Finite Group Databasebeta

← C3×#74359 #11802954

Subgroups — C3×#74359 #11802954

Order 1152.

1293 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#892C211yesmaximal, characteristic
2C2×C3×D3×C4⋊C4C211yesmaximal, characteristic
3C2×C3×#872C211yesmaximal, characteristic
4C32×#384C211yesmaximal, characteristic, complemented
5C3×#4989C214yesmaximal
6C3×#4977C211yesmaximal, characteristic, complemented
7C3×#4976C214yesmaximal, complemented
8C3×#4975C211yesmaximal, characteristic, complemented
9C3×#4936C211yesmaximal, characteristic, complemented
10C3×#2317—31nomaximal
11#74359C311yesmaximal, characteristic, direct product factor
12C22×C3×C4×D3C2212yes—
13C22×C3×C4×D3C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C4⋊C4C2214yes—
17C2×C32×C42C2211yescharacteristic
18C2×C3×(C2×D3)⋊C4C2212yes—
19C2×C3×(C2×D3)⋊C4C2212yes—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2214yes—
24C2×C3×Dic3⋊C4C2212yes—
25C2×C3×C4×Dic3C2212yes—
26C2×C3×C4×Dic3C2211yescharacteristic
27C3×#892—24no—
28C3×D3×C4⋊C4—24no—
29C3×#872C418yescomplemented
30C3×C3⋊#95C2214yes—
31C3×C3⋊#95C2214yes—
32C2×C3×#117—31no—
33C2×C3×#117—31no—
34C22×C3×C4⋊C4—31no—
35C2×#892C2×C311yescharacteristic
36C2×D3×C4⋊C4C2×C311yescharacteristic
37C2×#872C2×C311yescharacteristic
38C3×#389—34no—
39C3×#385—34no—
40C3×#384—21no—
41C3×#384—31no—
42C3×#384D311yescharacteristic, complemented
43C3×#384C2×C311yescharacteristic, complemented
44C3×#382—31no—
45C3×#382—31no—
46#4989C2×C314yes—
47#4977C2×C311yescharacteristic, complemented
48#4976C2×C314yescomplemented
49#4975C2×C311yescharacteristic, complemented
50#4936C2×C311yescharacteristic, complemented
51C23×C3×D3C2311yescharacteristic
52C22×C32×C4C2312yes—
53C22×C32×C4C2312yes—
54C22×C32×C4C2312yes—
55C22×C32×C4C2311yescharacteristic
56C22×C3×Dic3C2312yes—
57C22×C3×Dic3C2312yes—
58C22×C3×Dic3C2312yes—
59C22×C3×Dic3C2311yescharacteristic
60C2×C3×C4×D3—24no—
61C2×C3×C4×D3—44no—
62C2×C3×C4×D3C2×C414yes—
63C2×C3×C4×D3D414yes—
64C2×C3×C4×D3Q814yes—
65C32×C4⋊C4—22no—
66C32×C4⋊C4—22no—
67C32×C4⋊C4—44no—
68C32×C42C2×C414yescomplemented
69C3×(C2×D3)⋊C4—24no—
70C3×(C2×D3)⋊C4C2×C418yes—
71C3×C4⋊Dic3—22no—
72C3×C4⋊Dic3—22no—
73C3×Dic3⋊C4—24no—
74C3×Dic3⋊C4—44no—
75C3×Dic3⋊C4C2×C418yescomplemented
76C3×C4×Dic3—24no—
77C3×C4×Dic3C2×C414yescomplemented
78#2317—31noSylow
79C23×C3×C4—31no—
80C23×C3×C4—32no—
81C22×C4×D3C22×C312yes—
82C22×C4×D3C22×C311yescharacteristic
83C3×#117—38no—
84C3×#117—64no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4—32no—
89C2×C3×C4⋊C4—32no—
90C2×C3×C4⋊C4—31no—
91C2×C3×C4⋊C4—31no—
92C2×C3×C4⋊C4—34no—
93C2×C3×C4⋊C4—64no—
94C2×C3×C4⋊C4C2×D311yescharacteristic
95C2×C3×C4⋊C4C2×D311yescharacteristic
96C2×C3×C4⋊C4C2×D314yes—
97C2×C3×C4⋊C4C22×C311yescharacteristic
98C2×C3×C4⋊C4C22×C311yescharacteristic
99C2×C3×C4⋊C4C22×C314yes—
100C2×C3×C22⋊C4—32no—
101C2×C3×C22⋊C4—32no—
102C2×C3×C42—21no—
103C2×C3×C42—31no—
104C2×C3×C42—32no—
105C2×C3×C42C2×D311yescharacteristic
106C2×C3×C42C22×C311yescharacteristic
107C2×(C2×D3)⋊C4C22×C312yes—
108C2×(C2×D3)⋊C4C22×C312yes—
109C2×C4⋊Dic3C22×C311yescharacteristic
110C2×C4⋊Dic3C22×C311yescharacteristic
111C2×Dic3⋊C4C22×C312yes—
112C2×Dic3⋊C4C22×C314yes—
113C2×Dic3⋊C4C22×C312yes—
114C2×C4×Dic3C22×C312yes—
115C2×C4×Dic3C22×C311yescharacteristic
116#892—24no—
117D3×C4⋊C4—24no—
118#872C3×C418yescomplemented
119C3×#95—34no—
120C3×#95—34no—
121C3⋊#95C22×C314yes—
122C3⋊#95C22×C314yes—
