Finite Group Databasebeta

← C3×#10590390 #13157405

Subgroups — C3×#10590390 #13157405

Order 1728.

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

normal subgroups only

This group has 574 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×#63630C211yesmaximal, characteristic, complemented
4C32×#805C212yesmaximal, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6C3×#4823—32nomaximal
7#10590390C311yesmaximal, 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—
24#63630C2×C311yescharacteristic, complemented
25C3×#822—32no—
26C3×#805—22no—
27C3×#805D312yescomplemented
28C3×#805C2×C312yescomplemented
29C3×#803—32no—
30C3×#803—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)D412yes—
41C32×C3⋊C8—22no—
42C3×#444—91no—
43#4823—32no—
44C32×#41—32no—
45C3×#235—22no—
46C3×#235C2×D312yes—
47C3×#235C22×C312yes—
48C32×M16—32no—
49C32×C42—21no—
50C32×C42—22no—
51C32×C42—41no—
52C32×C42C2×D312yes—
53C32×C42C22×C311yescharacteristic
54C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
55C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
56C3×#216—32no—
57C3×#215—32no—
58C3×#214—32no—
59C3×#213—32no—
60C3×#208—22no—
61C3×#208—62no—
62C3×#208C2×D312yes—
63C3×#208C22×C312yes—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—61no—
69C3×ΓL(1,𝔽25)—62no—
70C3×D3×C8—62no—
71#3718C3×C411yescharacteristic, complemented
72#3715C3×C412yescomplemented
73#3713C3×C411yescharacteristic, complemented
74#3711—22no—
75D3×C3⋊C8—22no—
76C22×C33#4111yescharacteristic
77C2×C32×D3—22no—
78C33×C4—41no—
79C33×C4C22×C411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C32×Dic3—22no—
82C3×#135—91no—
83C3×#131—92no—
84#950—32no—
85#948—32no—
86#907—32no—
87C3×#108—91no—
88C3×#104—32no—
89C3×#104—32no—
90C3×#104—62no—
91C3×#97—91no—
92#822—32no—
93#805C3×D312yescomplemented
94#803—32no—
95#803—61no—
96C32×Q8—32no—
97C32×D4—32no—
98C32×D4—62no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—41no—
105C2×C32×C4—42no—
106C2×C32×C4—42no—
107C2×C32×C4—62no—
108C2×C32×C4C22×D312yes—
109C2×C32×C4C23×C311yescharacteristic
110C3×#67—22no—
111C3×#67—22no—
112C3×#67—42no—
113C3×D12—22no—
114C3×D12C4×D312yes—
115C3×D12C2×C3×C412yes—
116C3×C4×D3—22no—
117C3×C4×D3—22no—
118C3×C4×D3—42no—
119C3×Dic6—22no—
120C3×Dic6C4×D312yes—
121C3×Dic6C2×C3×C412yes—
122C32×C8—62no—
123C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
124C3⋊(C3⋊C8)C2×C3×C411yescharacteristic
125C3⋊(C3⋊C8)C3×D412yes—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—32no—
129C3×C3⋊C8—32no—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—32no—
132C3×C3⋊C8—42no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8—61no—
135C3×C3⋊C8—62no—
136#444—91noSylow
137C2×C33—21no—
138C2×C33C4×D411yescharacteristic
139C32×D3—42no—
140C3×#41—32no—
141C3×#41—32no—
142C3×#41—62no—
143#235C2×C3×D312yes—
144C3×Q16—91no—
145C3×SD16—92no—
146C3×D8—91no—
147C3×M16—32no—
148C3×M16—32no—
149C3×M16—62no—
150C3×M16—182no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C2×C3×C8—182no—
154C3×C42—21no—
155C3×C42—22no—
156C3×C42—41no—
157C3×C42D3211yescharacteristic
158C3×C42C2×C3×D312yes—
159#216—32no—
160#215—32no—
161#214—32no—
162#213—32no—
163#208—62no—
164#208C2×C3×D312yes—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—61no—
169C2×C3⋊C8—62no—
170ΓL(1,𝔽25)—62no—
171D3×C8—62no—
172C22×C32—21no—
173C22×C32—22no—
174C22×C32—41no—
175C22×C32—62no—
176C22×C32#23712yes—
177C22×C32C3×#4111yescharacteristic
178C2×C3×D3—22no—
179C2×C3×D3—22no—
