Finite Group Databasebeta

← C3×#10591166 #13158345

Subgroups — C3×#10591166 #13158345

Order 1728.

854 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
2C32×#930C212yesmaximal, complemented
3C2×C3×#3713C211yesmaximal, characteristic, complemented
4C3×#63576C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6C3×#5038—32nomaximal
7#10591166C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C2×C32×#67—22no—
10C2×C32×D12C2212yes—
11C2×C3×C3⋊(C3⋊C8)—22no—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C32×(C2×D3)⋊C4—22no—
15C32×C4⋊Dic3C2212yes—
16C3×#3713—22no—
17C32×#121—32no—
18C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
19C2×C3×#213—32no—
20C3×#930—22no—
21C3×#930D312yescomplemented
22C3×#930C2×C312yescomplemented
23C22×C3×C3⋊C8—32no—
24C22×C3×C3⋊C8—61no—
25C2×#3713C2×C311yescharacteristic, complemented
26C3×#832—32no—
27C3×#809—32no—
28C3×#807—32no—
29#63576C2×C311yescharacteristic, complemented
30#63562C2×C312yescomplemented
31C23×C33D411yescharacteristic
32C22×C32×D3—22no—
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4D411yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C32×#67—44no—
38C2×C32×Dic3—22no—
39C32×D12—22no—
40C32×D12—22no—
41C3×C3⋊(C3⋊C8)—21no—
42C3×C3⋊(C3⋊C8)D412yescomplemented
43C3×#466—91no—
44#5038—32no—
45C2×C32×D4—32no—
46C2×C32×D4—62no—
47C22×C32×C4—21no—
48C22×C32×C4—22no—
49C22×C32×C4—41no—
50C22×C32×C4C2×D312yes—
51C22×C32×C4C22×C311yescharacteristic
52C2×C3×#67—24no—
53C2×C3×#67—22no—
54C2×C3×#67—22no—
55C2×C3×D12—22no—
56C2×C3×D12C2×D312yes—
57C2×C3×D12C22×C312yes—
58C32×C4⋊C4—32no—
59C32×C22⋊C4—62no—
60C2×C3⋊(C3⋊C8)—22no—
61C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
62C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
63C3×#213—64no—
64C3×(C2×D3)⋊C4—24no—
65C3×(C2×D3)⋊C4—22no—
66C3×(C2×D3)⋊C4—22no—
67C3×C4⋊Dic3—22no—
68C3×C4⋊Dic3C2×D312yes—
69C3×C4⋊Dic3C22×C312yes—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—32no—
72C2×C3×C3⋊C8—64no—
73C2×C3×C3⋊C8—61no—
74C2×C3×C3⋊C8—61no—
75C2×C3×C3⋊C8—64no—
76#3713—22no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33D812yescomplemented
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—42no—
82C2×C32×D3—42no—
83C2×C32×D3—42no—
84C33×C4—21no—
85C33×C4C2×D411yescharacteristic
86C33×C4#4111yescharacteristic
87C32×Dic3—42no—
88C2×C3×D8—91no—
89C22×C3×C8—91no—
90C3×#121—32no—
91C3×#121—32no—
92C3×#121—62no—
93C2×#213—32no—
94#930C3×D312yescomplemented
95C22×C3⋊C8—32no—
96C22×C3⋊C8—61no—
97C3×#107—91no—
98C3×#102—92no—
99#832—32no—
100#809—32no—
101#807—32no—
102C23×C32—21no—
103C23×C32—22no—
104C23×C32—41no—
105C23×C32—62no—
106C23×C32#6712yes—
107C23×C32C3×D411yescharacteristic
108C22×C3×D3—24no—
109C22×C3×D3—22no—
110C22×C3×D3—22no—
111C32×D4—62no—
112C32×D4—62no—
113C32×D4—124no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—22no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4—42no—
127C2×C32×C4—42no—
128C2×C32×C4—42no—
129C2×C32×C4—42no—
130C2×C32×C4—62no—
131C2×C32×C4#6712yes—
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C22×D312yes—
134C2×C32×C4C23×C311yescharacteristic
135C3×#67—48no—
136C3×#67—44no—
