Finite Group Databasebeta

← C3×#68114 #11778022

Subgroups — C3×#68114 #11778022

Order 1152.

681 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5483C211yesmaximal, characteristic, complemented
2C3×#4960C211yesmaximal, characteristic
3C3×#4731C212yesmaximal, complemented
4C32×#323C211yesmaximal, characteristic, complemented
5C3×#4467C212yesmaximal, complemented
6C3×#1596—31nomaximal
7#68114C311yesmaximal, characteristic, direct product factor
8C22×C3×#67—21no—
9C22×C3×D12C2211yescharacteristic
10C2×C32×C22⋊C4C2211yescharacteristic
11C2×C3×#217—21no—
12C2×C3×(C2×D3)⋊C4—21no—
13C3×#924—21no—
14C3×#924C2212yes—
15C2×C3×#204—22no—
16C3×#878—22no—
17C3×#878—22no—
18C32×C22⋊C8C2212yes—
19C3×C3⋊#95C2211yescharacteristic
20C3×#821—22no—
21C3×#543—31no—
22#5483C2×C311yescharacteristic, complemented
23C3×#450—32no—
24C3×#392—31no—
25#4960C2×C311yescharacteristic
26#4731C2×C312yescomplemented
27C3×#328—32no—
28C3×#323—21no—
29C3×#323D311yescharacteristic, complemented
30C3×#323C2×C311yescharacteristic, complemented
31#4467C2×C312yescomplemented
32C24×C32D411yescharacteristic
33C23×C3×D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C2×C3×#67—42no—
37C2×C3×#67—42no—
38C2×C3×#67—42no—
39C22×C3×Dic3—21no—
40C22×C3×Dic3—21no—
41C22×C3×Dic3—21no—
42C2×C3×D12—22no—
43C2×C3×D12—42no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6D412yescomplemented
46C2×C32×C8—22no—
47C32×C22⋊C4—22no—
48C3×#217—22no—
49C3×#217—22no—
50C3×(C2×D3)⋊C4—42no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D412yescomplemented
53C3×Dic3⋊C4—22no—
54C3×Dic3⋊C4—22no—
55C3×#204—44no—
56#1596—31noSylow
57C22×C3×D4—31no—
58C22×C3×D4—61no—
59C22×#67—21no—
60C22×D12C22×C311yescharacteristic
61C2×C3×SD16—62no—
62C3×C22⋊Q8—32no—
63C3×C22⋊Q8—62no—
64C3×C22⋊Q8—62no—
65C3×C22⋊Q8—61no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—61no—
68C2×C3×C22⋊C4—61no—
69C2×C3×C22⋊C4C2×D311yescharacteristic
70C2×C3×C22⋊C4C22×C311yescharacteristic
71C2×#217—21no—
72C2×(C2×D3)⋊C4—21no—
73#924—21no—
74#924C22×C312yes—
75C2×#204—22no—
76#878—22no—
77#878—22no—
78C3×#102—62no—
79C3×C22⋊C8—22no—
80C3×C22⋊C8C2×D312yes—
81C3×C22⋊C8C22×C312yes—
82C3×#95—31no—
83C3⋊#95C22×C311yescharacteristic
84#821—22no—
85C23×C32—21no—
86C23×C32—21no—
87C23×C32—41no—
88C23×C32—41no—
89C23×C32SD1612yescomplemented
90C23×C32C2×D411yescharacteristic
91C22×C3×D3—22no—
92C22×C3×D3—41no—
93C22×C3×D3—41no—
94C22×C3×D3—41no—
95C22×C3×D3—41no—
96C22×C3×D3—41no—
97C22×C3×D3—41no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4C2×D412yes—
102C3×#67—48no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—81no—
109C2×C3×Dic3—81no—
110C3×D12—24no—
111C3×D12—42no—
112C3×Dic6—42no—
113C3×Dic6—81no—
114C32×C8—42no—
115#543—31no—
116#450—32no—
117#392—31no—
118#328—32no—
119#323C3×D311yescharacteristic, complemented
120C24×C3—21no—
121C24×C3—61no—
122C24×C3D1211yescharacteristic
123C24×C3C3×D411yescharacteristic
124C23×D3—21no—
125C2×C3×Q8—32no—
126C2×C3×Q8—61no—
127C2×C3×D4—62no—
128C2×C3×D4—122no—
129C2×C3×D4—122no—
130C2×C3×D4—122no—
131C2×C3×D4—122no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—61no—
