Finite Group Databasebeta

← C3×#10593742 #13173213

Subgroups — C3×#10593742 #13173213

Order 1728.

500 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C3×#3792C211yesmaximal, characteristic, complemented
2C2×C3×#3788C211yesmaximal, characteristic, complemented
3C3×#64090C211yesmaximal, characteristic
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5C3×#63779C211yesmaximal, characteristic, complemented
6C3×#63778C211yesmaximal, characteristic, complemented
7C3×C4×#622C211yesmaximal, characteristic, complemented
8#10593742C311yesmaximal, characteristic, direct product factor
9C2×C3×#638C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C3×#3792—22no—
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17C3×#3788—22no—
18C33×C22⋊C4C411yescharacteristic
19C3×#3752C411yescharacteristic
20C3×#3750C411yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×#3792C2×C311yescharacteristic, complemented
23C2×#3788C2×C311yescharacteristic, complemented
24#64090C2×C311yescharacteristic
25C3×#67⋊C4—62no—
26C22⋊#622C2×C311yescharacteristic, complemented
27#63779C2×C311yescharacteristic, complemented
28#63778C2×C311yescharacteristic, complemented
29C4×#622C2×C311yescharacteristic, complemented
30C23×C33C2×C411yescharacteristic
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C3×#638—22no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C2×C3×C3⋊Dic3C2×C411yescharacteristic
39C2×C3×C3⋊Dic3C2311yescharacteristic
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3—21no—
42C3×#622—21no—
43C3×#622—21no—
44C3×#622—21no—
45C3×#622D412yescomplemented
46C3×#436—91nomaximal
47C2×#638C3×C411yescharacteristic
48C22×C3⋊Dic3C22×C311yescharacteristic
49C2×C4×C3⋊D3C22×C311yescharacteristic
50C2×C3×#67—62no—
51C22×C3×Dic3—62no—
52C2×C3×C4×D3—62no—
53#3792—22no—
54C2×#622C22×C311yescharacteristic
55C2×#622C22×C311yescharacteristic
56C2×#622C22×C311yescharacteristic
57C2×#622C22×C311yescharacteristic
58#3788—22no—
59C32×C22⋊C4—22no—
60C32×C22⋊C4—42no—
61C32×C22⋊C4C3×C411yescharacteristic
62#3752C3×C411yescharacteristic
63#3750C3×C411yescharacteristic
64C4×C3⋊Dic3C22×C311yescharacteristic
65C3×(C2×D3)⋊C4—62no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—62no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33—21no—
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—21no—
75C33×C4—21no—
76C33×C4—21no—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3M1612yescomplemented
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C2×C3×M16—91no—
82C2×C3×M16—91no—
83C3×C4×D4—91no—
84#67⋊C4—62no—
85C3×C4⋊C8—91no—
86C3×C22⋊C8—91no—
87C3×C22⋊C8—91no—
88C3×C4×C8—91no—
89C23×C32—22no—
90C23×C32—42no—
91C23×C32C2×C3×C411yescharacteristic
92C22×C3⋊D3C2×C3×C411yescharacteristic
93C22×C3×D3—62no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4—42no—
98C2×C32×C4C2×C3×C411yescharacteristic
99C2×C32×C4C2×C3×C411yescharacteristic
100#638—22no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3C2×C3×C411yescharacteristic
104C2×C3⋊Dic3C2×C3×C411yescharacteristic
105C2×C3⋊Dic3C23×C311yescharacteristic
106C4×C3⋊D3—21no—
107C4×C3⋊D3—21no—
108C3×#67—68no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—122no—
113C2×C3×Dic3—122no—
114C3×C4×D3—122no—
115C3×C4×D3—122no—
116#622—21no—
117#622—21no—
118#622—21no—
119#622C3×D412yescomplemented
120#436—91noSylow
121C2×C33—41no—
122C2×C33C4×D411yescharacteristic
123C2×C33C2×M1611yescharacteristic
124C2×C33#13111yescharacteristic
125C3×C3⋊D3—41no—
126C2×C3×D4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C2×#67—62no—
130C22×Dic3—62no—
131C2×C4×D3—62no—
132C3×M16—182no—
133C3×M16—182no—
134C2×C3×C8—91no—
135C2×C3×C8—91no—
136C2×C3×C8—91no—
137C2×C3×C8—91no—
138C3×C4⋊C4—91no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—42no—
141C3×C22⋊C4—91no—
142C3×C22⋊C4#15711yescharacteristic
143C3×C42—91no—
144(C2×D3)⋊C4—62no—
145Dic3⋊C4—62no—
146C4×Dic3—62no—
147C22×C32—21no—
148C22×C32—24no—
149C22×C32—22no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—44no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32—42no—
