Finite Group Databasebeta

← C3×#10593569 #13167642

Subgroups — C3×#10593569 #13167642

Order 1728.

634 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C23×C3×#157C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3C2×C3×C22⋊#157C211yesmaximal, characteristic
4C2×C3×#3777C212yesmaximal, complemented
5C3×#63747C212yesmaximal, complemented
6#10593569C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×#157—22no—
9C22×C3×#157—24no—
10C22×C3×#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C22×C3×#157C2212yes—
13C22×C3×#157C2212yes—
14C3×C22⋊#157—22no—
15C3×C22⋊#157C414yescomplemented
16C3×#3777—24no—
17C23×#157C2×C311yescharacteristic
18C2×C22⋊#157C2×C311yescharacteristic
19C2×C22⋊#157C2×C311yescharacteristic
20C2×#3777C2×C312yescomplemented
21#63747C2×C312yescomplemented
22C23×C33Q811yescharacteristic
23C22×C3×C3⋊D3—22no—
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3C2×C412yes—
26C22×C3×C3⋊D3D412yes—
27C22×C3×C3⋊D3Q811yescharacteristic
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3×#157—28no—
30C2×C3×#157—22no—
31C2×C3×#157—22no—
32C2×C3×#157—22no—
33C2×C3×#157—22no—
34C2×C3×#157—22no—
35C2×C3×#157—21no—
36C2×C3×#157—21no—
37C2×C3×#157—22no—
38C2×C3×#157C2×C414yescomplemented
39C2×C3×#157D414yes—
40C3×#385—91nomaximal
41C23×C3⋊D3C22×C311yescharacteristic
42C23×C3×D3—121no—
43C22×#157—22no—
44C22×#157—24no—
45C22×#157C22×C311yescharacteristic
46C22×#157C22×C311yescharacteristic
47C22×#157C22×C312yes—
48C22×#157C22×C312yes—
49C22⋊#157—22no—
50C22⋊#157C3×C414yescomplemented
51#3777—24no—
52C22×C33—22no—
53C22×C33C4⋊C412yescomplemented
54C22×C33C2×Q811yescharacteristic
55C2×C3×C3⋊D3—22no—
56C2×C3×C3⋊D3—24no—
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3C4⋊C412yescomplemented
60C2×C3×C3⋊D3C22×C411yescharacteristic
61C2×C3×C3⋊D3C2×D412yes—
62C2×C3×C3⋊D3C2×D411yescharacteristic
63C2×C3×C3⋊D3#4112yes—
64C3×#157—24no—
65C3×#157—24no—
66C3×#157—41no—
67C3×#157—41no—
68C23×C3×C4—91no—
69C2×C3×C4⋊C4—92no—
70C2×C3×C22⋊C4—91no—
71C2×C3×C22⋊C4—91no—
72C3×#95—92no—
73C23×C32—41no—
74C23×C32—81no—
75C23×C32C3×Q811yescharacteristic
76C22×C3⋊D3—22no—
77C22×C3⋊D3—22no—
78C22×C3⋊D3C2×C3×C412yes—
79C22×C3⋊D3C3×D412yes—
80C22×C3⋊D3C3×Q811yescharacteristic
81C22×C3⋊D3C23×C311yescharacteristic
82C22×C3×D3—122no—
83C22×C3×D3—122no—
84C22×C3×D3—121no—
85C22×C3×D3—121no—
86C22×C3×D3—124no—
87C22×C3×D3—124no—
88C2×#157—28no—
89C2×#157—22no—
90C2×#157—22no—
91C2×#157—22no—
92C2×#157—22no—
93C2×#157—22no—
94C2×#157—21no—
95C2×#157—21no—
96C2×#157—22no—
97C2×#157C2×C3×C414yescomplemented
98C2×#157C3×D414yes—
99#385—91noSylow
100C2×C33—22no—
101C2×C33C2×C4⋊C411yescharacteristic
102C2×C33C22⋊Q812yes—
103C3×C3⋊D3—22no—
104C3×C3⋊D3C4×D412yes—
105C3×C3⋊D3#12011yescharacteristic
106C3×C3⋊D3#12311yescharacteristic
107C24×C3—91no—
108C23×D3—121no—
109C22×C3×C4—92no—
110C22×C3×C4—92no—
111C22×C3×C4—91no—
112C22×C3×C4—91no—
113C22×C3×C4—184no—
114C22×C3×C4—182no—
115C3×C4⋊C4—184no—
116C3×C22⋊C4—94no—
117C3×C22⋊C4—182no—
118C22×C32—22no—
119C22×C32—44no—
120C22×C32—41no—
121C22×C32—42no—
122C22×C32—84no—
123C22×C32—82no—
124C22×C32—81no—
125C22×C32C3×C4⋊C412yescomplemented
126C22×C32C2×C3×Q811yescharacteristic
127C2×C3⋊D3—22no—
128C2×C3⋊D3—24no—
129C2×C3⋊D3—22no—
130C2×C3⋊D3—22no—
131C2×C3⋊D3C3×C4⋊C412yescomplemented
132C2×C3⋊D3C22×C3×C411yescharacteristic
133C2×C3⋊D3C2×C3×D412yes—
134C2×C3⋊D3C2×C3×D411yescharacteristic
135C2×C3⋊D3C3×#4112yes—
136C2×C3×D3—122no—
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—122no—
140C2×C3×D3—122no—
141C2×C3×D3—124no—
142C2×C3×D3—128no—
143C2×C3×D3—124no—
144C2×C3×D3—124no—
145#157—24no—
146#157—24no—
147#157—41no—
148#157—41no—