180C2×C3×D3—42no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4—42no—
189C32×C4—42no—
190C32×C4—44no—
191C32×C4—44no—
192C32×C4—62no—
193C32×C4C2×C4×D312yes—
194C32×C4D3×D412yes—
195C32×C4C22×C3×C411yescharacteristic
196C32×C4C2×C3×D411yescharacteristic
197C3×Dic3—22no—
198C3×Dic3—22no—
199C3×Dic3—42no—
200#135—91no—
201#131—92no—
202#108—91no—
203#104—32no—
204#97—91no—
205C33#44411yescharacteristic, complemented, Sylow
206C3×Q8—32no—
207C3×Q8—32no—
208C3×Q8—62no—
209C3×D4—32no—
210C3×D4—32no—
211C3×D4—62no—
212C3×D4—62no—
213C3×D4—62no—
214C3×D4—122no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—21no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—122no—
226C2×C3×C4C2×D3211yescharacteristic
227C2×C3×C4C22×C3×D312yes—
228#67—22no—
229D12C3×C4×D312yes—
230C4×D3—22no—
231Dic6C3×C4×D312yes—
232C3×C8—62no—
233C3×C8—62no—
234C3×C8—91no—
235C3×C8—91no—
236C3×C8—92no—
237C3×C8—122no—
238C3⋊C8—22no—
239C3⋊C8—32no—
240C3⋊C8—32no—
241C3⋊C8—32no—
242C3⋊C8—32no—
243C3⋊C8—61no—
244C3⋊C8—61no—
245C3⋊C8—62no—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—22no—
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—41no—
252C2×C32—42no—
253C2×C32—42no—
254C2×C32—122no—
255C2×C32#67⋊C412yes—
256C2×C32C3×C4×D411yescharacteristic
257C3×D3—42no—
258C3×D3—42no—
259C3×D3—82no—
260#41—32no—
261Q16—91no—
262SD16—92no—
263D8—91no—
264M16—32no—
265M16—182no—
266C2×C8—91no—
267C2×C8—91no—
268C2×C8—182no—
269C42C3×D3211yescharacteristic
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—62no—
274C22×C3—62no—
275C22×C3—122no—
276C22×C3#380511yescharacteristic
277C22×C3C3×#23712yes—
278C2×D3—22no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—42no—
284C3×C4—42no—
285C3×C4—41no—
286C3×C4—44no—
287C3×C4—44no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—62no—
291C3×C4—62no—
292C3×C4—122no—
293C3×C4C4×D3211yescharacteristic
294C3×C4#380211yescharacteristic
295C3×C4C2×C3×C4×D312yes—
296C3×C4C3×D3×D412yes—
297Dic3—22no—
298C32—22no—
299C32—21no—
300C32—41no—
301C32#482312yescomplemented
302C32C3×#44411yescharacteristic, complemented
303Q8—32no—
304D4—32no—
305D4—62no—
306C2×C4—21no—
307C2×C4—62no—
308C2×C4C2×C3×D3211yescharacteristic
309C8—62no—
310C8—91no—
311C8—91no—
312C8—92no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—21no—
316C2×C3—22no—
317C2×C3—21no—
318C2×C3—42no—
319C2×C3—42no—
320C2×C3—41no—
321C2×C3—122no—
322C2×C3—122no—
323C2×C3—242no—
324C2×C3#6390211yescharacteristic
325C2×C3C3×#67⋊C412yes—
326D3—42no—
327C22—62no—
328C22C3×#380511yescharacteristic
329C4—41no—
330C4—62no—
331C4C3×C4×D3211yescharacteristic
332C4C3×#380211yescharacteristic
333C3—21no—
334C3—22no—
335C3—41no—
336C3#1059039011yescharacteristic, direct product factor
337C3C3×#482312yescomplemented
338C2—21no—
339C2—122no—
340C2C3×#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×#444
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×#63630C2semidirect
4. C32×#805C2semidirect
5. C3×#63578C2semidirect
7. #10590390C3direct (N × H)
13. C3×#3718C4semidirect
14. C3×#3715C4semidirect
15. C3×#3713C4semidirect
22. #63815C2×C3semidirect
24. #63630C2×C3semidirect
27. C3×#805D3semidirect
28. C3×#805C2×C3semidirect
31. #63578C2×C3semidirect
71. #3718C3×C4semidirect
72. #3715C3×C4semidirect
73. #3713C3×C4semidirect
93. #805C3×D3semidirect
205. C33#444semidirect
301. C32#4823semidirect
302. C32C3×#444semidirect
336. C3#10590390direct (N × H)
337. C3C3×#4823semidirect

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×#636302
C32×#8052
C3×#635782
C3×#48233
#105903903