Finite Group Databasebeta

← C3×#10593500 #13166873

Subgroups — C3×#10593500 #13166873

Order 1728.

462 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 70 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×#63982C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic
3C3×C22⋊#622C211yesmaximal, characteristic
4C3×#63743C212yesmaximal, complemented
5C3×#63740C212yesmaximal, complemented
6#10593500C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C2×C3×#627C2211yescharacteristic
10C2×C3×#622C2212yes—
11C32×#217—21no—
12C3×#3724—21no—
13C3×#3724C2212yes—
14C3×#3724C2211yescharacteristic
15C3×Dic3⋊Dic3—21no—
16C2×C3×Dic3⋊C4—61no—
17C3×#878—61no—
18#63982C2×C311yescharacteristic
19C2×#3724C2×C311yescharacteristic
20C22⋊#622C2×C311yescharacteristic
21#63743C2×C312yescomplemented
22#63740C2×C312yescomplemented
23C23×C33D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—21no—
28C2×C32×Dic3—21no—
29C2×C32×Dic3—22no—
30C2×C32×Dic3—42no—
31C3×#627—22no—
32C3×#622—22no—
33C3×#484—91nomaximal
34C22×C32×C4—61no—
35C22×C3⋊Dic3C22×C311yescharacteristic
36C22×C3×Dic3—21no—
37C22×C3×Dic3—21no—
38C22×C3×Dic3—41no—
39C22×C3×Dic3—61no—
40C22×C3×Dic3—61no—
41C2×C3×Dic6—61no—
42C2×#627C22×C311yescharacteristic
43C2×#622C22×C312yes—
44C32×C22⋊C4—61no—
45C3×#217—21no—
46C3×#217—21no—
47C3×#217—41no—
48C3×C4⋊Dic3—61no—
49C3×Dic3⋊C4—61no—
50C3×Dic3⋊C4—62no—
51C3×Dic3⋊C4—61no—
52C3×Dic3⋊C4—62no—
53#3724—21no—
54#3724C22×C312yes—
55#3724C22×C311yescharacteristic
56Dic3⋊Dic3—21no—
57C22×C33—21no—
58C22×C33—21no—
59C22×C33Q1612yes—
60C22×C33C2×D411yescharacteristic
61C3×C3⋊Dic3—21no—
62C3×C3⋊Dic3#4112yes—
63C32×Dic3—41no—
64C32×Dic3—42no—
65C32×Dic3—41no—
66C3×C22⋊Q8—91no—
67C2×C3×C4⋊C4—91no—
68C2×Dic3⋊C4—61no—
69#878—61no—
70C3×#107—92no—
71C3×#103—92no—
72C3×C22⋊C8—91no—
73C23×C32—21no—
74C23×C32—21no—
75C23×C32—41no—
76C23×C32—41no—
77C23×C32C3×D411yescharacteristic
78C2×C32×C4—62no—
79C2×C32×C4—61no—
80C2×C32×C4—61no—
81C2×C32×C4—122no—
82C2×C3⋊Dic3—22no—
83C2×C3⋊Dic3C3×D411yescharacteristic
84C2×C3⋊Dic3C23×C311yescharacteristic
85C2×C3×Dic3—21no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—21no—
88C2×C3×Dic3—21no—
89C2×C3×Dic3—21no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—42no—
96C2×C3×Dic3—44no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—122no—
103C3×Dic6—122no—
104#627—22no—
105#622—22no—
106#484—91noSylow
107C2×C33—22no—
108C2×C33#12311yescharacteristic
109C2×C33C2×Q1611yescharacteristic
110C2×C33AGL(1,ℤ/8)11yescharacteristic
111C2×C3×Q8—91no—
112C22×C3×C4—61no—
113C22×C3×C4—61no—
114C22×C3×C4—91no—
115C22×C3×C4—121no—
116C22×Dic3—21no—
117C22×Dic3—61no—
118C22×Dic3—61no—
119C2×Dic6—61no—
120C2×C3×C8—92no—
121C3×C4⋊C4—92no—
122C3×C4⋊C4—91no—
123C3×C4⋊C4—181no—
124C3×C4⋊C4—181no—
125C3×C22⋊C4—61no—
126C3×C22⋊C4—61no—
127C3×C22⋊C4—121no—
128#217—21no—
129C4⋊Dic3—61no—
130Dic3⋊C4—61no—
131Dic3⋊C4—62no—
132Dic3⋊C4—61no—
133Dic3⋊C4—62no—
134C22×C32—22no—
135C22×C32—22no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—42no—
143C22×C32—41no—
144C22×C32—41no—
145C22×C32—42no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32—42no—
149C22×C32—42no—
150C22×C32—81no—
