Finite Group Databasebeta

← C3×#10593751 #13173221

Subgroups — C3×#10593751 #13173221

Order 1728.

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

normal subgroups only

This group has 488 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×#3788C211yesmaximal, characteristic, complemented
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C3×#64102C211yesmaximal, characteristic
4C3×#63779C211yesmaximal, characteristic, complemented
5C3×#63778C211yesmaximal, characteristic, complemented
6C3×#63778C211yesmaximal, characteristic, complemented
7C3×#63777C211yesmaximal, characteristic, complemented
8#10593751C311yesmaximal, characteristic, direct product factor
9C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16C3×#3788—22no—
17C3×#3787—22no—
18C33×C4⋊C4C411yescharacteristic
19C3×#3752C411yescharacteristic
20C3×#3752C411yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×#3788C2×C311yescharacteristic, complemented
23C2×#3787C2×C311yescharacteristic, complemented
24#64102C2×C311yescharacteristic
25C3×#893—62no—
26#63779C2×C311yescharacteristic, complemented
27#63778C2×C311yescharacteristic, complemented
28#63778C2×C311yescharacteristic, complemented
29#63777C2×C311yescharacteristic, complemented
30C22×C3×C3⋊D3C2×C411yescharacteristic
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4×C3⋊D3—21no—
39C3×C4×C3⋊D3—21no—
40C3×C4×C3⋊D3—21no—
41C3×C4×C3⋊D3—21no—
42C3×#622—21no—
43C3×#622—21no—
44C3×#622—21no—
45C3×#622D412yescomplemented
46C3×#435—91nomaximal
47C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
48C2×C4×C3⋊D3C22×C311yescharacteristic
49C2×C4×C3⋊D3C22×C311yescharacteristic
50C2×C3×D12—62no—
51C2×C3×C4×D3—62no—
52C2×C3×C4×D3—62no—
53C2×#622C22×C311yescharacteristic
54C2×#622C22×C311yescharacteristic
55C2×#622C22×C311yescharacteristic
56C2×#622C22×C311yescharacteristic
57#3788—22no—
58#3787—22no—
59C32×C4⋊C4—22no—
60C32×C4⋊C4—42no—
61C32×C4⋊C4C3×C411yescharacteristic
62#3752C3×C411yescharacteristic
63#3752C3×C411yescharacteristic
64C4×C3⋊Dic3C22×C311yescharacteristic
65C3×(C2×D3)⋊C4—62no—
66C3×(C2×D3)⋊C4—62no—
67C3×C4×Dic3—62no—
68C22×C33C22×C411yescharacteristic
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3—21no—
71C2×C3×C3⋊D3—21no—
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3M1612yescomplemented
75C33×C4—21no—
76C33×C4—21no—
77C33×C4—21no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C2×C3×M16—91no—
82C22×C3×C8—91no—
83C3×C4×D4—91no—
84#893—62no—
85C3×C4⋊C8—91no—
86C3×C22⋊C8—91no—
87C3×C22⋊C8—91no—
88C3×#97—91no—
89C22×C3⋊D3C2×C3×C411yescharacteristic
90C22×C3⋊D3C2×C3×C411yescharacteristic
91C22×C3×D3—62no—
92C22×C3×D3—62no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4—42no—
98C2×C32×C4—42no—
99C2×C32×C4C2×C3×C411yescharacteristic
100C2×C32×C4C2×C3×C411yescharacteristic
101C2×C32×C4C2×C3×C411yescharacteristic
102C2×C3⋊Dic3C2×C3×C411yescharacteristic
103C2×C3⋊Dic3C23×C311yescharacteristic
104C4⋊(C3⋊D3)—22no—
105C4×C3⋊D3—21no—
106C4×C3⋊D3—21no—
107C4×C3⋊D3—21no—
108C4×C3⋊D3—21no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C3×D12—68no—
112C3×C4×D3—122no—
113C3×C4×D3—122no—
114C3×C4×D3—122no—
115C3×C4×D3—122no—
116#622—21no—
117#622—21no—
118#622—21no—
119#622C3×D412yescomplemented
120#435—91noSylow
121C2×C33C4×D411yescharacteristic
122C2×C33C2×M1611yescharacteristic
123C2×C33#13111yescharacteristic
124C3×C3⋊D3—22no—
125C3×C3⋊D3—41no—
126C2×C3×D4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C2×D12—62no—
130C2×C4×D3—62no—
131C2×C4×D3—62no—
132C3×M16—182no—
133C2×C3×C8—91no—
134C2×C3×C8—91no—
135C2×C3×C8—91no—
136C2×C3×C8—91no—
137C2×C3×C8—182no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—42no—
140C3×C4⋊C4#15711yescharacteristic
141C3×C22⋊C4—91no—
142C3×C22⋊C4—91no—
143C3×C42—91no—
144(C2×D3)⋊C4—62no—
145(C2×D3)⋊C4—62no—
146C4×Dic3—62no—
147C22×C32—22no—
148C22×C32—42no—
149C22×C32C22×C3×C411yescharacteristic
150C2×C3⋊D3—21no—
151C2×C3⋊D3—21no—
152C2×C3⋊D3—21no—
153C2×C3⋊D3—21no—
154C2×C3⋊D3—21no—