137C3×#67—44no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C3×D12—22no—
142C3×D12—22no—
143C3×D12—22no—
144C3×D12—22no—
145C3×D12—42no—
146C3×D12—42no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)C3×D412yescomplemented
149C3×C3⋊C8—34no—
150C3×C3⋊C8—62no—
151C3×C3⋊C8—62no—
152C3×C3⋊C8—62no—
153#466—91noSylow
154C2×C33—22no—
155C2×C33#12111yescharacteristic
156C2×C33C2×D811yescharacteristic
157C2×C33#13511yescharacteristic
158C32×D3—82no—
159C2×C3×D4—32no—
160C2×C3×D4—32no—
161C2×C3×D4—62no—
162C2×C3×D4—64no—
163C2×C3×D4—62no—
164C2×C3×D4—62no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—41no—
168C22×C3×C4D3211yescharacteristic
169C22×C3×C4C2×C3×D312yes—
170C2×#67—22no—
171C2×D12C2×C3×D312yes—
172C3×D8—182no—
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C2×C3×C8—182no—
176C3×C4⋊C4—32no—
177C3×C4⋊C4—32no—
178C3×C4⋊C4—62no—
179C3×C22⋊C4—64no—
180C3×C22⋊C4—62no—
181C3×C22⋊C4—62no—
182#213—64no—
183(C2×D3)⋊C4—22no—
184C4⋊Dic3C2×C3×D312yes—
185C2×C3⋊C8—32no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—64no—
190C2×C3⋊C8—64no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—24no—
199C22×C32—21no—
200C22×C32—22no—
201C22×C32—42no—
202C22×C32—41no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—42no—
207C22×C32—122no—
208C22×C32—122no—
209C22×C32—122no—
210C22×C32#21314yescomplemented
211C22×C32C3×D812yescomplemented
212C22×C32C2×#6712yes—
213C22×C32C2×C3×D411yescharacteristic
214C2×C3×D3—44no—
215C2×C3×D3—44no—
216C2×C3×D3—44no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—42no—
221C2×C3×D3—42no—
222C2×C3×D3—42no—
223C32×C4—21no—
224C32×C4—22no—
225C32×C4—22no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—22no—
229C32×C4—22no—
230C32×C4—41no—
231C32×C4—41no—
232C32×C4—42no—
233C32×C4—42no—
234C32×C4—122no—
235C32×C4D3×D412yes—
236C32×C4#23712yes—
237C32×C4C2×C3×D411yescharacteristic
238C32×C4C3×#4111yescharacteristic
239C3×Dic3—44no—
240C3×Dic3—42no—
241C3×Dic3—42no—
242C2×D8—91no—
243C22×C8—91no—
244#121—32no—
245#107—91no—
246#102—92no—
247C33#46611yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—41no—
251C23×C3—64no—
252C23×C3—62no—
253C23×C3—62no—
254C23×C3#62511yescharacteristic
255C23×C3C3×#6712yes—
256C22×D3—22no—
257C3×D4—62no—
258C3×D4—62no—
259C3×D4—62no—
260C3×D4—62no—
261C3×D4—122no—
262C3×D4—122no—
263C3×D4—128no—
264C3×D4—124no—
265C3×D4—124no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—64no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4#62511yescharacteristic
286C2×C3×C4C3×#6712yes—
287C2×C3×C4C2×D3211yescharacteristic
288C2×C3×C4C22×C3×D312yes—
289#67—44no—
290C2×Dic3—22no—
291D12—22no—
292D12—22no—
293C3×C8—92no—
294C3×C8—181no—
295C3⋊C8—34no—
296C3⋊C8—62no—
297C3⋊C8—62no—
298C3⋊C8—62no—
299C2×C32—22no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—22no—
303C2×C32—22no—
304C2×C32—22no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—44no—
312C2×C32—44no—
313C2×C32—242no—
314C2×C32C2×#21312yes—
315C2×C32#93912yes—
316C2×C32#95012yes—
317C2×C32C3×#12111yescharacteristic
318C2×C32C2×C3×D811yescharacteristic
319C2×C32C3×#13511yescharacteristic
320C3×D3—84no—