151C22×C32—81no—
152C22×C32C3×Q1612yes—
153C22×C32C2×C3×D411yescharacteristic
154C32×C4—122no—
155C32×C4—121no—
156C32×C4—121no—
157C3⋊Dic3—21no—
158C3⋊Dic3C3×#4112yes—
159C3×Dic3—41no—
160C3×Dic3—41no—
161C3×Dic3—42no—
162C3×Dic3—44no—
163C3×Dic3—41no—
164C3×Dic3—41no—
165C3×Dic3—42no—
166C3×Dic3—62no—
167C3×Dic3—62no—
168C3×Dic3—62no—
169C3×Dic3—81no—
170C3×Dic3—81no—
171C3×Dic3—121no—
172C22⋊Q8—91no—
173C2×C4⋊C4—91no—
174#107—92no—
175#103—92no—
176C22⋊C8—91no—
177C33#48411yescharacteristic, complemented, Sylow
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3GO+(4,𝔽2)11yescharacteristic
183C3×Q8—182no—
184C2×C3×C4—62no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—62no—
190C2×C3×C4—91no—
191C2×C3×C4—91no—
192C2×C3×C4—122no—
193C2×C3×C4—121no—
194C2×C3×C4—121no—
195C2×C3×C4—122no—
196C2×C3×C4—122no—
197C2×C3×C4—124no—
198C2×C3×C4—182no—
199C2×Dic3—21no—
200C2×Dic3—22no—
201C2×Dic3—21no—
202C2×Dic3—42no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—64no—
208C2×Dic3—122no—
209Dic6—122no—
210C3×C8—182no—
211C2×C32—24no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—42no—
226C2×C32—44no—
227C2×C32—82no—
228C2×C32C3×#12311yescharacteristic
229C2×C32C2×C3×Q1611yescharacteristic
230C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
231C2×Q8—91no—
232C22×C4—61no—
233C22×C4—91no—
234C2×C8—92no—
235C4⋊C4—92no—
236C4⋊C4—91no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C22⋊C4—61no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3#377612yes—
259C22×C3C2×GO+(4,𝔽2)11yescharacteristic
260C3×C4—92no—
261C3×C4—121no—
262C3×C4—122no—
263C3×C4—121no—
264C3×C4—124no—
265C3×C4—121no—
266C3×C4—122no—
267C3×C4—121no—
268C3×C4—181no—
269C3×C4—241no—
270C3×C4—241no—
271Dic3—41no—
272Dic3—42no—
273Dic3—41no—
274Dic3—62no—
275Dic3—62no—
276Dic3—62no—
277Dic3—121no—
278C32—21no—
279C32—21no—
280C32—41no—
281C32—41no—
282C32C3×#48411yescharacteristic, complemented
283C23C3×GO+(4,𝔽2)11yescharacteristic
284Q8—182no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—62no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—122no—
291C2×C4—182no—
292C8—182no—
293C2×C3—24no—
294C2×C3—22no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—42no—
302C2×C3—44no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—82no—
310C2×C3#6423311yescharacteristic
311C2×C3#6423611yescharacteristic
312C2×C3C2×#377611yescharacteristic
313C22—21no—
314C22—21no—
315C22C3×#377612yes—
316C22C2×C3×GO+(4,𝔽2)11yescharacteristic
317C4—92no—
318C4—121no—
319C4—122no—
320C4—121no—
321C4—181no—
322C3—21no—
323C3—21no—
324C3—41no—
325C3—41no—
326C3#1059350011yescharacteristic, direct product factor
327C2—22no—
328C2C3×#6423311yescharacteristic
329C2C3×#6423611yescharacteristic
330C2C2×C3×#377611yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#484
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C3×#63743C2semidirect
5. C3×#63740C2semidirect
6. #10593500C3direct (N × H)
21. #63743C2×C3semidirect
22. #63740C2×C3semidirect
177. C33#484semidirect
282. C32C3×#484semidirect
326. C3#10593500direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639822
C2×C3×#37242
C3×C22⋊#6222
C3×#637432
C3×#637402
#105935003
C3×#4849