Finite Group Databasebeta

← C2×C3×#4553 #11796029

Subgroups — C2×C3×#4553 #11796029

Order 1152.

621 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×M16C211yesmaximal, characteristic
2C2×C3×#812C212yesmaximal, complemented
3C3×#4553C214yesmaximal, direct product factor
4C2×C3×#349—31nomaximal
5C2×#4553C311yesmaximal, characteristic, direct product factor
6C23×C32×C4C411yescharacteristic
7C2×C32×M16—22no—
8C2×C32×M16—22no—
9C2×C32×M16C412yes—
10C2×C32×M16C2212yes—
11C22×C32×C8C411yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C3×#812—22no—
14C3×#812C2214yescomplemented
15C2×C3×C3⋊C16—22no—
16C22×C3×M16—21no—
17C22×C3×M16D311yescharacteristic
18C22×C3×M16C2×C311yescharacteristic
19C2×C3×M32—32no—
20C2×#812C2×C312yescomplemented
21C3×#349—34no—
22#4553C2×C314yesdirect product factor
23C24×C32C811yescharacteristic
24C22×C32×C4—21no—
25C22×C32×C4—21no—
26C22×C32×C4—21no—
27C22×C32×C4—21no—
28C22×C32×C4C811yescharacteristic
29C22×C32×C4C812yes—
30C22×C32×C4C2×C412yes—
31C22×C32×C4C2×C411yescharacteristic
32C32×M16—24no—
33C32×M16—24no—
34C2×C32×C8—21no—
35C2×C32×C8—21no—
36C2×C32×C8C2×C412yes—
37C2×C32×C8C2×C411yescharacteristic
38C2×C32×C8D412yes—
39C2×C32×C8D412yes—
40C2×C32×C8C2311yescharacteristic
41C3×C3⋊C16—24no—
42C2×#349—31noSylow
43C23×C3×C4—21no—
44C23×C3×C4Dic311yescharacteristic
45C23×C3×C4C3×C411yescharacteristic
46C2×C3×M16—24no—
47C2×C3×M16—24no—
48C2×C3×M16—22no—
49C2×C3×M16—22no—
50C2×C3×M16—22no—
51C2×C3×M16—22no—
52C2×C3×M16—22no—
53C2×C3×M16—22no—
54C2×C3×M16Dic312yes—
55C2×C3×M16C3×C412yes—
56C2×C3×M16C2×D312yes—
57C2×C3×M16C22×C312yes—
58C22×C3×C8—21no—
59C22×C3×C8—21no—
60C22×C3×C8Dic311yescharacteristic
61C22×C3×C8C3×C411yescharacteristic
62C22×C3×C8C2×D311yescharacteristic
63C22×C3×C8C22×C311yescharacteristic
64C3×M32—34no—
65C3×M32—62no—
66C2×C3×C16—62no—
67#812—22no—
68#812C22×C314yescomplemented
69C2×C3⋊C16—22no—
70C23×C32—21no—
71C23×C32—21no—
72C23×C32—42no—
73C23×C32C2×C811yescharacteristic
74C23×C32C2×C812yes—
75C2×C32×C4—21no—
76C2×C32×C4—21no—
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4C22⋊C411yescharacteristic
82C2×C32×C4C22⋊C411yescharacteristic
83C2×C32×C4C2×C812yes—
84C2×C32×C4C2×C811yescharacteristic
85C2×C32×C4M1611yescharacteristic
86C2×C32×C4M1611yescharacteristic
87C2×C32×C4C22×C411yescharacteristic
88C32×C8—22no—
89C32×C8C22⋊C412yes—
90C32×C8C2×D412yes—
91C22×M16C3×D311yescharacteristic
92C2×M32—32no—
93#349—34no—
94C24×C3—21no—
95C24×C3C3⋊C811yescharacteristic
96C24×C3C3×C811yescharacteristic
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4C3⋊C811yescharacteristic
114C22×C3×C4C3⋊C812yes—
115C22×C3×C4C3×C811yescharacteristic
116C22×C3×C4C3×C812yes—
117C22×C3×C4C2×Dic312yes—
118C22×C3×C4C2×Dic311yescharacteristic
119C22×C3×C4C2×C3×C412yes—
120C22×C3×C4C2×C3×C411yescharacteristic
121C3×M16—28no—
