Finite Group Databasebeta

← C3×#10591484 #13158523

Subgroups — C3×#10591484 #13158523

Order 1728.

962 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 140 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×#64149C211yesmaximal, characteristic, complemented
2C2×C32×#213C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4C2×C3×#3714C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6C3×#5151—32nomaximal
7#10591484C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12C2212yes—
12C2×C3×#625—22no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×#213—24no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3714—24no—
17C2×C32×D8—32no—
18#64149C2×C311yescharacteristic, complemented
19C3×#940—32no—
20C3×#940—32no—
21C3×#940—61no—
22C2×C3×#213—22no—
23C2×C3×#213—32no—
24C2×C3×#213D312yescomplemented
25C2×C3×#213C2×C312yescomplemented
26C2×C3×D24—32no—
27#63914C2×C311yescharacteristic, complemented
28C2×#3714C2×C312yescomplemented
29C3×C8×Dic3—32no—
30#63553C2×C311yescharacteristic, complemented
31C23×C33—21no—
32C22×C3×C3⋊D3—21no—
33C22×C32×D3—22no—
34C33×D4—22no—
35C2×C33×C4C2311yescharacteristic
36C3×#638—44no—
37C2×C3×C3⋊Dic3D412yes—
38C3×C4⋊(C3⋊D3)—22no—
39C32×D12—22no—
40C32×D12—22no—
41C3×#625—44no—
42C32×C3⋊C8D414yescomplemented
43C3×#493—91no—
44#5151—32no—
45C2×C32×D4—22no—
46C2×C32×D4—21no—
47C2×C32×D4—32no—
48C2×C32×D4—41no—
49C2×C32×D4C2×D312yes—
50C2×C32×D4C22×C311yescharacteristic
51C2×#638—22no—
52C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
53C2×C3×#67—64no—
54C2×C3×#67—64no—
55C2×C3×#67—64no—
56C2×C3×D12—22no—
57C2×C3×D12—32no—
58C2×C3×D12—61no—
59C2×C3×D12C2×D312yes—
60C2×C3×D12C22×C312yes—
61C2×#625—22no—
62C32×D8—64no—
63C2×C32×C8—32no—
64C4×C3⋊Dic3C22×C311yescharacteristic
65C3×#213—28no—
66C3×#213—24no—
67C3×#213—24no—
68C3×#213—64no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—61no—
71C2×C3×C3⋊C8—22no—
72C2×C3×C3⋊C8C2×D312yes—
73C2×C3×C3⋊C8C22×C312yes—
74C3×D24—64no—
75#3714—24no—
76C22×C33—41no—
77C22×C33—42no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—42no—
81C2×C32×D3—42no—
82C2×C32×D3—44no—
83C33×C4C2×D412yes—
84C3×C3⋊Dic3D814yescomplemented
85C2×C3×D8—32no—
86C2×C3×D8—32no—
87C2×C3×D8—62no—
88C2×C3×D8—92no—
89C3×#127—91no—
90C3×#127—91no—
91#940—32no—
92#940—32no—
93#940—61no—
94C2×#213—32no—
95C2×#213C3×D312yescomplemented
96C2×D24—32no—
97C3×C4×C8—91no—
98C8×Dic3—32no—
99C23×C32—21no—
100C23×C32—22no—
101C23×C32—22no—
102C23×C32—42no—
103C23×C32—42no—
104C23×C32—62no—
105C22×C3⋊D3—21no—
106C22×C3×D3—22no—
107C22×C3×D3—22no—
108C22×C3×D3—42no—
109C22×C3×D3—62no—
110C22×C3×D3—62no—
111C32×D4—22no—
112C32×D4—24no—
113C32×D4—22no—
114C32×D4—22no—
115C32×D4—42no—
116C32×D4—42no—
117C32×D4—44no—
118C32×D4—62no—
119C32×D4—62no—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4—41no—
123C2×C32×C4C22×D312yes—
124C2×C32×C4C23×C311yescharacteristic
125#638—44no—
