Finite Group Databasebeta

← C3×#10591493 #13158531

Subgroups — C3×#10591493 #13158531

Order 1728.

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

normal subgroups only

This group has 762 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
2C2×C32×#214C212yesmaximal
3C3×#63873C211yesmaximal, characteristic, complemented
4C2×C3×#3719C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#5167—32nomaximal
7#10591493C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C3×#3757—22no—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×#214—24no—
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—22no—
16C3×#3719—24no—
17C2×C32×SD16—32no—
18#64145C2×C311yescharacteristic
19C3×#944—32no—
20C2×C3×#216—32no—
21C3×#936—32no—
22C3×#936—61no—
23C2×C3×#214—22no—
24C2×C3×#214D312yes—
25C2×C3×#214C2×C312yes—
26C2×C3×Dic12—32no—
27#63873C2×C311yescharacteristic, complemented
28C2×#3719C2×C312yescomplemented
29C3×#815—32no—
30#63555C2×C311yescharacteristic, complemented
31C23×C33—21no—
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D412yes—
36C3×#634—22no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C32×C3⋊C8D414yes—
41C3×#497—91no—
42#5167—32no—
43C2×C32×Q8—32no—
44C2×C32×D4—22no—
45C2×C32×D4—21no—
46C2×C32×D4—41no—
47C2×C32×D4C2×D312yes—
48C2×C32×D4C22×C311yescharacteristic
49C2×#634C22×C311yescharacteristic
50C2×C3×Dic6—22no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—61no—
53C2×C3×Dic6C2×D312yes—
54C2×C3×Dic6C22×C312yes—
55C32×SD16—64no—
56C2×C32×C8—32no—
57#3757—22no—
58C4×C3⋊Dic3C22×C311yescharacteristic
59C3×#217—64no—
60C3×#217—64no—
61C3×#216—64no—
62C3×#214—28no—
63C3×#214—24no—
64C3×#214—24no—
65C3×Dic3⋊C4—64no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—61no—
68C2×C3×C3⋊C8—22no—
69C2×C3×C3⋊C8C2×D312yes—
70C2×C3×C3⋊C8C22×C312yes—
71C3×Dic12—64no—
72#3724—22no—
73#3719—24no—
74C22×C33—41no—
75C22×C33—42no—
76C22×C33C2×D411yescharacteristic
77C33×C4C2×D412yes—
78C3×C3⋊Dic3—22no—
79C3×C3⋊Dic3—41no—
80C32×Dic3—42no—
81C2×C3×Q16—92no—
82C2×C3×SD16—32no—
83C2×C3×SD16—32no—
84C2×C3×SD16—62no—
85C3×C4⋊Q8—91no—
86C3×#124—91no—
87#944—32no—
88C2×#216—32no—
89#936—32no—
90#936—61no—
91C2×#214C3×D312yes—
92C2×Dic12—32no—
93C3×#97—91no—
94#815—32no—
95C23×C32—21no—
96C23×C32—22no—
97C23×C32—22no—
98C23×C32—42no—
99C23×C32—42no—
100C32×Q8—62no—
101C32×Q8—62no—
102C32×D4—22no—
103C32×D4—24no—
104C32×D4—22no—
105C32×D4—22no—
106C32×D4—42no—
107C32×D4—42no—
108C32×D4—44no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—41no—
112C2×C32×C4—62no—
113C2×C32×C4C22×D312yes—
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3C3×D412yes—
117#634—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C3×Dic6—24no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—22no—
129C3×Dic6—22no—
130C3×Dic6—42no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133C3×Dic6—122no—
134C32×C8—34no—
135C3×C3⋊C8—24no—
136C3×C3⋊C8#6714yes—
137C3×C3⋊C8C3×D414yes—
138#497—91noSylow
139C2×C33—81no—
140C2×C33#12711yescharacteristic
141C2×C33#13712yes—
142C2×C3×Q8—32no—
143C2×C3×Q8—32no—
144C2×C3×Q8—62no—
145C2×C3×Q8—91no—
146C2×C3×D4—22no—
147C2×C3×D4—21no—
148C2×C3×D4—41no—
149C2×C3×D4D3211yescharacteristic
150C2×C3×D4C2×C3×D312yes—
151C2×Dic6—32no—
152C2×Dic6—61no—
153C2×Dic6C2×C3×D312yes—
154C3×Q16—184no—
155C3×SD16—64no—
156C3×SD16—64no—
157C3×SD16—68no—
158C2×C3×C8—32no—
159C2×C3×C8—32no—
160C2×C3×C8—62no—
161C3×C4⋊C4—182no—
162C3×C22⋊C4—182no—
163C3×C42—91no—
164#217—64no—
165#217—64no—
166#216—64no—
167#214—24no—
168Dic3⋊C4—64no—
169C4×Dic3—32no—
170C4×Dic3—61no—
171C2×C3⋊C8C2×C3×D312yes—
172Dic12—64no—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32—41no—
176C22×C32—44no—
177C22×C32—42no—
