Finite Group Databasebeta

← C3×#71617 #11785117

Subgroups — C3×#71617 #11785117

Order 1152.

885 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×C3×#833C212yesmaximal
3C3×#4924C212yesmaximal, complemented
4C3×#4530C212yesmaximal, complemented
5C3×#1800—31nomaximal
6#71617C311yesmaximal, characteristic, direct product factor
7C22×C32×D4C2211yescharacteristic
8C23×C32×C4C412yes—
9C32×C4×D4—22no—
10C32×C4×D4—22no—
11C2×C32×C4⋊C4C2211yescharacteristic
12C2×C32×C22⋊C4C412yes—
13C2×C32×C42C2211yescharacteristic
14C2×C3×#208C2214yes—
15C3×#833—24no—
16C3×C4⋊(C3⋊C8)—22no—
17C3×C4⋊(C3⋊C8)—22no—
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4D311yescharacteristic
20C2×C3×C4×D4C2×C311yescharacteristic
21C3×#423—32no—
22C2×C3×#100—32no—
23C2×#833C2×C312yes—
24#4924C2×C312yescomplemented
25C3×#343—32no—
26#4530C2×C312yescomplemented
27C24×C32C2×C412yes—
28C2×C32×D4—22no—
29C2×C32×D4—22no—
30C2×C32×D4D412yes—
31C2×C32×D4Q812yes—
32C22×C32×C4—22no—
33C22×C32×C4—22no—
34C22×C32×C4—44no—
35C22×C32×C4C2×C411yescharacteristic
36C22×C32×C4C2×C412yes—
37C22×C32×C4C2×C411yescharacteristic
38C22×C32×C4C2311yescharacteristic
39C32×C4⋊C4—22no—
40C32×C22⋊C4—24no—
41C32×C42D412yes—
42C32×C42D412yes—
43C3×#208—24no—
44C3×#208—24no—
45C2×C3×C3⋊C8—24no—
46#1800—31noSylow
47C22×C3×D4—21no—
48C22×C3×D4C2×D311yescharacteristic
49C22×C3×D4C22×C311yescharacteristic
50C23×C3×C4—22no—
51C23×C3×C4Dic312yes—
52C23×C3×C4C3×C412yes—
53C2×C3×M16—34no—
54C3×C4×D4—24no—
55C3×C4×D4—24no—
56C3×C4×D4—22no—
57C3×C4×D4—22no—
58C3×C4×D4—22no—
59C3×C4×D4—22no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4C2×D311yescharacteristic
62C2×C3×C4⋊C4C22×C311yescharacteristic
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4Dic312yes—
65C2×C3×C22⋊C4C3×C412yes—
66C2×C3×C42—21no—
67C2×C3×C42C2×D311yescharacteristic
68C2×C3×C42C22×C311yescharacteristic
69C2×#208C22×C314yes—
70C3×C4⋊C8—62no—
71C3×C4⋊C8—62no—
72C3×#100—64no—
73#833—24no—
74C4⋊(C3⋊C8)—22no—
75C4⋊(C3⋊C8)—22no—
76C23×C32—22no—
77C23×C32—22no—
78C23×C32—42no—
79C23×C32—42no—
80C23×C32C4⋊C414yes—
81C23×C32C22×C411yescharacteristic
82C32×D4—24no—
83C32×D4—24no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—41no—
87C2×C32×C4—44no—
88C2×C32×C4—44no—
89C2×C32×C4—42no—
90C2×C32×C4C22⋊C412yes—
91C2×C32×C4C22⋊C411yescharacteristic
92C2×C32×C4C22⋊C411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D412yes—
95C2×C32×C4C2×Q811yescharacteristic
96C2×C32×C4#4112yes—
97C3×C3⋊C8—44no—
98C2×C4×D4C3×D311yescharacteristic
99#423—32no—
100C2×#100—32no—
101#343—32no—
102C24×C3—22no—
103C24×C3C2×Dic312yes—
104C24×C3C2×C3×C412yes—
105C2×C3×D4—24no—
106C2×C3×D4—24no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4Dic612yes—
114C2×C3×D4D1212yes—
115C2×C3×D4C3×D412yes—
116C2×C3×D4C3×Q812yes—
117C22×C3×C4—24no—
118C22×C3×C4—24no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—48no—
128C22×C3×C4—44no—
129C22×C3×C4—44no—
