Finite Group Databasebeta

← C3×#10590398 #13157414

Subgroups — C3×#10590398 #13157414

Order 1728.

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

normal subgroups only

This group has 802 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×#64308C211yesmaximal, characteristic
2C3×#64084C211yesmaximal, characteristic, complemented
3C3×#63831C212yesmaximal, complemented
4C32×#805C212yesmaximal, complemented
5C3×#63559C211yesmaximal, characteristic, complemented
6C3×#4825—32nomaximal
7#10590398C311yesmaximal, characteristic, direct product factor
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic, complemented
10C32×#235C2212yes—
11C3×#3811—22no—
12C3×D3×D12—22no—
13C3×#3799—22no—
14C33×C42C2211yescharacteristic, complemented
15C3×#3752—22no—
16C32×#208C2212yescomplemented
17C3×#3719—22no—
18C3×#3716—22no—
19C3×#1009—32no—
20#64308C2×C311yescharacteristic
21#64084C2×C311yescharacteristic, complemented
22C3×#951—32no—
23C3×#909—32no—
24C3×#875—32no—
25C3×#875—61no—
26#63831C2×C312yescomplemented
27C32×#104—32no—
28C3×#823—32no—
29C3×#805—22no—
30C3×#805D312yescomplemented
31C3×#805C2×C312yescomplemented
32#63559C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3D412yes—
34C2×C3×D32—22no—
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—21no—
38C3×C4⋊(C3⋊D3)—22no—
39C3×#634—22no—
40C32×#67—22no—
41C32×D12D412yes—
42C32×C4×D3—22no—
43C32×Dic6D412yes—
44C3×#626—24no—
45C3×#624—22no—
46C32×C3⋊C8—22no—
47C3×#455—91no—
48#4825—32no—
49C32×#41—32no—
50C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
51C2×#634C22×C311yescharacteristic, complemented
52C3×#239—62no—
53C3×D3×D4—62no—
54C3×D3×D4—64no—
55C3×#235—22no—
56C3×#235—62no—
57C3×#235C2×D312yes—
58C3×#235C22×C312yes—
59C2×C3×D12—32no—
60C2×C3×D12—62no—
61C2×C3×D12—61no—
62C2×C3×Dic6—32no—
63C2×C3×Dic6—61no—
64#3811—22no—
65D3×D12—22no—
66#3799—22no—
67C32×M16—32no—
68C32×C42—21no—
69C32×C42—22no—
70C32×C42—41no—
71C32×C42C2×D312yescomplemented
72C32×C42C22×C311yescharacteristic, complemented
73#3752—22no—
74C3×#216—62no—
75C3×#214—62no—
76C3×(C2×D3)⋊C4—64no—
77C3×(C2×D3)⋊C4—64no—
78C3×#208—22no—
79C3×#208C2×D312yescomplemented
80C3×#208C22×C312yescomplemented
81C3×Dic12—62no—
82C3×#204—62no—
83#3719—22no—
84#3716—22no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—42no—
88C2×C32×D3—22no—
89C3×D32—82no—
90C33×C4—41no—
91C33×C4C2×D412yes—
92C3×C3⋊Dic3—41no—
93C32×Dic3—22no—
94C3×H(5,𝔽2)—91no—
95#1009—32no—
96C3×#137—92no—
97C3×#124—91no—
98#951—32no—
99#909—32no—
100#875—32no—
101#875—61no—
102C3×#104—32no—
103C3×#104—32no—
104C3×#104—62no—
105C3×#100—91no—
106#823—32no—
107#805C3×D312yescomplemented
108C22×C3⋊D3C3×D412yes—
109C22×C3×D3—34no—
110C22×C3×D3—62no—
111C22×C3×D3—64no—
112C2×D32—22no—
113C32×Q8—32no—
114C32×D4—32no—
115C32×D4—62no—
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—41no—
122C2×C32×C4—42no—
123C2×C32×C4—42no—
124C2×C32×C4—62no—
125C2×C32×C4C22×D312yes—
126C2×C32×C4C23×C311yescharacteristic
127C2×C3⋊Dic3—21no—
128C4⋊(C3⋊D3)—22no—
