic
56#99×C43—42no—
57#95×C43—22no—
58C24×C43C22⋊C411yescharacteristic
59C24×C43C2×D411yescharacteristic
60C2×D4×C43—22no—
61C2×D4×C43—21no—
62C2×D4×C43—22no—
63C2×D4×C43—21no—
64C22×C4×C43—21no—
65C22×C4×C43—21no—
66C22×C4×C43—42no—
67C22×C4×C43—42no—
68C22×C4×C43—42no—
69C22×C4×C43C22⋊C411yescharacteristic
70C22×C4×C43C22⋊C411yescharacteristic
71C22×C4×C43C22⋊C411yescharacteristic
72C22×C4×C43C22×C411yescharacteristic
73C22×C4×C43C2×D411yescharacteristic
74C22×C4×C43C2×D412yes—
75C22×C4×C43#4112yes—
76C22×Dic43—22no—
77C22×Dic43—22no—
78C22×Dic43—44no—
79C22×Dic43—42no—
80C2×Dic86—28no—
81C2×Dic86—24no—
82C2×Dic86—24no—
83C2×Dic86—44no—
84C2×Dic86—48no—
85C2×Dic86—44no—
86C2×Dic86—42no—
87C2×Dic86—41no—
88C2×Dic86—41no—
89C4⋊C4×C43—22no—
90C4⋊C4×C43—44no—
91C22⋊C4×C43—22no—
92C22⋊C4×C43—22no—
93C22⋊C4×C43—42no—
94C22⋊C4×C43—42no—
95C42×C43—21no—
96C42×C43—21no—
97C22⋊Dic43—44no—
98C22⋊Dic43—41no—
99C22⋊Dic43—41no—
100C4⋊Dic43—42no—
101Dic43⋊C4—44no—
102#10621877—82no—
103C2×C43⋊C8—24no—
104C2×C43⋊C8—44no—
105C23×C43—21no—
106C23×C43—21no—
107C23×C43—21no—
108C23×C43—41no—
109C23×C43—41no—
110C23×C43—41no—
111C23×C43—41no—
112C23×C43C2×C22⋊C411yescharacteristic
113C23×C43#12112yes—
114C23×C43AGL(1,ℤ/8)11yescharacteristic
115C23×C43#13712yes—
116C23×C43#13711yescharacteristic
117D4×C43—42no—
118D4×C43—42no—
119C2×C4×C43—24no—
120C2×C4×C43—22no—
121C2×C4×C43—21no—
122C2×C4×C43—21no—
123C2×C4×C43—21no—
124C2×C4×C43—21no—
125C2×C4×C43—21no—
126C2×C4×C43—41no—
127C2×C4×C43—42no—
128C2×C4×C43—42no—
129C2×C4×C43—41no—
130C2×C4×C43—41no—
131C2×C4×C43—81no—
132C2×C4×C43—82no—
133C2×C4×C43—82no—
134C2×C4×C43—82no—
135C2×C4×C43C4×D412yes—
136C2×C4×C43#12011yescharacteristic
137C2×C4×C43#12311yescharacteristic
138C2×Dic43—24no—
139C2×Dic43—48no—
140C2×Dic43—42no—
141C2×Dic43—42no—
142C2×Dic43—42no—
143C2×Dic43—42no—
144C2×Dic43—42no—
145Dic86—28no—
146Dic86—44no—
147Dic86—44no—
148Dic86—44no—
149C43⋊C8—44no—
150#9471—431nomaximal, Sylow
151C22×C43—21no—
152C22×C43—22no—
153C22×C43—21no—
154C22×C43—22no—
155C22×C43—41no—
156C22×C43—42no—
157C22×C43—41no—
158C22×C43—42no—
159C22×C43—41no—
160C22×C43—41no—
161C22×C43—41no—
162C22×C43—81no—
163C22×C43#38611yescharacteristic
164C22×C43#41711yescharacteristic
165C22×C43#41911yescharacteristic
166C22×C43#46812yes—
167C22×C43#47012yes—
168C4×C43—22no—
169C4×C43—21no—
170C4×C43—41no—
171C4×C43—41no—
172C4×C43—41no—
173C4×C43—162no—
174C4×C43B(4,𝔽2)12yes—
175Dic43—44no—
176Dic43—82no—
177#2973—431no—
178#2409—431no—
179#1799D4311yescharacteristic
180#1787—432no—
181#1289—432no—
182C2×C43—21no—
183C2×C43—22no—
184C2×C43—21no—
185C2×C43—22no—
186C2×C43—81no—
187C2×C43#188911yescharacteristic
188C2×C43#189411yescharacteristic
189C2×C43#190911yescharacteristic
190C23×Q8—431no—
191C2×C22⋊Q8—861no—
192C2×C4×D4C2×D4311yescharacteristic
193#435—864no—
194#432—861no—
195C2×#103—864no—
196C2×#99—21no—
197#407—861no—
198C2×C22⋊C8—432no—
199#395—432no—
200#393—432no—
201#391—861no—
202#386—861no—
203#385—861no—
204#380—431no—
205#380C2×D4312yes—
206#342—21no—
207C43#947111yescharacteristic, complemented, Sylow