135C22×C3×C4—61no—
136C22×C3×C4—61no—
137C22×C3×C4D1211yescharacteristic
138C22×C3×C4C3×D411yescharacteristic
139C22×C3×C4C22×D311yescharacteristic
140C22×C3×C4C23×C311yescharacteristic
141C2×#67—42no—
142C2×#67—42no—
143C2×#67—42no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C2×D12—22no—
148C2×D12—42no—
149C2×Dic6—21no—
150C2×Dic6C3×D412yescomplemented
151C3×SD16—124no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—42no—
155C3×C4⋊C4—32no—
156C3×C4⋊C4—62no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—62no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—42no—
162C3×C22⋊C4—62no—
163C3×C22⋊C4—62no—
164C3×C22⋊C4—122no—
165#217—22no—
166#217—22no—
167(C2×D3)⋊C4—42no—
168C4⋊Dic3—21no—
169C4⋊Dic3C3×D412yescomplemented
170Dic3⋊C4—22no—
171Dic3⋊C4—22no—
172#204—44no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—22no—
176C22×C32—42no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—81no—
180C22×C32#12011yescharacteristic
181C22×C32C2×SD1611yescharacteristic
182C22×C32#13711yescharacteristic
183C2×C3×D3—42no—
184C2×C3×D3—42no—
185C2×C3×D3—42no—
186C2×C3×D3—81no—
187C2×C3×D3—81no—
188C2×C3×D3—81no—
189C2×C3×D3—81no—
190C32×C4—22no—
191C32×C4—41no—
192C3×Dic3—42no—
193C3×Dic3—42no—
194C3×Dic3—42no—
195C22×D4—31no—
196C22×D4—61no—
197C2×SD16—62no—
198C22⋊Q8—32no—
199C22⋊Q8—62no—
200C22⋊Q8—62no—
201C22⋊Q8—61no—
202C2×C22⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4C2×C3×D311yescharacteristic
205#102—62no—
206C22⋊C8C2×C3×D312yes—
207#95—31no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—22no—
213C23×C3—21no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—62no—
221C23×C3—81no—
222C23×C3—81no—
223C23×C3—121no—
224C23×C3—121no—
225C23×C3—121no—
226C23×C3—121no—
227C23×C3—121no—
228C23×C3—121no—
229C23×C3#20412yescomplemented
230C23×C3C3×SD1612yescomplemented
231C23×C3C2×D1211yescharacteristic
232C23×C3C2×C3×D411yescharacteristic
233C22×D3—22no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C3×Q8—122no—
241C3×Q8—241no—
242C3×D4—64no—
243C3×D4—122no—
244C3×D4—128no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—62no—
254C2×C3×C4—62no—
255C2×C3×C4—62no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4—121no—
259C2×C3×C4—121no—
260C2×C3×C4—241no—
261C2×C3×C4—241no—
262C2×C3×C4D3×D412yes—
263C2×C3×C4C2×C3×D412yes—
264#67—48no—
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272D12—24no—
273D12—42no—
274Dic6—42no—
275Dic6—81no—
276C3×C8—42no—
277C3×C8—42no—
278C3×C8—82no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—42no—
282C2×C32#44811yescharacteristic
283C2×C32#45311yescharacteristic
284C2×C32B(4,𝔽2)11yescharacteristic
285C3×D3—82no—
286C24—61no—
287C24C3×D1211yescharacteristic
288C2×Q8—32no—
289C2×Q8—61no—
290C2×D4—62no—
291C2×D4—122no—
292C2×D4—122no—
293C2×D4—122no—
294C2×D4—122no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4C3×D1211yescharacteristic
299C22×C4C22×C3×D311yescharacteristic
300SD16—124no—
301C2×C8—22no—
302C4⋊C4—32no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—62no—
