Finite Group Databasebeta

← C3×#72129 #11784638

Subgroups — C3×#72129 #11784638

Order 1152.

2321 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×D4C211yesmaximal, characteristic
2C2×C3×#833C214yesmaximal, complemented
3C3×#5207C212yesmaximal, complemented
4C3×#1783—31nomaximal
5#72129C311yesmaximal, characteristic, direct product factor
6C25×C32C412yes—
7C22×C32×D4—28no—
8C22×C32×D4—24no—
9C22×C32×D4C412yes—
10C22×C32×D4C2212yes—
11C23×C32×C4C2211yescharacteristic
12C2×C3×#208C2214yescomplemented
13C3×#833—28no—
14C3×#830—24no—
15C23×C3×D4—21no—
16C23×C3×D4D311yescharacteristic
17C23×C3×D4C2×C311yescharacteristic
18C2×C3×#100—34no—
19C3×#407—32no—
20C2×#833C2×C314yescomplemented
21#5207C2×C312yescomplemented
22C24×C32—28no—
23C24×C32—24no—
24C24×C32—48no—
25C24×C32C2×C411yescharacteristic
26C24×C32C2×C414yes—
27C2×C32×D4—232no—
28C2×C32×D4—28no—
29C2×C32×D4—24no—
30C2×C32×D4—24no—
31C2×C32×D4—24no—
32C2×C32×D4D418yes—
33C22×C32×C4—24no—
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4D414yes—
36C22×C32×C4C2311yescharacteristic
37C3×#208—28no—
38C2×C3×C3⋊C8—24no—
39#1783—31noSylow
40C25×C3—22no—
41C25×C3Dic312yes—
42C25×C3C3×C412yes—
43C22×C3×D4—216no—
44C22×C3×D4—28no—
45C22×C3×D4—28no—
46C22×C3×D4—28no—
47C22×C3×D4—24no—
48C22×C3×D4—24no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4Dic312yes—
52C22×C3×D4C3×C412yes—
53C22×C3×D4C2×D312yes—
54C22×C3×D4C22×C312yes—
55C23×C3×C4—21no—
56C23×C3×C4C2×D311yescharacteristic
57C23×C3×C4C22×C311yescharacteristic
58C2×C3×M16—34no—
59C2×#208C22×C314yescomplemented
60C3×#100—68no—
61C3×C22⋊C8—64no—
62#833—28no—
63#830—24no—
64C23×C32—24no—
65C23×C32—22no—
66C23×C32—216no—
67C23×C32—24no—
68C23×C32—24no—
69C23×C32—48no—
70C23×C32—432no—
71C23×C32—48no—
72C23×C32—48no—
73C23×C32C22⋊C412yes—
74C23×C32C22⋊C418yes—
75C23×C32C22×C411yescharacteristic
76C32×D4—216no—
77C32×D4—216no—
78C2×C32×C4—22no—
79C2×C32×C4—28no—
80C2×C32×C4C22⋊C412yes—
81C2×C32×C4C2×D412yes—
82C2×C32×C4C2×D414yes—
83C3×C3⋊C8—44no—
84C23×D4C3×D311yescharacteristic
85C2×#100—34no—
86#407—32no—
87C24×C3—216no—
88C24×C3—28no—
89C24×C3—28no—
90C24×C3—24no—
91C24×C3—28no—
92C24×C3—24no—
93C24×C3—21no—
94C24×C3—24no—
95C24×C3—48no—
96C24×C3—48no—
97C24×C3—416no—
98C24×C3C2×Dic311yescharacteristic
99C24×C3C2×Dic314yes—
100C24×C3C2×C3×C411yescharacteristic
101C24×C3C2×C3×C414yes—
102C2×C3×D4—264no—
103C2×C3×D4—232no—
104C2×C3×D4—232no—
105C2×C3×D4—216no—
106C2×C3×D4—28no—
107C2×C3×D4—28no—
108C2×C3×D4—28no—
109C2×C3×D4—28no—
110C2×C3×D4—28no—
111C2×C3×D4—24no—
112C2×C3×D4—24no—
113C2×C3×D4—24no—
114C2×C3×D4—24no—
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—28no—
118C2×C3×D4#6718yes—
119C2×C3×D4C3×D418yes—
120C22×C3×C4—28no—
121C22×C3×C4—24no—
122C22×C3×C4—24no—
123C22×C3×C4—21no—
124C22×C3×C4—24no—
125C22×C3×C4—21no—
126C22×C3×C4C2×Dic311yescharacteristic
127C22×C3×C4#6714yes—
128C22×C3×C4C2×C3×C411yescharacteristic
129C22×C3×C4C3×D414yes—
130C22×C3×C4C22×D311yescharacteristic
131C22×C3×C4C23×C311yescharacteristic
132C3×M16—68no—
133C2×C3×C8—64no—
134#208—28no—
135C2×C3⋊C8—24no—
136C22×C32—28no—
137C22×C32—22no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—28no—
141C22×C32—48no—
142C22×C32—432no—
143C22×C32—48no—
144C22×C32—48no—
145C22×C32#10014yescomplemented
146C22×C32C2×C22⋊C412yes—
147C22×C32C2×C22⋊C411yescharacteristic
148C32×C4—22no—
149C32×C4#12014yes—
150C25C3×Dic312yes—
151C22×D4—28no—
152C22×D4—24no—
153C22×D4C3×Dic312yes—
154C22×D4C2×C3×D312yes—
155C23×C4C2×C3×D311yescharacteristic
156C2×M16—34no—
157#100—68no—
158C22⋊C8—64no—
159C23×C3—24no—
160C23×C3—28no—
161C23×C3—24no—
162C23×C3—24no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—232no—
166C23×C3—216no—
167C23×C3—216no—
168C23×C3—28no—
169C23×C3—28no—
170C23×C3—24no—
171C23×C3—24no—
172C23×C3—24no—
173C23×C3—24no—
174C23×C3—22no—
175C23×C3—28no—
