Finite Group Databasebeta

← C3×#69008 #11792240

Subgroups — C3×#69008 #11792240

Order 1152.

861 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#908C212yesmaximal, complemented
2C3×#5510C211yesmaximal, characteristic, complemented
3C2×C3×#825C212yesmaximal, complemented
4C3×#4988C211yesmaximal, characteristic, complemented
5C32×#343C211yesmaximal, characteristic, complemented
6C3×#2006—31nomaximal
7#69008C311yesmaximal, characteristic, direct product factor
8C2×C3×#235C2212yes—
9C22×C3×D12C2211yescharacteristic
10C2×C32×M16C2212yes—
11C2×C32×C4⋊C4C2211yescharacteristic
12C3×#930—22no—
13C2×C3×(C2×D3)⋊C4—22no—
14C2×C3×C4×Dic3C2211yescharacteristic
15C3×#908—24no—
16C3×#908—24no—
17C2×C3×D24—22no—
18C2×C3×#204—22no—
19C3×#897—22no—
20C3×#895—22no—
21C3×#825—24no—
22C2×C3×AGL(1,ℤ/8)—32no—
23C3×#537—31no—
24C2×#908C2×C312yescomplemented
25#5510C2×C311yescharacteristic, complemented
26C2×C3×#104—32no—
27C3×#390—31no—
28C2×#825C2×C312yescomplemented
29#4988C2×C311yescharacteristic, complemented
30C3×#343—21no—
31C3×#343D311yescharacteristic, complemented
32C3×#343C2×C311yescharacteristic, complemented
33C23×C3×D3—21no—
34C22×C32×C4D412yes—
35C22×C32×C4C2311yescharacteristic
36C2×C3×#67—22no—
37C22×C3×Dic3—21no—
38C3×#235—22no—
39C3×#235—22no—
40C3×#235—22no—
41C3×#235—22no—
42C2×C3×D12—22no—
43C2×C3×D12—22no—
44C2×C3×D12—21no—
45C2×C3×D12—22no—
46C2×C3×D12D412yes—
47C2×C3×D12D412yescomplemented
48C2×C3×C4×D3—22no—
49C2×C3×Dic6D412yes—
50C32×M16—22no—
51C32×M16D414yescomplemented
52C2×C32×C8—22no—
53C32×C4⋊C4—22no—
54C3×(C2×D3)⋊C4—24no—
55C3×(C2×D3)⋊C4—44no—
56C3×C4⋊Dic3—22no—
57C3×C4×Dic3—22no—
58C3×D24—28no—
59C3×#204—28no—
60#2006—31noSylow
61C2×C3×#41—32no—
62C22×C3×D4—31no—
63C2×#235C22×C312yes—
64C22×D12C22×C311yescharacteristic
65C3×AGL(1,ℤ/8)—64no—
66C3×AGL(1,ℤ/8)—64no—
67C2×C3×SD16—62no—
68C2×C3×D8—62no—
69C2×C3×M16—22no—
70C2×C3×M16C2×D312yes—
71C2×C3×M16C22×C312yes—
72C3×C22⋊Q8—62no—
73C3×#121—62no—
74C3×#121—62no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2×D311yescharacteristic
77C2×C3×C4⋊C4C22×C311yescharacteristic
78C2×C3×C22⋊C4—62no—
79C2×C3×C42—31no—
80#930—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4×Dic3C22×C311yescharacteristic
83#908—24no—
84#908—24no—
85C2×D24—22no—
86C2×#204—22no—
87#897—22no—
88#895—22no—
89C3×#104—64no—
90#825—24no—
91C23×C32C2×D411yescharacteristic
92C22×C3×D3—22no—
93C22×C3×D3—22no—
94C22×C3×D3—42no—
95C22×C3×D3—42no—
96C22×C3×D3—41no—
97C22×C3×D3—41no—
98C2×C32×C4—22no—
99C2×C32×C4—42no—
100C2×C32×C4C2×D412yes—
101C2×C32×C4C2×D411yescharacteristic
102C2×C32×C4C2×D412yes—
103C2×C32×C4#4111yescharacteristic
104C3×#67—42no—
105C3×#67—42no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—41no—
110C3×D12—24no—
111C3×D12—22no—
112C3×D12—22no—
113C3×D12—42no—
114C3×D12—82no—
115C3×C4×D3—42no—
116C3×C4×D3—42no—
117C3×Dic6—22no—
118C3×Dic6—22no—
119C32×C8—24no—
120C2×AGL(1,ℤ/8)—32no—
121#537—31no—
122C2×#104—32no—
123#390—31no—
124#343C3×D311yescharacteristic, complemented
125C24×C3—61no—
126C23×D3—21no—
127C3×#41—62no—
128C3×#41—62no—
129C3×#41—62no—
130C3×#41—62no—
131C2×C3×Q8—32no—
132C2×C3×D4—32no—
133C2×C3×D4—32no—
134C2×C3×D4—62no—