126C2×C3⋊Dic3C3×D412yes—
127C4⋊(C3⋊D3)—22no—
128C3×#67—128no—
129C3×#67—128no—
130C3×#67—128no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—62no—
133C3×D12—24no—
134C3×D12—22no—
135C3×D12—22no—
136C3×D12—22no—
137C3×D12—22no—
138C3×D12—42no—
139C3×D12—62no—
140C3×D12—62no—
141C3×D12—122no—
142#625—44no—
143C32×C8—34no—
144C3×C3⋊C8—24no—
145C3×C3⋊C8#6714yescomplemented
146C3×C3⋊C8C3×D414yescomplemented
147#493—91noSylow
148C2×C33—81no—
149C2×C33#12711yescharacteristic
150C2×C33C2×D812yes—
151C3×C3⋊D3—81no—
152C32×D3—82no—
153C2×C3×D4—22no—
154C2×C3×D4—21no—
155C2×C3×D4—32no—
156C2×C3×D4—32no—
157C2×C3×D4—41no—
158C2×C3×D4—62no—
159C2×C3×D4—91no—
160C2×C3×D4—182no—
161C2×C3×D4—182no—
162C2×C3×D4D3211yescharacteristic
163C2×C3×D4C2×C3×D312yes—
164C2×#67—64no—
165C2×#67—64no—
166C2×#67—64no—
167C2×D12—32no—
168C2×D12—61no—
169C2×D12C2×C3×D312yes—
170C3×D8—64no—
171C3×D8—64no—
172C3×D8—68no—
173C3×D8—184no—
174C2×C3×C8—32no—
175C2×C3×C8—32no—
176C2×C3×C8—62no—
177C3×C42—91no—
178#213—24no—
179#213—64no—
180C4×Dic3—32no—
181C4×Dic3—61no—
182C2×C3⋊C8C2×C3×D312yes—
183D24—64no—
184C22×C32—21no—
185C22×C32—22no—
186C22×C32—41no—
187C22×C32—44no—
188C22×C32—42no—
189C22×C32—44no—
190C22×C32—42no—
191C22×C32—41no—
192C22×C32—42no—
193C22×C32—44no—
194C22×C32—44no—
195C22×C32—84no—
196C22×C32—84no—
197C22×C32—122no—
198C22×C32—124no—
199C22×C32D3×D412yes—
200C22×C32C2×C3×D411yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—42no—
203C2×C3×D3—44no—
204C2×C3×D3—44no—
205C2×C3×D3—42no—
206C2×C3×D3—42no—
207C2×C3×D3—44no—
208C2×C3×D3—84no—
209C2×C3×D3—122no—
210C2×C3×D3—124no—
211C2×C3×D3—124no—
212C2×C3×D3—122no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—22no—
216C32×C4—42no—
217C32×C4D3×D412yes—
218C32×C4C2×#6712yes—
219C32×C4C2×C3×D412yes—
220C3⋊Dic3C3×D814yescomplemented
221C3×Dic3—38no—
222C3×Dic3—64no—
223C2×D8—32no—
224C2×D8—92no—
225#127—91no—
226#127—91no—
227C4×C8—91no—
228C33#49311yescharacteristic, complemented, Sylow
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—62no—
235C23×C3—62no—
236C23×C3—122no—
237C23×C3—181no—
238C22×D3—22no—
239C22×D3—62no—
240C22×D3—62no—
241C3×D4—24no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—44no—
248C3×D4—64no—
249C3×D4—62no—
250C3×D4—62no—
251C3×D4—62no—
252C3×D4—62no—
253C3×D4—122no—
254C3×D4—182no—
255C3×D4—364no—
256C3×D4—364no—
257C2×C3×C4—22no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—92no—
261C2×C3×C4C2×D3211yescharacteristic
262C2×C3×C4C22×C3×D312yes—
263#67—128no—
264#67—128no—
265#67—128no—
266C2×Dic3—34no—
267C2×Dic3—62no—
268D12—22no—
269D12—22no—
270D12—62no—
271D12—62no—
272D12—122no—
273C3×C8—34no—
274C3×C8—34no—
275C3×C8—64no—
276C3⋊C8C3×#6714yescomplemented
277C2×C32—24no—
278C2×C32—22no—
279C2×C32—22no—
280C2×C32—21no—
281C2×C32—41no—
282C2×C32—42no—
283C2×C32—82no—
284C2×C32—84no—
285C2×C32—84no—
286C2×C32—82no—
287C2×C32—81no—
288C2×C32—242no—
