Finite Group Databasebeta

← C3×#71872 #11796525

Subgroups — C3×#71872 #11796525

Order 1152.

595 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#536C211yesmaximal, characteristic
2C2×C3×#836C212yesmaximal, complemented
3C3×#4548C212yesmaximal, complemented
4C3×#4542C212yesmaximal, complemented
5C3×#2133—31nomaximal
6#71872C311yesmaximal, characteristic, direct product factor
7C2×C32×#41C411yescharacteristic
8C22×C32×Q8C2211yescharacteristic
9C32×C4⋊Q8C411yescharacteristic
10C32×C4⋊Q8C2211yescharacteristic
11C32×#124C411yescharacteristic
12C32×#124C2211yescharacteristic
13C32×#123—22no—
14C32×C22⋊Q8—22no—
15C32×#117C411yescharacteristic
16C2×C3×#208—22no—
17C3×#836—22no—
18C3×#836C2214yes—
19C3×#536—21no—
20C3×#536D311yescharacteristic
21C3×#536C2×C311yescharacteristic
22C2×C3×#101—32no—
23C2×#836C2×C312yescomplemented
24C3×#356—32no—
25C3×#355—32no—
26#4548C2×C312yescomplemented
27#4542C2×C312yescomplemented
28C32×#41—42no—
29C2×C32×Q8—42no—
30C2×C32×Q8C2×C411yescharacteristic
31C2×C32×Q8C2×C411yescharacteristic
32C2×C32×Q8D412yes—
33C2×C32×Q8C2311yescharacteristic
34C2×C32×D4—21no—
35C2×C32×D4C2×C411yescharacteristic
36C22×C32×C4—21no—
37C22×C32×C4C2×C411yescharacteristic
38C22×C32×C4D412yes—
39C32×C4⋊C4—21no—
40C32×C4⋊C4—22no—
41C32×C4⋊C4—22no—
42C32×C22⋊C4—21no—
43C32×C22⋊C4—22no—
44C32×C22⋊C4—22no—
45C32×C42C2×C411yescharacteristic
46C32×C42C2×C411yescharacteristic
47C3×#208—24no—
48C3×#208—42no—
49C2×C3×C3⋊C8—42no—
50#2133—31noSylow
51C2×C3×#41—21no—
52C2×C3×#41Dic311yescharacteristic
53C2×C3×#41C3×C411yescharacteristic
54C22×C3×Q8—21no—
55C22×C3×Q8C2×D311yescharacteristic
56C22×C3×Q8C22×C311yescharacteristic
57C2×C3×M16—62no—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8Dic311yescharacteristic
61C3×C4⋊Q8C3×C411yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4⋊Q8C22×C311yescharacteristic
64C3×#124—21no—
65C3×#124—21no—
66C3×#124Dic311yescharacteristic
67C3×#124C3×C411yescharacteristic
68C3×#124C2×D311yescharacteristic
69C3×#124C22×C311yescharacteristic
70C3×#123—24no—
71C3×#123—22no—
72C3×#123—22no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—22no—
76C3×#117—21no—
77C3×#117Dic311yescharacteristic
78C3×#117C3×C411yescharacteristic
79C2×#208—22no—
80C3×#101—34no—
81C3×#101—62no—
82#836—22no—
83#836C22×C314yes—
84C23×C32—21no—
85C23×C32C22⋊C411yescharacteristic
86C32×Q8—41no—
87C32×Q8—42no—
88C32×Q8—41no—
89C32×Q8—41no—
90C32×D4—41no—
91C32×D4—42no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—42no—
99C2×C32×C4C22⋊C412yes—
100C2×C32×C4C22⋊C411yescharacteristic
101C2×C32×C4C22×C411yescharacteristic
102C2×C32×C4C2×D412yes—
103C3×C3⋊C8—44no—
104#536C3×D311yescharacteristic
105C2×#101—32no—
106#356—32no—
107#355—32no—
108C3×#41—44no—
109C3×#41—42no—
110C3×#41—42no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—21no—
115C2×C3×Q8—42no—
116C2×C3×Q8—42no—
117C2×C3×Q8—44no—
118C2×C3×Q8C2×Dic311yescharacteristic
119C2×C3×Q8C2×Dic311yescharacteristic
120C2×C3×Q8#6712yes—
121C2×C3×Q8C2×C3×C411yescharacteristic
122C2×C3×Q8C2×C3×C411yescharacteristic
123C2×C3×Q8C3×D412yes—
124C2×C3×Q8C22×D311yescharacteristic
125C2×C3×Q8C23×C311yescharacteristic
126C2×C3×D4—22no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4C2×Dic311yescharacteristic
131C2×C3×D4C2×C3×C411yescharacteristic
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4C2×Dic311yescharacteristic
138C22×C3×C4#6712yes—
139C22×C3×C4C2×C3×C411yescharacteristic
140C22×C3×C4C3×D412yes—
141C3×M16—64no—
142C3×M16—122no—
143C2×C3×C8—122no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C22⋊C4—22no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C42—21no—
163C3×C42—21no—
164C3×C42C2×Dic311yescharacteristic
165C3×C42C2×Dic311yescharacteristic
166C3×C42C2×C3×C411yescharacteristic
167C3×C42C2×C3×C411yescharacteristic
168#208—24no—
169#208—42no—
170C2×C3⋊C8—42no—
171C22×C32—41no—
172C22×C32—42no—
173C22×C32—41no—
174C22×C32#9912yes—
175C22×C32C2×C22⋊C411yescharacteristic