135C2×C3×D4—62no—
136C2×C3×D4—62no—
137C2×C3×D4—62no—
138C2×C3×D4—61no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—61no—
142C22×C3×C4—62no—
143C22×C3×C4D1212yes—
144C22×C3×C4C3×D412yes—
145C22×C3×C4C22×D311yescharacteristic
146C22×C3×C4C23×C311yescharacteristic
147C2×#67—22no—
148C22×Dic3—21no—
149#235—22no—
150#235—22no—
151#235—22no—
152#235—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—21no—
156C2×D12—22no—
157C2×D12C3×D412yes—
158C2×D12C3×D412yescomplemented
159C2×C4×D3—22no—
160C2×Dic6C3×D412yes—
161C3×SD16—68no—
162C3×D8—68no—
163C3×M16—22no—
164C3×M16—22no—
165C3×M16—24no—
166C3×M16—42no—
167C3×M16D1214yescomplemented
168C3×M16C3×D414yescomplemented
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—42no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—62no—
176C3×C22⋊C4—64no—
177C3×C22⋊C4—124no—
178C3×C42—62no—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—44no—
181C4⋊Dic3—22no—
182C4×Dic3—22no—
183D24—28no—
184#204—28no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32#12011yescharacteristic
188C22×C32#12111yescharacteristic
189C22×C32#12711yescharacteristic
190C2×C3×D3—44no—
191C2×C3×D3—42no—
192C2×C3×D3—42no—
193C2×C3×D3—42no—
194C2×C3×D3—42no—
195C2×C3×D3—82no—
196C2×C3×D3—81no—
197C2×C3×D3—81no—
198C32×C4—42no—
199C32×C4#12012yes—
200C32×C4#12112yes—
201C3×Dic3—42no—
202C3×Dic3—42no—
203C2×#41—32no—
204C22×D4—31no—
205AGL(1,ℤ/8)—64no—
206AGL(1,ℤ/8)—64no—
207C2×SD16—62no—
208C2×D8—62no—
209C2×M16C2×C3×D312yes—
210C22⋊Q8—62no—
211#121—62no—
212#121—62no—
213C2×C4⋊C4C2×C3×D311yescharacteristic
214C2×C22⋊C4—62no—
215C2×C42—31no—
216#104—64no—
217C23×C3—21no—
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—122no—
221C23×C3—122no—
222C23×C3—121no—
223C23×C3—121no—
224C23×C3C2×D1211yescharacteristic
225C23×C3C2×C3×D411yescharacteristic
226C22×D3—22no—
227C22×D3—22no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—41no—
231C22×D3—41no—
232C3×Q8—62no—
233C3×Q8—62no—
234C3×D4—62no—
235C3×D4—64no—
236C3×D4—62no—
237C3×D4—122no—
238C3×D4—122no—
239C3×D4—122no—
240C3×D4—242no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—62no—
251C2×C3×C4—62no—
252C2×C3×C4—82no—
253C2×C3×C4—121no—
254C2×C3×C4—121no—
255C2×C3×C4—122no—
256C2×C3×C4—122no—
257C2×C3×C4C2×D1212yes—
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4D3×D412yes—
260C2×C3×C4#23911yescharacteristic
261C2×C3×C4C2×C3×D412yes—
262C2×C3×C4C2×C3×D411yescharacteristic
263C2×C3×C4C2×C3×D412yes—
264C2×C3×C4C3×#4111yescharacteristic
265#67—42no—
266#67—42no—
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271D12—24no—
272D12—22no—
273D12—22no—
274D12—42no—
275D12—82no—
276C4×D3—42no—
277C4×D3—42no—
278Dic6—22no—
279Dic6—22no—
280C3×C8—24no—
281C3×C8—24no—
282C3×C8—44no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32#39211yescharacteristic
286C2×C32#45311yescharacteristic
287C2×C32#45411yescharacteristic
288C3×D3—82no—
289C3×D3—82no—
290C24—61no—
291#41—62no—
292#41—62no—
293#41—62no—
294#41—62no—
295C2×Q8—32no—
296C2×D4—32no—
297C2×D4—32no—
298C2×D4—62no—
299C2×D4—62no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—61no—
303C22×C4—61no—
304C22×C4—62no—
305C22×C4C3×D1212yes—