178C22×C32—44no—
179C22×C32—42no—
180C22×C32—41no—
181C22×C32—42no—
182C22×C32—44no—
183C22×C32—44no—
184C22×C32—84no—
185C22×C32—84no—
186C22×C32D3×D412yes—
187C22×C32C2×C3×D411yescharacteristic
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—42no—
192C32×C4—122no—
193C32×C4D3×D412yes—
194C32×C4C2×#6712yes—
195C32×C4C2×C3×D412yes—
196C3⋊Dic3—22no—
197C3⋊Dic3—41no—
198C3×Dic3—44no—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C3×Dic3—64no—
202C3×Dic3—122no—
203C3×Dic3—122no—
204C3×Dic3—122no—
205C2×Q16—92no—
206C2×SD16—32no—
207C4⋊Q8—91no—
208#124—91no—
209#97—91no—
210C33#49711yescharacteristic, complemented, Sylow
211C23×C3—22no—
212C23×C3—21no—
213C23×C3—22no—
214C23×C3—42no—
215C23×C3—42no—
216C3×Q8—64no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8—62no—
220C3×Q8—62no—
221C3×Q8—122no—
222C3×Q8—182no—
223C3×D4—24no—
224C3×D4—22no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—42no—
228C3×D4—42no—
229C3×D4—44no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—92no—
236C2×C3×C4—122no—
237C2×C3×C4—181no—
238C2×C3×C4C2×D3211yescharacteristic
239C2×C3×C4C22×C3×D312yes—
240C2×Dic3—22no—
241C2×Dic3—34no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245Dic6—22no—
246Dic6—22no—
247Dic6—62no—
248Dic6—62no—
249Dic6—122no—
250C3×C8—34no—
251C3×C8—34no—
252C3×C8—64no—
253C3⋊C8C3×#6714yes—
254C2×C32—24no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—41no—
259C2×C32—42no—
260C2×C32—82no—
261C2×C32—84no—
262C2×C32—84no—
263C2×C32—82no—
264C2×C32—81no—
265C2×C32#91514yes—
266C2×C32#94012yes—
267C2×C32C3×#12711yescharacteristic
268C2×C32C3×#13712yes—
269C2×Q8—32no—
270C2×Q8—91no—
271C2×D4C3×D3211yescharacteristic
272Q16—184no—
273SD16—64no—
274C2×C8—32no—
275C4⋊C4—182no—
276C22⋊C4—182no—
277C42—91no—
278C22×C3—21no—
279C22×C3—22no—
280C22×C3—44no—
281C22×C3—42no—
282C22×C3—44no—
283C22×C3—44no—
284C22×C3—44no—
285C22×C3—42no—
286C22×C3—42no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—84no—
290C22×C3—84no—
291C22×C3#381111yescharacteristic
292C22×C3C3×D3×D412yes—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—42no—
297C3×C4—124no—
298C3×C4—122no—
299C3×C4—122no—
300C3×C4—182no—
301C3×C4—361no—
302C3×C4D3×#6712yes—
303C3×C4C3×D3×D412yes—
304C3×C4C2×C3×#6712yes—
305Dic3—42no—
306Dic3—64no—
307Dic3—122no—
308Dic3—122no—
309Dic3—122no—
310C32—21no—
311C32—22no—
312C32—41no—
313C32#516712yescomplemented
314C32C3×#49711yescharacteristic, complemented
315C23—21no—
316Q8—62no—
317Q8—62no—
318Q8—182no—
319D4—22no—
320C2×C4—62no—
321C2×C4—92no—
322C2×C4—181no—
323C2×C4C2×C3×D3211yescharacteristic
324C8—34no—
325C2×C3—21no—
326C2×C3—22no—
327C2×C3—22no—
328C2×C3—24no—
329C2×C3—41no—
330C2×C3—42no—
331C2×C3—82no—
332C2×C3—84no—
333C2×C3—84no—
334C2×C3—81no—
335C2×C3—82no—
336C2×C3#6393112yes—
337C2×C3#6397911yescharacteristic
338C2×C3C3×#91514yes—
339C2×C3C3×#94012yes—
340C22—41no—
341C22—42no—
342C22C3×#381111yescharacteristic
343C4—122no—
344C4—182no—
345C4—361no—
346C4C3×D3×#6712yes—
347C3—22no—
348C3—21no—
349C3—41no—
350C3#1059149311yescharacteristic, direct product factor
351C3C3×#516712yescomplemented
352C2—81no—
353C2C3×#6393112yes—
354C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#497
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
3. C3×#63873C2semidirect
4. C2×C3×#3719C2semidirect
5. C3×#63555C2semidirect
7. #10591493C3direct (N × H)
27. #63873C2×C3semidirect
28. C2×#3719C2×C3semidirect
30. #63555C2×C3semidirect
210. C33#497semidirect
313. C32#5167semidirect
314. C32C3×#497semidirect
350. C3#10591493direct (N × H)
351. C3C3×#5167semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C2×C32×#2142
C3×#638732
C2×C3×#37192
C3×#635552
C3×#51673
#105914933