129#634—22no—
130C3×#67—22no—
131C3×#67—22no—
132C3×#67—42no—
133C3×#67—64no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C3×D12—22no—
137C3×D12—64no—
138C3×D12—62no—
139C3×D12—62no—
140C3×D12—122no—
141C3×D12D1212yes—
142C3×D12C3×D412yes—
143C3×C4×D3—22no—
144C3×C4×D3—22no—
145C3×C4×D3—42no—
146C3×C4×D3—64no—
147C3×Dic6—22no—
148C3×Dic6—62no—
149C3×Dic6—62no—
150C3×Dic6—122no—
151C3×Dic6D1212yes—
152C3×Dic6C3×D412yes—
153#626—24no—
154#624—22no—
155C32×C8—62no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—22no—
158C3×C3⋊C8—42no—
159#455—91noSylow
160C2×C33—21no—
161C2×C33#12011yescharacteristic
162C3×C3⋊D3—42no—
163C32×D3—42no—
164C3×#41—32no—
165C3×#41—32no—
166C3×#41—62no—
167C3×#41—182no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—182no—
171C2×C3×D4—182no—
172#239—62no—
173D3×D4—62no—
174D3×D4—64no—
175#235—62no—
176#235C2×C3×D312yes—
177C2×D12—32no—
178C2×D12—61no—
179C2×D12—62no—
180C2×Dic6—32no—
181C2×Dic6—61no—
182C3×Q16—182no—
183C3×SD16—182no—
184C3×M16—32no—
185C3×M16—32no—
186C3×M16—62no—
187C3×C22⋊C4—182no—
188C3×C42—21no—
189C3×C42—22no—
190C3×C42—41no—
191C3×C42D3211yescharacteristic, complemented
192C3×C42C2×C3×D312yescomplemented
193#216—62no—
194#214—62no—
195(C2×D3)⋊C4—64no—
196(C2×D3)⋊C4—64no—
197#208C2×C3×D312yescomplemented
198Dic12—62no—
199#204—62no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32—41no—
203C22×C32—62no—
204C22×C32D3×D412yes—
205C22×C32C2×C3×D411yescharacteristic
206C2×C3⋊D3—22no—
207C2×C3⋊D3—42no—
208C2×C3×D3—22no—
209C2×C3×D3—22no—
210C2×C3×D3—42no—
211C2×C3×D3—64no—
212C2×C3×D3—122no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C2×C3×D3—244no—
216D32—82no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—22no—
220C32×C4—42no—
221C32×C4—41no—
222C32×C4—42no—
223C32×C4—42no—
224C32×C4—44no—
225C32×C4—44no—
226C32×C4—62no—
227C32×C4C2×D1212yes—
228C32×C4D3×D412yes—
229C32×C4C2×C3×D412yes—
230C3⋊Dic3—41no—
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—42no—
234C3×Dic3—122no—
235C3×Dic3—122no—
236H(5,𝔽2)—91no—
237#137—92no—
238#124—91no—
239#104—32no—
240#100—91no—
241C33#45511yescharacteristic, complemented, Sylow
242C23×C3—92no—
243C23×C3—182no—
244C22×D3—34no—
245C22×D3—62no—
246C22×D3—64no—
247C3×Q8—32no—
248C3×Q8—32no—
249C3×Q8—62no—
250C3×Q8—182no—
251C3×D4—32no—
252C3×D4—32no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—122no—
257C3×D4—184no—
258C3×D4—182no—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—122no—
270C2×C3×C4—182no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3211yescharacteristic
273C2×C3×C4C22×C3×D312yes—
274#67—22no—
275#67—64no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278D12—64no—
279D12—62no—
280D12—62no—
281D12—122no—
282D12C3×D1212yes—
283C4×D3—22no—
284C4×D3—64no—
285Dic6—62no—
286Dic6—62no—
287Dic6—122no—
288Dic6C3×D1212yes—
289C3×C8—62no—
290C3×C8—62no—
291C3×C8—122no—
292C3⋊C8—22no—
293C2×C32—21no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32—41no—
299C2×C32—42no—