156C22×C32—82no—
157C22×C32—82no—
158C22×C32C22×C3×C411yescharacteristic
159C2×C3⋊D3—21no—
160C2×C3⋊D3—21no—
161C2×C3⋊D3—21no—
162C2×C3×D3—64no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4—82no—
170C32×C4—82no—
171C3⋊Dic3—21no—
172C3⋊Dic3C3×M1612yescomplemented
173C3⋊Dic3C2×C3×D411yescharacteristic
174C3⋊Dic3C3×#4111yescharacteristic
175C3×Dic3—64no—
176C3×Dic3—62no—
177C3×Dic3—62no—
178C3×Dic3—122no—
179C2×M16—91no—
180C2×M16—91no—
181C4×D4—91no—
182C4⋊C8—91no—
183C22⋊C8—91no—
184C22⋊C8—91no—
185C4×C8—91no—
186C33#43611yescharacteristic, complemented, Sylow
187C23×C3—22no—
188C23×C3—42no—
189C23×C3—91no—
190C23×C3C2×#15711yescharacteristic
191C22×D3—62no—
192C3×D4—182no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×C3×C4C2×#15711yescharacteristic
205C2×C3×C4C2×#15711yescharacteristic
206#67—68no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—122no—
211C2×Dic3—122no—
212C4×D3—122no—
213C4×D3—122no—
214C3×C8—92no—
215C3×C8—181no—
216C3×C8—181no—
217C3×C8—181no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—42no—
222C2×C32—42no—
223C2×C32—42no—
224C2×C32—41no—
225C2×C32—44no—
226C2×C32—84no—
227C2×C32C3×C4×D411yescharacteristic
228C2×C32C2×C3×M1611yescharacteristic
229C2×C32C3×#13111yescharacteristic
230C3⋊D3—41no—
231C3×D3—124no—
232C2×D4—91no—
233C22×C4—91no—
234C22×C4—91no—
235M16—182no—
236M16—182no—
237C2×C8—91no—
238C2×C8—91no—
239C2×C8—91no—
240C2×C8—91no—
241C4⋊C4—91no—
242C22⋊C4—91no—
243C22⋊C4C3×#15711yescharacteristic
244C42—91no—
245C22×C3—24no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—22no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—42no—
254C22×C3—82no—
255C22×C3—82no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3—181no—
259C22×C3C22×#15711yescharacteristic
260C2×D3—64no—
261C2×D3—122no—
262C2×D3—122no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—42no—
266C3×C4—42no—
267C3×C4—82no—
268C3×C4—82no—
269C3×C4—92no—
270C3×C4—91no—
271C3×C4—91no—
272C3×C4—181no—
273Dic3—62no—
274Dic3—62no—
275Dic3—64no—
276Dic3—122no—
277C32—22no—
278C32—42no—
279C32C3×#43611yescharacteristic, complemented
280C23—91no—
281C23C2×C3×#15711yescharacteristic
282D4—182no—
283C2×C4—91no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4C2×C3×#15711yescharacteristic
291C2×C4C2×C3×#15711yescharacteristic
292C8—92no—
293C8—181no—
294C8—181no—
295C8—181no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—22no—
299C2×C3—44no—
300C2×C3—41no—
301C2×C3—42no—
302C2×C3—42no—
303C2×C3—42no—
304C2×C3—84no—
305C2×C3—361no—
306C2×C3#6428311yescharacteristic
307C2×C3#6428711yescharacteristic
308C2×C3D4×#15711yescharacteristic
309D3—124no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—181no—
314C22—181no—
315C22—181no—
316C22C22×C3×#15711yescharacteristic
317C4—21no—
318C4—21no—
319C4—92no—
320C4—91no—
321C4—91no—
322C4—181no—
323C3—22no—
324C3—42no—
325C3#1059374211yescharacteristic, direct product factor
326C2—41no—
327C2—361no—
328C2C3×#6428311yescharacteristic
329C2C3×#6428711yescharacteristic
330C2C3×D4×#15711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC22×C32C22×C3×C4
FrattiniC22C22×C3×#157
FittingC33×C22⋊C4C4
socleC22×C33C22×C4
prime radicalC2×C3×#638C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C22⋊C4#157
hypocenterC32C3×#436
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3792C2semidirect
2. C2×C3×#3788C2semidirect
4. C3×C22⋊#622C2semidirect
5. C3×#63779C2semidirect
6. C3×#63778C2semidirect
7. C3×C4×#622C2semidirect
8. #10593742C3direct (N × H)
22. C2×#3792C2×C3semidirect
23. C2×#3788C2×C3semidirect
26. C22⋊#622C2×C3semidirect
27. #63779C2×C3semidirect
28. #63778C2×C3semidirect
29. C4×#622C2×C3semidirect
45. C3×#622D4semidirect
78. C3×C3⋊Dic3M16semidirect
119. #622C3×D4semidirect
172. C3⋊Dic3C3×M16semidirect
186. C33#436semidirect
279. C32C3×#436semidirect
325. C3#10593742direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C2×C3×#37882
C3×#640902
C3×C22⋊#6222
C3×#637792
C3×#637782
C3×C4×#6222
#105937423
C3×#4369