Finite Group Databasebeta

← C3×#72021 #11787768

Subgroups — C3×#72021 #11787768

Order 1152.

793 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#568C211yesmaximal, characteristic, complemented
2C3×#5832C211yesmaximal, characteristic
3C3×#5240C211yesmaximal, characteristic, complemented
4C3×#5139C212yesmaximal, complemented
5C3×#4555C212yesmaximal, complemented
6C3×#1884—31nomaximal
7#72021C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C2×C3×#235C2211yescharacteristic
10C32×#131—24no—
11C2×C32×M16—22no—
12C2×C32×M16C2212yes—
13C3×#947C411yescharacteristic, complemented
14C3×#944C411yescharacteristic, complemented
15C3×#940C411yescharacteristic, complemented
16C3×#939—22no—
17C3×#936C411yescharacteristic, complemented
18C3×C4×#67—22no—
19C3×#926C2212yes—
20C2×C3×C4×Dic3C2211yescharacteristic
21C3×#834—22no—
22C3×#825—24no—
23C3×#568—21no—
24C3×#568D311yescharacteristic, complemented
25C3×#568C2×C311yescharacteristic, complemented
26C3×#532—31no—
27#5832C2×C311yescharacteristic
28C3×#421—32no—
29C3×#410—31no—
30#5240C2×C311yescharacteristic, complemented
31#5139C2×C312yescomplemented
32C3×#339—32no—
33#4555C2×C312yescomplemented
34C32×#41—22no—
35C32×#41—22no—
36C2×C32×Q8C2×C411yescharacteristic
37C2×C32×D4C2×C411yescharacteristic
38C2×C32×D4D412yes—
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C2×C3×#67—22no—
42C2×C3×#67—21no—
43C22×C3×Dic3—21no—
44C3×#235—22no—
45C3×#235—22no—
46C2×C3×D12C2×C411yescharacteristic
47C2×C3×C4×D3—21no—
48C2×C3×Dic6C2×C411yescharacteristic
49C32×M16—24no—
50C32×M16—22no—
51C32×M16D414yescomplemented
52C2×C32×C8—24no—
53C2×C32×C8—22no—
54C3×#217—22no—
55C3×#217—22no—
56C3×(C2×D3)⋊C4—22no—
57C3×C4⋊Dic3—22no—
58C3×Dic3⋊C4—22no—
59C3×C4×Dic3—21no—
60C3×C4×Dic3—22no—
61C3×C4×Dic3C2×C411yescharacteristic
62C3×C4×Dic3C2×C411yescharacteristic
63#1884—31noSylow
64C2×C3×#41—21no—
65C2×C3×#41—31no—
66C2×C3×#41C2×D311yescharacteristic
67C2×C3×#41C22×C311yescharacteristic
68C2×#235C22×C311yescharacteristic
69C3×#131—28no—
70C3×#131—24no—
71C3×#131—24no—
72C2×C3×M16—24no—
73C2×C3×M16—22no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C2×C3×M16C2×D312yes—
77C2×C3×M16C22×C312yes—
78C3×C4⋊Q8—31no—
79C3×#127—31no—
80C3×#124—31no—
81C3×#124—31no—
82C3×#121—62no—
83C3×C4×D4—62no—
84C3×#117—32no—
85C2×C3×C42—31no—
86#947C3×C411yescharacteristic, complemented
87#944C3×C411yescharacteristic, complemented
88#940C3×C411yescharacteristic, complemented
89#939—22no—
90#936C3×C411yescharacteristic, complemented
91C4×#67—22no—
92#926C22×C312yes—
93C2×C4×Dic3C22×C311yescharacteristic
94C3×#104—64no—
95C3×#99—62no—
96#834—22no—
97#825—24no—
98C23×C32C22⋊C412yes—
99C23×C32C2×D411yescharacteristic
100C22×C3×D3—21no—
101C32×Q8—22no—
102C32×D4—24no—
103C32×D4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4C22⋊C412yes—
107C2×C32×C4C22×C411yescharacteristic
108C2×C32×C4C2×D412yes—
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4#4112yes—
111C3×#67—44no—
112C3×#67—42no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—42no—
117C2×C3×Dic3—42no—
118C3×D12—22no—
119C3×C4×D3—41no—
120C3×C4×D3—41no—
121C3×Dic6—22no—
122C32×C8—24no—
123#568C3×D311yescharacteristic, complemented
124#532—31no—
125#421—32no—
126#410—31no—
127#339—32no—
128C3×#41—24no—
129C3×#41—24no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—22no—
