Finite Group Databasebeta

← C3×#10590381 #13157394

Subgroups — C3×#10590381 #13157394

Order 1728.

642 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C32×C4×Dic6C212yesmaximal
2C2×C3×#3718C211yesmaximal, characteristic
3C3×#63630C211yesmaximal, characteristic
4C3×#63574C211yesmaximal, characteristic
5C3×#63565C212yesmaximal
6C3×#5030—32nomaximal
7#10590381C311yesmaximal, characteristic, direct product factor
8C2×C32×Dic6C2212yes—
9C33×C42C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C32×Dic3⋊C4—22no—
14C32×C4×Dic3—22no—
15C3×#3718C414yescomplemented
16C32×C4×Q8—32no—
17C2×C3×#216—32no—
18C3×C4×Dic6—22no—
19C3×C4×Dic6D312yes—
20C3×C4×Dic6C2×C312yes—
21C2×#3718C2×C311yescharacteristic
22#63630C2×C311yescharacteristic
23C3×#835—32no—
24C3×#810—32no—
25C3×#808—32no—
26C3×#803—32no—
27C3×#803—61no—
28#63574C2×C311yescharacteristic
29#63565C2×C312yes—
30C2×C33×C4D412yes—
31C2×C33×C4C2311yescharacteristic
32C2×C32×Dic3—22no—
33C2×C32×Dic3—22no—
34C32×Dic6—22no—
35C32×Dic6C2×C414yes—
36C3×C3⋊(C3⋊C8)—21no—
37C3×C3⋊(C3⋊C8)C2×C412yes—
38C3×#441—91no—
39#5030—32no—
40C2×C32×Q8—32no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6C2×D312yes—
43C2×C3×Dic6C22×C312yes—
44C32×C4⋊C4—32no—
45C32×C4⋊C4—62no—
46C32×C42—21no—
47C32×C42—22no—
48C32×C42—41no—
49C32×C42—62no—
50C32×C42C2×D312yes—
51C32×C42C22×C311yescharacteristic
52C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
53C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
54C3×#216—38no—
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3C2×D312yes—
57C3×C4⋊Dic3C22×C312yes—
58C3×Dic3⋊C4—24no—
59C3×Dic3⋊C4—22no—
60C3×Dic3⋊C4—22no—
61C3×C4×Dic3—24no—
62C3×C4×Dic3—22no—
63C3×C4×Dic3—22no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—61no—
67C2×C3×C3⋊C8—61no—
68#3718C3×C414yescomplemented
69C22×C33C2×D411yescharacteristic
70C33×C4—22no—
71C33×C4C22×C411yescharacteristic
72C33×C4#4111yescharacteristic
73C32×Dic3—24no—
74C32×Dic3—42no—
75C2×C3×Q16—91no—
76C3×C4×Q8—32no—
77C3×C4×Q8—32no—
78C3×C4×Q8—62no—
79C2×#216—32no—
80C4×Dic6C3×D312yes—
81C3×#106—91no—
82C3×#103—92no—
83C3×#97—91no—
84#835—32no—
85#810—32no—
86#808—32no—
87#803—32no—
88#803—61no—
89C32×Q8—34no—
90C32×Q8—62no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4—41no—
98C2×C32×C4—62no—
99C2×C32×C4—62no—
100C2×C32×C4#6712yes—
101C2×C32×C4#6712yes—
102C2×C32×C4C3×D412yes—
103C2×C32×C4C22×D312yes—
104C2×C32×C4C23×C311yescharacteristic
105C2×C3×Dic3—24no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—24no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C3×Dic6—24no—
112C3×Dic6—22no—
113C3×Dic6—22no—
114C3×Dic6—24no—
115C3×Dic6C4×D314yes—
116C3×Dic6C2×C3×C414yes—
117C3⋊(C3⋊C8)—21no—
118C3⋊(C3⋊C8)C2×C3×C412yes—
119C3×C3⋊C8—34no—
120C3×C3⋊C8—62no—
121C3×C3⋊C8—62no—
122C3×C3⋊C8—62no—
123#441—91noSylow
124C2×C33C4×D411yescharacteristic
125C2×C33#13712yes—
126C2×C3×Q8—32no—
127C2×C3×Q8—32no—
128C2×C3×Q8—62no—
129C2×Dic6C2×C3×D312yes—
130C3×Q16—94no—
131C2×C3×C8—91no—
132C2×C3×C8—91no—
133C3×C4⋊C4—32no—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—62no—
136C3×C4⋊C4—64no—
137C3×C4⋊C4—62no—
138C3×C4⋊C4—62no—
139C3×C42—21no—
140C3×C42—22no—
141C3×C42—41no—
142C3×C42—64no—
143C3×C42—62no—
144C3×C42—62no—
145C3×C42D3211yescharacteristic
146C3×C42C2×C3×D312yes—
147#216—38no—
148C4⋊Dic3C2×C3×D312yes—
149Dic3⋊C4—22no—
