Finite Group Databasebeta

← C3×#10590030 #13154880

Subgroups — C3×#10590030 #13154880

Order 1728.

358 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 54 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×#63902C211yesmaximal, characteristic
2C3×#63779C211yesmaximal, characteristic, complemented
3C3×#63553C211yesmaximal, characteristic, complemented
4#10590030C311yesmaximal, characteristic, direct product factor
5C2×C3×#625C411yescharacteristic
6C2×C3×D3×Dic3—21no—
7C2×C3×#622—21no—
8C3×C4×C3⋊Dic3C2211yescharacteristic
9C32×(C2×D3)⋊C4—21no—
10C2×C32×C3⋊C8—21no—
11C3×#3724C411yescharacteristic
12C3×#67⋊C4—61no—
13#63902C2×C311yescharacteristic
14#63779C2×C311yescharacteristic, complemented
15C3×C8×Dic3—61no—
16#63553C2×C311yescharacteristic, complemented
17C22×C32×D3—21no—
18C2×C33×C4D411yescharacteristic
19C2×C3×C3⋊Dic3C2×C411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C2×C32×Dic3—21no—
22C3×#625—21no—
23C3×#625C812yescomplemented
24C3×D3×Dic3—41no—
25C3×D3×Dic3—41no—
26C3×#622—41no—
27C32×C3⋊C8—22no—
28C3×D4⋊C8—91nomaximal
29C2×C3×#67—61no—
30C22×C3×Dic3—61no—
31C2×C3×C4×D3—61no—
32C2×#625C3×C411yescharacteristic
33C2×D3×Dic3—21no—
34C2×#622—21no—
35C2×C32×C8—61no—
36C32×C22⋊C4—61no—
37C4×C3⋊Dic3C22×C311yescharacteristic
38C3×(C2×D3)⋊C4—21no—
39C3×(C2×D3)⋊C4—21no—
40C3×(C2×D3)⋊C4—41no—
41C3×Dic3⋊C4—61no—
42C3×C4×Dic3—61no—
43C3×C4×Dic3—61no—
44C2×C3×C3⋊C8—21no—
45C2×C3×C3⋊C8—21no—
46C2×C3×C3⋊C8—41no—
47#3724C3×C411yescharacteristic
48C22×C33C22⋊C411yescharacteristic
49C2×C32×D3—22no—
50C2×C32×D3—41no—
51C2×C32×D3—41no—
52C33×C4—21no—
53C3×C3⋊Dic3C2×C811yescharacteristic
54C3×C3⋊Dic3M1611yescharacteristic
55C3×C3⋊Dic3D811yescharacteristic
56C3×C3⋊Dic3SD1611yescharacteristic
57C32×Dic3—41no—
58C3×C4×D4—91no—
59#67⋊C4—61no—
60C3×C4⋊C8—91no—
61C3×C4×C8—91no—
62C8×Dic3—61no—
63C23×C32—61no—
64C22×C3×D3—21no—
65C22×C3×D3—21no—
66C22×C3×D3—41no—
67C2×C32×C4—21no—
68C2×C32×C4—21no—
69C2×C32×C4—41no—
70C2×C32×C4—41no—
71C2×C32×C4—61no—
72C2×C32×C4C3×D411yescharacteristic
73C2×C3⋊Dic3C2×C3×C411yescharacteristic
74C2×C3⋊Dic3C3×D411yescharacteristic
75C3×#67—62no—
76C3×#67—62no—
77C2×C3×Dic3—21no—
78C2×C3×Dic3—21no—
79C2×C3×Dic3—41no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—121no—
85C2×C3×Dic3—121no—
86C3×C4×D3—121no—
87C3×C4×D3—121no—
88#625—21no—
89#625C3×C812yescomplemented
90D3×Dic3—41no—
91D3×Dic3—41no—
92#622—41no—
93C32×C8—62no—
94C3×C3⋊C8—22no—
95C3×C3⋊C8—22no—
96C3×C3⋊C8—42no—
97D4⋊C8—91noSylow
98C2×C33C22⋊C811yescharacteristic
99C2×C33#10211yescharacteristic
100C2×C33#10411yescharacteristic
101C32×D3—42no—
102C2×C3×D4—91no—
103C22×C3×C4—181no—
104C2×#67—61no—
105C22×Dic3—61no—
106C2×C4×D3—61no—
107C2×C3×C8—61no—
108C2×C3×C8—61no—
109C2×C3×C8—121no—
110C2×C3×C8—181no—
111C3×C4⋊C4—91no—
112C3×C22⋊C4—61no—
113C3×C22⋊C4—61no—
114C3×C22⋊C4—121no—
115C3×C42—91no—
116(C2×D3)⋊C4—21no—
117Dic3⋊C4—61no—
118C4×Dic3—61no—
119C4×Dic3—61no—
120C2×C3⋊C8—21no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—41no—
124C22×C32—41no—
125C22×C32—62no—
126C22×C32—121no—
127C22×C32—121no—
128C22×C32C3×C22⋊C411yescharacteristic
129C2×C3×D3—22no—
130C2×C3×D3—22no—
131C2×C3×D3—42no—
132C2×C3×D3—41no—
133C2×C3×D3—41no—
134C2×C3×D3—41no—
135C2×C3×D3—41no—
136C2×C3×D3—81no—
