Finite Group Databasebeta

← C3×#10590389 #13157404

Subgroups — C3×#10590389 #13157404

Order 1728.

586 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 128 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×#63822C211yesmaximal, characteristic
2C3×#63815C212yesmaximal, complemented
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C32×#805C212yesmaximal, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6C3×#4824—32nomaximal
7#10590389C311yesmaximal, characteristic, direct product factor
8C32×#235C2212yes—
9C33×C42C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×#208C2212yes—
13C3×#3718C411yescharacteristic, complemented
14C3×#3715C412yescomplemented
15C3×#3713C411yescharacteristic, complemented
16C3×#3711—22no—
17C3×D3×C3⋊C8—22no—
18C3×#950—32no—
19C3×#948—32no—
20C3×#907—32no—
21#63822C2×C311yescharacteristic
22#63815C2×C312yescomplemented
23C32×#104—32no—
24C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
25C3×#822—32no—
26C3×#805—22no—
27C3×#805D312yescomplemented
28C3×#805C2×C312yescomplemented
29C3×C4×C3⋊C8—32no—
30C3×C4×C3⋊C8—61no—
31#63578C2×C311yescharacteristic, complemented
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C32×#67—22no—
35C32×D12C2×C412yes—
36C32×C4×D3—22no—
37C32×Dic6C2×C412yes—
38C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
39C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C3×C3⋊(C3⋊C8)D411yescharacteristic
41C3×C3⋊(C3⋊C8)D411yescharacteristic
42C32×C3⋊C8—22no—
43C3×#443—91no—
44#4824—32no—
45C32×#41—32no—
46C3×#235—22no—
47C3×#235C2×D312yes—
48C3×#235C22×C312yes—
49C32×M16—32no—
50C32×C42—21no—
51C32×C42—22no—
52C32×C42—41no—
53C32×C42C2×D312yes—
54C32×C42C22×C311yescharacteristic
55C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
56C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
57C3×#216—32no—
58C3×#215—32no—
59C3×#214—32no—
60C3×#213—32no—
61C3×#208—22no—
62C3×#208—62no—
63C3×#208C2×D312yes—
64C3×#208C22×C312yes—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—62no—
68C2×C3×C3⋊C8—61no—
69C2×C3×C3⋊C8—61no—
70C3×ΓL(1,𝔽25)—62no—
71C3×D3×C8—62no—
72#3718C3×C411yescharacteristic, complemented
73#3715C3×C412yescomplemented
74#3713C3×C411yescharacteristic, complemented
75#3711—22no—
76D3×C3⋊C8—22no—
77C22×C33#4111yescharacteristic
78C2×C32×D3—22no—
79C33×C4—22no—
80C33×C4C22×C411yescharacteristic
81C33×C4C2×D411yescharacteristic
82C32×Dic3—22no—
83C3×#135—91no—
84C3×#131—92no—
85#950—32no—
86#948—32no—
87#907—32no—
88C3×#108—91no—
89C3×#104—32no—
90C3×#104—32no—
91C3×#104—62no—
92C3×C4×C8—91no—
93#822—32no—
94#805C3×D312yescomplemented
95C4×C3⋊C8—32no—
96C4×C3⋊C8—61no—
97C32×Q8—32no—
98C32×D4—32no—
99C32×D4—62no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—41no—
106C2×C32×C4—42no—
107C2×C32×C4—42no—
108C2×C32×C4—62no—
109C2×C32×C4C22×D312yes—
110C2×C32×C4C23×C311yescharacteristic
111C3×#67—22no—
112C3×#67—22no—
113C3×#67—42no—
114C3×D12—22no—
115C3×D12C4×D312yes—
116C3×D12C2×C3×C412yes—
117C3×C4×D3—22no—
118C3×C4×D3—22no—
119C3×C4×D3—42no—
120C3×Dic6—22no—
121C3×Dic6C4×D312yes—
122C3×Dic6C2×C3×C412yes—
123C32×C8—62no—
124C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
125C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
126C3⋊(C3⋊C8)C3×D411yescharacteristic
127C3⋊(C3⋊C8)C3×D411yescharacteristic
128C3×C3⋊C8—22no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—32no—
132C3×C3⋊C8—32no—
133C3×C3⋊C8—32no—
134C3×C3⋊C8—42no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139#443—91noSylow
140C2×C33—21no—
141C2×C33C4×D411yescharacteristic
142C32×D3—42no—
143C3×#41—32no—
144C3×#41—32no—
145C3×#41—62no—
146#235C2×C3×D312yes—
147C3×Q16—91no—
148C3×SD16—92no—
149C3×D8—91no—
150C3×M16—32no—
151C3×M16—32no—
152C3×M16—62no—
153C3×M16—182no—
154C2×C3×C8—91no—
155C2×C3×C8—91no—
156C2×C3×C8—182no—
157C3×C42—21no—
158C3×C42—22no—
159C3×C42—41no—
160C3×C42D3211yescharacteristic
161C3×C42C2×C3×D312yes—
162#216—32no—
163#215—32no—
164#214—32no—
165#213—32no—
166#208—62no—
167#208C2×C3×D312yes—
