Finite Group Databasebeta

← C3×#10591492 #13158528

Subgroups — C3×#10591492 #13158528

Order 1728.

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

normal subgroups only

This group has 866 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
2C2×C32×#214C212yesmaximal, complemented
3C3×#63873C211yesmaximal, characteristic, complemented
4C2×C3×#3717C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6C3×#5175—32nomaximal
7#10591492C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×Dic6C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×#214—24no—
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—22no—
16C3×#3717—24no—
17C2×C32×SD16—32no—
18#64149C2×C311yescharacteristic
19C3×#944—32no—
20C2×C3×#215—32no—
21C3×#940—32no—
22C3×#940—61no—
23C2×C3×#214—22no—
24C2×C3×#214D312yescomplemented
25C2×C3×#214C2×C312yescomplemented
26C2×C3×#204—32no—
27#63873C2×C311yescharacteristic, complemented
28C2×#3717C2×C312yescomplemented
29C3×C8×Dic3—32no—
30#63553C2×C311yescharacteristic, complemented
31C23×C33—21no—
32C22×C3×C3⋊D3—21no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35C3×#638—44no—
36C2×C3×C3⋊Dic3D412yes—
37C3×C4⋊(C3⋊D3)—22no—
38C2×C32×Dic3—22no—
39C32×Dic6—22no—
40C32×Dic6—22no—
41C32×C3⋊C8D414yescomplemented
42C3×#492—91no—
43#5175—32no—
44C2×C32×Q8—32no—
45C2×C32×D4—22no—
46C2×C32×D4—21no—
47C2×C32×D4—41no—
48C2×C32×D4C2×D312yes—
49C2×C32×D4C22×C311yescharacteristic
50C2×#638—22no—
51C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
52C2×C3×#67—64no—
53C2×C3×#67—64no—
54C2×C3×D12—32no—
55C2×C3×D12—61no—
56C2×C3×Dic6—22no—
57C2×C3×Dic6C2×D312yes—
58C2×C3×Dic6C22×C312yes—
59C32×SD16—64no—
60C2×C32×C8—32no—
61C4×C3⋊Dic3C22×C311yescharacteristic
62C3×#215—64no—
63C3×#214—28no—
64C3×#214—24no—
65C3×#214—24no—
66C3×Dic3⋊C4—64no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—61no—
69C2×C3×C3⋊C8—22no—
70C2×C3×C3⋊C8C2×D312yes—
71C2×C3×C3⋊C8C22×C312yes—
72C3×#204—64no—
73#3724—22no—
74#3717—24no—
75C22×C33—41no—
76C22×C33—42no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—42no—
80C33×C4C2×D412yes—
81C3×C3⋊Dic3SD1614yescomplemented
82C32×Dic3—42no—
83C2×C3×SD16—32no—
84C2×C3×SD16—32no—
85C2×C3×SD16—62no—
86C2×C3×SD16—92no—
87C3×C4⋊Q8—91no—
88C3×#127—91no—
89#944—32no—
90C2×#215—32no—
91#940—32no—
92#940—61no—
93C2×#214C3×D312yescomplemented
94C2×#204—32no—
95C3×C4×C8—91no—
96C8×Dic3—32no—
97C23×C32—21no—
98C23×C32—22no—
99C23×C32—22no—
100C23×C32—42no—
101C23×C32—42no—
102C22×C3⋊D3—21no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C32×Q8—62no—
106C32×Q8—62no—
107C32×D4—22no—
108C32×D4—24no—
109C32×D4—22no—
110C32×D4—22no—
111C32×D4—42no—
112C32×D4—42no—
113C32×D4—44no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4—41no—
117C2×C32×C4—62no—