150C4×Dic3—22no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—32no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—41no—
158C22×C32C2×#6712yes—
159C22×C32C2×C3×D411yescharacteristic
160C32×C4—22no—
161C32×C4—24no—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—41no—
169C32×C4—41no—
170C32×C4—42no—
171C32×C4—42no—
172C32×C4—44no—
173C32×C4—64no—
174C32×C4—122no—
175C32×C4C2×C4×D312yes—
176C32×C4#23512yes—
177C32×C4C22×C3×C411yescharacteristic
178C32×C4C3×#4111yescharacteristic
179C3×Dic3—28no—
180C3×Dic3—24no—
181C3×Dic3—24no—
182C3×Dic3—44no—
183C3×Dic3—42no—
184C3×Dic3—42no—
185C2×Q16—91no—
186C4×Q8—32no—
187#106—91no—
188#103—92no—
189#97—91no—
190C33#44111yescharacteristic, complemented, Sylow
191C3×Q8—34no—
192C3×Q8—34no—
193C3×Q8—64no—
194C3×Q8—64no—
195C3×Q8—62no—
196C3×Q8—62no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—64no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—64no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4#62512yes—
211C2×C3×C4C3×#6712yes—
212C2×C3×C4C3×#6712yes—
213C2×C3×C4C2×D3211yescharacteristic
214C2×C3×C4C22×C3×D312yes—
215C2×Dic3—22no—
216C2×Dic3—22no—
217Dic6—22no—
218Dic6C3×C4×D314yes—
219C3×C8—92no—
220C3×C8—181no—
221C3⋊C8—34no—
222C3⋊C8—62no—
223C3⋊C8—62no—
224C3⋊C8—62no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32C4×#6712yes—
233C2×C32#94212yes—
234C2×C32#95112yes—
235C2×C32C3×C4×D411yescharacteristic
236C2×C32C3×#13712yes—
237C2×Q8—32no—
238Q16—94no—
239C2×C8—91no—
240C2×C8—91no—
241C4⋊C4—32no—
242C4⋊C4—62no—
243C42—62no—
244C42C3×D3211yescharacteristic
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—41no—
248C22×C3C2×#62511yescharacteristic
249C22×C3C2×C3×#6712yes—
250C3×C4—24no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—21no—
255C3×C4—22no—
256C3×C4—21no—
257C3×C4—22no—
258C3×C4—42no—
259C3×C4—42no—
260C3×C4—44no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—68no—
264C3×C4—64no—
265C3×C4—64no—
266C3×C4—124no—
267C3×C4—122no—
268C3×C4—122no—
269C3×C4#379811yescharacteristic
270C3×C4C4×D3211yescharacteristic
271C3×C4C2×C3×C4×D312yes—
272C3×C4C3×#23512yes—
273Dic3—24no—
274Dic3—42no—
275C32—22no—
276C32—21no—
277C32—41no—
278C32#503012yescomplemented
279C32C3×#44111yescharacteristic, complemented
280Q8—34no—
281Q8—62no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4C3×#62512yes—
285C2×C4C2×C3×D3211yescharacteristic
286C8—92no—
287C8—181no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—22no—
291C2×C3—22no—
292C2×C3—21no—
293C2×C3—42no—
294C2×C3—41no—
295C2×C3#6382712yes—
296C2×C3C4×#62511yescharacteristic
297C2×C3C3×C4×#6712yes—
298C2×C3C3×#94212yes—
299C2×C3C3×#95112yes—
300C22C2×C3×#62511yescharacteristic
301C4—22no—
302C4—64no—
303C4—122no—
304C4C3×#379811yescharacteristic
305C4C3×C4×D3211yescharacteristic
306C3—22no—
307C3—21no—
308C3—41no—
309C3#1059038111yescharacteristic, direct product factor
310C3C3×#503012yescomplemented
311C2C3×#6382712yes—
312C2C3×C4×#62511yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#441
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
7. #10590381C3direct (N × H)
15. C3×#3718C4semidirect
68. #3718C3×C4semidirect
190. C33#441semidirect
278. C32#5030semidirect
279. C32C3×#441semidirect
309. C3#10590381direct (N × H)
310. C3C3×#5030semidirect

Maximal subgroups

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

subgroupindex
C32×C4×Dic62
C2×C3×#37182
C3×#636302
C3×#635742
C3×#635652
C3×#50303
#105903813