289C2×C32D3×D814yes—
290C2×C32#94012yes—
291C2×C32C3×#12711yescharacteristic
292C2×C32C2×C3×D812yes—
293C3⋊D3—81no—
294C3×D3—84no—
295C3×D3—82no—
296C3×D3—82no—
297C3×D3—242no—
298C3×D3—242no—
299C2×D4—32no—
300C2×D4—91no—
301C2×D4—182no—
302C2×D4—182no—
303C2×D4C3×D3211yescharacteristic
304D8—64no—
305D8—184no—
306C2×C8—32no—
307C42—91no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—44no—
311C22×C3—44no—
312C22×C3—42no—
313C22×C3—44no—
314C22×C3—44no—
315C22×C3—42no—
316C22×C3—42no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—84no—
320C22×C3—84no—
321C22×C3—124no—
322C22×C3—124no—
323C22×C3—122no—
324C22×C3—124no—
325C22×C3—122no—
326C22×C3—244no—
327C22×C3—362no—
328C22×C3—361no—
329C22×C3#381111yescharacteristic
330C22×C3C3×D3×D412yes—
331C2×D3—44no—
332C2×D3—42no—
333C2×D3—122no—
334C2×D3—124no—
335C2×D3—124no—
336C2×D3—122no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—42no—
341C3×C4—94no—
342C3×C4D3×#6712yes—
343C3×C4C3×D3×D412yes—
344C3×C4C2×C3×#6712yes—
345Dic3—38no—
346Dic3—64no—
347C32—22no—
348C32—21no—
349C32—41no—
350C32#515112yescomplemented
351C32C3×#49311yescharacteristic, complemented
352C23—21no—
353C23—62no—
354C23—181no—
355D4—22no—
356D4—62no—
357D4—62no—
358D4—182no—
359D4—364no—
360D4—364no—
361C2×C4—92no—
362C2×C4C2×C3×D3211yescharacteristic
363C8—34no—
364C2×C3—21no—
365C2×C3—22no—
366C2×C3—22no—
367C2×C3—24no—
368C2×C3—41no—
369C2×C3—42no—
370C2×C3—82no—
371C2×C3—84no—
372C2×C3—84no—
373C2×C3—82no—
374C2×C3—81no—
375C2×C3—244no—
376C2×C3—242no—
377C2×C3—242no—
378C2×C3—721no—
379C2×C3#6392612yes—
380C2×C3#6397911yescharacteristic
381C2×C3C3×D3×D814yes—
382C2×C3C3×#94012yes—
383D3—82no—
384D3—242no—
385D3—242no—
386C22—41no—
387C22—42no—
388C22—122no—
389C22—124no—
390C22—361no—
391C22—362no—
392C22C3×#381111yescharacteristic
393C4—94no—
394C4C3×D3×#6712yes—
395C3—21no—
396C3—22no—
397C3—41no—
398C3#1059148411yescharacteristic, direct product factor
399C3C3×#515112yescomplemented
400C2—81no—
401C2—242no—
402C2—721no—
403C2C3×#6392612yes—
404C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#493
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64149C2semidirect
2. C2×C32×#213C2semidirect
3. C3×#63914C2semidirect
4. C2×C3×#3714C2semidirect
5. C3×#63553C2semidirect
7. #10591484C3direct (N × H)
18. #64149C2×C3semidirect
24. C2×C3×#213D3semidirect
25. C2×C3×#213C2×C3semidirect
27. #63914C2×C3semidirect
28. C2×#3714C2×C3semidirect
30. #63553C2×C3semidirect
42. C32×C3⋊C8D4semidirect
84. C3×C3⋊Dic3D8semidirect
95. C2×#213C3×D3semidirect
145. C3×C3⋊C8#67semidirect
146. C3×C3⋊C8C3×D4semidirect
220. C3⋊Dic3C3×D8semidirect
228. C33#493semidirect
276. C3⋊C8C3×#67semidirect
350. C32#5151semidirect
351. C32C3×#493semidirect
398. C3#10591484direct (N × H)
399. C3C3×#5151semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C2×C32×#2132
C3×#639142
C2×C3×#37142
C3×#635532
C3×#51513
#105914843