321C3×D3—82no—
322C3×D3—82no—
323C2×D4—32no—
324C2×D4—62no—
325C22×C4C3×D3211yescharacteristic
326D8—182no—
327C2×C8—91no—
328C2×C8—91no—
329C2×C8—182no—
330C4⋊C4—32no—
331C22⋊C4—62no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—24no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—41no—
347C22×C3—42no—
348C22×C3—124no—
349C22×C3—124no—
350C22×C3—124no—
351C22×C3—122no—
352C22×C3—122no—
353C22×C3—122no—
354C22×C3—122no—
355C22×C3—122no—
356C22×C3—122no—
357C22×C3#371312yescomplemented
358C22×C3C3×#21314yescomplemented
359C22×C3C2×#62511yescharacteristic
360C22×C3C2×C3×#6712yes—
361C2×D3—42no—
362C2×D3—42no—
363C2×D3—42no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—22no—
369C3×C4—21no—
370C3×C4—22no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—124no—
376C3×C4—122no—
377C3×C4—122no—
378C3×C4#380511yescharacteristic
379C3×C4#381111yescharacteristic
380C3×C4C3×D3×D412yes—
381C3×C4C3×#23712yes—
382Dic3—42no—
383C32—22no—
384C32—21no—
385C32—41no—
386C32#503812yescomplemented
387C32C3×#46611yescharacteristic, complemented
388C23—62no—
389C23C3×#62511yescharacteristic
390D4—62no—
391D4—62no—
392D4—124no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—62no—
396C2×C4C3×#62511yescharacteristic
397C2×C4C2×C3×D3211yescharacteristic
398C8—92no—
399C8—181no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—22no—
403C2×C3—21no—
404C2×C3—22no—
405C2×C3—22no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—44no—
410C2×C3—44no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—244no—
415C2×C3—242no—
416C2×C3—242no—
417C2×C3C2×#371311yescharacteristic
418C2×C3#6382211yescharacteristic
419C2×C3#6398011yescharacteristic
420C2×C3C2×C3×#21312yes—
421C2×C3C3×#93912yes—
422C2×C3C3×#95012yes—
423D3—82no—
424C22—21no—
425C22—21no—
426C22—122no—
427C22—122no—
428C22—122no—
429C22C3×#371312yescomplemented
430C22C2×C3×#62511yescharacteristic
431C4—21no—
432C4—122no—
433C4C3×#380511yescharacteristic
434C4C3×#381111yescharacteristic
435C3—21no—
436C3—22no—
437C3—41no—
438C3#1059116611yescharacteristic, direct product factor
439C3C3×#503812yescomplemented
440C2—22no—
441C2—242no—
442C2C2×C3×#371311yescharacteristic
443C2C3×#6382211yescharacteristic
444C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#466
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊(C3⋊C8)C2semidirect
2. C32×#930C2semidirect
3. C2×C3×#3713C2semidirect
4. C3×#63576C2semidirect
5. C3×#63562C2semidirect
7. #10591166C3direct (N × H)
18. C22×C3⋊(C3⋊C8)C2×C3semidirect
21. C3×#930D3semidirect
22. C3×#930C2×C3semidirect
25. C2×#3713C2×C3semidirect
29. #63576C2×C3semidirect
30. #63562C2×C3semidirect
42. C3×C3⋊(C3⋊C8)D4semidirect
79. C22×C33D8semidirect
94. #930C3×D3semidirect
148. C3⋊(C3⋊C8)C3×D4semidirect
210. C22×C32#213semidirect
211. C22×C32C3×D8semidirect
247. C33#466semidirect
357. C22×C3#3713semidirect
358. C22×C3C3×#213semidirect
386. C32#5038semidirect
387. C32C3×#466semidirect
429. C22C3×#3713semidirect
438. C3#10591166direct (N × H)
439. C3C3×#5038semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C32×#9302
C2×C3×#37132
C3×#635762
C3×#635622
C3×#50383
#105911663