Finite Group Databasebeta

← C3×#10590416 #13157553

Subgroups — C3×#10590416 #13157553

Order 1728.

558 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 140 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×#64097C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3C32×#808C212yesmaximal
4C3×#63573C212yesmaximal
5C3×#63553C211yesmaximal, characteristic
6C3×#4855—32nomaximal
7#10590416C311yesmaximal, characteristic, direct product factor
8C2×C3×#634—21no—
9C2×C3×#627—21no—
10C33×C4⋊C4C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
12C3×#3750—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64097C2×C311yescharacteristic
18C3×#888—32no—
19C3×#888—32no—
20C3×#888—61no—
21#63919C2×C311yescharacteristic
22C32×#106—32no—
23C3×#817—32no—
24C3×C8×Dic3—32no—
25C3×#808—22no—
26C3×#808—32no—
27C3×#808D312yes—
28C3×#808C2×C312yes—
29#63573C2×C312yes—
30#63553C2×C311yescharacteristic
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C3×#634—42no—
37C2×C32×Dic3—22no—
38C3×#627—42no—
39C32×C3⋊C8Q814yescomplemented
40C3×#498—91no—
41#4855—32no—
42C2×#634—21no—
43C2×C3×Dic6—62no—
44C2×C3×Dic6—62no—
45C2×C3×Dic6—62no—
46C2×#627—21no—
47C2×C32×C8—32no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—22no—
50C32×C4⋊C4—32no—
51C32×C4⋊C4—41no—
52C32×C4⋊C4C2×D312yes—
53C32×C4⋊C4C22×C311yescharacteristic
54C4⋊(C3⋊Dic3)C22×C311yescharacteristic
55#3750—21no—
56C4×C3⋊Dic3C22×C311yescharacteristic
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4⋊Dic3C22×C312yes—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—32no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—22no—
68C2×C3×C3⋊C8C2×D312yes—
69C2×C3×C3⋊C8C22×C312yes—
70#3724—21no—
71C22×C33C2×D411yescharacteristic
72C33×C4—41no—
73C33×C4C2×Q812yes—
74C3×C3⋊Dic3—41no—
75C3×C3⋊Dic3SD1612yes—
76C3×C3⋊Dic3SD1612yes—
77C32×Dic3—42no—
78C3×C4⋊Q8—91no—
79C3×C4⋊Q8—91no—
80#888—32no—
81#888—32no—
82#888—61no—
83C3×#106—32no—
84C3×#106—32no—
85C3×#106—62no—
86C3×#106—92no—
87C3×C4×C8—91no—
88#817—32no—
89C8×Dic3—32no—
90#808—32no—
91#808C3×D312yes—
92C2×C32×C4—21no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4—41no—
98C2×C32×C4—42no—
99C2×C32×C4—42no—
100C2×C32×C4—62no—
101C2×C32×C4C22×D312yes—
102C2×C32×C4C23×C311yescharacteristic
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3C3×D411yescharacteristic
105C2×C3⋊Dic3C3×D411yescharacteristic
106#634—42no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—42no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C3×Dic6—124no—
117C3×Dic6—124no—
118C3×Dic6—124no—
119#627—42no—
120C32×C8—34no—
121C3×C3⋊C8—24no—
122C3×C3⋊C8Dic614yescomplemented
123C3×C3⋊C8C3×Q814yescomplemented
124#498—91noSylow
125C2×C33C4⋊Q811yescharacteristic
126C2×C33C2×SD1611yescharacteristic
127C2×C33C2×SD1611yescharacteristic
128C2×C3×Q8—181no—
129C2×C3×Q8—181no—
130C2×Dic6—62no—
131C2×Dic6—62no—
132C2×Dic6—62no—
133C2×C3×C8—32no—
134C2×C3×C8—32no—
135C2×C3×C8—62no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—32no—
139C3×C4⋊C4—32no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—62no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4D3211yescharacteristic
146C3×C4⋊C4C2×C3×D312yes—
147C3×C42—91no—
148C4⋊Dic3—32no—
149C4⋊Dic3—61no—
150C4⋊Dic3C2×C3×D312yes—
151Dic3⋊C4—62no—
152Dic3⋊C4—62no—
153Dic3⋊C4—62no—
154C4×Dic3—32no—
