Finite Group Databasebeta

← #24989238

Subgroups — #24989238

Order 11008.

1166 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 147 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#19134675C212yesmaximal, complemented
2#19132922C212yesmaximal, complemented
3#19132922C212yesmaximal, complemented
4C22×#123×C43C211yesmaximal, characteristic
5#14582863C2214yes—
6#14582668—24no—
7#14582623—24no—
8#14582623—24no—
9C23×D4×C43C411yescharacteristic
10C24×C4×C43C411yescharacteristic
11C2×#123×C43—24no—
12C2×#123×C43—24no—
13C2×#123×C43—24no—
14C22×C4⋊C4×C43C411yescharacteristic
15C22×C4⋊C4×C43C2211yescharacteristic
16C22×C22⋊C4×C43C411yescharacteristic
17C22×C22⋊C4×C43C2211yescharacteristic
18C22×C22⋊C4×C43C2211yescharacteristic
19C2×#10621877—24no—
20C22⋊(C43⋊C8)—24no—
21C22⋊(C43⋊C8)—24no—
22C25×C43C2×C411yescharacteristic
23C25×C43C2×C411yescharacteristic
24C22×D4×C43—22no—
25C22×D4×C43—22no—
26C22×D4×C43—22no—
27C22×D4×C43—44no—
28C23×C4×C43—22no—
29C23×C4×C43—22no—
30C23×C4×C43—22no—
31C23×C4×C43—44no—
32C23×C4×C43C2×C411yescharacteristic
33C23×C4×C43C2×C411yescharacteristic
34C23×C4×C43C2×C411yescharacteristic
35C23×C4×C43C2×C411yescharacteristic
36C23×C4×C43C2311yescharacteristic
37#123×C43—232no—
38C2×C4⋊C4×C43—22no—
39C2×C4⋊C4×C43—22no—
40C2×C4⋊C4×C43—22no—
41C2×C4⋊C4×C43—22no—
42C2×C4⋊C4×C43—22no—
43C2×C4⋊C4×C43D414yes—
44C2×C22⋊C4×C43—22no—
45C2×C22⋊C4×C43—22no—
46C2×C22⋊C4×C43—22no—
47C2×C22⋊C4×C43—22no—
48C2×C22⋊C4×C43—22no—
49C2×C22⋊C4×C43—22no—
50C2×C22⋊C4×C43—22no—
51C2×C22⋊C4×C43D414yes—
52C2×C22⋊C4×C43D414yes—
53C24×C43—24no—
54C24×C43—21no—
55C24×C43—21no—
56C24×C43—21no—
57C24×C43—21no—
58C24×C43—42no—
59C24×C43—42no—
60C24×C43—42no—
61C24×C43—42no—
62C24×C43—42no—
63C24×C43C22⋊C412yes—
64C24×C43C22⋊C412yes—
65C24×C43C22×C411yescharacteristic
66C2×D4×C43—28no—
67C2×D4×C43—48no—
68C2×D4×C43—48no—
69C2×D4×C43—44no—
70C2×D4×C43—44no—
71C22×C4×C43—28no—
72C22×C4×C43—21no—
73C22×C4×C43—21no—
74C22×C4×C43—21no—
75C22×C4×C43—21no—
76C22×C4×C43—21no—
77C22×C4×C43—21no—
78C22×C4×C43—21no—
79C22×C4×C43—21no—
80C22×C4×C43—42no—
81C22×C4×C43—42no—
82C22×C4×C43—42no—
83C22×C4×C43—42no—
84C22×C4×C43—42no—
85C22×C4×C43—48no—
86C22×C4×C43—48no—
87C22×C4×C43—44no—
88C22×C4×C43—44no—
89C22×C4×C43C22⋊C412yes—
90C22×C4×C43C22⋊C412yes—
91C22×C4×C43C22⋊C412yes—
92C22×C4×C43C22⋊C412yes—
93C22×C4×C43C2×D412yes—
94C22×C4×C43C2×D412yes—
95C22×C4×C43C2×D412yes—
96C4⋊C4×C43—28no—
97C4⋊C4×C43—28no—
98C22⋊C4×C43—28no—
99C22⋊C4×C43—28no—
100C22⋊C4×C43—28no—
101#10621877—48no—
102C2×C43⋊C8—44no—
103C23×C43—22no—
104C23×C43—22no—
105C23×C43—22no—
106C23×C43—24no—
107C23×C43—24no—
108C23×C43—22no—
109C23×C43—22no—
110C23×C43—22no—
111C23×C43—24no—
112C23×C43—42no—
113C23×C43—42no—
114C23×C43—416no—
115C23×C43—42no—
116C23×C43—42no—
117C23×C43—42no—
118C23×C43—42no—
119C23×C43—42no—
120C23×C43—42no—
121C23×C43—42no—
122C23×C43—42no—
123C23×C43—42no—
124C23×C43—42no—
125C23×C43—42no—
126C23×C43—42no—
127C23×C43—88no—
128C23×C43#10014yes—
129C23×C43C2×C22⋊C411yescharacteristic
130C23×C43C2×C22⋊C411yescharacteristic
131C23×C43C2×C22⋊C411yescharacteristic
132D4×C43—416no—
133C2×C4×C43—24no—
134C2×C4×C43—22no—
135C2×C4×C43—22no—
136C2×C4×C43—22no—
137C2×C4×C43—22no—
138C2×C4×C43—42no—
139C2×C4×C43—42no—
140C2×C4×C43—42no—
141C2×C4×C43—42no—
142C2×C4×C43—42no—
143C2×C4×C43—416no—
144C2×C4×C43—42no—
145C2×C4×C43—42no—
146C2×C4×C43—42no—
147C2×C4×C43—42no—
148C2×C4×C43—42no—
149C2×C4×C43—42no—
150C2×C4×C43—42no—
151C2×C4×C43—42no—
152C2×C4×C43—42no—
153C2×C4×C43#12014yes—
154C43⋊C8—84no—
155#8553—431nomaximal, Sylow
