Finite Group Databasebeta

← C3×#81802 #11878876

Subgroups — C3×#81802 #11878876

Order 1152.

829 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5728C211yesmaximal, characteristic
2C3×D3×C4⋊Q8C211yesmaximal, characteristic
3C32×#494C211yesmaximal, characteristic, complemented
4C3×#5186C214yesmaximal
5C3×#5182C214yesmaximal, complemented
6C3×#5092C211yesmaximal, characteristic, complemented
7C3×#5079C211yesmaximal, characteristic, complemented
8C3×#4685C211yesmaximal, characteristic
9C3×#4681C211yesmaximal, characteristic, complemented
10C3×#3209—31nomaximal
11#81802C311yesmaximal, characteristic, direct product factor
12C2×C3×D3×Q8C2212yes—
13C2×C32×Q16C2214yes—
14C32×C4⋊Q8C2211yescharacteristic
15C32×C4⋊Q8C2211yescharacteristic
16C3×#946C2212yes—
17C3×Q8×Dic3C2212yes—
18C3×#944C2212yes—
19C2×C3×#216C2214yes—
20C3×#897—21no—
21C3×#892—21no—
22C3×D3×C4⋊C4—21no—
23C3×D3×C4⋊C4—21no—
24C3×#890—21no—
25C3×#888—21no—
26C3×#888—21no—
27C3×#887—21no—
28C3×C42×D3C2211yescharacteristic
29C3×#869C2211yescharacteristic
30C3×#869C2211yescharacteristic
31C32×C4×C8C2211yescharacteristic
32C3×#835C2214yes—
33C3×(C2×D3)⋊C8C2212yes—
34C3×#816C2214yes—
35C3×Dic3⋊C8C2212yes—
36C3×C4×C3⋊C8C2211yescharacteristic
37C3×#533—31no—
38C2×C3×C4⋊Q8—31no—
39#5728C2×C311yescharacteristic
40D3×C4⋊Q8C2×C311yescharacteristic
41C3×#494—21no—
42C3×#494—31no—
43C3×#494D311yescharacteristic, complemented
44C3×#494C2×C311yescharacteristic, complemented
45C3×#487—31no—
46C3×#487—31no—
47C3×#462—34no—
48C3×#451—34no—
49C3×#431—31no—
50#5186C2×C314yes—
51#5182C2×C314yescomplemented
52#5092C2×C311yescharacteristic, complemented
53#5079C2×C311yescharacteristic, complemented
54#4685C2×C311yescharacteristic
55#4681C2×C311yescharacteristic, complemented
56C2×C32×Q8C2312yes—
57C2×C32×Q8C2312yes—
58C3×D3×Q8—44no—
59C2×C3×C4×D3—22no—
60C2×C3×C4×D3D411yescharacteristic
61C2×C3×C4×D3D411yescharacteristic
62C2×C3×C4×D3C2311yescharacteristic
63C2×C3×Dic6—21no—
64C2×C3×Dic6—22no—
65C2×C3×Dic6C2312yes—
66C32×Q16—44no—
67C2×C32×C8C2312yes—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4—21no—
72C32×C42C2311yescharacteristic
73C3×#216—44no—
74C3×(C2×D3)⋊C4—22no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—22no—
79C3×C4⋊Dic3C2312yes—
80C3×Dic3⋊C4—22no—
81C3×Dic3⋊C4—22no—
82C3×Dic3⋊C4—22no—
83C3×C4×Dic3—22no—
84C3×C4×Dic3D411yescharacteristic
85C3×C4×Dic3D411yescharacteristic
86C3×C4×Dic3C2311yescharacteristic
87C2×C3×C3⋊C8C2312yes—
88#3209—31noSylow
89C22×C3×Q8—32no—
90C2×D3×Q8C22×C312yes—
91C2×C3×Q16—24no—
92C2×C3×Q16—34no—
93C2×C3×Q16C2×D314yes—
94C2×C3×Q16C22×C314yes—
95C3×C4⋊Q8—21no—
96C3×C4⋊Q8—21no—
97C3×C4⋊Q8—32no—
98C3×C4⋊Q8—31no—
99C3×C4⋊Q8—31no—
100C3×C4⋊Q8—61no—
101C3×C4⋊Q8—61no—
102C3×C4⋊Q8C2×D311yescharacteristic
103C3×C4⋊Q8C2×D311yescharacteristic
104C3×C4⋊Q8C22×C311yescharacteristic
105C3×C4⋊Q8C22×C311yescharacteristic
106C3×#125—61no—
107C3×C22⋊Q8—32no—
108C3×C22⋊Q8—61no—
109C3×C4×Q8—32no—
110C3×C4×Q8—61no—
111C3×#117—61no—
112C2×C3×C4⋊C4—61no—
113C2×C3×C4⋊C4—61no—
114C2×C3×C42—31no—
115#946C22×C312yes—
116Q8×Dic3C22×C312yes—
117#944C22×C312yes—
118C2×#216C22×C314yes—
119#897—21no—
120#892—21no—
121D3×C4⋊C4—21no—
122D3×C4⋊C4—21no—
123#890—21no—
124#888—21no—
125#888—21no—
126#887—21no—
127C42×D3C22×C311yescharacteristic
128#869C22×C311yescharacteristic
129#869C22×C311yescharacteristic
130C3×C4⋊C8—32no—
131C3×#103—34no—
132C3×#103—34no—
133C3×C22⋊C8—32no—
134C3×C4×C8—21no—
135C3×C4×C8—31no—
136C3×C4×C8C2×D311yescharacteristic
137C3×C4×C8C22×C311yescharacteristic