118C2×C32×C4C22×D312yes—
119C2×C32×C4C23×C311yescharacteristic
120#638—44no—
121C2×C3⋊Dic3C3×D412yes—
122C4⋊(C3⋊D3)—22no—
123C3×#67—128no—
124C3×#67—128no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—42no—
129C2×C3×Dic3—62no—
130C3×D12—62no—
131C3×D12—62no—
132C3×D12—122no—
133C3×Dic6—24no—
134C3×Dic6—22no—
135C3×Dic6—22no—
136C3×Dic6—22no—
137C3×Dic6—22no—
138C3×Dic6—42no—
139C32×C8—34no—
140C3×C3⋊C8—24no—
141C3×C3⋊C8#6714yescomplemented
142C3×C3⋊C8C3×D414yescomplemented
143#492—91noSylow
144C2×C33—81no—
145C2×C33#12711yescharacteristic
146C2×C33C2×SD1612yes—
147C3×C3⋊D3—81no—
148C2×C3×Q8—32no—
149C2×C3×Q8—32no—
150C2×C3×Q8—62no—
151C2×C3×D4—22no—
152C2×C3×D4—21no—
153C2×C3×D4—41no—
154C2×C3×D4—91no—
155C2×C3×D4—182no—
156C2×C3×D4D3211yescharacteristic
157C2×C3×D4C2×C3×D312yes—
158C2×#67—64no—
159C2×#67—64no—
160C2×D12—32no—
161C2×D12—61no—
162C2×Dic6C2×C3×D312yes—
163C3×SD16—64no—
164C3×SD16—64no—
165C3×SD16—68no—
166C3×SD16—184no—
167C2×C3×C8—32no—
168C2×C3×C8—32no—
169C2×C3×C8—62no—
170C3×C4⋊C4—182no—
171C3×C42—91no—
172#215—64no—
173#214—24no—
174Dic3⋊C4—64no—
175C4×Dic3—32no—
176C4×Dic3—61no—
177C2×C3⋊C8C2×C3×D312yes—
178#204—64no—
179C22×C32—21no—
180C22×C32—22no—
181C22×C32—41no—
182C22×C32—44no—
183C22×C32—42no—
184C22×C32—44no—
185C22×C32—42no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—44no—
189C22×C32—44no—
190C22×C32—84no—
191C22×C32—84no—
192C22×C32D3×D412yes—
193C22×C32C2×C3×D411yescharacteristic
194C2×C3⋊D3—41no—
195C2×C3⋊D3—42no—
196C2×C3×D3—122no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—122no—
200C32×C4—22no—
201C32×C4—22no—
202C32×C4—22no—
203C32×C4—42no—
204C32×C4—122no—
205C32×C4D3×D412yes—
206C32×C4C2×#6712yes—
207C32×C4C2×C3×D412yes—
208C3⋊Dic3C3×SD1614yescomplemented
209C3×Dic3—38no—
210C3×Dic3—44no—
211C3×Dic3—42no—
212C3×Dic3—42no—
213C3×Dic3—64no—
214C2×SD16—32no—
215C2×SD16—92no—
216C4⋊Q8—91no—
217#127—91no—
218C4×C8—91no—
219C33#49211yescharacteristic, complemented, Sylow
220C23×C3—22no—
221C23×C3—21no—
222C23×C3—22no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—181no—
226C22×D3—62no—
227C22×D3—62no—
228C3×Q8—64no—
229C3×Q8—62no—
230C3×Q8—62no—
231C3×Q8—62no—
232C3×Q8—62no—
233C3×Q8—122no—
234C3×D4—24no—
235C3×D4—22no—
236C3×D4—22no—
237C3×D4—22no—
238C3×D4—42no—
239C3×D4—42no—
240C3×D4—44no—
241C3×D4—182no—
242C3×D4—364no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—41no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—92no—
249C2×C3×C4—122no—
250C2×C3×C4C2×D3211yescharacteristic
251C2×C3×C4C22×C3×D312yes—
252#67—128no—
253#67—128no—
254C2×Dic3—22no—
255C2×Dic3—34no—
256C2×Dic3—62no—
257D12—62no—
258D12—62no—
259D12—122no—
260Dic6—22no—
261Dic6—22no—
262C3×C8—34no—
263C3×C8—34no—
264C3×C8—64no—