156C22×C43—28no—
157C22×C43—21no—
158C22×C43—21no—
159C22×C43—21no—
160C22×C43—21no—
161C22×C43—21no—
162C22×C43—21no—
163C22×C43—24no—
164C22×C43—44no—
165C22×C43—48no—
166C22×C43—44no—
167C22×C43—48no—
168C22×C43—42no—
169C22×C43—42no—
170C22×C43—42no—
171C22×C43—42no—
172C22×C43—42no—
173C22×C43—84no—
174C22×C43—84no—
175C22×C43—84no—
176C22×C43#37911yescharacteristic
177C22×C43#41012yes—
178C22×C43C2×#10012yes—
179C22×C43#41312yes—
180C4×C43—24no—
181C4×C43—42no—
182C4×C43—42no—
183C4×C43—42no—
184C4×C43—42no—
185C22×#123D4311yescharacteristic
186#1821—432no—
187#1462—432no—
188#1462—432no—
189C2×C43—22no—
190C2×C43—22no—
191C2×C43—22no—
192C2×C43—44no—
193C2×C43—82no—
194C2×C43#178211yescharacteristic
195C2×C43#178311yescharacteristic
196C2×C43#178511yescharacteristic
197C23×D4Dic4311yescharacteristic
198C24×C4Dic4311yescharacteristic
199C2×#123—24no—
200C2×#123—24no—
201C2×#123—24no—
202C22×C4⋊C4Dic4311yescharacteristic
203C22×C4⋊C4C2×D4311yescharacteristic
204C22×C22⋊C4Dic4311yescharacteristic
205C22×C22⋊C4C2×D4311yescharacteristic
206C22×C22⋊C4C2×D4311yescharacteristic
207#407—434no—
208#343—864no—
209#323—864no—
210#323—864no—
211C43#855311yescharacteristic, complemented, Sylow
212C25C2×Dic4311yescharacteristic
213C25C2×Dic4311yescharacteristic
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—44no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4C2×Dic4311yescharacteristic
223C23×C4C2×Dic4311yescharacteristic
224C23×C4C2×Dic4311yescharacteristic
225C23×C4C2×Dic4311yescharacteristic
226C23×C4C22×D4311yescharacteristic
227C2×M16—864no—
228#123—232no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C22⋊D4314yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4C22⋊D4314yes—
243C2×C22⋊C4C22⋊D4314yes—
244C22⋊C8—864no—
245C22⋊C8—864no—
246C24—24no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C22⋊Dic4312yes—
257C24C22⋊Dic4312yes—
258C24C22×Dic4311yescharacteristic
259C2×D4—28no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C22×C4—28no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C22⋊Dic4312yes—
283C22×C4C22⋊Dic4312yes—
284C22×C4C22⋊Dic4312yes—
285C22×C4C22⋊Dic4312yes—
286C22×C4C2×C22⋊D4312yes—
287C22×C4C2×C22⋊D4312yes—
288C22×C4C2×C22⋊D4312yes—
289M16—1728no—
290C2×C8—1724no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—88no—
321C23#1306324214yes—
322C23C2×C22⋊Dic4311yescharacteristic
323C23C2×C22⋊Dic4311yescharacteristic
324C23C2×C22⋊Dic4311yescharacteristic
325D4—416no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—416no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4#1306329414yes—
347C8—3444no—
348C22—28no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—24no—
356C22—44no—
357C22—48no—
358C22—44no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22#1458275511yescharacteristic
369C22#1458287312yes—
370C22C2×#1306324212yes—
371C22#1458288312yes—
372C4—24no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—44no—
381C2—82no—
382C2#1913451311yescharacteristic
383C2#1913451611yescharacteristic
384C2#1913452311yescharacteristic
subgroupG/H
centerC22#14582755
commutatorC24×C43C22×C4
FrattiniC23×C4C22×D43
FittingC22×#123×C43C2
socleC22×C43#379
prime radicalC23×D4×C43C4
cosocleC23×C4×C43C23
hypercenterC22×#123D43
hypocenterC43#8553
LCSG ▷ C24×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. #19134675C2semidirect
2. #19132922C2semidirect
3. #19132922C2semidirect
211. C43#8553semidirect

Maximal subgroups

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

subgroupindex
#191346752
#191329222
C22×#123×C432
#855343