Finite Group Databasebeta

← C3×#10591485 #13158526

Subgroups — C3×#10591485 #13158526

Order 1728.

858 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 132 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×#64145C211yesmaximal, characteristic, complemented
2C2×C32×#213C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4C2×C3×#3716C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#5154—32nomaximal
7#10591485C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×C32×D12C2212yes—
11C2×C3×#625—22no—
12C3×#3757—22no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×#213—24no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3716—24no—
17C2×C32×D8—32no—
18#64145C2×C311yescharacteristic, complemented
19C3×#940—32no—
20C3×#936—32no—
21C3×#936—61no—
22C2×C3×#214—32no—
23C2×C3×#213—22no—
24C2×C3×#213D312yescomplemented
25C2×C3×#213C2×C312yescomplemented
26C2×C3×#204—32no—
27#63914C2×C311yescharacteristic, complemented
28C2×#3716C2×C312yescomplemented
29C3×#815—32no—
30#63555C2×C311yescharacteristic, complemented
31C23×C33—21no—
32C22×C32×D3—22no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D412yes—
37C3×#634—22no—
38C32×D12—22no—
39C32×D12—22no—
40C3×#625—44no—
41C32×C3⋊C8D414yescomplemented
42C3×#496—91no—
43#5154—32no—
44C2×C32×D4—22no—
45C2×C32×D4—21no—
46C2×C32×D4—32no—
47C2×C32×D4—41no—
48C2×C32×D4C2×D312yes—
49C2×C32×D4C22×C311yescharacteristic
50C2×#634C22×C311yescharacteristic
51C2×C3×#67—64no—
52C2×C3×D12—22no—
53C2×C3×D12C2×D312yes—
54C2×C3×D12C22×C312yes—
55C2×C3×Dic6—32no—
56C2×C3×Dic6—61no—
57C2×#625—22no—
58C32×D8—64no—
59C2×C32×C8—32no—
60#3757—22no—
61C4×C3⋊Dic3C22×C311yescharacteristic
62C3×#217—64no—
63C3×#217—64no—
64C3×#214—64no—
65C3×#213—28no—
66C3×#213—24no—
67C3×#213—24no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C22×C312yes—
73C3×#204—64no—
74#3716—24no—
75C22×C33—41no—
76C22×C33—42no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—42no—
79C2×C32×D3—44no—
80C33×C4C2×D412yes—
81C3×C3⋊Dic3—22no—
82C3×C3⋊Dic3—41no—
83C2×C3×SD16—92no—
84C2×C3×D8—32no—
85C2×C3×D8—32no—
86C2×C3×D8—62no—
87C3×#127—91no—
88C3×#124—91no—
89#940—32no—
90#936—32no—
91#936—61no—
92C2×#214—32no—
93C2×#213C3×D312yescomplemented
94C2×#204—32no—
95C3×#97—91no—
96#815—32no—
97C23×C32—21no—
98C23×C32—22no—
99C23×C32—22no—
100C23×C32—42no—
101C23×C32—42no—
102C23×C32—62no—
103C22×C3×D3—22no—
104C22×C3×D3—22no—
105C22×C3×D3—42no—
106C32×D4—22no—
107C32×D4—24no—
108C32×D4—22no—
109C32×D4—22no—
110C32×D4—42no—
111C32×D4—42no—
112C32×D4—44no—
113C32×D4—62no—
114C32×D4—62no—
115C2×C32×C4—21no—
116C2×C32×C4—22no—
117C2×C32×C4—41no—
118C2×C32×C4C22×D312yes—
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3C3×D412yes—
122#634—22no—
123C3×#67—128no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C3×D12—24no—
129C3×D12—22no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12—42no—
134C3×Dic6—62no—
135C3×Dic6—62no—
136C3×Dic6—122no—
137#625—44no—
138C32×C8—34no—
139C3×C3⋊C8—24no—
140C3×C3⋊C8#6714yescomplemented
141C3×C3⋊C8C3×D414yescomplemented
142#496—91noSylow
143C2×C33—81no—
144C2×C33#12711yescharacteristic
145C2×C33AGL(1,ℤ/8)12yes—
146C32×D3—82no—
147C2×C3×Q8—91no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—32no—
151C2×C3×D4—32no—
152C2×C3×D4—41no—
153C2×C3×D4—62no—
154C2×C3×D4—182no—
155C2×C3×D4D3211yescharacteristic
156C2×C3×D4C2×C3×D312yes—
157C2×#67—64no—
158C2×D12C2×C3×D312yes—
159C2×Dic6—32no—
160C2×Dic6—61no—
161C3×SD16—184no—
162C3×D8—64no—
163C3×D8—64no—
164C3×D8—68no—
165C2×C3×C8—32no—
166C2×C3×C8—32no—
167C2×C3×C8—62no—
168C3×C22⋊C4—182no—
169C3×C42—91no—
170#217—64no—
171#217—64no—
172#214—64no—
173#213—24no—
174C4×Dic3—32no—
175C4×Dic3—61no—
176C2×C3⋊C8C2×C3×D312yes—
177#204—64no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—41no—
181C22×C32—44no—
182C22×C32—42no—
183C22×C32—44no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—42no—
187C22×C32—44no—
188C22×C32—44no—
189C22×C32—84no—
190C22×C32—84no—
191C22×C32—122no—
192C22×C32—124no—
193C22×C32D3×D412yes—
194C22×C32C2×C3×D411yescharacteristic