176C32×C4—21no—
177C32×C4—21no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—41no—
181C32×C4—41no—
182C32×C4—41no—
183C2×#41C3×Dic311yescharacteristic
184C22×Q8C2×C3×D311yescharacteristic
185C2×M16—62no—
186C4⋊Q8C3×Dic311yescharacteristic
187C4⋊Q8C2×C3×D311yescharacteristic
188#124C3×Dic311yescharacteristic
189#124C2×C3×D311yescharacteristic
190#123—22no—
191C22⋊Q8—22no—
192#117C3×Dic311yescharacteristic
193#101—34no—
194#101—62no—
195C23×C3—22no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3#21711yescharacteristic
200C23×C3C3×C22⋊C411yescharacteristic
201C3×Q8—41no—
202C3×Q8—42no—
203C3×Q8—41no—
204C3×Q8—42no—
205C3×Q8—41no—
206C3×Q8—41no—
207C3×Q8—41no—
208C3×Q8—41no—
209C3×Q8—42no—
210C3×Q8—44no—
211C3×Q8—42no—
212C3×Q8—42no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—42no—
216C3×D4—44no—
217C3×D4—42no—
218C3×D4—42no—
219C2×C3×C4—24no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—44no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4#21712yes—
245C2×C3×C4#21711yescharacteristic
246C2×C3×C4C3×C22⋊C412yes—
247C2×C3×C4C3×C22⋊C411yescharacteristic
248C2×C3×C4C22×Dic311yescharacteristic
249C2×C3×C4C2×#6712yes—
250C2×C3×C4C22×C3×C411yescharacteristic
251C2×C3×C4C2×C3×D412yes—
252C3×C8—124no—
253C3⋊C8—44no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—81no—
257C2×C32C2×#9911yescharacteristic
258#41—42no—
259C2×Q8—42no—
260C2×Q8C2×C3×Dic311yescharacteristic
261C2×Q8C2×C3×Dic311yescharacteristic
262C2×Q8C3×#6712yes—
263C2×Q8C22×C3×D311yescharacteristic
264C2×D4—21no—
265C2×D4C2×C3×Dic311yescharacteristic
266C22×C4—21no—
267C22×C4C2×C3×Dic311yescharacteristic
268C22×C4C3×#6712yes—
269M16—64no—
270M16—122no—
271C2×C8—122no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C22⋊C4—21no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C42C2×C3×Dic311yescharacteristic
279C42C2×C3×Dic311yescharacteristic
280C22×C3—21no—
281C22×C3—22no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3—44no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3#83412yes—
292C22×C3C3×#9912yes—
293C22×C3C2×#21711yescharacteristic
294C22×C3C2×C3×C22⋊C411yescharacteristic
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—22no—
304C3×C4—21no—
305C3×C4—42no—
306C3×C4—42no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—41no—
316C32#213311yescharacteristic, complemented, Sylow
317C23—21no—
318C23C3×#21711yescharacteristic
319Q8—41no—
320Q8—42no—
321Q8—41no—
322Q8—41no—
323D4—41no—
324D4—42no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4C3×#21712yes—
333C2×C4C3×#21711yescharacteristic
334C2×C4C22×C3×Dic311yescharacteristic
335C2×C4C2×C3×#6712yes—
336C8—124no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—21no—
343C2×C3—42no—
344C2×C3—82no—
345C2×C3—81no—
346C2×C3—81no—
347C2×C3C2×#83411yescharacteristic
348C2×C3C2×C3×#9911yescharacteristic
349C22—41no—
350C22—42no—
351C22—41no—
352C22C3×#83412yes—
353C22C2×C3×#21711yescharacteristic
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C3—21no—
362C3#7187211yescharacteristic, direct product factor
363C3C3×#213311yescharacteristic, complemented
364C2—21no—
365C2—22no—
366C2—81no—
367C2C2×C3×#83411yescharacteristic
subgroupG/H
centerC2×C3C2×#834
commutatorC2×C3×C4C22×C3×C4
FrattiniC2×Q8C22×C3×D3
FittingC32×#536C2
socleC2×C32C2×#99
prime radicalC2×C32×#41C4
cosocleC2×C3×Q8C23×C3
hypercenterC3×#536D3
hypocenterC3C3×#2133
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ C3×#536

Complements

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

NH ≅ G/Ntype
2. C2×C3×#836C2semidirect
3. C3×#4548C2semidirect
4. C3×#4542C2semidirect
6. #71872C3direct (N × H)
23. C2×#836C2×C3semidirect
26. #4548C2×C3semidirect
27. #4542C2×C3semidirect
316. C32#2133semidirect
362. C3#71872direct (N × H)
363. C3C3×#2133semidirect

Maximal subgroups

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

subgroupindex
C32×#5362
C2×C3×#8362
C3×#45482
C3×#45422
C3×#21333
#718723