137C2×C3×D3—81no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—41no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4—81no—
145C32×C4—121no—
146C3⋊Dic3C2×C3×C811yescharacteristic
147C3⋊Dic3C3×M1611yescharacteristic
148C3⋊Dic3C3×D811yescharacteristic
149C3⋊Dic3C3×SD1611yescharacteristic
150C3×Dic3—41no—
151C3×Dic3—41no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C3×Dic3—81no—
161C4×D4—91no—
162C4⋊C8—91no—
163C4×C8—91no—
164C33D4⋊C811yescharacteristic, complemented, Sylow
165C23×C3—61no—
166C23×C3—61no—
167C23×C3—121no—
168C22×D3—21no—
169C3×D4—92no—
170C3×D4—181no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—121no—
180C2×C3×C4—361no—
181C2×C3×C4—361no—
182C2×C3×C4GO+(4,𝔽2)11yescharacteristic
183#67—62no—
184#67—62no—
185C2×Dic3—21no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—121no—
191C2×Dic3—121no—
192C4×D3—121no—
193C4×D3—121no—
194C3×C8—62no—
195C3×C8—62no—
196C3×C8—122no—
197C3×C8—361no—
198C3⋊C8—22no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—122no—
212C2×C32C3×C22⋊C811yescharacteristic
213C2×C32C3×#10211yescharacteristic
214C2×C32C3×#10411yescharacteristic
215C3×D3—42no—
216C3×D3—42no—
217C3×D3—82no—
218C2×D4—91no—
219C22×C4—181no—
220C2×C8—61no—
221C2×C8—181no—
222C4⋊C4—91no—
223C22⋊C4—61no—
224C42—91no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—62no—
230C22×C3—62no—
231C22×C3—122no—
232C22×C3—121no—
233C22×C3—121no—
234C22×C3—121no—
235C22×C3—121no—
236C22×C3—241no—
237C22×C3—241no—
238C22×C3D32⋊C411yescharacteristic
239C2×D3—22no—
240C2×D3—41no—
241C2×D3—41no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—81no—
249C3×C4—91no—
250C3×C4—91no—
251C3×C4—91no—
252C3×C4—91no—
253C3×C4—121no—
254C3×C4—121no—
255C3×C4—241no—
256Dic3—41no—
257Dic3—61no—
258Dic3—61no—
259Dic3—61no—
260Dic3—61no—
261Dic3—61no—
262Dic3—61no—
263Dic3—61no—
264Dic3—61no—
265C32—21no—
266C32—21no—
267C32—41no—
268C32—41no—
269C32C3×D4⋊C811yescharacteristic, complemented
270C23—61no—
271D4—92no—
272D4—181no—
273C2×C4—61no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—361no—
277C2×C4—361no—
278C2×C4C3×GO+(4,𝔽2)11yescharacteristic
279C8—62no—
280C8—361no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—122no—
294C2×C3—122no—
295C2×C3—242no—
296C2×C3D32⋊C811yescharacteristic
297C2×C3#6372911yescharacteristic
298C2×C3#6374111yescharacteristic
299D3—42no—
300C22—62no—
301C22—121no—
302C22—121no—
303C22C3×D32⋊C411yescharacteristic
304C4—21no—
305C4—91no—
306C4—91no—
307C4—91no—
308C4—91no—
309C4—121no—
310C3—21no—
311C3—21no—
312C3—41no—
313C3—41no—
314C3#1059003011yescharacteristic, direct product factor
315C2—122no—
316C2C3×D32⋊C811yescharacteristic
317C2C3×#6372911yescharacteristic
318C2C3×#6374111yescharacteristic
subgroupG/H
centerC22×C3D32⋊C4
commutatorC3⋊Dic3C2×C3×C8
FrattiniC2×C4C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC2×C3×#625C4
cosocleC4×C3⋊Dic3C22×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×D4⋊C8
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63779C2semidirect
3. C3×#63553C2semidirect
4. #10590030C3direct (N × H)
14. #63779C2×C3semidirect
16. #63553C2×C3semidirect
23. C3×#625C8semidirect
89. #625C3×C8semidirect
164. C33D4⋊C8semidirect
269. C32C3×D4⋊C8semidirect
314. C3#10590030direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639022
C3×#637792
C3×#635532
#105900303
C3×D4⋊C89