168C2×C3⋊C8—32no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—61no—
171C2×C3⋊C8—61no—
172C2×C3⋊C8—62no—
173ΓL(1,𝔽25)—62no—
174D3×C8—62no—
175C22×C32—21no—
176C22×C32—22no—
177C22×C32—41no—
178C22×C32—62no—
179C22×C32#23712yes—
180C22×C32C3×#4111yescharacteristic
181C2×C3×D3—22no—
182C2×C3×D3—22no—
183C2×C3×D3—42no—
184C32×C4—22no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4—44no—
194C32×C4—44no—
195C32×C4—62no—
196C32×C4C2×C4×D312yes—
197C32×C4D3×D412yes—
198C32×C4C22×C3×C411yescharacteristic
199C32×C4C2×C3×D411yescharacteristic
200C3×Dic3—22no—
201C3×Dic3—22no—
202C3×Dic3—42no—
203#135—91no—
204#131—92no—
205#108—91no—
206#104—32no—
207C4×C8—91no—
208C33#44311yescharacteristic, complemented, Sylow
209C3×Q8—32no—
210C3×Q8—32no—
211C3×Q8—62no—
212C3×D4—32no—
213C3×D4—32no—
214C3×D4—62no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—122no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—122no—
229C2×C3×C4C2×D3211yescharacteristic
230C2×C3×C4C22×C3×D312yes—
231#67—22no—
232D12C3×C4×D312yes—
233C4×D3—22no—
234Dic6C3×C4×D312yes—
235C3×C8—62no—
236C3×C8—62no—
237C3×C8—91no—
238C3×C8—91no—
239C3×C8—91no—
240C3×C8—91no—
241C3×C8—122no—
242C3⋊C8—22no—
243C3⋊C8—32no—
244C3⋊C8—32no—
245C3⋊C8—32no—
246C3⋊C8—32no—
247C3⋊C8—61no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C3⋊C8—61no—
251C2×C32—21no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32—42no—
259C2×C32—122no—
260C2×C32#67⋊C412yes—
261C2×C32C3×C4×D411yescharacteristic
262C3×D3—42no—
263C3×D3—42no—
264C3×D3—82no—
265#41—32no—
266Q16—91no—
267SD16—92no—
268D8—91no—
269M16—32no—
270M16—182no—
271C2×C8—91no—
272C2×C8—91no—
273C2×C8—182no—
274C42C3×D3211yescharacteristic
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—41no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3—122no—
281C22×C3#380511yescharacteristic
282C22×C3C3×#23712yes—
283C2×D3—22no—
284C3×C4—22no—
285C3×C4—24no—
286C3×C4—24no—
287C3×C4—21no—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—22no—
291C3×C4—44no—
292C3×C4—44no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—62no—
296C3×C4—62no—
297C3×C4—122no—
298C3×C4C4×D3211yescharacteristic
299C3×C4#380211yescharacteristic
300C3×C4C2×C3×C4×D312yes—
301C3×C4C3×D3×D412yes—
302Dic3—22no—
303C32—22no—
304C32—21no—
305C32—41no—
306C32#482412yescomplemented
307C32C3×#44311yescharacteristic, complemented
308Q8—32no—
309D4—32no—
310D4—62no—
311C2×C4—21no—
312C2×C4—62no—
313C2×C4C2×C3×D3211yescharacteristic
314C8—62no—
315C8—91no—
316C8—91no—
317C8—91no—
318C8—91no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—42no—
325C2×C3—42no—
326C2×C3—41no—
327C2×C3—122no—
328C2×C3—122no—
329C2×C3—242no—
330C2×C3#6390211yescharacteristic
331C2×C3C3×#67⋊C412yes—
332D3—42no—
333C22—62no—
334C22C3×#380511yescharacteristic
335C4—22no—
336C4—62no—
337C4C3×C4×D3211yescharacteristic
338C4C3×#380211yescharacteristic
339C3—21no—
340C3—22no—
341C3—41no—
342C3#1059038911yescharacteristic, direct product factor
343C3C3×#482412yescomplemented
344C2—21no—
345C2—122no—
346C2C3×#6390211yescharacteristic
subgroupG/H
centerC3×C4#3802
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33C4×D4
prime radicalC3×#63822C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#443
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×#63815C2semidirect
3. C3×C4×C3⋊(C3⋊C8)C2semidirect
4. C32×#805C2semidirect
5. C3×#63578C2semidirect
7. #10590389C3direct (N × H)
13. C3×#3718C4semidirect
14. C3×#3715C4semidirect
15. C3×#3713C4semidirect
22. #63815C2×C3semidirect
24. C4×C3⋊(C3⋊C8)C2×C3semidirect
27. C3×#805D3semidirect
28. C3×#805C2×C3semidirect
31. #63578C2×C3semidirect
72. #3718C3×C4semidirect
73. #3715C3×C4semidirect
74. #3713C3×C4semidirect
94. #805C3×D3semidirect
208. C33#443semidirect
306. C32#4824semidirect
307. C32C3×#443semidirect
342. C3#10590389direct (N × H)
343. C3C3×#4824semidirect

Maximal subgroups

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

subgroupindex
C3×#638222
C3×#638152
C3×C4×C3⋊(C3⋊C8)2
C32×#8052
C3×#635782
C3×#48243
#105903893