133C3×#41—22no—
134C3×#41—62no—
135C3×#41—62no—
136C2×C3×Q8—21no—
137C2×C3×Q8—31no—
138C2×C3×Q8C4×D311yescharacteristic
139C2×C3×Q8C2×C3×C411yescharacteristic
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—31no—
143C2×C3×D4—62no—
144C2×C3×D4—61no—
145C2×C3×D4C4×D311yescharacteristic
146C2×C3×D4D1212yes—
147C2×C3×D4C2×C3×C411yescharacteristic
148C2×C3×D4C3×D412yes—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—61no—
152C22×C3×C4—61no—
153C22×C3×C4#6712yes—
154C22×C3×C4C3×D412yes—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C23×C311yescharacteristic
157C2×#67—22no—
158C2×#67—21no—
159C22×Dic3—21no—
160#235—22no—
161#235—22no—
162C2×D12C2×C3×C411yescharacteristic
163C2×C4×D3—21no—
164C2×Dic6C2×C3×C411yescharacteristic
165C3×M16—24no—
166C3×M16—28no—
167C3×M16—24no—
168C3×M16—24no—
169C3×M16—22no—
170C3×M16—22no—
171C3×M16—24no—
172C3×M16#6714yescomplemented
173C3×M16C3×D414yescomplemented
174C2×C3×C8—28no—
175C2×C3×C8—24no—
176C2×C3×C8—24no—
177C2×C3×C8—24no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—62no—
181C3×C4⋊C4—62no—
182C3×C22⋊C4—62no—
183C3×C22⋊C4—62no—
184C3×C22⋊C4—62no—
185C3×C42—31no—
186C3×C42—31no—
187C3×C42—61no—
188C3×C42—62no—
189#217—22no—
190#217—22no—
191(C2×D3)⋊C4—22no—
192C4⋊Dic3—22no—
193Dic3⋊C4—22no—
194C4×Dic3—21no—
195C4×Dic3—22no—
196C4×Dic3C2×C3×C411yescharacteristic
197C4×Dic3C2×C3×C411yescharacteristic
198C22×C32—22no—
199C22×C32—41no—
200C22×C32—42no—
201C22×C32C2×C22⋊C411yescharacteristic
202C22×C32#12112yes—
203C2×C3×D3—42no—
204C2×C3×D3—41no—
205C32×C4—22no—
206C32×C4C4×D412yes—
207C32×C4#12011yescharacteristic
208C32×C4#12311yescharacteristic
209C3×Dic3—22no—
210C3×Dic3—41no—
211C3×Dic3—41no—
212C3×Dic3—82no—
213C2×#41—31no—
214C2×#41C2×C3×D311yescharacteristic
215#131—24no—
216C2×M16—22no—
217C2×M16C2×C3×D312yes—
218C4⋊Q8—31no—
219#127—31no—
220#124—31no—
221#124—31no—
222#121—62no—
223C4×D4—62no—
224#117—32no—
225C2×C42—31no—
226#104—64no—
227#99—62no—
228C23×C3—22no—
229C23×C3—21no—
230C23×C3—61no—
231C23×C3(C2×D3)⋊C412yes—
232C23×C3C3×C22⋊C412yes—
233C23×C3C2×D1211yescharacteristic
234C23×C3C2×C3×D411yescharacteristic
235C22×D3—21no—
236C3×Q8—24no—
237C3×Q8—22no—
238C3×Q8—22no—
239C3×Q8—62no—
240C3×D4—28no—
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—24no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—62no—
247C3×D4—124no—
248C3×D4—122no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—122no—
264C2×C3×C4—121no—
265C2×C3×C4—121no—
266C2×C3×C4—122no—
267C2×C3×C4(C2×D3)⋊C412yes—
268C2×C3×C4C3×C22⋊C412yes—
269C2×C3×C4C2×C4×D311yescharacteristic
270C2×C3×C4#23512yes—
271C2×C3×C4C2×#6712yes—
272C2×C3×C4C2×#6711yescharacteristic
273C2×C3×C4C22×C3×C411yescharacteristic
274C2×C3×C4C2×C3×D412yes—
275C2×C3×C4C2×C3×D411yescharacteristic
276C2×C3×C4C3×#4112yes—
277#67—44no—
278#67—42no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284D12—22no—
285C4×D3—41no—
286C4×D3—41no—
287Dic6—22no—
288C3×C8—28no—
289C3×C8—24no—
290C3×C8—24no—
291C2×C32—21no—
292C2×C32—22no—
293C2×C32—42no—
294C2×C32#38611yescharacteristic
295C3×D3—81no—
296#41—22no—
297#41—22no—
298#41—62no—
299#41—62no—
300C2×Q8—31no—
301C2×Q8C3×C4×D311yescharacteristic
302C2×D4—31no—
303C2×D4—62no—
304C2×D4—61no—
305C2×D4C3×C4×D311yescharacteristic
306C2×D4C3×D1212yes—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4C3×#6712yes—