123C23×C32C2411yescharacteristic
124C22×C3×D3—22no—
125C22×C3×D3—22no—
126C22×C3×D3C22×C412yes—
127C22×C3×D3C2×D412yes—
128C22×C3×D3C2×Q812yes—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4C22×C414yes—
136C2×C32×C4C22×C412yes—
137C2×C32×C4C2×D412yes—
138C2×C32×C4C2×Q812yes—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3C22×C414yes—
146C2×C3×Dic3C22×C412yes—
147C2×C3×Dic3C2×D412yes—
148C2×C3×Dic3C2×Q812yes—
149C3×C4×D3—48no—
150C3×C4×D3C4⋊C4116yescomplemented
151C2×#117—31no—
152C2×#117—31no—
153C22×C4⋊C4—31no—
154#389—34no—
155#385—34no—
156#384—31no—
157#384C3×D311yescharacteristic, complemented
158#382—31no—
159#382—31no—
160C24×C3—31no—
161C23×D3C23×C311yescharacteristic
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—31no—
167C22×C3×C4—34no—
168C22×C3×C4—34no—
169C22×C3×C4—34no—
170C22×C3×C4—32no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—64no—
174C22×C3×C4—124no—
175C22×C3×C4C22×D312yes—
176C22×C3×C4C22×D312yes—
177C22×C3×C4C22×D312yes—
178C22×C3×C4C22×D311yescharacteristic
179C22×C3×C4C23×C312yes—
180C22×C3×C4C23×C312yes—
181C22×C3×C4C23×C312yes—
182C22×C3×C4C23×C311yescharacteristic
183C22×Dic3C23×C312yes—
184C22×Dic3C23×C312yes—
185C22×Dic3C23×C312yes—
186C22×Dic3C23×C311yescharacteristic
187C2×C4×D3—24no—
188C2×C4×D3—44no—
189C2×C4×D3C2×C3×C414yes—
190C2×C4×D3C3×D414yes—
191C2×C4×D3C3×Q814yes—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—38no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—44no—
200C3×C4⋊C4—44no—
201C3×C4⋊C4—48no—
202C3×C4⋊C4—64no—
203C3×C4⋊C4—62no—
204C3×C4⋊C4—62no—
205C3×C4⋊C4—124no—
206C3×C22⋊C4—38no—
207C3×C22⋊C4—64no—
208C3×C42—24no—
209C3×C42—34no—
210C3×C42—64no—
211C3×C42C4×D314yescomplemented
212C3×C42C2×C3×C414yescomplemented
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4C2×C3×C418yes—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217Dic3⋊C4—24no—
218Dic3⋊C4—44no—
219Dic3⋊C4C2×C3×C418yescomplemented
220C4×Dic3—24no—
221C4×Dic3C2×C3×C414yescomplemented
222C22×C32C23×C411yescharacteristic
223C22×C32C22×D411yescharacteristic
224C22×C32C22×Q811yescharacteristic
225C22×C32H(5,𝔽2)12yes—
226C22×C322−1+412yes—
227C2×C3×D3—24no—
228C2×C3×D3—22no—
229C2×C3×D3—22no—
230C2×C3×D3—24no—
231C2×C3×D3C2×C4⋊C414yes—
232C32×C4—24no—
233C32×C4—42no—
234C32×C4—42no—
235C32×C4C2×C4⋊C414yes—
236C3×Dic3—24no—
237C3×Dic3—42no—
238C3×Dic3—42no—
239C3×Dic3C2×C4⋊C414yes—
240C23×C4—31no—
241C23×C4—32no—
242#117—38no—
243#117—64no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4C2×C3×D311yescharacteristic
251C2×C4⋊C4C2×C3×D311yescharacteristic
252C2×C4⋊C4C2×C3×D314yes—
253C2×C22⋊C4—32no—
254C2×C22⋊C4—32no—
255C2×C42—31no—
256C2×C42—32no—
257C2×C42C2×C3×D311yescharacteristic
258#95—34no—
259#95—34no—
260C23×C3—21no—
261C23×C3—32no—
262C23×C3—32no—
263C23×C3—32no—
264C23×C3—62no—
265C23×C3—62no—
266C23×C3C23×D311yescharacteristic
267C23×C3C24×C311yescharacteristic
268C22×D3—22no—
269C22×D3—22no—
270C22×D3C22×C3×C412yes—
271C22×D3C2×C3×D412yes—
272C22×D3C2×C3×Q812yes—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—32no—
292C2×C3×C4—32no—
293C2×C3×C4—32no—
294C2×C3×C4—316no—
295C2×C3×C4—34no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4—44no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—64no—
305C2×C3×C4—64no—
306C2×C3×C4—128no—
307C2×C3×C4C2×C4×D314yes—
308C2×C3×C4C2×C4×D312yes—
309C2×C3×C4D3×D412yes—
310C2×C3×C4D3×Q812yes—