306C22⋊C4—22no—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—122no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—82no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—82no—
334C22×C3—122no—
335C22×C3—122no—
336C22×C3—122no—
337C22×C3—241no—
338C22×C3—241no—
339C22×C3—241no—
340C22×C3—241no—
341C22×C3#88211yescharacteristic
342C22×C3C2×#20411yescharacteristic
343C22×C3#90911yescharacteristic
344C22×C3C3×#12011yescharacteristic
345C22×C3C2×C3×SD1611yescharacteristic
346C22×C3C3×#13711yescharacteristic
347C2×D3—42no—
348C2×D3—42no—
349C2×D3—42no—
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—42no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—81no—
360C3×C4—122no—
361C3×C4—122no—
362C3×C4—122no—
363Dic3—42no—
364Dic3—42no—
365Dic3—42no—
366C32#159611yescharacteristic, complemented, Sylow
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—62no—
372C23—121no—
373C23—121no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23—121no—
378C23C3×#20412yescomplemented
379C23C2×C3×D1211yescharacteristic
380Q8—122no—
381Q8—241no—
382D4—64no—
383D4—122no—
384D4—128no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4C3×D3×D412yes—
396C8—42no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—41no—
405C2×C3—41no—
406C2×C3—42no—
407C2×C3—42no—
408C2×C3—82no—
409C2×C3—242no—
410C2×C3#473811yescharacteristic
411C2×C3#474111yescharacteristic
412C2×C3#474711yescharacteristic
413C2×C3C3×#44811yescharacteristic
414C2×C3C3×#45311yescharacteristic
415C2×C3C3×B(4,𝔽2)11yescharacteristic
416D3—82no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—81no—
424C22—122no—
425C22—122no—
426C22—122no—
427C22—241no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22C3×#88211yescharacteristic
432C22C2×C3×#20411yescharacteristic
433C22C3×#90911yescharacteristic
434C4—22no—
435C4—41no—
436C4—122no—
437C4—122no—
438C4—122no—
439C3—21no—
440C3#6811411yescharacteristic, direct product factor
441C3C3×#159611yescharacteristic, complemented
442C2—21no—
443C2—21no—
444C2—42no—
445C2—242no—
446C2C3×#473811yescharacteristic
447C2C3×#474111yescharacteristic
448C2C3×#474711yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
prime radicalC3×#4960C2
hypercenterC3×#323D3
hypocenterC3C3×#1596
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#323

Complements

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

NH ≅ G/Ntype
1. C3×#5483C2semidirect
3. C3×#4731C2semidirect
4. C32×#323C2semidirect
5. C3×#4467C2semidirect
7. #68114C3direct (N × H)
22. #5483C2×C3semidirect
26. #4731C2×C3semidirect
29. C3×#323D3semidirect
30. C3×#323C2×C3semidirect
31. #4467C2×C3semidirect
45. C2×C3×Dic6D4semidirect
52. C3×C4⋊Dic3D4semidirect
89. C23×C32SD16semidirect
119. #323C3×D3semidirect
150. C2×Dic6C3×D4semidirect
169. C4⋊Dic3C3×D4semidirect
229. C23×C3#204semidirect
230. C23×C3C3×SD16semidirect
366. C32#1596semidirect
378. C23C3×#204semidirect
440. C3#68114direct (N × H)
441. C3C3×#1596semidirect

Maximal subgroups

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

subgroupindex
C3×#54832
C3×#49602
C3×#47312
C32×#3232
C3×#44672
C3×#15963
#681143