265C3⋊C8C3×#6714yescomplemented
266C2×C32—24no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—42no—
272C2×C32—82no—
273C2×C32—84no—
274C2×C32—84no—
275C2×C32—82no—
276C2×C32—81no—
277C2×C32D3×SD1614yes—
278C2×C32#94012yes—
279C2×C32C3×#12711yescharacteristic
280C2×C32C2×C3×SD1612yes—
281C3⋊D3—81no—
282C3×D3—242no—
283C3×D3—242no—
284C2×Q8—32no—
285C2×D4—91no—
286C2×D4—182no—
287C2×D4C3×D3211yescharacteristic
288SD16—64no—
289SD16—184no—
290C2×C8—32no—
291C4⋊C4—182no—
292C42—91no—
293C22×C3—21no—
294C22×C3—22no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—42no—
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—362no—
307C22×C3—361no—
308C22×C3#381111yescharacteristic
309C22×C3C3×D3×D412yes—
310C2×D3—122no—
311C2×D3—124no—
312C2×D3—124no—
313C2×D3—122no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—42no—
318C3×C4—94no—
319C3×C4—124no—
320C3×C4—122no—
321C3×C4—122no—
322C3×C4D3×#6712yes—
323C3×C4C3×D3×D412yes—
324C3×C4C2×C3×#6712yes—
325Dic3—38no—
326Dic3—42no—
327Dic3—64no—
328C32—21no—
329C32—22no—
330C32—41no—
331C32#517512yescomplemented
332C32C3×#49211yescharacteristic, complemented
333C23—21no—
334C23—181no—
335Q8—62no—
336Q8—62no—
337D4—22no—
338D4—182no—
339D4—364no—
340C2×C4—62no—
341C2×C4—92no—
342C2×C4C2×C3×D3211yescharacteristic
343C8—34no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3—22no—
347C2×C3—24no—
348C2×C3—41no—
349C2×C3—42no—
350C2×C3—82no—
351C2×C3—84no—
352C2×C3—84no—
353C2×C3—81no—
354C2×C3—82no—
355C2×C3—721no—
356C2×C3#6393012yes—
357C2×C3#6397911yescharacteristic
358C2×C3C3×D3×SD1614yes—
359C2×C3C3×#94012yes—
360D3—242no—
361D3—242no—
362C22—41no—
363C22—42no—
364C22—361no—
365C22—362no—
366C22C3×#381111yescharacteristic
367C4—94no—
368C4—122no—
369C4C3×D3×#6712yes—
370C3—21no—
371C3—22no—
372C3—41no—
373C3#1059149211yescharacteristic, direct product factor
374C3C3×#517512yescomplemented
375C2—81no—
376C2—721no—
377C2C3×#6393012yes—
378C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#492
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
2. C2×C32×#214C2semidirect
3. C3×#63873C2semidirect
4. C2×C3×#3717C2semidirect
5. C3×#63553C2semidirect
7. #10591492C3direct (N × H)
24. C2×C3×#214D3semidirect
25. C2×C3×#214C2×C3semidirect
27. #63873C2×C3semidirect
28. C2×#3717C2×C3semidirect
30. #63553C2×C3semidirect
41. C32×C3⋊C8D4semidirect
81. C3×C3⋊Dic3SD16semidirect
93. C2×#214C3×D3semidirect
141. C3×C3⋊C8#67semidirect
142. C3×C3⋊C8C3×D4semidirect
208. C3⋊Dic3C3×SD16semidirect
219. C33#492semidirect
265. C3⋊C8C3×#67semidirect
331. C32#5175semidirect
332. C32C3×#492semidirect
373. C3#10591492direct (N × H)
374. C3C3×#5175semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C2×C32×#2142
C3×#638732
C2×C3×#37172
C3×#635532
C3×#51753
#105914923