122C3×M16—28no—
123C3×M16—24no—
124C3×M16—24no—
125C3×M16—24no—
126C3×M16—24no—
127C2×C3×C8—22no—
128C2×C3×C8—22no—
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—21no—
133C2×C3×C8—22no—
134C2×C3×C8—21no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—21no—
138C2×C3×C8C2×Dic312yes—
139C2×C3×C8C2×Dic311yescharacteristic
140C2×C3×C8#6712yes—
141C2×C3×C8#6712yes—
142C2×C3×C8C2×C3×C412yes—
143C2×C3×C8C2×C3×C411yescharacteristic
144C2×C3×C8C3×D412yes—
145C2×C3×C8C3×D412yes—
146C2×C3×C8C22×D311yescharacteristic
147C2×C3×C8C23×C311yescharacteristic
148C3×C16—64no—
149C3⋊C16—24no—
150C22×C32—22no—
151C22×C32—22no—
152C22×C32—42no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32C22⋊C811yescharacteristic
156C22×C32C22⋊C811yescharacteristic
157C22×C32C22×C811yescharacteristic
158C32×C4—22no—
159C32×C4C22⋊C811yescharacteristic
160C32×C4C22⋊C811yescharacteristic
161C32×C4C2×C22⋊C411yescharacteristic
162C32×C4C2×M1611yescharacteristic
163C23×C4C3×Dic311yescharacteristic
164C2×M16—22no—
165C2×M16—22no—
166C2×M16C3×Dic312yes—
167C2×M16C2×C3×D312yes—
168C22×C8C3×Dic311yescharacteristic
169C22×C8C2×C3×D311yescharacteristic
170M32—34no—
171M32—62no—
172C2×C16—62no—
173C23×C3—22no—
174C23×C3—22no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—22no—
181C23×C3—42no—
182C23×C3—42no—
183C23×C3—44no—
184C23×C3C2×C3⋊C811yescharacteristic
185C23×C3C2×C3⋊C812yes—
186C23×C3C2×C3×C811yescharacteristic
187C23×C3C2×C3×C812yes—
188C2×C3×C4—24no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4C2×C3⋊C812yes—
214C2×C3×C4C2×C3⋊C811yescharacteristic
215C2×C3×C4#20811yescharacteristic
216C2×C3×C4#20811yescharacteristic
217C2×C3×C4#21711yescharacteristic
218C2×C3×C4#21711yescharacteristic
219C2×C3×C4C3×C22⋊C411yescharacteristic
220C2×C3×C4C3×C22⋊C411yescharacteristic
221C2×C3×C4C2×C3×C812yes—
222C2×C3×C4C2×C3×C811yescharacteristic
223C2×C3×C4C3×M1611yescharacteristic
224C2×C3×C4C3×M1611yescharacteristic
225C2×C3×C4C22×Dic311yescharacteristic
226C2×C3×C4C22×C3×C411yescharacteristic
227C3×C8—24no—
228C3×C8—22no—
229C3×C8—22no—
230C3×C8—22no—
231C3×C8—22no—
232C3×C8#21712yes—
233C3×C8C3×C22⋊C412yes—
234C3×C8C2×#6712yes—
235C3×C8C2×C3×D412yes—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—42no—
239C2×C32#34912yescomplemented
240C2×C32C2×C22⋊C811yescharacteristic
241C24C3×C3⋊C811yescharacteristic
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C3×C3⋊C811yescharacteristic
247C22×C4C3×C3⋊C812yes—
248C22×C4C2×C3×Dic312yes—
249C22×C4C2×C3×Dic311yescharacteristic
250M16—24no—
251M16—24no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8C2×C3×Dic312yes—
255C2×C8C2×C3×Dic311yescharacteristic
256C2×C8C3×#6712yes—
257C2×C8C3×#6712yes—
258C2×C8C22×C3×D311yescharacteristic
259C16—64no—
260C22×C3—24no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—24no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—44no—
