Finite Group Databasebeta

← C3×#10590564 #13157557

Subgroups — C3×#10590564 #13157557

Order 1728.

514 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 132 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×#64099C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3C32×#808C212yesmaximal
4C3×#63575C212yesmaximal
5C3×#63555C211yesmaximal, characteristic
6C3×#4856—32nomaximal
7#10590564C311yesmaximal, characteristic, direct product factor
8C2×C3×#627—21no—
9C33×C4⋊C4C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)—21no—
11C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
12C3×#3750—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C2×C32×C3⋊C8C2212yes—
16C3×#3724—21no—
17#64099C2×C311yescharacteristic
18C3×#890—32no—
19C3×#890—61no—
20C3×#888—32no—
21#63919C2×C311yescharacteristic
22C32×#106—32no—
23C3×#818—32no—
24C3×#815—32no—
25C3×#808—22no—
26C3×#808D312yes—
27C3×#808C2×C312yes—
28C3×#807—32no—
29#63575C2×C312yes—
30#63555C2×C311yescharacteristic
31C2×C33×C4—21no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—22no—
37C3×#627—42no—
38C32×C3⋊C8Q814yescomplemented
39C3×#501—91no—
40#4856—32no—
41C2×C3×Dic6—62no—
42C2×#627—21no—
43C2×C32×C8—32no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—22no—
46C32×C4⋊C4—32no—
47C32×C4⋊C4—41no—
48C32×C4⋊C4C2×D312yes—
49C32×C4⋊C4C22×C311yescharacteristic
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)C22×C311yescharacteristic
52#3750—21no—
53C4×C3⋊Dic3C22×C311yescharacteristic
54C3×C4⋊Dic3—22no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3—61no—
58C3×C4⋊Dic3—62no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C22×C312yes—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×C4×Dic3—32no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—22no—
67C2×C3×C3⋊C8C2×D312yes—
68C2×C3×C3⋊C8C22×C312yes—
69#3724—21no—
70C22×C33C2×D411yescharacteristic
71C33×C4—41no—
72C33×C4C2×Q812yes—
73C3×C3⋊Dic3—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—41no—
76C32×Dic3—42no—
77C3×C4⋊Q8—91no—
78C3×#125—91no—
79#890—32no—
80#890—61no—
81#888—32no—
82C3×#107—92no—
83C3×#106—32no—
84C3×#106—32no—
85C3×#106—62no—
86C3×#97—91no—
87#818—32no—
88#815—32no—
89#808C3×D312yes—
90#807—32no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—41no—
97C2×C32×C4—42no—
98C2×C32×C4—42no—
99C2×C32×C4—62no—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C23×C311yescharacteristic
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3C3×D411yescharacteristic
104C2×C3⋊Dic3C3×D411yescharacteristic
105C2×C3×Dic3—22no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—32no—
109C2×C3×Dic3—42no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C3×Dic6—124no—
115#627—42no—
116C32×C8—34no—
117C3×C3⋊C8—24no—
118C3×C3⋊C8Dic614yescomplemented
119C3×C3⋊C8C3×Q814yescomplemented
120#501—91noSylow
121C2×C33C4⋊Q811yescharacteristic
122C2×C33AGL(1,ℤ/8)11yescharacteristic
123C2×C33#13711yescharacteristic
124C2×C3×Q8—181no—
125C2×Dic6—62no—
126C2×C3×C8—32no—
127C2×C3×C8—32no—
128C2×C3×C8—62no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—32no—
132C3×C4⋊C4—32no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—62no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4D3211yescharacteristic
140C3×C4⋊C4C2×C3×D312yes—
141C3×C42—91no—
142C4⋊Dic3—32no—
143C4⋊Dic3—61no—
144C4⋊Dic3—62no—
145C4⋊Dic3—62no—
146C4⋊Dic3C2×C3×D312yes—
147Dic3⋊C4—62no—
148Dic3⋊C4—62no—
149Dic3⋊C4—62no—
150C4×Dic3—32no—
