Finite Group Databasebeta

← C3×#10592567 #13162905

Subgroups — C3×#10592567 #13162905

Order 1728.

720 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 170 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×#64106C211yesmaximal, characteristic
2C3×#64101C211yesmaximal, characteristic
3C33×#106C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic
5C3×#63643C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7C3×#63634C211yesmaximal, characteristic, complemented
8#10592567C311yesmaximal, characteristic, direct product factor
9C3×#4868—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752—21no—
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#64106C2×C311yescharacteristic
22#64101C2×C311yescharacteristic
23C3×#897—34no—
24C3×#892—34no—
25C32×#106—24no—
26C32×#106D314yescomplemented
27C32×#106C2×C311yescharacteristic, complemented
28#63647C2×C311yescharacteristic
29#63643C2×C311yescharacteristic, complemented
30#63637C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×(C2×D3)⋊C8—34no—
33C3×#816—34no—
34C3×#810—34no—
35C3×#807—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3—21no—
44C3×C4×C3⋊D3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C33×C8—21no—
48C3×C3⋊(C3⋊C8)—21no—
49C3×#485—91no—
50#4868—34no—
51C2×C4×C3⋊D3C22×C311yescharacteristic
52C2×#634C22×C311yescharacteristic
53C2×C3×C4×D3—34no—
54C2×C3×Dic6—34no—
55C2×C32×C8—24no—
56C2×C32×C8C2×D314yes—
57C2×C32×C8C22×C311yescharacteristic
58C32×C4⋊C4—24no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C22×C311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64#3752—21no—
65#3752—21no—
66C4⋊(C3⋊Dic3)—21no—
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68C4×C3⋊Dic3—21no—
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×(C2×D3)⋊C4—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—64no—
74C3×C4×Dic3—64no—
75C2×C3×C3⋊C8—34no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C33×C4—41no—
81C33×C4—41no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—41no—
86C3×C3⋊Dic3—41no—
87C3×C22⋊Q8—91no—
88C3×#117—91no—
89#897—34no—
90#892—34no—
91C3×#107—91no—
92C3×#106—24no—
93C3×#106C3×D314yescomplemented
94C3×#106C3⋊D311yescharacteristic, complemented
95C3×#103—91no—
96C3×#103—91no—
97C3×C22⋊C8—91no—
98(C2×D3)⋊C8—34no—
99#816—34no—
100#810—34no—
101#807—34no—
102C22×C3⋊D3C3×D411yescharacteristic
103C22×C3×D3—34no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—44no—
110C2×C32×C4—44no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C23×C311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3C3×D411yescharacteristic
116C4×C3⋊D3—21no—
117C4×C3⋊D3—21no—
118#634—21no—
119#634—21no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C3×C4×D3—64no—
124C3×C4×D3—64no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127C32×C8—28no—
128C32×C8—24no—
129C32×C8—21no—
130C3⋊(C3⋊C8)—21no—
131C3×C3⋊C8—64no—
132#485—91noSylow
133C2×C33#12311yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—41no—
137C2×C3×Q8—91no—
138C22×C3×C4—91no—
139C2×C4×D3—34no—
140C2×Dic6—34no—
141C2×C3×C8—24no—
142C2×C3×C8—91no—
143C2×C3×C8C2×C3×D314yes—
144C2×C3×C8C2×C3⋊D311yescharacteristic
145C3×C4⋊C4—24no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4C2×C3×D314yes—
150C3×C4⋊C4C2×C3×D314yes—
151C3×C4⋊C4C2×C3⋊D311yescharacteristic
152C3×C4⋊C4C2×C3⋊D311yescharacteristic
153C3×C22⋊C4—181no—
154C3×C22⋊C4—181no—
155C3×C42—181no—
156(C2×D3)⋊C4—64no—
157(C2×D3)⋊C4—64no—
158C4⋊Dic3—34no—
159C4⋊Dic3—64no—
160C4×Dic3—64no—
161C2×C3⋊C8—34no—
162C22×C32—24no—
163C22×C32D3×D414yes—
164C22×C32C2×C3×D411yescharacteristic
165C2×C3⋊D3—21no—