176C23×C3—21no—
177C23×C3—48no—
178C23×C3—48no—
179C23×C3—416no—
180C23×C3—432no—
181C23×C3—432no—
182C23×C3—464no—
183C23×C3—416no—
184C23×C3—48no—
185C23×C3—48no—
186C23×C3—416no—
187C23×C3—48no—
188C23×C3—48no—
189C23×C3#21712yes—
190C23×C3#21718yes—
191C23×C3C3×C22⋊C412yes—
192C23×C3C3×C22⋊C418yes—
193C23×C3C22×Dic311yescharacteristic
194C23×C3C22×C3×C411yescharacteristic
195C3×D4—232no—
196C3×D4—216no—
197C3×D4—216no—
198C3×D4—232no—
199C3×D4—216no—
200C3×D4—216no—
201C2×C3×C4—24no—
202C2×C3×C4—216no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—28no—
206C2×C3×C4—28no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—24no—
210C2×C3×C4#21712yes—
211C2×C3×C4C3×C22⋊C412yes—
212C2×C3×C4C2×#6714yes—
213C2×C3×C4C2×#6712yes—
214C2×C3×C4C2×C3×D412yes—
215C2×C3×C4C2×C3×D414yes—
216C3×C8—124no—
217C3⋊C8—44no—
218C2×C32—24no—
219C2×C32—22no—
220C2×C32—48no—
221C2×C32#37911yescharacteristic
222C2×C32C2×#10012yes—
223C24—28no—
224C24—24no—
225C24—48no—
226C24C2×C3×Dic311yescharacteristic
227C24C2×C3×Dic314yes—
228C2×D4—232no—
229C2×D4—28no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4C3×#6718yes—
234C22×C4—24no—
235C22×C4C2×C3×Dic311yescharacteristic
236C22×C4C3×#6714yes—
237C22×C4C22×C3×D311yescharacteristic
238M16—68no—
239C2×C8—64no—
240C22×C3—24no—
241C22×C3—216no—
242C22×C3—28no—
243C22×C3—28no—
244C22×C3—24no—
245C22×C3—24no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—216no—
253C22×C3—28no—
254C22×C3—28no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—24no—
258C22×C3—48no—
259C22×C3—48no—
260C22×C3—416no—
261C22×C3—416no—
262C22×C3—464no—
263C22×C3—432no—
264C22×C3—48no—
265C22×C3—432no—
266C22×C3—48no—
267C22×C3—416no—
268C22×C3—48no—
269C22×C3—48no—
270C22×C3#83314yescomplemented
271C22×C3C3×#10014yescomplemented
272C22×C3C2×#21712yes—
273C22×C3C2×#21711yescharacteristic
274C22×C3C2×C3×C22⋊C412yes—
275C22×C3C2×C3×C22⋊C411yescharacteristic
276C3×C4—24no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—24no—
280C3×C4#95314yes—
281C3×C4C3×#12014yes—
282C32#178311yescharacteristic, complemented, Sylow
283C23—22no—
284C23—24no—
285C23—216no—
286C23—24no—
287C23—24no—
288C23—48no—
289C23—432no—
290C23—48no—
291C23—48no—
292C23C3×#21712yes—
293C23C3×#21718yes—
294C23C22×C3×Dic311yescharacteristic
295D4—216no—
296D4—216no—
297C2×C4—22no—
298C2×C4—28no—
299C2×C4C3×#21712yes—
300C2×C4C2×C3×#6712yes—
301C2×C4C2×C3×#6714yes—
302C8—124no—
303C2×C3—24no—
304C2×C3—28no—
305C2×C3—24no—
306C2×C3—22no—
307C2×C3—24no—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—416no—
312C2×C3—48no—
313C2×C3—48no—
314C2×C3C2×#83312yes—
315C2×C3#524711yescharacteristic
316C2×C3C3×#37911yescharacteristic
317C2×C3C2×C3×#10012yes—
318C22—28no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—28no—
323C22—48no—
324C22—48no—
325C22—432no—
326C22—48no—
327C22C3×#83314yescomplemented
328C22C2×C3×#21712yes—
329C22C2×C3×#21711yescharacteristic
330C4—22no—
331C4C3×#95314yes—
332C3—21no—
333C3#7212911yescharacteristic, direct product factor
334C3C3×#178311yescharacteristic, complemented
335C2—22no—
336C2—24no—
337C2—48no—
338C2C2×C3×#83312yes—
339C2C3×#524711yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC23×C32×D4C2
socleC22×C32C2×C22⋊C4
cosocleC22×C3×C4C23×C3
hypercenterC23×C3×D4D3
hypocenterC3C3×#1783
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#833C2semidirect
3. C3×#5207C2semidirect
5. #72129C3direct (N × H)
12. C2×C3×#208C22semidirect
20. C2×#833C2×C3semidirect
21. #5207C2×C3semidirect
59. C2×#208C22×C3semidirect
145. C22×C32#100semidirect
270. C22×C3#833semidirect
271. C22×C3C3×#100semidirect
282. C32#1783semidirect
327. C22C3×#833semidirect
333. C3#72129direct (N × H)
334. C3C3×#1783semidirect

Maximal subgroups

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

subgroupindex
C23×C32×D42
C2×C3×#8332
C3×#52072
C3×#17833
#721293