130C22×C3×C4C2×Dic311yescharacteristic
131C22×C3×C4C2×Dic312yes—
132C22×C3×C4C2×Dic311yescharacteristic
133C22×C3×C4C2×C3×C411yescharacteristic
134C22×C3×C4C2×C3×C412yes—
135C22×C3×C4C2×C3×C411yescharacteristic
136C22×C3×C4C22×D311yescharacteristic
137C22×C3×C4C23×C311yescharacteristic
138C3×M16—64no—
139C3×M16—64no—
140C2×C3×C8—64no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C22⋊C4—28no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—24no—
147C3×C42—22no—
148C3×C42—22no—
149C3×C42#6712yes—
150C3×C42#6712yes—
151C3×C42C3×D412yes—
152C3×C42C3×D412yes—
153#208—24no—
154#208—24no—
155C2×C3⋊C8—24no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32—24no—
159C22×C32—44no—
160C22×C32—44no—
161C22×C32—42no—
162C22×C32—42no—
163C22×C32C2×C22⋊C411yescharacteristic
164C22×C32C2×C4⋊C411yescharacteristic
165C22×C32#11711yescharacteristic
166C32×C4—21no—
167C32×C4—22no—
168C32×C4—42no—
169C32×C4#10012yescomplemented
170C32×C4#12112yes—
171C32×C4C22⋊Q812yes—
172C22×D4C2×C3×D311yescharacteristic
173C23×C4C3×Dic312yes—
174C2×M16—34no—
175C4×D4—22no—
176C4×D4—22no—
177C2×C4⋊C4C2×C3×D311yescharacteristic
178C2×C22⋊C4C3×Dic312yes—
179C2×C42C2×C3×D311yescharacteristic
180C4⋊C8—62no—
181C4⋊C8—62no—
182#100—64no—
183C23×C3—24no—
184C23×C3—24no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—24no—
190C23×C3—21no—
191C23×C3—42no—
192C23×C3—42no—
193C23×C3—42no—
194C23×C3—42no—
195C23×C3—44no—
196C23×C3—44no—
197C23×C3C4⋊Dic314yes—
198C23×C3C3×C4⋊C414yes—
199C23×C3C22×Dic311yescharacteristic
200C23×C3C22×C3×C411yescharacteristic
201C3×D4—28no—
202C3×D4—28no—
203C3×D4—24no—
204C3×D4—24no—
205C3×D4—24no—
206C3×D4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—24no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—48no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4—48no—
226C2×C3×C4—42no—
227C2×C3×C4—44no—
228C2×C3×C4—44no—
229C2×C3×C4—44no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4#21712yes—
233C2×C3×C4#21711yescharacteristic
234C2×C3×C4#21711yescharacteristic
235C2×C3×C4C3×C22⋊C412yes—
236C2×C3×C4C3×C22⋊C411yescharacteristic
237C2×C3×C4C3×C22⋊C411yescharacteristic
238C2×C3×C4C2×Dic611yescharacteristic
239C2×C3×C4C2×D1211yescharacteristic
240C2×C3×C4#23512yes—
241C2×C3×C4C2×#6712yes—
242C2×C3×C4C2×C3×D411yescharacteristic
243C2×C3×C4C2×C3×D412yes—
244C2×C3×C4C2×C3×Q811yescharacteristic
245C2×C3×C4C3×#4112yes—
246C3×C8—124no—
247C3⋊C8—44no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—44no—
251C2×C32#38011yescharacteristic
252C2×C32C2×#10011yescharacteristic
253C2×C32#41311yescharacteristic
254C24C2×C3×Dic312yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C3×Dic612yes—
258C2×D4C3×D1212yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—44no—
262C22×C4C2×C3×Dic311yescharacteristic
263C22×C4C2×C3×Dic312yes—
264C22×C4C2×C3×Dic311yescharacteristic
265C22×C4C22×C3×D311yescharacteristic
266M16—64no—
267M16—64no—
268C2×C8—64no—
269C4⋊C4—22no—