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3×D3—64no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C32×C4—24no—
172C32×C4—24no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4—44no—
176C32×C4—44no—
177C32×C4—84no—
178C32×C4—84no—
179C32×C4#23514yes—
180C32×C4#23914yes—
181C32×C4C3×#4111yescharacteristic
182C32×C4C3×#4111yescharacteristic
183C3⋊Dic3—21no—
184C3⋊Dic3—41no—
185C3⋊Dic3—41no—
186C3×Dic3—64no—
187C3×Dic3—124no—
188C3×Dic3—124no—
189C22⋊Q8—91no—
190#117—91no—
191#107—91no—
192#106C3×C3⋊D311yescharacteristic, complemented
193#103—91no—
194#103—91no—
195C22⋊C8—91no—
196C33#48511yescharacteristic, complemented, Sylow
197C23×C3—91no—
198C22×D3—34no—
199C3×Q8—181no—
200C3×Q8—181no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4—91no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4C22×C3×D314yes—
214C2×C3×C4C22×C3⋊D311yescharacteristic
215C2×Dic3—34no—
216C2×Dic3—64no—
217C2×Dic3—64no—
218C4×D3—64no—
219C4×D3—64no—
220Dic6—64no—
221Dic6—64no—
222C3×C8—28no—
223C3×C8—24no—
224C3×C8—21no—
225C3×C8—181no—
226C3⋊C8—64no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#89414yes—
231C2×C32#91514yes—
232C2×C32#91614yes—
233C2×C32C3×#12311yescharacteristic
234C2×C32C3×#13511yescharacteristic
235C2×C32C3×#13711yescharacteristic
236C3⋊D3—41no—
237C3×D3—124no—
238C2×Q8—91no—
239C22×C4—91no—
240C2×C8—91no—
241C2×C8C2×C3×C3⋊D311yescharacteristic
242C4⋊C4—91no—
243C4⋊C4—181no—
244C4⋊C4C2×C3×C3⋊D311yescharacteristic
245C4⋊C4C2×C3×C3⋊D311yescharacteristic
246C22⋊C4—181no—
247C22⋊C4—181no—
248C42—181no—
249C22×C3—24no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3C3×D3×D414yes—
254C22×C3D4×C3⋊D311yescharacteristic
255C2×D3—64no—
256C2×D3—64no—
257C2×D3—64no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—44no—
261C3×C4—44no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—84no—
265C3×C4—84no—
266C3×C4—181no—
267C3×C4—361no—
268C3×C4—361no—
269C3×C4C3×#23514yes—
270C3×C4C3×#23914yes—
271C3×C4#382811yescharacteristic
272C3×C4#383211yescharacteristic
273Dic3—64no—
274Dic3—124no—
275Dic3—124no—
276C32—24no—
277C32#486814yescomplemented
278C32C3×#48511yescharacteristic, complemented
279C23—91no—
280Q8—181no—
281Q8—181no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—91no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4C22×C3×C3⋊D311yescharacteristic
290C8—21no—
291C8—181no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—361no—
296C2×C3C3×#89414yes—
297C2×C3C3×#91514yes—
298C2×C3C3×#91614yes—
299C2×C3#6410311yescharacteristic
300C2×C3#6412411yescharacteristic
301C2×C3#6412511yescharacteristic
302D3—124no—
303C22—181no—
304C22—181no—
305C22—181no—
306C22C3×D4×C3⋊D311yescharacteristic
307C4—41no—
308C4—41no—
309C4—181no—
310C4—361no—
311C4—361no—
312C4C3×#382811yescharacteristic
313C4C3×#383211yescharacteristic
314C3—24no—
315C3C3×#486814yescomplemented
316C3#1059256711yescharacteristic, direct product factor
317C2—361no—
318C2C3×#6410311yescharacteristic
319C2C3×#6412411yescharacteristic
320C2C3×#6412511yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
3. C33×#106C2semidirect
5. C3×#63643C2semidirect
6. C3×#63637C2semidirect
7. C3×#63634C2semidirect
8. #10592567C3direct (N × H)
26. C32×#106D3semidirect
27. C32×#106C2×C3semidirect
29. #63643C2×C3semidirect
30. #63637C2×C3semidirect
31. #63634C2×C3semidirect
93. C3×#106C3×D3semidirect
94. C3×#106C3⋊D3semidirect
192. #106C3×C3⋊D3semidirect
196. C33#485semidirect
277. C32#4868semidirect
278. C32C3×#485semidirect
315. C3C3×#4868semidirect
316. C3#10592567direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641062
C3×#641012
C33×#1062
C3×#636472
C3×#636432
C3×#636372
C3×#636342
#105925673
C3×#48683