Finite Group Databasebeta

← C2×M16×C22⋊C9 #11919777

Subgroups — C2×M16×C22⋊C9 #11919777

Order 1152.

732 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C22×C4×C22⋊C9C211yesmaximal, characteristic
2M16×C22⋊C9C214yesmaximal, direct product factor
3C2×C8×C22⋊C9C212yesmaximal, complemented
4C23×C3×M16C311yesmaximal, characteristic
5C23×C22⋊C9C411yescharacteristic
6C2×C4×C22⋊C9C412yes—
7C2×C4×C22⋊C9C411yescharacteristic
8C2×C4×C22⋊C9C2211yescharacteristic
9C2×C4×C22⋊C9C2212yes—
10C2×C9×M16—41nomaximal
11C8×C22⋊C9C2214yescomplemented
12C24×C3×C4C2×C311yescharacteristic
13C22×C3×M16—31no—
14C22×C3×M16—32no—
15C22×C3×M16—31no—
16C22×C3×M16—34no—
17C22×C3×M16C2×C314yes—
18C23×C3×C8C2×C312yes—
19C22×C22⋊C9—22no—
20C22×C22⋊C9C2×C412yes—
21C22×C22⋊C9C2×C411yescharacteristic
22C22×C4×C9—41no—
23C4×C22⋊C9C2×C411yescharacteristic
24C4×C22⋊C9C2×C412yes—
25C4×C22⋊C9C2311yescharacteristic
26C9×M16—44no—
27C2×C8×C9—42no—
28C23×M16C911yescharacteristic, complemented, Sylow
29C25×C3C3×C411yescharacteristic
30C23×C3×C4—31no—
31C23×C3×C4—32no—
32C23×C3×C4—31no—
33C23×C3×C4—31no—
34C23×C3×C4—31no—
35C23×C3×C4—32no—
36C23×C3×C4C3×C412yes—
37C23×C3×C4C3×C411yescharacteristic
38C23×C3×C4C22×C311yescharacteristic
39C23×C3×C4C22×C312yes—
40C2×C3×M16—34no—
41C2×C3×M16—38no—
42C2×C3×M16—32no—
43C2×C3×M16—31no—
44C2×C3×M16—34no—
45C2×C3×M16—34no—
46C2×C3×M16—38no—
47C2×C3×M16—34no—
48C2×C3×M16—32no—
49C2×C3×M16A411yescharacteristic
50C22×C3×C8—32no—
51C22×C3×C8—32no—
52C22×C3×C8—34no—
53C22×C3×C8C22×C314yes—
54C23×C9—41no—
55C2×C22⋊C9—22no—
56C2×C22⋊C9M1612yesdirect product factor
57C2×C22⋊C9C22×C411yescharacteristic
58C2×C4×C9—41no—
59C2×C4×C9—42no—
60C2×C4×C9—42no—
61C2×C4×C9—41no—
62C8×C9—44no—
63C24×C4C2×C911yescharacteristic
64C22×M16—31no—
65C22×M16—32no—
66C22×M16—31no—
67C22×M16—34no—
68C22×M16C2×C914yescomplemented
69C23×C8C2×C912yescomplemented
70C24×C3—22no—
71C24×C3—32no—
72C24×C3—31no—
73C24×C3—31no—
74C24×C3—62no—
75C24×C3C2×C3×C412yes—
76C24×C3C2×C3×C411yescharacteristic
77C22×C3×C4—32no—
78C22×C3×C4—34no—
79C22×C3×C4—32no—
80C22×C3×C4—32no—
81C22×C3×C4—34no—
82C22×C3×C4—31no—
83C22×C3×C4—32no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—32no—
87C22×C3×C4—32no—
88C22×C3×C4—32no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—32no—
93C22×C3×C4—31no—
94C22×C3×C4—32no—
95C22×C3×C4—32no—
96C22×C3×C4—32no—
97C22×C3×C4—32no—
98C22×C3×C4—31no—
99C22×C3×C4—32no—
100C22×C3×C4—31no—
101C22×C3×C4—32no—
102C22×C3×C4C2×C3×C411yescharacteristic
103C22×C3×C4C2×C3×C412yes—
104C22×C3×C4C2×A411yescharacteristic
105C22×C3×C4C23×C311yescharacteristic
106C3×M16—38no—
107C3×M16—34no—
108C3×M16—38no—
109C3×M16C2×A414yes—