149C23×C4—91no—
150C2×C4⋊C4—92no—
151C2×C22⋊C4—91no—
152C2×C22⋊C4—91no—
153#95—92no—
154C33#38511yescharacteristic, complemented, Sylow
155C23×C3—41no—
156C23×C3—81no—
157C23×C3—91no—
158C23×C3—91no—
159C23×C3—92no—
160C23×C3—92no—
161C23×C3—182no—
162C23×C3—182no—
163C23×C3PSU(3,𝔽2)11yescharacteristic
164C22×D3—122no—
165C22×D3—122no—
166C22×D3—121no—
167C22×D3—121no—
168C22×D3—124no—
169C22×D3—124no—
170C2×C3×C4—94no—
171C2×C3×C4—94no—
172C2×C3×C4—182no—
173C2×C3×C4—182no—
174C2×C3×C4—182no—
175C2×C3×C4—182no—
176C2×C3×C4—188no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—182no—
180C2×C3×C4—182no—
181C2×C32—22no—
182C2×C32—44no—
183C2×C32—41no—
184C2×C32—42no—
185C2×C32—84no—
186C2×C32—81no—
187C2×C32—82no—
188C2×C32C2×C3×C4⋊C411yescharacteristic
189C2×C32C3×C22⋊Q812yes—
190C3⋊D3—22no—
191C3⋊D3C3×C4×D412yes—
192C3⋊D3C3×#12011yescharacteristic
193C3⋊D3C3×#12311yescharacteristic
194C3×D3—122no—
195C3×D3—121no—
196C3×D3—121no—
197C3×D3—124no—
198C24—91no—
199C22×C4—92no—
200C22×C4—92no—
201C22×C4—91no—
202C22×C4—91no—
203C22×C4—184no—
204C22×C4—182no—
205C4⋊C4—184no—
206C22⋊C4—94no—
207C22⋊C4—182no—
208C22×C3—22no—
209C22×C3—44no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—84no—
213C22×C3—81no—
214C22×C3—82no—
215C22×C3—91no—
216C22×C3—91no—
217C22×C3—92no—
218C22×C3—92no—
219C22×C3—92no—
220C22×C3—182no—
221C22×C3—184no—
222C22×C3—182no—
223C22×C3—182no—
224C22×C3#377712yescomplemented
225C22×C3C2×PSU(3,𝔽2)11yescharacteristic
226C2×D3—122no—
227C2×D3—121no—
228C2×D3—121no—
229C2×D3—122no—
230C2×D3—122no—
231C2×D3—124no—
232C2×D3—128no—
233C2×D3—124no—
234C2×D3—124no—
235C3×C4—184no—
236C3×C4—184no—
237C3×C4—361no—
238C3×C4—361no—
239C32—41no—
240C32—81no—
241C32C3×#38511yescharacteristic, complemented
242C23—91no—
243C23—91no—
244C23—92no—
245C23—92no—
246C23—182no—
247C23—182no—
248C23C3×PSU(3,𝔽2)11yescharacteristic
249C2×C4—94no—
250C2×C4—94no—
251C2×C4—182no—
252C2×C4—182no—
253C2×C4—182no—
254C2×C4—182no—
255C2×C4—188no—
256C2×C4—181no—
257C2×C4—181no—
258C2×C4—182no—
259C2×C4—182no—
260C2×C3—22no—
261C2×C3—44no—
262C2×C3—41no—
263C2×C3—42no—
264C2×C3—84no—
265C2×C3—81no—
266C2×C3—82no—
267C2×C3—91no—
268C2×C3—92no—
269C2×C3—91no—
270C2×C3—182no—
271C2×C3C2×#377711yescharacteristic
272C2×C3C22⋊PSU(3,𝔽2)12yes—
273D3—122no—
274D3—121no—
275D3—121no—
276D3—124no—
277C22—22no—
278C22—91no—
279C22—91no—
280C22—92no—
281C22—92no—
282C22—92no—
283C22—182no—
284C22—184no—
285C22—182no—
286C22—182no—
287C22C3×#377712yescomplemented
288C22C2×C3×PSU(3,𝔽2)11yescharacteristic
289C4—184no—
290C4—184no—
291C4—361no—
292C4—361no—
293C3—41no—
294C3—81no—
295C3#1059356911yescharacteristic, direct product factor
296C2—22no—
297C2—91no—
298C2—92no—
299C2—91no—
300C2—182no—
301C2C2×C3×#377711yescharacteristic
302C2C3×C22⋊PSU(3,𝔽2)12yes—
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC23×C33Q8
socleC22×C33C2×Q8
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3⋊D3C23×C3
hypercenterC23×C3PSU(3,𝔽2)
hypocenterC32C3×#385
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C2×C3×#3777C2semidirect
5. C3×#63747C2semidirect
6. #10593569C3direct (N × H)
15. C3×C22⋊#157C4semidirect
20. C2×#3777C2×C3semidirect
21. #63747C2×C3semidirect
38. C2×C3×#157C2×C4semidirect
50. C22⋊#157C3×C4semidirect
53. C22×C33C4⋊C4semidirect
59. C2×C3×C3⋊D3C4⋊C4semidirect
97. C2×#157C2×C3×C4semidirect
125. C22×C32C3×C4⋊C4semidirect
131. C2×C3⋊D3C3×C4⋊C4semidirect
154. C33#385semidirect
224. C22×C3#3777semidirect
241. C32C3×#385semidirect
287. C22C3×#3777semidirect
295. C3#10593569direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C2×C3×C22⋊#1572
C2×C3×#37772
C3×#637472
#105935693
C3×#3859