Finite Group Databasebeta

← C3×#10590401 #13157419

Subgroups — C3×#10590401 #13157419

Order 1728.

754 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 116 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×#64298C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic, complemented
3C3×#63831C212yesmaximal, complemented
4C32×#805C212yesmaximal, complemented
5C3×#63561C211yesmaximal, characteristic, complemented
6C3×#4826—32nomaximal
7#10590401C311yesmaximal, characteristic, direct product factor
8C2×C3×#634C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C32×#235C2212yes—
11C3×#3805—22no—
12C3×#3795—22no—
13C3×D3×Dic6—22no—
14C33×C42C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)—22no—
16C32×#208C2212yes—
17C3×#3719—22no—
18C3×#3716—22no—
19C3×#1010—32no—
20#64298C2×C311yescharacteristic
21#64078C2×C311yescharacteristic, complemented
22C3×#951—32no—
23C3×#909—32no—
24C3×#869—32no—
25C3×#869—61no—
26#63831C2×C312yescomplemented
27C32×#104—32no—
28C3×#824—32no—
29C3×#805—22no—
30C3×#805D312yescomplemented
31C3×#805C2×C312yescomplemented
32#63561C2×C311yescharacteristic, complemented
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D412yes—
37C3×#634—22no—
38C3×#634—22no—
39C32×#67—22no—
40C32×D12D412yes—
41C32×C4×D3—22no—
42C32×Dic6D412yes—
43C3×#627—22no—
44C3×#625—22no—
45C3×D3×Dic3—24no—
46C32×C3⋊C8—22no—
47C3×#456—91no—
48#4826—32no—
49C32×#41—32no—
50C2×#634C22×C311yescharacteristic
51C2×#634C22×C311yescharacteristic
52C3×D3×Q8—62no—
53C3×#237—62no—
54C3×#237—64no—
55C3×#235—22no—
56C3×#235—62no—
57C3×#235C2×D312yes—
58C3×#235C22×C312yes—
59C2×C3×Dic6—32no—
60C2×C3×Dic6—32no—
61C2×C3×Dic6—62no—
62C2×C3×Dic6—61no—
63C2×C3×Dic6—61no—
64#3805—22no—
65#3795—22no—
66D3×Dic6—22no—
67C32×M16—32no—
68C32×C42—21no—
69C32×C42—22no—
70C32×C42—41no—
71C32×C42C2×D312yes—
72C32×C42C22×C311yescharacteristic
73C4⋊(C3⋊Dic3)—22no—
74C3×#216—62no—
75C3×#214—62no—
76C3×C4⋊Dic3—64no—
77C3×C4⋊Dic3—64no—
78C3×#208—22no—
79C3×#208C2×D312yes—
80C3×#208C22×C312yes—
81C3×Dic12—62no—
82C3×#204—62no—
83#3719—22no—
84#3716—22no—
85C22×C33C2×D411yescharacteristic
86C2×C32×D3—22no—
87C33×C4—22no—
88C33×C4C2×D412yes—
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3—41no—
91C32×Dic3—22no—
92C3×2−1+4—91no—
93#1010—32no—
94C3×#137—92no—
95C3×C4⋊Q8—91no—
96#951—32no—
97#909—32no—
98#869—32no—
99#869—61no—
100C3×#104—32no—
101C3×#104—32no—
102C3×#104—62no—
103C3×#101—91no—
104#824—32no—
105#805C3×D312yescomplemented
106C32×Q8—32no—
107C32×D4—32no—
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×C4—62no—
118C2×C32×C4C22×D312yes—
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3C3×D412yes—
122#634—22no—
123#634—22no—
124C3×#67—22no—
125C3×#67—22no—
126C3×#67—42no—
127C3×#67—64no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C3×D12—22no—
134C3×D12D1212yes—
135C3×D12C3×D412yes—
136C3×C4×D3—22no—
137C3×C4×D3—22no—
138C3×C4×D3—42no—
139C3×C4×D3—64no—
140C3×Dic6—22no—
141C3×Dic6—64no—
142C3×Dic6—62no—
143C3×Dic6—62no—
144C3×Dic6—62no—
145C3×Dic6—62no—
146C3×Dic6—122no—
147C3×Dic6—122no—
148C3×Dic6D1212yes—
149C3×Dic6C3×D412yes—
150#627—22no—
151#625—22no—
152D3×Dic3—24no—
153C32×C8—62no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—42no—
157#456—91noSylow
158C2×C33—21no—
159C2×C33#12011yescharacteristic
160C32×D3—42no—
161C3×#41—32no—
162C3×#41—32no—
163C3×#41—62no—
164C3×#41—182no—
165C3×#41—182no—
166C2×C3×Q8—91no—
167C2×C3×Q8—91no—
168C2×C3×Q8—182no—
169D3×Q8—62no—
170#237—62no—
171#237—64no—
172#235—62no—
173#235C2×C3×D312yes—
174C2×Dic6—32no—
175C2×Dic6—32no—
176C2×Dic6—61no—
177C2×Dic6—61no—
178C2×Dic6—62no—
179C3×Q16—182no—
180C3×SD16—182no—
181C3×M16—32no—
182C3×M16—32no—
183C3×M16—62no—
184C3×C4⋊C4—182no—
185C3×C42—21no—
186C3×C42—22no—
187C3×C42—41no—
188C3×C42D3211yescharacteristic
189C3×C42C2×C3×D312yes—
190#216—62no—
191#214—62no—
192C4⋊Dic3—64no—
193C4⋊Dic3—64no—
194#208C2×C3×D312yes—
195Dic12—62no—
196#204—62no—
197C22×C32—21no—
198C22×C32—22no—
199C22×C32—41no—
200C22×C32—62no—