311C2×C3×C4C22×C3×C414yes—
312C2×C3×C4C22×C3×C412yes—
313C2×C3×C4C2×C3×D412yes—
314C2×C3×C4C2×C3×Q812yes—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3C22×C3×C414yes—
322C2×Dic3C22×C3×C412yes—
323C2×Dic3C2×C3×D412yes—
324C2×Dic3C2×C3×Q812yes—
325C4×D3—48no—
326C4×D3C3×C4⋊C4116yescomplemented
327C2×C32C22×C4⋊C411yescharacteristic
328C2×C32#52012yes—
329C2×C32#53511yescharacteristic
330C2×C32#53611yescharacteristic
331C2×C32#54412yes—
332C3×D3—24no—
333C24—31no—
334C22×C4—31no—
335C22×C4—34no—
336C22×C4—34no—
337C22×C4—34no—
338C22×C4—32no—
339C22×C4—32no—
340C22×C4—32no—
341C22×C4—64no—
342C22×C4—124no—
343C22×C4C22×C3×D312yes—
344C22×C4C22×C3×D312yes—
345C22×C4C22×C3×D312yes—
346C22×C4C22×C3×D311yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—38no—
350C4⋊C4—44no—
351C4⋊C4—64no—
352C4⋊C4—62no—
353C4⋊C4—62no—
354C4⋊C4—124no—
355C22⋊C4—38no—
356C22⋊C4—64no—
357C42—34no—
358C42—64no—
359C42C3×C4×D314yescomplemented
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—34no—
366C22×C3—62no—
367C22×C3—62no—
368C22×C3—64no—
369C22×C3—64no—
370C22×C3C22×C4×D311yescharacteristic
371C22×C3C2×D3×D411yescharacteristic
372C22×C3#100412yes—
373C22×C3C2×D3×Q811yescharacteristic
374C22×C3#100712yes—
375C22×C3C23×C3×C411yescharacteristic
376C22×C3C22×C3×D411yescharacteristic
377C22×C3C22×C3×Q811yescharacteristic
378C22×C3C3×H(5,𝔽2)12yes—
379C22×C3C3×2−1+412yes—
380C2×D3—24no—
381C2×D3—22no—
382C2×D3—22no—
383C2×D3—24no—
384C2×D3C2×C3×C4⋊C414yes—
385C3×C4—28no—
386C3×C4—24no—
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—34no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—64no—
395C3×C4—82no—
396C3×C4—82no—
397C3×C4—122no—
398C3×C4—122no—
399C3×C4D3×C4⋊C414yes—
400C3×C4C2×C3×C4⋊C414yes—
401Dic3—24no—
402Dic3—42no—
403Dic3—42no—
404Dic3C2×C3×C4⋊C414yes—
405C32#231711yescharacteristic, complemented, Sylow
406C23—32no—
407C23—32no—
408C23—32no—
409C23—62no—
410C23—62no—
411C23C23×C3×D311yescharacteristic
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—316no—
422C2×C4—34no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—64no—
428C2×C4—64no—
429C2×C4—128no—
430C2×C4C2×C3×C4×D314yes—
431C2×C4C2×C3×C4×D312yes—
432C2×C4C3×D3×D412yes—
433C2×C4C3×D3×Q812yes—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—64no—
440C2×C3C2×D3×C4⋊C411yescharacteristic
441C2×C3#550412yes—
442C2×C3#571511yescharacteristic
443C2×C3#572411yescharacteristic
444C2×C3#572912yes—
445C2×C3C22×C3×C4⋊C411yescharacteristic
446C2×C3C3×#52012yes—
447C2×C3C3×#53511yescharacteristic
448C2×C3C3×#53611yescharacteristic
449C2×C3C3×#54412yes—
450D3—24no—
451C22—34no—
452C22—64no—
453C22—62no—
454C22—62no—
455C22—64no—
456C22C22×C3×C4×D311yescharacteristic
457C22C2×C3×D3×D411yescharacteristic
458C22C3×#100412yes—
459C22C2×C3×D3×Q811yescharacteristic
460C22C3×#100712yes—
461C4—24no—
462C4—34no—
463C4—42no—
464C4—42no—
465C4—64no—
466C4—122no—
467C4—122no—
468C4C3×D3×C4⋊C414yes—
469C3—21no—
470C3#7435911yescharacteristic, direct product factor
471C3C3×#231711yescharacteristic, complemented
472C2—64no—
473C2C2×C3×D3×C4⋊C411yescharacteristic
474C2C3×#550412yes—
475C2C3×#571511yescharacteristic
476C2C3×#572411yescharacteristic
477C2C3×#572912yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#384C2
socleC23×C32C24
prime radicalC23×C3×D3C23
cosocleC23×C3C24×C3
hypercenterC3×#384D3
hypocenterC3C3×#2317
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#384