138#835C22×C314yes—
139(C2×D3)⋊C8C22×C312yes—
140#816C22×C314yes—
141Dic3⋊C8C22×C312yes—
142C4×C3⋊C8C22×C311yescharacteristic
143C22×C3×D3C2×D411yescharacteristic
144C32×Q8—24no—
145C32×Q8—24no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2411yescharacteristic
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3C2×D411yescharacteristic
154C2×C3×Dic3C2×D411yescharacteristic
155C2×C3×Dic3C2×D411yescharacteristic
156C3×C4×D3—22no—
157C3×C4×D3—22no—
158C3×C4×D3—44no—
159C3×C4×D3Q1614yescomplemented
160C3×Dic6—24no—
161C3×Dic6—44no—
162C3×Dic6—42no—
163C32×C8—22no—
164C3×C3⋊C8—22no—
165#533—31no—
166C2×C4⋊Q8—31no—
167#494—31no—
168#494C3×D311yescharacteristic, complemented
169#487—31no—
170#487—31no—
171#462—34no—
172#451—34no—
173#431—31no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—32no—
177C2×C3×Q8—62no—
178C2×C3×Q8—61no—
179C2×C3×Q8—124no—
180C2×C3×Q8C22×D312yes—
181C2×C3×Q8C22×D312yes—
182C2×C3×Q8C23×C312yes—
183C2×C3×Q8C23×C312yes—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—62no—
188D3×Q8—44no—
189C2×C4×D3—22no—
190C2×C4×D3C3×D411yescharacteristic
191C2×C4×D3C3×D411yescharacteristic
192C2×C4×D3C23×C311yescharacteristic
193C2×Dic6—21no—
194C2×Dic6—22no—
195C2×Dic6C23×C312yes—
196C3×Q16—44no—
197C3×Q16—44no—
198C3×Q16—48no—
199C3×Q16—124no—
200C2×C3×C8—22no—
201C2×C3×C8—32no—
202C2×C3×C8C22×D312yes—
203C2×C3×C8C23×C312yes—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—32no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4—41no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4—62no—
218C3×C4⋊C4—62no—
219C3×C4⋊C4—62no—
220C3×C4⋊C4—62no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—61no—
224C3×C22⋊C4—62no—
225C3×C42—21no—
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—31no—
229C3×C42—62no—
230C3×C42C22×D311yescharacteristic
231C3×C42C23×C311yescharacteristic
232#216—44no—
233(C2×D3)⋊C4—22no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—22no—
238C4⋊Dic3C23×C312yes—
239Dic3⋊C4—22no—
240Dic3⋊C4—22no—
241Dic3⋊C4—22no—
242C4×Dic3—22no—
243C4×Dic3C3×D411yescharacteristic
244C4×Dic3C3×D411yescharacteristic
245C4×Dic3C23×C311yescharacteristic
246C2×C3⋊C8C23×C312yes—
247C22×C32C22×D411yescharacteristic
248C2×C3×D3—21no—
249C2×C3×D3—21no—
250C2×C3×D3C2×Q1612yes—
251C32×C4—42no—
252C32×C4—42no—
253C32×C4C2×Q1612yes—
254C32×C4#13712yes—
255C32×C4H(5,𝔽2)12yes—
256C3×Dic3—21no—
257C3×Dic3—21no—
258C3×Dic3—42no—
259C3×Dic3—42no—
260C3×Dic3C2×Q1612yes—
261C22×Q8—32no—
262C2×Q16—34no—
263C2×Q16C2×C3×D314yes—
264C4⋊Q8—32no—
265C4⋊Q8—31no—
266C4⋊Q8—31no—
267C4⋊Q8—61no—
268C4⋊Q8—61no—
269C4⋊Q8C2×C3×D311yescharacteristic
270C4⋊Q8C2×C3×D311yescharacteristic
271#125—61no—
272C22⋊Q8—32no—
273C22⋊Q8—61no—
274C4×Q8—32no—
275C4×Q8—61no—
276#117—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C42—31no—
280C4⋊C8—32no—
281#103—34no—
282#103—34no—
283C22⋊C8—32no—
284C4×C8—31no—
285C4×C8C2×C3×D311yescharacteristic
286C23×C3—31no—
287C22×D3C2×C3×D411yescharacteristic
288C3×Q8—24no—
289C3×Q8—24no—
290C3×Q8—24no—
291C3×Q8—24no—
292C3×Q8—44no—
293C3×Q8—44no—
294C3×Q8—64no—
295C3×Q8—124no—
296C3×Q8—122no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—31no—
305C2×C3×C4—34no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—124no—
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4D3×D411yescharacteristic
317C2×C3×C4C2×C3×D411yescharacteristic
318C2×C3×C4C2×C3×D411yescharacteristic
319C2×C3×C4C23×D311yescharacteristic
320C2×C3×C4C24×C311yescharacteristic
