Finite Group Databasebeta

← C3×#10590553 #13157536

Subgroups — C3×#10590553 #13157536

Order 1728.

708 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 124 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×#64096C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic
3C2×C3×#3716C212yesmaximal, complemented
4C32×#809C212yesmaximal, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#4762—32nomaximal
7#10590553C311yesmaximal, characteristic, direct product factor
8C2×C3×#634C2211yescharacteristic
9C2×C32×D12C2212yes—
10C2×C3×#625—21no—
11C2×C3×#625—21no—
12C33×C4⋊C4C2211yescharacteristic
13C3×#3750—21no—
14C3×C4×C3⋊Dic3—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2212yes—
17C3×#3716—24no—
18#64096C2×C311yescharacteristic, complemented
19C3×#940—32no—
20C2×C3×#214—32no—
21C2×C3×#204—32no—
22C3×#887—32no—
23C3×#887—61no—
24#63914C2×C311yescharacteristic
25C2×#3716C2×C312yescomplemented
26C32×#102—32no—
27C3×Dic3⋊C8—32no—
28C3×#809—22no—
29C3×#809D312yescomplemented
30C3×#809C2×C312yescomplemented
31#63572C2×C311yescharacteristic, complemented
32C22×C32×D3—22no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×#634—21no—
39C3×#634D412yescomplemented
40C32×D12—22no—
41C32×D12—22no—
42C3×#625—42no—
43C3×#625—42no—
44C32×C3⋊C8—22no—
45C3×#460—91no—
46#4762—32no—
47C2×C32×D4—32no—
48C2×#634C22×C311yescharacteristic
49C2×C3×#67—62no—
50C2×C3×#67—62no—
51C2×C3×D12—22no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C22×C312yes—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—61no—
56C2×#625—21no—
57C2×#625—21no—
58C2×C32×C8—32no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—22no—
61C32×C4⋊C4—41no—
62C32×C4⋊C4C2×D312yes—
63C32×C4⋊C4C22×C311yescharacteristic
64#3750—21no—
65C4×C3⋊Dic3—21no—
66C4×C3⋊Dic3C22×C311yescharacteristic
67C3×#214—64no—
68C3×Dic3⋊C4—62no—
69C3×Dic3⋊C4—62no—
70C3×C4×Dic3—32no—
71C3×C4×Dic3—62no—
72C3×C4×Dic3—61no—
73C3×C4×Dic3—62no—
74C2×C3×C3⋊C8—22no—
75C2×C3×C3⋊C8C2×D312yes—
76C2×C3×C3⋊C8C22×C312yes—
77C3×#204—64no—
78#3716—24no—
79C22×C33C2×D411yescharacteristic
80C2×C32×D3—42no—
81C2×C32×D3—42no—
82C2×C32×D3—42no—
83C33×C4—41no—
84C33×C4C2×D411yescharacteristic
85C33×C4#4111yescharacteristic
86C3×C3⋊Dic3—22no—
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3SD1612yescomplemented
89C2×C3×SD16—92no—
90C3×#127—91no—
91C3×C4×Q8—91no—
92#940—32no—
93C2×#214—32no—
94C2×#204—32no—
95#887—32no—
96#887—61no—
97C3×C4⋊C8—91no—
98C3×#102—32no—
99C3×#102—32no—
100C3×#102—62no—
101Dic3⋊C8—32no—
102#809C3×D312yescomplemented
103C23×C32—62no—
104C22×C3×D3—22no—
105C22×C3×D3—22no—
106C22×C3×D3—42no—
107C32×D4—62no—
108C32×D4—62no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4—41no—
115C2×C32×C4—42no—
116C2×C32×C4—42no—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C23×C311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3C3×D411yescharacteristic
121C2×C3⋊Dic3C3×D411yescharacteristic
122#634—21no—
123#634C3×D412yescomplemented
124C3×#67—124no—
125C3×#67—124no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C3×D12—24no—
133C3×D12—22no—
134C3×D12—22no—
135C3×D12—22no—
136C3×D12—22no—
137C3×D12—42no—
138C3×Dic6—34no—
139C3×Dic6—62no—
140C3×Dic6—62no—
141C3×Dic6—62no—
142#625—42no—
143#625—42no—
144C32×C8—62no—
145C3×C3⋊C8—22no—
146C3×C3⋊C8—22no—
147C3×C3⋊C8—42no—
148#460—91noSylow
149C2×C33#12111yescharacteristic
150C2×C33C2×SD1611yescharacteristic
151C2×C33AGL(1,ℤ/8)11yescharacteristic
152C32×D3—82no—
153C2×C3×Q8—91no—
154C2×C3×D4—32no—
155C2×C3×D4—32no—
156C2×C3×D4—62no—
157C2×C3×D4—181no—
158C2×C3×D4—181no—
159C2×#67—62no—
160C2×#67—62no—
161C2×D12C2×C3×D312yes—
162C2×Dic6—32no—
163C2×Dic6—61no—
164C3×SD16—184no—
165C2×C3×C8—32no—
166C2×C3×C8—32no—
167C2×C3×C8—62no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4D3211yescharacteristic
173C3×C4⋊C4C2×C3×D312yes—
174C3×C42—91no—
175C3×C42—181no—
176#214—64no—
177Dic3⋊C4—62no—
178Dic3⋊C4—62no—
179C4×Dic3—32no—
180C4×Dic3—61no—
181C4×Dic3—62no—
182C4×Dic3—62no—
183C2×C3⋊C8C2×C3×D312yes—
184#204—64no—
185C22×C32—21no—
186C22×C32—22no—
187C22×C32—41no—
188C22×C32—122no—
189C22×C32—122no—
190C22×C32—122no—
191C22×C32D3×D412yes—
192C22×C32C2×C3×D411yescharacteristic
193C2×C3×D3—44no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—42no—
197C2×C3×D3—42no—
198C2×C3×D3—42no—
199C2×C3×D3—42no—
200C2×C3×D3—82no—