110C2×C3×C8—34no—
111C2×C3×C8—32no—
112C2×C3×C8—34no—
113C2×C3×C8—34no—
114C2×C3×C8—34no—
115C2×C3×C8C2×A412yes—
116C22×C9—41no—
117C22×C9—42no—
118C22×C9—82no—
119C22⋊C9C2×M1611yescharacteristic, direct product factor
120C4×C9—42no—
121C4×C9—41no—
122C4×C9—41no—
123C25C4×C911yescharacteristic
124C23×C4—31no—
125C23×C4—31no—
126C23×C4—32no—
127C23×C4—31no—
128C23×C4—31no—
129C23×C4—32no—
130C23×C4C4×C912yes—
131C23×C4C4×C911yescharacteristic
132C23×C4C22×C911yescharacteristic
133C23×C4C22×C912yes—
134C2×M16—34no—
135C2×M16—38no—
136C2×M16—32no—
137C2×M16—31no—
138C2×M16—34no—
139C2×M16—34no—
140C2×M16—38no—
141C2×M16—34no—
142C2×M16—32no—
143C2×M16C22⋊C911yescharacteristic, direct product factor
144C22×C8—32no—
145C22×C8—32no—
146C22×C8—34no—
147C22×C8C22×C914yescomplemented
148C23×C3—22no—
149C23×C3—31no—
150C23×C3—32no—
151C23×C3—31no—
152C23×C3—32no—
153C23×C3—32no—
154C23×C3—32no—
155C23×C3—31no—
156C23×C3—32no—
157C23×C3—62no—
158C23×C3—62no—
159C23×C3—62no—
160C23×C3—64no—
161C23×C3—62no—
162C23×C3—62no—
163C23×C3—64no—
164C23×C3C3×M1612yes—
165C23×C3C4×A411yescharacteristic
166C23×C3C22×C3×C411yescharacteristic
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—34no—
170C2×C3×C4—34no—
171C2×C3×C4—32no—
172C2×C3×C4—32no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—32no—
185C2×C3×C4—31no—
186C2×C3×C4—32no—
187C2×C3×C4—31no—
188C2×C3×C4C4×A411yescharacteristic
189C2×C3×C4C4×A412yes—
190C2×C3×C4C22×A412yes—
191C2×C3×C4C22×A411yescharacteristic
192C3×C8—34no—
193C3×C8C22×A414yes—
194C2×C9—41no—
195C2×C9—42no—
196C2×C9—82no—
197C24—22no—
198C24—32no—
199C24—31no—
200C24—31no—
201C24—62no—
202C24C2×C4×C912yes—
203C24C2×C4×C911yescharacteristic
204C22×C4—32no—
205C22×C4—34no—
206C22×C4—32no—
207C22×C4—32no—
208C22×C4—34no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—32no—
212C22×C4—32no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—32no—
216C22×C4—32no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—32no—
220C22×C4—31no—
221C22×C4—32no—
222C22×C4—32no—
223C22×C4—32no—
224C22×C4—32no—
225C22×C4—31no—
226C22×C4—32no—
227C22×C4—31no—
228C22×C4—32no—
229C22×C4C2×C4×C911yescharacteristic
230C22×C4C2×C4×C912yes—
231C22×C4C2×C22⋊C911yescharacteristic
232C22×C4C23×C911yescharacteristic
233M16—38no—
234M16—34no—
235M16—38no—
236M16C2×C22⋊C914yesdirect product factor
237C2×C8—34no—
238C2×C8—32no—
239C2×C8—34no—
240C2×C8—34no—
241C2×C8—34no—
242C2×C8C2×C22⋊C912yescomplemented
243C22×C3—22no—
244C22×C3—32no—
245C22×C3—31no—
246C22×C3—32no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—62no—
253C22×C3—64no—
254C22×C3—62no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—64no—
258C22×C3—62no—