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3C2×C3×D411yescharacteristic
324C2×Dic3C2×C3×D411yescharacteristic
325C2×Dic3C2×C3×D411yescharacteristic
326C4×D3—22no—
327C4×D3—22no—
328C4×D3—44no—
329C4×D3C3×Q1614yescomplemented
330Dic6—24no—
331Dic6—44no—
332Dic6—42no—
333C3×C8—24no—
334C3×C8—22no—
335C3×C8—22no—
336C3×C8—62no—
337C3⋊C8—22no—
338C2×C32#53511yescharacteristic
339C2×C32C22×Q1611yescharacteristic
340C2×C32C2×#13711yescharacteristic
341C3×D3—22no—
342C2×Q8—32no—
343C2×Q8—62no—
344C2×Q8—61no—
345C2×Q8—124no—
346C2×Q8C22×C3×D312yes—
347C2×Q8C22×C3×D312yes—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—62no—
352Q16—44no—
353Q16—124no—
354C2×C8—32no—
355C2×C8C22×C3×D312yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—32no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C22⋊C4—62no—
369C42—31no—
370C42—31no—
371C42—31no—
372C42—62no—
373C42C22×C3×D311yescharacteristic
374C22×C3—21no—
375C22×C3—32no—
376C22×C3—61no—
377C22×C3—61no—
378C22×C3C2×D3×D411yescharacteristic
379C22×C3C22×C3×D411yescharacteristic
380C2×D3—21no—
381C2×D3—21no—
382C2×D3C2×C3×Q1612yes—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—32no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—82no—
394C3×C4—82no—
395C3×C4—122no—
396C3×C4—122no—
397C3×C4D3×Q1612yes—
398C3×C4#91812yes—
399C3×C4C2×C3×Q1612yes—
400C3×C4C3×#13712yes—
401C3×C4#100412yes—
402C3×C4C3×H(5,𝔽2)12yes—
403Dic3—21no—
404Dic3—21no—
405Dic3—42no—
406Dic3—42no—
407Dic3C2×C3×Q1612yes—
408C32#320911yescharacteristic, complemented, Sylow
409C23—31no—
410Q8—24no—
411Q8—24no—
412Q8—64no—
413Q8—124no—
414Q8—122no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—31no—
418C2×C4—34no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—124no—
426C2×C4C3×D3×D411yescharacteristic
427C2×C4C3×D3×D411yescharacteristic
428C2×C4C23×C3×D311yescharacteristic
429C8—22no—
430C8—62no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—62no—
435C2×C3#571511yescharacteristic
436C2×C3C2×D3×Q1611yescharacteristic
437C2×C3C2×#91811yescharacteristic
438C2×C3C3×#53511yescharacteristic
439C2×C3C22×C3×Q1611yescharacteristic
440C2×C3C2×C3×#13711yescharacteristic
441D3—22no—
442C22—32no—
443C22—61no—
444C22—61no—
445C22C2×C3×D3×D411yescharacteristic
446C4—32no—
447C4—42no—
448C4—42no—
449C4—61no—
450C4—61no—
451C4—122no—
452C4—122no—
453C4C3×D3×Q1612yes—
454C4C3×#91812yes—
455C4C3×#100412yes—
456C3—21no—
457C3#8180211yescharacteristic, direct product factor
458C3C3×#320911yescharacteristic, complemented
459C2—62no—
460C2C3×#571511yescharacteristic
461C2C2×C3×D3×Q1611yescharacteristic
462C2C2×C3×#91811yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#494C2
socleC22×C32C22×D4
prime radicalC22×C3×D3C2×D4
hypercenterC3×#494D3
hypocenterC3C3×#3209
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#494

Complements

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

NH ≅ G/Ntype
3. C32×#494C2semidirect
5. C3×#5182C2semidirect
6. C3×#5092C2semidirect
7. C3×#5079C2semidirect
9. C3×#4681C2semidirect
11. #81802C3direct (N × H)
43. C3×#494D3semidirect
44. C3×#494C2×C3semidirect
51. #5182C2×C3semidirect
52. #5092C2×C3semidirect
53. #5079C2×C3semidirect
55. #4681C2×C3semidirect
159. C3×C4×D3Q16semidirect
168. #494C3×D3semidirect
329. C4×D3C3×Q16semidirect
408. C32#3209semidirect
457. C3#81802direct (N × H)
458. C3C3×#3209semidirect

Maximal subgroups

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

subgroupindex
C3×#57282
C3×D3×C4⋊Q82
C32×#4942
C3×#51862
C3×#51822
C3×#50922
C3×#50792
C3×#46852
C3×#46812
C3×#32093
#818023