300C2×C32—42no—
301C2×C32—122no—
302C2×C32#88212yes—
303C2×C32C3×#12011yescharacteristic
304C3⋊D3—42no—
305C3×D3—42no—
306C3×D3—42no—
307C3×D3—82no—
308C3×D3—124no—
309C3×D3—242no—
310#41—32no—
311#41—182no—
312C2×Q8—91no—
313C2×D4—91no—
314C2×D4—182no—
315C2×D4—182no—
316Q16—182no—
317SD16—182no—
318M16—32no—
319C22⋊C4—182no—
320C42C3×D3211yescharacteristic, complemented
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—62no—
325C22×C3—62no—
326C22×C3—122no—
327C22×C3—182no—
328C22×C3—362no—
329C22×C3—722no—
330C22×C3#381111yescharacteristic
331C22×C3C3×D3×D412yes—
332C2×D3—22no—
333C2×D3—64no—
334C2×D3—122no—
335C2×D3—124no—
336C2×D3—124no—
337C2×D3—244no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—41no—
344C3×C4—44no—
345C3×C4—44no—
346C3×C4—42no—
347C3×C4—62no—
348C3×C4—62no—
349C3×C4—122no—
350C3×C4—361no—
351C3×C4D3×D1212yes—
352C3×C4C2×C3×D1212yes—
353C3×C4C3×D3×D412yes—
354Dic3—22no—
355Dic3—122no—
356Dic3—122no—
357C32—22no—
358C32—21no—
359C32—41no—
360C32#482512yescomplemented
361C32C3×#45511yescharacteristic, complemented
362C23—92no—
363C23—182no—
364Q8—32no—
365Q8—182no—
366D4—32no—
367D4—62no—
368D4—184no—
369D4—182no—
370C2×C4—21no—
371C2×C4—62no—
372C2×C4—182no—
373C2×C4—181no—
374C2×C4C2×C3×D3211yescharacteristic
375C8—62no—
376C2×C3—22no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—42no—
382C2×C3—42no—
383C2×C3—41no—
384C2×C3—122no—
385C2×C3—122no—
386C2×C3—242no—
387C2×C3—362no—
388C2×C3#6392311yescharacteristic
389C2×C3C3×#88212yes—
390D3—42no—
391D3—124no—
392D3—242no—
393C22—62no—
394C22—182no—
395C22—362no—
396C22—722no—
397C22C3×#381111yescharacteristic
398C4—41no—
399C4—62no—
400C4—361no—
401C4C3×D3×D1212yes—
402C3—21no—
403C3—22no—
404C3—41no—
405C3#1059039811yescharacteristic, direct product factor
406C3C3×#482512yescomplemented
407C2—21no—
408C2—122no—
409C2—362no—
410C2C3×#6392311yescharacteristic
subgroupG/H
centerC2×C3#63923
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33#120
prime radicalC3×#64308C2
hypercenterC3×C42D32
hypocenterC32C3×#455
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×#64084C2semidirect
3. C3×#63831C2semidirect
4. C32×#805C2semidirect
5. C3×#63559C2semidirect
7. #10590398C3direct (N × H)
9. C2×C3×#634C22semidirect
14. C33×C42C22semidirect
16. C32×#208C22semidirect
21. #64084C2×C3semidirect
26. #63831C2×C3semidirect
30. C3×#805D3semidirect
31. C3×#805C2×C3semidirect
32. #63559C2×C3semidirect
51. C2×#634C22×C3semidirect
71. C32×C42C2×D3semidirect
72. C32×C42C22×C3semidirect
79. C3×#208C2×D3semidirect
80. C3×#208C22×C3semidirect
107. #805C3×D3semidirect
191. C3×C42D32semidirect
192. C3×C42C2×C3×D3semidirect
197. #208C2×C3×D3semidirect
241. C33#455semidirect
320. C42C3×D32semidirect
360. C32#4825semidirect
361. C32C3×#455semidirect
405. C3#10590398direct (N × H)
406. C3C3×#4825semidirect

Maximal subgroups

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

subgroupindex
C3×#643082
C3×#640842
C3×#638312
C32×#8052
C3×#635592
C3×#48253
#105903983