Finite Group Databasebeta

← #24991612

Subgroups — #24991612

Order 11008.

928 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 135 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#19137198C211yesmaximal, characteristic, complemented
2#19134526C212yesmaximal
3C2×#14582625C211yesmaximal, characteristic, complemented
4#19132474C212yesmaximal, complemented
5C2×C43×#394C211yesmaximal, characteristic
6C2×#13063246—24no—
7C2×C22⋊(C43⋊C8)C2212yes—
8#14582785—24no—
9#14582762—21no—
10#14582759C2212yes—
11C4×C22⋊Dic43—21no—
12#14582625—22no—
13C23×D4×C43C2211yescharacteristic
14C22×C4⋊C4×C43C2211yescharacteristic
15C22×C22⋊C4×C43—21no—
16C22×C22⋊C4×C43—21no—
17C43×#394—22no—
18C43×#394C414yescomplemented
19C2×#95×C43C2211yescharacteristic
20C22×C43⋊C8—22no—
21C2×C4⋊Dic43D414yes—
22C2×C22⋊Dic43—22no—
23C2×C4×Dic43—22no—
24#13063246—48no—
25C22⋊(C43⋊C8)—22no—
26C22⋊(C43⋊C8)—22no—
27C43⋊#95—22no—
28C43⋊#95—24no—
29C43⋊#95—22no—
30C25×C43—21no—
31C22×D4×C43—22no—
32C22×D4×C43—21no—
33C22×D4×C43—22no—
34C22×D4×C43—44no—
35C22×D4×C43C2×C412yes—
36C23×C4×C43—21no—
37C23×C4×C43D411yescharacteristic
38C23×C4×C43D411yescharacteristic
39C23×C4×C43C2311yescharacteristic
40C2×C4⋊C4×C43—22no—
41C2×C4⋊C4×C43—22no—
42C2×C4⋊C4×C43C2×C412yes—
43C2×C4⋊C4×C43D412yes—
44C2×C22⋊C4×C43—24no—
45C2×C22⋊C4×C43—24no—
46C2×C22⋊C4×C43—44no—
47C2×C22⋊C4×C43—44no—
48#95×C43—21no—
49#95×C43C2×C412yes—
50C24×C43—24no—
51C24×C43—22no—
52C24×C43—42no—
53C24×C43—42no—
54C24×C43—42no—
55C24×C43—41no—
56C24×C43—41no—
57C24×C43—42no—
58C24×C43—42no—
59C24×C43C2×D411yescharacteristic
60C2×D4×C43—28no—
61C2×D4×C43—24no—
62C2×D4×C43—44no—
63C2×D4×C43—44no—
64C2×D4×C43—44no—
65C2×D4×C43—48no—
66C2×D4×C43—42no—
67C22×C4×C43—22no—
68C22×C4×C43—21no—
69C22×C4×C43—21no—
70C22×C4×C43—21no—
71C22×C4×C43—21no—
72C22×C4×C43—22no—
73C22×C4×C43—22no—
74C22×C4×C43—42no—
75C22×C4×C43—42no—
76C22×C4×C43—42no—
77C22×C4×C43—42no—
78C22×C4×C43—42no—
79C22×C4×C43C22⋊C412yes—
80C22×C4×C43C22⋊C412yes—
81C22×C4×C43C22×C411yescharacteristic
82C22×C4×C43C2×D411yescharacteristic
83C22×C4×C43C2×D412yes—
84C22×C4×C43#4112yes—
85C22×Dic43—24no—
86C4⋊C4×C43—24no—
87C4⋊C4×C43—24no—
88C22⋊C4×C43—48no—
89C22⋊C4×C43—48no—
90C22⋊Dic43—42no—
91C22⋊Dic43—42no—
92C4⋊Dic43—44no—
93C4×Dic43—82no—
94C2×C43⋊C8—44no—
95C2×C43⋊C8—42no—
96C23×C43—22no—
97C23×C43—22no—
98C23×C43—24no—
99C23×C43—42no—
100C23×C43—44no—
101C23×C43—48no—
102C23×C43—48no—
103C23×C43—42no—
104C23×C43—42no—
105C23×C43—42no—
106C23×C43—42no—
107C23×C43—42no—
108C23×C43—44no—
109C23×C43—84no—
110C23×C43—84no—
111C23×C43—82no—
112C23×C43—82no—
113C23×C43—82no—
114C23×C43—82no—
115C23×C43C2×C22⋊C411yescharacteristic
116C23×C43#12011yescharacteristic
117C23×C43#12311yescharacteristic
118C23×C43#13512yes—
119C23×C43AGL(1,ℤ/8)12yes—
120D4×C43—28no—
121D4×C43—48no—
122D4×C43—44no—
123C2×C4×C43—24no—
124C2×C4×C43—22no—
125C2×C4×C43—22no—
126C2×C4×C43—22no—
127C2×C4×C43—22no—
128C2×C4×C43—41no—
129C2×C4×C43—41no—
130C2×C4×C43—42no—
131C2×C4×C43—42no—
132C2×C4×C43—42no—
133C2×C4×C43—42no—
134C2×C4×C43—42no—
135C2×C4×C43—42no—
136C2×C4×C43—42no—
137C2×C4×C43—42no—
138C2×C4×C43—81no—
139C2×C4×C43—84no—
140C2×C4×C43C4×D412yes—
141C2×C4×C43#12112yes—
142C2×Dic43—44no—
143C2×Dic43—44no—
144C2×Dic43—44no—
145C43⋊C8—82no—
146#9018—431nomaximal, Sylow
147C22×C43—24no—
148C22×C43—24no—
149C22×C43—21no—
150C22×C43—21no—
151C22×C43—21no—
152C22×C43—21no—
153C22×C43—44no—
154C22×C43—44no—
155C22×C43—44no—
156C22×C43—82no—
157C22×C43—82no—
158C22×C43—88no—
159C22×C43—88no—
160C22×C43—82no—
161C22×C43—82no—
162C22×C43#38611yescharacteristic
163C22×C43#41611yescharacteristic
164C22×C43#41811yescharacteristic
165C22×C43#44912yes—
166C22×C43#45412yes—
167C22×C43B(4,𝔽2)12yes—