306C22×C4C22×C3×D311yescharacteristic
307SD16—68no—
308D8—68no—
309M16—22no—
310M16C3×D1214yescomplemented
311C2×C8—22no—
312C4⋊C4—22no—
313C4⋊C4—62no—
314C22⋊C4—64no—
315C22⋊C4—124no—
316C42—62no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—124no—
327C22×C3—122no—
328C22×C3—122no—
329C22×C3—122no—
330C22×C3—122no—
331C22×C3—242no—
332C22×C3—241no—
333C22×C3—241no—
334C22×C3#87411yescharacteristic
335C22×C3#88211yescharacteristic
336C22×C3#89511yescharacteristic
337C22×C3C3×#12011yescharacteristic
338C22×C3C3×#12111yescharacteristic
339C22×C3C3×#12711yescharacteristic
340C2×D3—44no—
341C2×D3—42no—
342C2×D3—42no—
343C2×D3—42no—
344C2×D3—42no—
345C2×D3—82no—
346C2×D3—81no—
347C2×D3—81no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—42no—
351C3×C4—42no—
352C3×C4—82no—
353C3×C4—122no—
354C3×C4—122no—
355C3×C4#88212yes—
356C3×C4#89512yes—
357C3×C4C3×#12012yes—
358C3×C4C3×#12112yes—
359Dic3—42no—
360Dic3—42no—
361C32#200611yescharacteristic, complemented, Sylow
362C23—62no—
363C23—62no—
364C23—122no—
365C23—122no—
366C23—121no—
367C23—121no—
368C23C2×C3×D1211yescharacteristic
369Q8—62no—
370Q8—62no—
371D4—62no—
372D4—64no—
373D4—62no—
374D4—122no—
375D4—122no—
376D4—122no—
377D4—242no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4C2×C3×D1212yes—
387C2×C4C3×D3×D411yescharacteristic
388C2×C4C3×D3×D412yes—
389C2×C4C3×#23911yescharacteristic
390C8—24no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—242no—
401C2×C3—242no—
402C2×C3#467111yescharacteristic
403C2×C3#474711yescharacteristic
404C2×C3#475311yescharacteristic
405C2×C3C3×#39211yescharacteristic
406C2×C3C3×#45311yescharacteristic
407C2×C3C3×#45411yescharacteristic
408D3—82no—
409D3—82no—
410C22—21no—
411C22—21no—
412C22—124no—
413C22—122no—
414C22—122no—
415C22—122no—
416C22—122no—
417C22—242no—
418C22—241no—
419C22—241no—
420C22C3×#87411yescharacteristic
421C22C3×#88211yescharacteristic
422C22C3×#89511yescharacteristic
423C4—42no—
424C4—122no—
425C4—122no—
426C4C3×#88212yes—
427C4C3×#89512yes—
428C3—21no—
429C3#6900811yescharacteristic, direct product factor
430C3C3×#200611yescharacteristic, complemented
431C2—21no—
432C2—21no—
433C2—242no—
434C2—242no—
435C2C3×#467111yescharacteristic
436C2C3×#474711yescharacteristic
437C2C3×#475311yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#343C2
socleC22×C32#120
hypercenterC3×#343D3
hypocenterC3C3×#2006
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#343

Complements

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

NH ≅ G/Ntype
1. C2×C3×#908C2semidirect
2. C3×#5510C2semidirect
3. C2×C3×#825C2semidirect
4. C3×#4988C2semidirect
5. C32×#343C2semidirect
7. #69008C3direct (N × H)
24. C2×#908C2×C3semidirect
25. #5510C2×C3semidirect
28. C2×#825C2×C3semidirect
29. #4988C2×C3semidirect
31. C3×#343D3semidirect
32. C3×#343C2×C3semidirect
47. C2×C3×D12D4semidirect
51. C32×M16D4semidirect
124. #343C3×D3semidirect
158. C2×D12C3×D4semidirect
167. C3×M16D12semidirect
168. C3×M16C3×D4semidirect
310. M16C3×D12semidirect
361. C32#2006semidirect
429. C3#69008direct (N × H)
430. C3C3×#2006semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9082
C3×#55102
C2×C3×#8252
C3×#49882
C32×#3432
C3×#20063
#690083