272C22×C3—44no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—44no—
276C22×C3—42no—
277C22×C3—42no—
278C22×C3#83011yescharacteristic
279C22×C3#83011yescharacteristic
280C22×C3C3×C22⋊C811yescharacteristic
281C22×C3C3×C22⋊C811yescharacteristic
282C22×C3C22×C3⋊C811yescharacteristic
283C22×C3C22×C3×C811yescharacteristic
284C3×C4—24no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4#83011yescharacteristic
292C3×C4#83011yescharacteristic
293C3×C4C3×C22⋊C811yescharacteristic
294C3×C4C3×C22⋊C811yescharacteristic
295C3×C4C2×#20811yescharacteristic
296C3×C4C2×#21711yescharacteristic
297C3×C4C2×C3×C22⋊C411yescharacteristic
298C3×C4C2×C3×M1611yescharacteristic
299C32C2×#34911yescharacteristic, complemented, Sylow
300C23—21no—
301C23—21no—
302C23—42no—
303C23C2×C3×C3⋊C811yescharacteristic
304C23C2×C3×C3⋊C812yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4C2×C3×C3⋊C812yes—
312C2×C4C2×C3×C3⋊C811yescharacteristic
313C2×C4C3×#20811yescharacteristic
314C2×C4C3×#20811yescharacteristic
315C2×C4C3×#21711yescharacteristic
316C2×C4C3×#21711yescharacteristic
317C2×C4C22×C3×Dic311yescharacteristic
318C8—22no—
319C8C3×#21712yes—
320C8C2×C3×#6712yes—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—44no—
330C2×C3—42no—
331C2×C3—42no—
332C2×C3#455312yesdirect product factor
333C2×C3C3×#34912yescomplemented
334C2×C3C2×#83011yescharacteristic
335C2×C3C2×C3×C22⋊C811yescharacteristic
336C22—22no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22C3×#83011yescharacteristic
342C22C3×#83011yescharacteristic
343C22C22×C3×C3⋊C811yescharacteristic
344C4—22no—
345C4C3×#83011yescharacteristic
346C4C3×#83011yescharacteristic
347C4C2×C3×#20811yescharacteristic
348C4C2×C3×#21711yescharacteristic
349C3—21no—
350C3C2×#455311yescharacteristic, direct product factor
351C3C2×C3×#34911yescharacteristic, complemented
352C2—21no—
353C2—21no—
354C2—42no—
355C2C3×#455312yesdirect product factor
356C2C2×C3×#83011yescharacteristic
subgroupG/H
centerC2×C3×C4#217
commutatorC22×C3C22×C3×C8
FrattiniC2×C8C22×C3×D3
FittingC22×C32×M16C2
socleC22×C32C22⋊C8
prime radicalC24×C32C8
cosocleC2×C3×C8C23×C3
hypercenterC22×C3×M16D3
hypocenterC3C2×C3×#349
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C8 ◁ C22×C3×M16

Complements

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

NH ≅ G/Ntype
2. C2×C3×#812C2semidirect
3. C3×#4553C2direct (N × H)
5. C2×#4553C3direct (N × H)
14. C3×#812C22semidirect
20. C2×#812C2×C3semidirect
22. #4553C2×C3direct (N × H)
68. #812C22×C3semidirect
239. C2×C32#349semidirect
299. C32C2×#349semidirect
332. C2×C3#4553direct (N × H)
333. C2×C3C3×#349semidirect
350. C3C2×#4553direct (N × H)
351. C3C2×C3×#349semidirect
355. C2C3×#4553direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×M162
C2×C3×#8122
C3×#45532
C2×C3×#3493
C2×#45533