Finite Group Databasebeta

← C3×#10590155 #13157080

Subgroups — C3×#10590155 #13157080

Order 1728.

616 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×C3×C22⋊#157C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic
3C3×C22⋊C4×C3⋊D3C211yesmaximal, characteristic
4#10590155C311yesmaximal, characteristic, direct product factor
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3×#157—21no—
7C22×C3×#157C412yescomplemented
8C2×C3×C4×C3⋊D3—21no—
9C3×C22⋊#157—21no—
10C3×C22⋊#157—21no—
11C3×C22⋊#157—21no—
12C3×C22⋊#157C412yescomplemented
13C33×C22⋊C4C411yescharacteristic, complemented
14C3×#3757C411yescharacteristic, complemented
15C3×#3752—21no—
16C2×C22⋊#157C2×C311yescharacteristic
17C2×C22⋊#157C2×C311yescharacteristic
18C22⋊C4×C3⋊D3C2×C311yescharacteristic
19C3×D3×C22⋊C4—62no—
20C23×C33C2×C411yescharacteristic
21C22×C3×C3⋊D3—42no—
22C22×C3×C3⋊D3C2×C411yescharacteristic
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C22×C3×C3⋊D3D411yescharacteristic
25C22×C3×C3⋊D3D411yescharacteristic
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C3×C3⋊D3Q811yescharacteristic
28C2×C3×#157—22no—
29C2×C3×#157—22no—
30C2×C3×#157—42no—
31C2×C3×#157—42no—
32C2×C33×C4—21no—
33C2×C3×C3⋊Dic3—21no—
34C3×C4×C3⋊D3—42no—
35C3×#342—91nomaximal
36C23×C3⋊D3C22×C311yescharacteristic
37C23×C3×D3—62no—
38C22×#157—21no—
39C22×#157C3×C412yescomplemented
40C2×C4×C3⋊D3—21no—
41C2×C3×C4×D3—64no—
42C22⋊#157—21no—
43C22⋊#157—21no—
44C22⋊#157—21no—
45C22⋊#157C3×C412yescomplemented
46C32×C22⋊C4—22no—
47C32×C22⋊C4—42no—
48C32×C22⋊C4C3×C411yescharacteristic, complemented
49#3757C3×C411yescharacteristic, complemented
50#3752—21no—
51C3×#217—62no—
52C3×(C2×D3)⋊C4—64no—
53C22×C33—41no—
54C22×C33C4211yescharacteristic
55C22×C33C22⋊C411yescharacteristic
56C22×C33C4⋊C411yescharacteristic
57C2×C3×C3⋊D3—22no—
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3—22no—
60C2×C3×C3⋊D3—41no—
61C2×C3×C3⋊D3—41no—
62C2×C3×C3⋊D3—41no—
63C2×C3×C3⋊D3C22⋊C411yescharacteristic
64C2×C3×C3⋊D3C22⋊C411yescharacteristic
65C2×C3×C3⋊D3C4⋊C411yescharacteristic
66C2×C3×C3⋊D3C4⋊C411yescharacteristic
67C3×#157—42no—
68C3×#157—42no—
69C33×C4—41no—
70C3×C3⋊Dic3—41no—
71C2×C3×C22⋊C4—91no—
72C2×C3×C22⋊C4—91no—
73C2×C3×C22⋊C4—91no—
74D3×C22⋊C4—62no—
75C23×C32—22no—
76C23×C32—42no—
77C23×C32C2×C3×C411yescharacteristic
78C22×C3⋊D3—42no—
79C22×C3⋊D3C2×C3×C411yescharacteristic
80C22×C3⋊D3C2×C3×C411yescharacteristic
81C22×C3⋊D3C3×D411yescharacteristic
82C22×C3⋊D3C3×D411yescharacteristic
83C22×C3⋊D3C3×D411yescharacteristic
84C22×C3⋊D3C3×Q811yescharacteristic
85C22×C3×D3—62no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—62no—
89C22×C3×D3—62no—
90C22×C3×D3—62no—
91C22×C3×D3—128no—
92C2×#157—22no—
93C2×#157—22no—
94C2×#157—42no—
95C2×#157—42no—
96C2×C32×C4—21no—
97C2×C32×C4—24no—
98C2×C32×C4—44no—
99C2×C3⋊Dic3—21no—
100C4×C3⋊D3—42no—
101C2×C3×Dic3—64no—
102C3×C4×D3—128no—
103#342—91noSylow
104C2×C33—21no—
105C2×C33—21no—
106C2×C33—21no—
107C2×C33#9511yescharacteristic
108C3×C3⋊D3—21no—
109C3×C3⋊D3—21no—
110C3×C3⋊D3—21no—
111C3×C3⋊D3#9912yes—
112C24×C3—91no—
113C23×D3—62no—
114C22×C3×C4—92no—
115C22×C3×C4—181no—
116C22×C3×C4—181no—
117C2×C4×D3—64no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—92no—
121C3×C22⋊C4—91no—
122C3×C22⋊C4—181no—
123C3×C22⋊C4—181no—
124C3×C22⋊C4—181no—
125C3×C22⋊C4—181no—
126C3×C22⋊C4#15711yescharacteristic, complemented
127#217—62no—
128(C2×D3)⋊C4—64no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—42no—
133C22×C32—42no—
134C22×C32—42no—
135C22×C32—41no—
136C22×C32—44no—
137C22×C32—84no—
138C22×C32C3×C4211yescharacteristic
139C22×C32C3×C22⋊C411yescharacteristic
140C22×C32C3×C4⋊C411yescharacteristic
141C2×C3⋊D3—22no—
142C2×C3⋊D3—22no—
143C2×C3⋊D3—22no—
144C2×C3⋊D3—41no—
145C2×C3⋊D3—41no—
146C2×C3⋊D3—41no—
147C2×C3⋊D3C3×C22⋊C411yescharacteristic
148C2×C3⋊D3C3×C22⋊C411yescharacteristic