201C2×C3×D3—82no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—41no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—42no—
210C32×C4—42no—
211C32×C4—82no—
212C32×C4—82no—
213C32×C4#23512yes—
214C32×C4D3×D412yes—
215C32×C4C2×C3×D411yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—22no—
218C3⋊Dic3—21no—
219C3⋊Dic3C3×SD1612yescomplemented
220C3×Dic3—34no—
221C3×Dic3—62no—
222C3×Dic3—64no—
223C3×Dic3—64no—
224C3×Dic3—62no—
225C3×Dic3—121no—
226C2×SD16—92no—
227#127—91no—
228C4×Q8—91no—
229C4⋊C8—91no—
230#102—32no—
231C33#46011yescharacteristic, complemented, Sylow
232C23×C3—62no—
233C23×C3—62no—
234C23×C3—122no—
235C22×D3—22no—
236C3×Q8—92no—
237C3×Q8—181no—
238C3×D4—64no—
239C3×D4—62no—
240C3×D4—62no—
241C3×D4—62no—
242C3×D4—62no—
243C3×D4—122no—
244C3×D4—362no—
245C3×D4—362no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—181no—
257C2×C3×C4C2×D3211yescharacteristic
258C2×C3×C4C22×C3×D312yes—
259#67—124no—
260#67—124no—
261C2×Dic3—32no—
262C2×Dic3—32no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—62no—
266C2×Dic3—62no—
267D12—22no—
268D12—22no—
269Dic6—34no—
270Dic6—62no—
271Dic6—62no—
272Dic6—62no—
273C3×C8—62no—
274C3×C8—62no—
275C3×C8—122no—
276C3⋊C8—22no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—22no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—41no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—242no—
287C2×C32#88412yes—
288C2×C32#91112yes—
289C2×C32D3×SD1612yes—
290C2×C32C3×#12111yescharacteristic
291C2×C32C2×C3×SD1611yescharacteristic
292C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
293C3×D3—84no—
294C3×D3—82no—
295C3×D3—82no—
296C2×Q8—91no—
297C2×D4—32no—
298C2×D4—181no—
299C2×D4—181no—
300SD16—184no—
301C2×C8—32no—
302C4⋊C4—181no—
303C4⋊C4C3×D3211yescharacteristic
304C42—91no—
305C42—181no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—41no—
309C22×C3—124no—
310C22×C3—122no—
311C22×C3—122no—
312C22×C3—122no—
313C22×C3—122no—
314C22×C3—122no—
315C22×C3—122no—
316C22×C3—242no—
317C22×C3—242no—
318C22×C3#381111yescharacteristic
319C22×C3C3×D3×D412yes—
320C2×D3—42no—
321C2×D3—42no—
322C2×D3—42no—
323C3×C4—21no—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—21no—
327C3×C4—42no—
328C3×C4—42no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—82no—
333C3×C4—82no—
334C3×C4—92no—
335C3×C4—182no—
336C3×C4—181no—
337C3×C4#379811yescharacteristic
338C3×C4#380211yescharacteristic
339C3×C4C3×#23512yes—
340C3×C4C3×D3×D412yes—
341Dic3—34no—
342Dic3—62no—
343Dic3—64no—
344Dic3—64no—
345Dic3—62no—
346Dic3—121no—
347C32—21no—
348C32—22no—
349C32—41no—
350C32#476212yescomplemented
351C32C3×#46011yescharacteristic, complemented
352C23—62no—
353Q8—92no—
354Q8—181no—
355D4—62no—
356D4—62no—
357D4—362no—
358D4—362no—
359C2×C4—21no—
360C2×C4—91no—
361C2×C4—91no—
362C2×C4—181no—
363C2×C4C2×C3×D3211yescharacteristic
364C8—62no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—22no—
369C2×C3—21no—
370C2×C3—22no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—244no—
375C2×C3—242no—
376C2×C3—242no—
377C2×C3#6391211yescharacteristic
378C2×C3#6392711yescharacteristic
379C2×C3#6394011yescharacteristic
380C2×C3C3×#88412yes—
381C2×C3C3×#91112yes—
382C2×C3C3×D3×SD1612yes—
383D3—82no—
384C22—122no—
385C22—122no—
386C22—122no—
387C22C3×#381111yescharacteristic
388C4—41no—
389C4—92no—
390C4—182no—
391C4—181no—
392C4C3×#379811yescharacteristic
393C4C3×#380211yescharacteristic
394C3—22no—
395C3—21no—
396C3—41no—
397C3#1059055311yescharacteristic, direct product factor
398C3C3×#476212yescomplemented
399C2—242no—
400C2C3×#6391211yescharacteristic
401C2C3×#6392711yescharacteristic
402C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#460
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64096C2semidirect
3. C2×C3×#3716C2semidirect
4. C32×#809C2semidirect
5. C3×#63572C2semidirect
7. #10590553C3direct (N × H)
18. #64096C2×C3semidirect
25. C2×#3716C2×C3semidirect
29. C3×#809D3semidirect
30. C3×#809C2×C3semidirect
31. #63572C2×C3semidirect
39. C3×#634D4semidirect
88. C3×C3⋊Dic3SD16semidirect
102. #809C3×D3semidirect
123. #634C3×D4semidirect
219. C3⋊Dic3C3×SD16semidirect
231. C33#460semidirect
350. C32#4762semidirect
351. C32C3×#460semidirect
397. C3#10590553direct (N × H)
398. C3C3×#4762semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#639142
C2×C3×#37162
C32×#8092
C3×#635722
C3×#47623
#105905533