168C22×C43#48212yes—
169C4×C43—22no—
170C4×C43—22no—
171C4×C43—42no—
172C4×C43—42no—
173C4×C43—82no—
174Dic43—84no—
175C2×#394D4311yescharacteristic
176#2313—431no—
177#1786—432no—
178C2×#324—431no—
179#1294—432no—
180C2×C43—21no—
181C2×C43—21no—
182C2×C43—22no—
183C2×C43—84no—
184C2×C43#159714yes—
185C2×C43#187511yescharacteristic
186C2×C43#188811yescharacteristic
187C2×C43#188911yescharacteristic
188C23×D4C2×D4311yescharacteristic
189C22×C4⋊C4C2×D4311yescharacteristic
190C22×C22⋊C4—21no—
191C22×C22⋊C4—21no—
192C2×#102—864no—
193C2×C22⋊C8—432no—
194#394—22no—
195#394Dic4314yescomplemented
196#387—864no—
197#381—861no—
198#380—432no—
199C4×C22⋊C4—861no—
200C2×#95C2×D4311yescharacteristic
201#324—862no—
202C43#901811yescharacteristic, complemented, Sylow
203C25—21no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—44no—
208C22×D4C2×Dic4312yes—
209C23×C4—21no—
210C23×C4C22⋊D4311yescharacteristic
211C23×C4C22⋊D4311yescharacteristic
212C23×C4C22×D4311yescharacteristic
213C22×C8—862no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—434no—
217C2×C4⋊C4C2×Dic4312yes—
218C2×C4⋊C4C22⋊D4312yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—862no—
224C2×C42—862no—
225#102—1728no—
226C22⋊C8—862no—
227C22⋊C8—862no—
228#95—21no—
229#95—862no—
230#95—864no—
231#95—862no—
232#95C2×Dic4312yes—
233C24—24no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—41no—
239C24—41no—
240C24—42no—
241C24—42no—
242C24C2×C22⋊D4311yescharacteristic
243C2×D4—28no—
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—864no—
263C22×C4C22⋊Dic4312yes—
264C22×C4C22⋊Dic4312yes—
265C22×C4D4×D4312yes—
266C22×C4#1062190012yes—
267C22×C4C22×Dic4311yescharacteristic
268C22×C4C2×C22⋊D4311yescharacteristic
269C2×C8—1724no—
270C2×C8—1722no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—1724no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4—1722no—
277C22⋊C4—1722no—
278C42—3442no—
279C23—22no—
280C23—22no—
281C23—24no—
282C23—42no—
283C23—44no—
284C23—42no—
285C23—42no—
286C23—48no—
287C23—48no—
288C23—44no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—82no—
293C23—84no—
294C23—84no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23C2×C22⋊Dic4311yescharacteristic
299C23#1306329311yescharacteristic
300C23#1306330611yescharacteristic
301C23#1306334512yes—
302C23#1306335112yes—
303D4—28no—
304D4—48no—
305D4—44no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—84no—
323C2×C4—1724no—
324C2×C4—1724no—
325C2×C4—1724no—
326C2×C4D4×Dic4312yes—
327C2×C4#1306329712yes—
328C8—3442no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—24no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—82no—
339C22—82no—
340C22—88no—
341C22—82no—
342C22—88no—
343C22—82no—
344C22#1458277911yescharacteristic
345C22#1458289511yescharacteristic
346C22#1458290411yescharacteristic
347C22#1458304212yes—
348C22#1458307112yes—
349C22#1458308512yes—
350C22#1458322012yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—82no—
356C4—3444no—
357C2—21no—
358C2—21no—
359C2—22no—
360C2—84no—
361C2#1913352214yes—
362C2#1913491811yescharacteristic
363C2#1913499911yescharacteristic
364C2#1913500411yescharacteristic
subgroupG/H
centerC23#13063293
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC2×C43×#394C2
socleC23×C43#120
prime radicalC23×D4×C43C22
cosocleC23×C4×C43C23
hypercenterC2×#394D43
hypocenterC43#9018
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. #19137198C2semidirect
3. C2×#14582625C2semidirect
4. #19132474C2semidirect
18. C43×#394C4semidirect
195. #394Dic43semidirect
202. C43#9018semidirect

Maximal subgroups

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

subgroupindex
#191371982
#191345262
C2×#145826252
#191324742
C2×C43×#3942
#901843