208C22×Q8—432no—
209C22×Q8—861no—
210C22×Q8—862no—
211C22×Q8—862no—
212C22×Q8—1724no—
213C22×D4C22⋊D4311yescharacteristic
214C23×C4—861no—
215C23×C4D17211yescharacteristic
216C23×C4C22×D4311yescharacteristic
217C2×M16—862no—
218C22×C8—434no—
219C22⋊Q8—1724no—
220C4×D4—22no—
221C4×D4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—862no—
224C2×C4⋊C4—861no—
225C2×C4⋊C4C4×D4311yescharacteristic
226C2×C22⋊C4—22no—
227C2×C22⋊C4—862no—
228C2×C22⋊C4—861no—
229C2×C22⋊C4C4×D4311yescharacteristic
230C2×C22⋊C4C22⋊D4311yescharacteristic
231C2×C42D17211yescharacteristic
232C4⋊C8—862no—
233#103—1728no—
234#99—42no—
235C22⋊C8—862no—
236C22⋊C8—862no—
237C22⋊C8—862no—
238#97—862no—
239#95—22no—
240#95—432no—
241#95—862no—
242#95—862no—
243C24(C2×D43)⋊C411yescharacteristic
244C24C2×C22⋊D4311yescharacteristic
245C2×Q8—864no—
246C2×Q8—868no—
247C2×Q8—864no—
248C2×Q8—1724no—
249C2×Q8—1722no—
250C2×Q8—1724no—
251C2×Q8—1728no—
252C2×Q8—1721no—
253C2×Q8—1721no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—862no—
264C22×C4—862no—
265C22×C4—1724no—
266C22×C4—1722no—
267C22×C4(C2×D43)⋊C411yescharacteristic
268C22×C4(C2×D43)⋊C411yescharacteristic
269C22×C4(C2×D43)⋊C411yescharacteristic
270C22×C4C2×C4×D4311yescharacteristic
271C22×C4C2×D17211yescharacteristic
272C22×C4D4×D4312yes—
273C22×C4#1062190012yes—
274M16—3442no—
275C2×C8—864no—
276C2×C8—1724no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C4⋊C4—1722no—
280C4⋊C4—1724no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—1724no—
286C22⋊C4—1721no—
287C22⋊C4—1721no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×(C2×D43)⋊C411yescharacteristic
298C23#1306329712yes—
299C23#1306335111yescharacteristic
300C23#1306335512yes—
301C23#1306335611yescharacteristic
302Q8—868no—
303Q8—1724no—
304Q8—1724no—
305Q8—1724no—
306D4—42no—
307D4—42no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—864no—
325C2×C4—1722no—
326C2×C4—1722no—
327C2×C4—1722no—
328C2×C4—1722no—
329C2×C4—1728no—
330C2×C4—1722no—
331C2×C4(C22⋊D43)⋊C412yes—
332C2×C4#1306329511yescharacteristic
333C2×C4#1306330911yescharacteristic
334C8—1724no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1458277611yescharacteristic
348C22#1458289811yescharacteristic
349C22#1458290711yescharacteristic
350C22#1458314312yes—
351C22#1458315512yes—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—162no—
358C4—3442no—
359C4—1724no—
360C4#1458308712yes—
361C2—22no—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#1913500111yescharacteristic
367C2#1913502611yescharacteristic
368C2#1913511511yescharacteristic
subgroupG/H
centerC22#14582776
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#1799C2
socleC22×C43#386
prime radicalC22×D4×C43D4
cosocleC23×C4×C43C23
hypercenter#1799D43
hypocenterC43#9471
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #19134529C2semidirect
4. #19132462C2semidirect
13. #14582819C4semidirect
14. #14582811C4semidirect
207. C43#9471semidirect

Maximal subgroups

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

subgroupindex
#191436242
#191379602
#191345292
#191324622
C43×#17992
#947143