155C4×Dic3—61no—
156C2×C3⋊C8C2×C3×D312yes—
157C22×C32—21no—
158C22×C32—22no—
159C22×C32—41no—
160C22×C32D3×D412yes—
161C22×C32C2×C3×D411yescharacteristic
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—42no—
166C32×C4—41no—
167C32×C4—42no—
168C32×C4—42no—
169C32×C4—82no—
170C32×C4—82no—
171C32×C4—122no—
172C32×C4C2×Dic612yes—
173C32×C4D3×Q812yes—
174C32×C4C2×C3×Q812yes—
175C3⋊Dic3—41no—
176C3⋊Dic3C3×SD1612yes—
177C3⋊Dic3C3×SD1612yes—
178C3×Dic3—34no—
179C3×Dic3—34no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—82no—
185C3×Dic3—122no—
186C3×Dic3—122no—
187C4⋊Q8—91no—
188C4⋊Q8—91no—
189#106—32no—
190#106—92no—
191C4×C8—91no—
192C33#49811yescharacteristic, complemented, Sylow
193C3×Q8—362no—
194C3×Q8—362no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—122no—
208C2×C3×C4—181no—
209C2×C3×C4C2×D3211yescharacteristic
210C2×C3×C4C22×C3×D312yes—
211C2×Dic3—22no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218Dic6—124no—
219Dic6—124no—
220Dic6—124no—
221C3×C8—34no—
222C3×C8—34no—
223C3×C8—64no—
224C3⋊C8C3×Dic614yescomplemented
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32#88812yes—
235C2×C32D3×SD1612yes—
236C2×C32D3×SD1612yes—
237C2×C32C3×C4⋊Q811yescharacteristic
238C2×C32C2×C3×SD1611yescharacteristic
239C2×C32C2×C3×SD1611yescharacteristic
240C2×Q8—181no—
241C2×Q8—181no—
242C2×C8—32no—
243C4⋊C4—32no—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C4⋊C4—181no—
247C4⋊C4C3×D3211yescharacteristic
248C42—91no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—41no—
252C22×C3#381111yescharacteristic
253C22×C3C3×D3×D412yes—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—42no—
259C3×C4—41no—
260C3×C4—42no—
261C3×C4—82no—
262C3×C4—82no—
263C3×C4—92no—
264C3×C4—92no—
265C3×C4—122no—
266C3×C4—122no—
267C3×C4—242no—
268C3×C4—361no—
269C3×C4D3×Dic612yes—
270C3×C4C2×C3×Dic612yes—
271C3×C4C3×D3×Q812yes—
272Dic3—34no—
273Dic3—34no—
274Dic3—42no—
275Dic3—62no—
276Dic3—62no—
277Dic3—122no—
278Dic3—122no—
279C32—21no—
280C32—22no—
281C32—41no—
282C32#485512yescomplemented
283C32C3×#49811yescharacteristic, complemented
284Q8—362no—
285Q8—362no—
286C2×C4—21no—
287C2×C4—62no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—181no—
291C2×C4C2×C3×D3211yescharacteristic
292C8—34no—
293C2×C3—21no—
294C2×C3—22no—
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3#6384011yescharacteristic
303C2×C3#6393011yescharacteristic
304C2×C3#6394011yescharacteristic
305C2×C3C3×#88812yes—
306C2×C3C3×D3×SD1612yes—
307C2×C3C3×D3×SD1612yes—
308C22C3×#381111yescharacteristic
309C4—41no—
310C4—92no—
311C4—92no—
312C4—122no—
313C4—361no—
314C4C3×D3×Dic612yes—
315C3—22no—
316C3—21no—
317C3—41no—
318C3#1059041611yescharacteristic, direct product factor
319C3C3×#485512yescomplemented
320C2C3×#6384011yescharacteristic
321C2C3×#6393011yescharacteristic
322C2C3×#6394011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#498
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10590416C3direct (N × H)
39. C32×C3⋊C8Q8semidirect
122. C3×C3⋊C8Dic6semidirect
123. C3×C3⋊C8C3×Q8semidirect
192. C33#498semidirect
224. C3⋊C8C3×Dic6semidirect
282. C32#4855semidirect
283. C32C3×#498semidirect
318. C3#10590416direct (N × H)
319. C3C3×#4855semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
C3×#639192
C32×#8082
C3×#635732
C3×#635532
C3×#48553
#105904163