151C4×Dic3—61no—
152C2×C3⋊C8C2×C3×D312yes—
153C22×C32—21no—
154C22×C32—22no—
155C22×C32—41no—
156C22×C32D3×D412yes—
157C22×C32C2×C3×D411yescharacteristic
158C32×C4—22no—
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—42no—
162C32×C4—41no—
163C32×C4—42no—
164C32×C4—42no—
165C32×C4—82no—
166C32×C4—82no—
167C32×C4—122no—
168C32×C4C2×Dic612yes—
169C32×C4D3×Q812yes—
170C32×C4C2×C3×Q812yes—
171C3⋊Dic3—21no—
172C3⋊Dic3—21no—
173C3⋊Dic3—41no—
174C3×Dic3—42no—
175C3×Dic3—42no—
176C3×Dic3—62no—
177C3×Dic3—62no—
178C3×Dic3—82no—
179C3×Dic3—121no—
180C3×Dic3—121no—
181C3×Dic3—122no—
182C3×Dic3—122no—
183C4⋊Q8—91no—
184#125—91no—
185#107—92no—
186#106—32no—
187#97—91no—
188C33#50111yescharacteristic, complemented, Sylow
189C3×Q8—362no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—62no—
199C2×C3×C4—62no—
200C2×C3×C4—91no—
201C2×C3×C4—91no—
202C2×C3×C4—122no—
203C2×C3×C4—181no—
204C2×C3×C4C2×D3211yescharacteristic
205C2×C3×C4C22×C3×D312yes—
206C2×Dic3—22no—
207C2×Dic3—32no—
208C2×Dic3—32no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213Dic6—124no—
214C3×C8—34no—
215C3×C8—34no—
216C3×C8—64no—
217C3⋊C8C3×Dic614yescomplemented
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32#88812yes—
228C2×C32#91412yes—
229C2×C32#91512yes—
230C2×C32C3×C4⋊Q811yescharacteristic
231C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
232C2×C32C3×#13711yescharacteristic
233C2×Q8—181no—
234C2×C8—32no—
235C4⋊C4—32no—
236C4⋊C4—91no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4C3×D3211yescharacteristic
241C42—91no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—41no—
245C22×C3#381111yescharacteristic
246C22×C3C3×D3×D412yes—
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—42no—
251C3×C4—42no—
252C3×C4—41no—
253C3×C4—42no—
254C3×C4—82no—
255C3×C4—82no—
256C3×C4—122no—
257C3×C4—122no—
258C3×C4—181no—
259C3×C4—181no—
260C3×C4—242no—
261C3×C4—361no—
262C3×C4D3×Dic612yes—
263C3×C4C2×C3×Dic612yes—
264C3×C4C3×D3×Q812yes—
265Dic3—42no—
266Dic3—62no—
267Dic3—62no—
268Dic3—121no—
269Dic3—121no—
270Dic3—122no—
271Dic3—122no—
272C32—21no—
273C32—22no—
274C32—41no—
275C32#485612yescomplemented
276C32C3×#50111yescharacteristic, complemented
277Q8—362no—
278C2×C4—21no—
279C2×C4—62no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—181no—
283C2×C4C2×C3×D3211yescharacteristic
284C8—34no—
285C2×C3—21no—
286C2×C3—22no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3#6384011yescharacteristic
295C2×C3#6393111yescharacteristic
296C2×C3#6394111yescharacteristic
297C2×C3C3×#88812yes—
298C2×C3C3×#91412yes—
299C2×C3C3×#91512yes—
300C22C3×#381111yescharacteristic
301C4—41no—
302C4—122no—
303C4—181no—
304C4—181no—
305C4—361no—
306C4C3×D3×Dic612yes—
307C3—22no—
308C3—21no—
309C3—41no—
310C3#1059056411yescharacteristic, direct product factor
311C3C3×#485612yescomplemented
312C2C3×#6384011yescharacteristic
313C2C3×#6393111yescharacteristic
314C2C3×#6394111yescharacteristic
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×#501
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. #10590564C3direct (N × H)
38. C32×C3⋊C8Q8semidirect
118. C3×C3⋊C8Dic6semidirect
119. C3×C3⋊C8C3×Q8semidirect
188. C33#501semidirect
217. C3⋊C8C3×Dic6semidirect
275. C32#4856semidirect
276. C32C3×#501semidirect
310. C3#10590564direct (N × H)
311. C3C3×#4856semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
C3×#639192
C32×#8082
C3×#635752
C3×#635552
C3×#48563
#105905643