259C22×C3C2×C3×M1611yescharacteristic
260C22×C3C2×C4×A412yes—
261C22×C3C2×C4×A411yescharacteristic
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—32no—
265C3×C4C2×C4×A412yes—
266C3×C4C2×C4×A411yescharacteristic
267C3×C4C23×A411yescharacteristic
268C9—41noSylow
269C23—22no—
270C23—31no—
271C23—32no—
272C23—31no—
273C23—32no—
274C23—32no—
275C23—32no—
276C23—32no—
277C23—31no—
278C23—62no—
279C23—62no—
280C23—62no—
281C23—64no—
282C23—62no—
283C23—62no—
284C23—64no—
285C23C9×M1612yescomplemented
286C23C4×C22⋊C911yescharacteristic
287C23C22×C4×C911yescharacteristic
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—34no—
291C2×C4—34no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—32no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—31no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4C4×C22⋊C912yes—
310C2×C4C4×C22⋊C911yescharacteristic
311C2×C4C22×C22⋊C911yescharacteristic
312C2×C4C22×C22⋊C912yes—
313C8—34no—
314C8C22×C22⋊C914yescomplemented
315C2×C3—22no—
316C2×C3—32no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—62no—
320C2×C3A4×M1612yes—
321C2×C3C22×C4×A411yescharacteristic
322C22—22no—
323C22—32no—
324C22—31no—
325C22—32no—
326C22—31no—
327C22—31no—
328C22—32no—
329C22—32no—
330C22—32no—
331C22—62no—
332C22—62no—
333C22—64no—
334C22—62no—
335C22—62no—
336C22—64no—
337C22—62no—
338C22C2×C9×M1611yescharacteristic, complemented
339C22C2×C4×C22⋊C912yes—
340C22C2×C4×C22⋊C911yescharacteristic
341C4—31no—
342C4—31no—
343C4—32no—
344C4C2×C4×C22⋊C911yescharacteristic
345C4C2×C4×C22⋊C912yes—
346C4C23×C22⋊C911yescharacteristic
347C3C2×A4×M1611yescharacteristic
348C2—22no—
349C2—31no—
350C2—32no—
351C2—31no—
352C2—62no—
353C2M16×C22⋊C912yesdirect product factor
354C2C22×C4×C22⋊C911yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutatorC23C22×C4×C9
FrattiniC3×C4C23×A4
FittingC23×C3×M16C3
socleC24×C3C2×C3×C4
prime radicalC25×C3C3×C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×M16A4
hypocenterC22C2×C9×M16
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×M16

Complements

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

NH ≅ G/Ntype
2. M16×C22⋊C9C2direct (N × H)
3. C2×C8×C22⋊C9C2semidirect
11. C8×C22⋊C9C22semidirect
28. C23×M16C9semidirect
56. C2×C22⋊C9M16direct (N × H)
68. C22×M16C2×C9semidirect
69. C23×C8C2×C9semidirect
119. C22⋊C9C2×M16direct (N × H)
143. C2×M16C22⋊C9direct (N × H)
147. C22×C8C22×C9semidirect
236. M16C2×C22⋊C9direct (N × H)
242. C2×C8C2×C22⋊C9semidirect
285. C23C9×M16semidirect
314. C8C22×C22⋊C9semidirect
338. C22C2×C9×M16semidirect
353. C2M16×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊C92
M16×C22⋊C92
C2×C8×C22⋊C92
C23×C3×M163
C2×C9×M164