201C22×C32D3×D412yes—
202C22×C32C2×C3×D411yescharacteristic
203C2×C3×D3—22no—
204C2×C3×D3—22no—
205C2×C3×D3—42no—
206C32×C4—22no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—42no—
213C32×C4—44no—
214C32×C4—44no—
215C32×C4—62no—
216C32×C4C2×D1212yes—
217C32×C4D3×D412yes—
218C32×C4C2×C3×D412yes—
219C3⋊Dic3—22no—
220C3⋊Dic3—41no—
221C3×Dic3—22no—
222C3×Dic3—22no—
223C3×Dic3—42no—
224C3×Dic3—64no—
225C3×Dic3—122no—
226C3×Dic3—122no—
227C3×Dic3—122no—
2282−1+4—91no—
229#137—92no—
230C4⋊Q8—91no—
231#104—32no—
232#101—91no—
233C33#45611yescharacteristic, complemented, Sylow
234C3×Q8—32no—
235C3×Q8—32no—
236C3×Q8—62no—
237C3×Q8—182no—
238C3×Q8—182no—
239C3×Q8—182no—
240C3×D4—32no—
241C3×D4—32no—
242C3×D4—62no—
243C3×D4—62no—
244C3×D4—62no—
245C3×D4—122no—
246C3×D4—182no—
247C2×C3×C4—22no—
248C2×C3×C4—21no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—92no—
258C2×C3×C4—122no—
259C2×C3×C4—184no—
260C2×C3×C4—181no—
261C2×C3×C4C2×D3211yescharacteristic
262C2×C3×C4C22×C3×D312yes—
263#67—22no—
264#67—64no—
265C2×Dic3—34no—
266C2×Dic3—62no—
267C2×Dic3—64no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270D12C3×D1212yes—
271C4×D3—22no—
272C4×D3—64no—
273Dic6—64no—
274Dic6—62no—
275Dic6—62no—
276Dic6—62no—
277Dic6—62no—
278Dic6—122no—
279Dic6—122no—
280Dic6C3×D1212yes—
281C3×C8—62no—
282C3×C8—62no—
283C3×C8—122no—
284C3⋊C8—22no—
285C2×C32—21no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—41no—
291C2×C32—42no—
292C2×C32—42no—
293C2×C32—122no—
294C2×C32#88212yes—
295C2×C32C3×#12011yescharacteristic
296C3×D3—42no—
297C3×D3—42no—
298C3×D3—82no—
299#41—32no—
300#41—182no—
301#41—182no—
302C2×Q8—91no—
303C2×Q8—91no—
304C2×Q8—182no—
305Q16—182no—
306SD16—182no—
307M16—32no—
308C4⋊C4—182no—
309C42C3×D3211yescharacteristic
310C22×C3—21no—
311C22×C3—22no—
312C22×C3—41no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—122no—
316C22×C3#381111yescharacteristic
317C22×C3C3×D3×D412yes—
318C2×D3—22no—
319C3×C4—22no—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4—22no—
323C3×C4—22no—
324C3×C4—22no—
325C3×C4—44no—
326C3×C4—44no—
327C3×C4—42no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—122no—
331C3×C4—182no—
332C3×C4—361no—
333C3×C4D3×D1212yes—
334C3×C4C2×C3×D1212yes—
335C3×C4C3×D3×D412yes—
336Dic3—22no—
337Dic3—64no—
338Dic3—122no—
339Dic3—122no—
340Dic3—122no—
341C32—22no—
342C32—21no—
343C32—41no—
344C32#482612yescomplemented
345C32C3×#45611yescharacteristic, complemented
346Q8—32no—
347Q8—182no—
348Q8—182no—
349Q8—182no—
350D4—32no—
351D4—62no—
352D4—182no—
353C2×C4—21no—
354C2×C4—62no—
355C2×C4—92no—
356C2×C4—184no—
357C2×C4—181no—
358C2×C4C2×C3×D3211yescharacteristic
359C8—62no—
360C2×C3—22no—
361C2×C3—22no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—22no—
365C2×C3—42no—
366C2×C3—42no—
367C2×C3—41no—
368C2×C3—122no—
369C2×C3—122no—
370C2×C3—242no—
371C2×C3#6392311yescharacteristic
372C2×C3C3×#88212yes—
373D3—42no—
374C22—62no—
375C22C3×#381111yescharacteristic
376C4—22no—
377C4—62no—
378C4—182no—
379C4—361no—
380C4C3×D3×D1212yes—
381C3—21no—
382C3—22no—
383C3—41no—
384C3#1059040111yescharacteristic, direct product factor
385C3C3×#482612yescomplemented
386C2—21no—
387C2—122no—
388C2C3×#6392311yescharacteristic
subgroupG/H
centerC2×C3#63923
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33#120
prime radicalC3×#64298C2
hypercenterC3×C42D32
hypocenterC32C3×#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×#64078C2semidirect
3. C3×#63831C2semidirect
4. C32×#805C2semidirect
5. C3×#63561C2semidirect
7. #10590401C3direct (N × H)
21. #64078C2×C3semidirect
26. #63831C2×C3semidirect
30. C3×#805D3semidirect
31. C3×#805C2×C3semidirect
32. #63561C2×C3semidirect
105. #805C3×D3semidirect
233. C33#456semidirect
344. C32#4826semidirect
345. C32C3×#456semidirect
384. C3#10590401direct (N × H)
385. C3C3×#4826semidirect

Maximal subgroups

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

subgroupindex
C3×#642982
C3×#640782
C3×#638312
C32×#8052
C3×#635612
C3×#48263
#105904013