310C22×C4C22×C3×D311yescharacteristic
311M16—24no—
312M16—22no—
313M16C3×#6714yescomplemented
314C2×C8—24no—
315C2×C8—22no—
316C4⋊C4—62no—
317C4⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C42—31no—
322C42—31no—
323C42—61no—
324C42—62no—
325C22×C3—24no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—21no—
329C22×C3—22no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—81no—
335C22×C3—82no—
336C22×C3—122no—
337C22×C3—121no—
338C22×C3C2×(C2×D3)⋊C411yescharacteristic
339C22×C3#93012yes—
340C22×C3C2×C3×C22⋊C411yescharacteristic
341C22×C3C3×#12112yes—
342C2×D3—42no—
343C2×D3—41no—
344C3×C4—24no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—62no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—242no—
354C3×C4C4×#6712yes—
355C3×C4#92911yescharacteristic
356C3×C4#95311yescharacteristic
357C3×C4C3×C4×D412yes—
358C3×C4C3×#12011yescharacteristic
359C3×C4C3×#12311yescharacteristic
360Dic3—22no—
361Dic3—41no—
362Dic3—41no—
363Dic3—82no—
364C32#188411yescharacteristic, complemented, Sylow
365C23—61no—
366C23C3×(C2×D3)⋊C412yes—
367C23C2×C3×D1211yescharacteristic
368Q8—22no—
369Q8—62no—
370D4—24no—
371D4—22no—
372D4—62no—
373D4—124no—
374D4—122no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4C3×(C2×D3)⋊C412yes—
385C2×C4C2×C3×C4×D311yescharacteristic
386C2×C4C3×#23512yes—
387C2×C4C2×C3×#6712yes—
388C2×C4C2×C3×#6711yescharacteristic
389C8—24no—
390C2×C3—21no—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—22no—
394C2×C3—21no—
395C2×C3—41no—
396C2×C3—42no—
397C2×C3—44no—
398C2×C3—42no—
399C2×C3—42no—
400C2×C3—241no—
401C2×C3#521311yescharacteristic
402C2×C3C3×#38611yescharacteristic
403D3—81no—
404C22—22no—
405C22—41no—
406C22—42no—
407C22—122no—
408C22—121no—
409C22C2×C3×(C2×D3)⋊C411yescharacteristic
410C22C3×#93012yes—
411C4—22no—
412C4—62no—
413C4—121no—
414C4—121no—
415C4—242no—
416C4C3×C4×#6712yes—
417C4C3×#92911yescharacteristic
418C4C3×#95311yescharacteristic
419C3—21no—
420C3#7202111yescharacteristic, direct product factor
421C3C3×#188411yescharacteristic, complemented
422C2—21no—
423C2—22no—
424C2—42no—
425C2—241no—
426C2C3×#521311yescharacteristic
subgroupG/H
centerC3×C4#953
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#568C2
socleC2×C32#386
prime radicalC3×#5832C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#568D3
hypocenterC3C3×#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#568

Complements

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

NH ≅ G/Ntype
1. C32×#568C2semidirect
3. C3×#5240C2semidirect
4. C3×#5139C2semidirect
5. C3×#4555C2semidirect
7. #72021C3direct (N × H)
13. C3×#947C4semidirect
14. C3×#944C4semidirect
15. C3×#940C4semidirect
17. C3×#936C4semidirect
24. C3×#568D3semidirect
25. C3×#568C2×C3semidirect
30. #5240C2×C3semidirect
31. #5139C2×C3semidirect
33. #4555C2×C3semidirect
51. C32×M16D4semidirect
86. #947C3×C4semidirect
87. #944C3×C4semidirect
88. #940C3×C4semidirect
90. #936C3×C4semidirect
123. #568C3×D3semidirect
172. C3×M16#67semidirect
173. C3×M16C3×D4semidirect
313. M16C3×#67semidirect
364. C32#1884semidirect
420. C3#72021direct (N × H)
421. C3C3×#1884semidirect

Maximal subgroups

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

subgroupindex
C32×#5682
C3×#58322
C3×#52402
C3×#51392
C3×#45552
C3×#18843
#720213