Complements

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

NH ≅ G/Ntype
4. C32×#384C2semidirect
6. C3×#4977C2semidirect
7. C3×#4976C2semidirect
8. C3×#4975C2semidirect
9. C3×#4936C2semidirect
11. #74359C3direct (N × H)
29. C3×#872C4semidirect
42. C3×#384D3semidirect
43. C3×#384C2×C3semidirect
47. #4977C2×C3semidirect
48. #4976C2×C3semidirect
49. #4975C2×C3semidirect
50. #4936C2×C3semidirect
68. C32×C42C2×C4semidirect
75. C3×Dic3⋊C4C2×C4semidirect
77. C3×C4×Dic3C2×C4semidirect
118. #872C3×C4semidirect
150. C3×C4×D3C4⋊C4semidirect
157. #384C3×D3semidirect
211. C3×C42C4×D3semidirect
212. C3×C42C2×C3×C4semidirect
219. Dic3⋊C4C2×C3×C4semidirect
221. C4×Dic3C2×C3×C4semidirect
326. C4×D3C3×C4⋊C4semidirect
359. C42C3×C4×D3semidirect
405. C32#2317semidirect
470. C3#74359direct (N × H)
471. C3C3×#2317semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8922
C2×C3×D3×C4⋊C42
C2×C3×#8722
C32×#3842
C3×#49892
C3×#49772
C3×#49762
C3×#49752
C3×#49362
C3×#23173
#743593