195C2×C3×D3—44no—
196C2×C3×D3—44no—
197C2×C3×D3—42no—
198C2×C3×D3—42no—
199C2×C3×D3—44no—
200C2×C3×D3—84no—
201C32×C4—22no—
202C32×C4—22no—
203C32×C4—22no—
204C32×C4—42no—
205C32×C4D3×D412yes—
206C32×C4C2×#6712yes—
207C32×C4C2×C3×D412yes—
208C3⋊Dic3—22no—
209C3⋊Dic3—41no—
210C3×Dic3—64no—
211C3×Dic3—122no—
212C3×Dic3—122no—
213C3×Dic3—122no—
214C2×SD16—92no—
215C2×D8—32no—
216#127—91no—
217#124—91no—
218#97—91no—
219C33#49611yescharacteristic, complemented, Sylow
220C23×C3—22no—
221C23×C3—21no—
222C23×C3—22no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—62no—
226C23×C3—62no—
227C23×C3—122no—
228C22×D3—22no—
229C3×Q8—182no—
230C3×D4—24no—
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—22no—
234C3×D4—42no—
235C3×D4—42no—
236C3×D4—44no—
237C3×D4—64no—
238C3×D4—62no—
239C3×D4—62no—
240C3×D4—62no—
241C3×D4—62no—
242C3×D4—122no—
243C3×D4—364no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—92no—
248C2×C3×C4—181no—
249C2×C3×C4C2×D3211yescharacteristic
250C2×C3×C4C22×C3×D312yes—
251#67—128no—
252C2×Dic3—34no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256D12—22no—
257D12—22no—
258Dic6—62no—
259Dic6—62no—
260Dic6—122no—
261C3×C8—34no—
262C3×C8—34no—
263C3×C8—64no—
264C3⋊C8C3×#6714yescomplemented
265C2×C32—24no—
266C2×C32—22no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—41no—
270C2×C32—42no—
271C2×C32—82no—
272C2×C32—84no—
273C2×C32—84no—
274C2×C32—82no—
275C2×C32—81no—
276C2×C32—242no—
277C2×C32#91114yes—
278C2×C32#94012yes—
279C2×C32C3×#12711yescharacteristic
280C2×C32C3×AGL(1,ℤ/8)12yes—
281C3×D3—84no—
282C3×D3—82no—
283C3×D3—82no—
284C2×Q8—91no—
285C2×D4—32no—
286C2×D4—182no—
287C2×D4C3×D3211yescharacteristic
288SD16—184no—
289D8—64no—
290C2×C8—32no—
291C22⋊C4—182no—
292C42—91no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—44no—
296C22×C3—42no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—84no—
305C22×C3—84no—
306C22×C3—124no—
307C22×C3—124no—
308C22×C3—122no—
309C22×C3—124no—
310C22×C3—122no—
311C22×C3—244no—
312C22×C3#381111yescharacteristic
313C22×C3C3×D3×D412yes—
314C2×D3—44no—
315C2×D3—42no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—42no—
320C3×C4—182no—
321C3×C4—361no—
322C3×C4D3×#6712yes—
323C3×C4C3×D3×D412yes—
324C3×C4C2×C3×#6712yes—
325Dic3—64no—
326Dic3—122no—
327Dic3—122no—
328Dic3—122no—
329C32—22no—
330C32—21no—
331C32—41no—
332C32#515412yescomplemented
333C32C3×#49611yescharacteristic, complemented
334C23—21no—
335C23—62no—
336Q8—182no—
337D4—22no—
338D4—62no—
339D4—62no—
340D4—364no—
341C2×C4—92no—
342C2×C4—181no—
343C2×C4C2×C3×D3211yescharacteristic
344C8—34no—
345C2×C3—21no—
346C2×C3—22no—
347C2×C3—22no—
348C2×C3—24no—
349C2×C3—41no—
350C2×C3—42no—
351C2×C3—82no—
352C2×C3—84no—
353C2×C3—84no—
354C2×C3—82no—
355C2×C3—81no—
356C2×C3—244no—
357C2×C3—242no—
358C2×C3—242no—
359C2×C3#6392712yes—
360C2×C3#6397911yescharacteristic
361C2×C3C3×#91114yes—
362C2×C3C3×#94012yes—
363D3—82no—
364C22—41no—
365C22—42no—
366C22—122no—
367C22—124no—
368C22C3×#381111yescharacteristic
369C4—182no—
370C4—361no—
371C4C3×D3×#6712yes—
372C3—22no—
373C3—21no—
374C3—41no—
375C3#1059148511yescharacteristic, direct product factor
376C3C3×#515412yescomplemented
377C2—81no—
378C2—242no—
379C2C3×#6392712yes—
380C2C3×#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×#496
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×#64145C2semidirect
2. C2×C32×#213C2semidirect
3. C3×#63914C2semidirect
4. C2×C3×#3716C2semidirect
5. C3×#63555C2semidirect
7. #10591485C3direct (N × H)
18. #64145C2×C3semidirect
24. C2×C3×#213D3semidirect
25. C2×C3×#213C2×C3semidirect
27. #63914C2×C3semidirect
28. C2×#3716C2×C3semidirect
30. #63555C2×C3semidirect
41. C32×C3⋊C8D4semidirect
93. C2×#213C3×D3semidirect
140. C3×C3⋊C8#67semidirect
141. C3×C3⋊C8C3×D4semidirect
219. C33#496semidirect
264. C3⋊C8C3×#67semidirect
332. C32#5154semidirect
333. C32C3×#496semidirect
375. C3#10591485direct (N × H)
376. C3C3×#5154semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C2×C32×#2132
C3×#639142
C2×C3×#37162
C3×#635552
C3×#51543
#105914853