270C22⋊C4—24no—
271C42C3×#6712yes—
272C42C3×#6712yes—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—28no—
280C22×C3—24no—
281C22×C3—24no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—44no—
286C22×C3—44no—
287C22×C3—44no—
288C22×C3—48no—
289C22×C3—48no—
290C22×C3—44no—
291C22×C3—44no—
292C22×C3—44no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3C2×C4⋊Dic311yescharacteristic
298C22×C3#92611yescharacteristic
299C22×C3C2×#21711yescharacteristic
300C22×C3C2×C3×C22⋊C411yescharacteristic
301C22×C3C2×C3×C4⋊C411yescharacteristic
302C22×C3C3×#11711yescharacteristic
303C3×C4—22no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—21no—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—22no—
310C3×C4—22no—
311C3×C4—42no—
312C3×C4—44no—
313C3×C4—42no—
314C3×C4—42no—
315C3×C4#83312yescomplemented
316C3×C4C3×#10012yescomplemented
317C3×C4#92412yes—
318C3×C4#93012yes—
319C3×C4C3×#12112yes—
320C3×C4C3×C22⋊Q812yes—
321C32#180011yescharacteristic, complemented, Sylow
322C23—22no—
323C23—22no—
324C23—42no—
325C23—42no—
326C23C3×C4⋊Dic314yes—
327C23C22×C3×Dic311yescharacteristic
328D4—24no—
329D4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4C3×#21712yes—
337C2×C4C3×#21711yescharacteristic
338C2×C4C3×#21711yescharacteristic
339C2×C4C2×C3×Dic611yescharacteristic
340C2×C4C2×C3×D1211yescharacteristic
341C2×C4C3×#23512yes—
342C2×C4C2×C3×#6712yes—
343C8—124no—
344C2×C3—22no—
345C2×C3—22no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—48no—
354C2×C3—44no—
355C2×C3—44no—
356C2×C3#521211yescharacteristic
357C2×C3C2×#83311yescharacteristic
358C2×C3#523711yescharacteristic
359C2×C3C3×#38011yescharacteristic
360C2×C3C2×C3×#10011yescharacteristic
361C2×C3C3×#41311yescharacteristic
362C22—21no—
363C22—21no—
364C22—24no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22C2×C3×C4⋊Dic311yescharacteristic
370C22C3×#92611yescharacteristic
371C22C2×C3×#21711yescharacteristic
372C4—21no—
373C4—22no—
374C4—42no—
375C4C3×#83312yescomplemented
376C4C3×#92412yes—
377C4C3×#93012yes—
378C3—21no—
379C3#7161711yescharacteristic, direct product factor
380C3C3×#180011yescharacteristic, complemented
381C2—21no—
382C2—21no—
383C2—44no—
384C2C3×#521211yescharacteristic
385C2C2×C3×#83311yescharacteristic
386C2C3×#523711yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×C4×D4C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C32×D4C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×C4×D4D3
hypocenterC3C3×#1800
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C3×#4924C2semidirect
4. C3×#4530C2semidirect
6. #71617C3direct (N × H)
24. #4924C2×C3semidirect
26. #4530C2×C3semidirect
169. C32×C4#100semidirect
315. C3×C4#833semidirect
316. C3×C4C3×#100semidirect
321. C32#1800semidirect
375. C4C3×#833semidirect
379. C3#71617direct (N × H)
380. C3C3×#1800semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C2×C3×#8332
C3×#49242
C3×#45302
C3×#18003
#716173