155C2×C3⋊D3C3×M1612yescomplemented
156C2×C3×D3—64no—
157C2×C3×D3—64no—
158C2×C3×D3—122no—
159C2×C3×D3—122no—
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C32×C4—21no—
163C32×C4—24no—
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—44no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4—82no—
170C32×C4—82no—
171C3⋊Dic3—21no—
172C3⋊Dic3C2×C3×D411yescharacteristic
173C3⋊Dic3C3×#4111yescharacteristic
174C3×Dic3—62no—
175C3×Dic3—62no—
176C3×Dic3—122no—
177C2×M16—91no—
178C22×C8—91no—
179C4×D4—91no—
180C4⋊C8—91no—
181C22⋊C8—91no—
182C22⋊C8—91no—
183#97—91no—
184C33#43511yescharacteristic, complemented, Sylow
185C23×C3—91no—
186C23×C3—91no—
187C22×D3—62no—
188C22×D3—62no—
189C3×D4—182no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—181no—
199C2×C3×C4—181no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4C2×#15711yescharacteristic
203C2×C3×C4C2×#15711yescharacteristic
204C2×C3×C4C2×#15711yescharacteristic
205C2×Dic3—62no—
206C2×Dic3—62no—
207D12—68no—
208C4×D3—122no—
209C4×D3—122no—
210C4×D3—122no—
211C4×D3—122no—
212C3×C8—92no—
213C3×C8—181no—
214C3×C8—181no—
215C3×C8—181no—
216C2×C32—22no—
217C2×C32—22no—
218C2×C32—22no—
219C2×C32—42no—
220C2×C32—42no—
221C2×C32—42no—
222C2×C32C3×C4×D411yescharacteristic
223C2×C32C2×C3×M1611yescharacteristic
224C2×C32C3×#13111yescharacteristic
225C3⋊D3—22no—
226C3⋊D3—41no—
227C3×D3—124no—
228C3×D3—124no—
229C2×D4—91no—
230C22×C4—91no—
231C22×C4—91no—
232M16—182no—
233C2×C8—91no—
234C2×C8—91no—
235C2×C8—91no—
236C2×C8—91no—
237C2×C8—182no—
238C4⋊C4C3×#15711yescharacteristic
239C22⋊C4—91no—
240C22⋊C4—91no—
241C42—91no—
242C22×C3—22no—
243C22×C3—42no—
244C22×C3—92no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3—181no—
248C22×C3—181no—
249C22×C3—181no—
250C22×C3C22×#15711yescharacteristic
251C2×D3—64no—
252C2×D3—64no—
253C2×D3—122no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3—122no—
257C3×C4—24no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—42no—
262C3×C4—42no—
263C3×C4—44no—
264C3×C4—82no—
265C3×C4—82no—
266C3×C4—91no—
267C3×C4—91no—
268C3×C4—181no—
269Dic3—62no—
270Dic3—62no—
271Dic3—122no—
272C32—22no—
273C32—42no—
274C32C3×#43511yescharacteristic, complemented
275C23—91no—
276C23—91no—
277D4—182no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4C2×C3×#15711yescharacteristic
285C2×C4C2×C3×#15711yescharacteristic
286C2×C4C2×C3×#15711yescharacteristic
287C8—92no—
288C8—181no—
289C8—181no—
290C8—181no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—22no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—182no—
298C2×C3—361no—
299C2×C3#6428311yescharacteristic
300C2×C3D4×#15711yescharacteristic
301C2×C3#6428911yescharacteristic
302D3—124no—
303D3—124no—
304C22—92no—
305C22—181no—
306C22—181no—
307C22—181no—
308C22—181no—
309C22—181no—
310C22C22×C3×#15711yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—91no—
315C4—91no—
316C4—181no—
317C3—22no—
318C3—42no—
319C3#1059375111yescharacteristic, direct product factor
320C2—182no—
321C2—361no—
322C2C3×#6428311yescharacteristic
323C2C3×D4×#15711yescharacteristic
324C2C3×#6428911yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC22×C32C22×C3×C4
FrattiniC22C22×C3×#157
FittingC33×C4⋊C4C4
socleC22×C33C22×C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×#435
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3788C2semidirect
2. C2×C3×#3787C2semidirect
4. C3×#63779C2semidirect
5. C3×#63778C2semidirect
6. C3×#63778C2semidirect
7. C3×#63777C2semidirect
8. #10593751C3direct (N × H)
22. C2×#3788C2×C3semidirect
23. C2×#3787C2×C3semidirect
26. #63779C2×C3semidirect
27. #63778C2×C3semidirect
28. #63778C2×C3semidirect
29. #63777C2×C3semidirect
45. C3×#622D4semidirect
74. C2×C3×C3⋊D3M16semidirect
119. #622C3×D4semidirect
155. C2×C3⋊D3C3×M16semidirect
184. C33#435semidirect
274. C32C3×#435semidirect
319. C3#10593751direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C2×C3×#37872
C3×#641022
C3×#637792
C3×#637782
C3×#637772
#105937513
C3×#4359