149C2×C3⋊D3C3×C4⋊C411yescharacteristic
150C2×C3⋊D3C3×C4⋊C411yescharacteristic
151C2×C3×D3—62no—
152C2×C3×D3—62no—
153C2×C3×D3—62no—
154C2×C3×D3—62no—
155C2×C3×D3—68no—
156C2×C3×D3—124no—
157C2×C3×D3—124no—
158C2×C3×D3—124no—
159C2×C3×D3—124no—
160C2×C3×D3—124no—
161#157—42no—
162#157—42no—
163C32×C4—41no—
164C32×C4—44no—
165C32×C4—84no—
166C3⋊Dic3—41no—
167C3×Dic3—124no—
168C2×C22⋊C4—91no—
169C2×C22⋊C4—91no—
170C2×C22⋊C4—91no—
171C33#34211yescharacteristic, complemented, Sylow
172C23×C3—22no—
173C23×C3—42no—
174C23×C3—91no—
175C23×C3—91no—
176C23×C3—91no—
177C23×C3—91no—
178C23×C3—91no—
179C23×C3—91no—
180C23×C3—362no—
181C23×C3C2×#15711yescharacteristic
182C22×D3—62no—
183C22×D3—62no—
184C22×D3—62no—
185C22×D3—62no—
186C22×D3—62no—
187C22×D3—62no—
188C22×D3—128no—
189C2×C3×C4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—44no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—181no—
195C2×C3×C4—362no—
196C2×C3×C4—362no—
197C2×C3×C4—362no—
198C2×Dic3—64no—
199C4×D3—128no—
200C2×C32—21no—
201C2×C32—24no—
202C2×C32—22no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—44no—
206C2×C32—42no—
207C2×C32—42no—
208C2×C32—42no—
209C2×C32—82no—
210C2×C32—82no—
211C2×C32C3×#9511yescharacteristic
212C3⋊D3—21no—
213C3⋊D3—21no—
214C3⋊D3—21no—
215C3⋊D3C3×#9912yes—
216C3×D3—64no—
217C3×D3—64no—
218C3×D3—122no—
219C3×D3—122no—
220C24—91no—
221C22×C4—92no—
222C22×C4—181no—
223C22×C4—181no—
224C22⋊C4—92no—
225C22⋊C4—91no—
226C22⋊C4—181no—
227C22⋊C4—181no—
228C22⋊C4—181no—
229C22⋊C4—181no—
230C22⋊C4C3×#15711yescharacteristic, complemented
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—41no—
235C22×C3—44no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—84no—
240C22×C3—91no—
241C22×C3—91no—
242C22×C3—91no—
243C22×C3—91no—
244C22×C3—182no—
245C22×C3—182no—
246C22×C3—182no—
247C22×C3—361no—
248C22×C3—361no—
249C22×C3—361no—
250C22×C3C4×#15711yescharacteristic
251C22×C3C4⋊#15711yescharacteristic
252C22×C3C22⋊#15711yescharacteristic
253C2×D3—62no—
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—62no—
257C2×D3—68no—
258C2×D3—124no—
259C2×D3—124no—
260C2×D3—124no—
261C2×D3—124no—
262C2×D3—124no—
263C3×C4—44no—
264C3×C4—41no—
265C3×C4—84no—
266C3×C4—361no—
267C3×C4—362no—
268C3×C4—362no—
269Dic3—124no—
270C32—22no—
271C32—42no—
272C32C3×#34211yescharacteristic, complemented
273C23—91no—
274C23—91no—
275C23—91no—
276C23—91no—
277C23—91no—
278C23—91no—
279C23—362no—
280C23C2×C3×#15711yescharacteristic
281C2×C4—21no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4—181no—
285C2×C4—362no—
286C2×C4—362no—
287C2×C4—362no—
288C2×C3—24no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—22no—
293C2×C3—42no—
294C2×C3—42no—
295C2×C3—44no—
296C2×C3—42no—
297C2×C3—82no—
298C2×C3—82no—
299C2×C3—92no—
300C2×C3—181no—
301C2×C3—181no—
302C2×C3—181no—
303C2×C3#6378111yescharacteristic
304D3—64no—
305D3—64no—
306D3—122no—
307D3—122no—
308C22—41no—
309C22—91no—
310C22—91no—
311C22—91no—
312C22—91no—
313C22—182no—
314C22—182no—
315C22—182no—
316C22—361no—
317C22—361no—
318C22—361no—
319C22C3×C4×#15711yescharacteristic
320C22C3×C4⋊#15711yescharacteristic
321C22C3×C22⋊#15711yescharacteristic
322C4—41no—
323C4—361no—
324C4—362no—
325C4—362no—
326C3—22no—
327C3—42no—
328C3#1059015511yescharacteristic, direct product factor
329C2—21no—
330C2—21no—
331C2—21no—
332C2—92no—
333C2—181no—
334C2—181no—
335C2—181no—
336C2C3×#6378111yescharacteristic
subgroupG/H
centerC2×C3#63781
commutatorC22×C32C3×C42
FrattiniC23C2×C3×#157
FittingC33×C22⋊C4C4
socleC2×C33#95
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C3⋊D3C22×C3
hypercenterC3×C22⋊C4#157
hypocenterC32C3×#342
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #10590155C3direct (N × H)
7. C22×C3×#157C4semidirect
12. C3×C22⋊#157C4semidirect
13. C33×C22⋊C4C4semidirect
14. C3×#3757C4semidirect
39. C22×#157C3×C4semidirect
45. C22⋊#157C3×C4semidirect
48. C32×C22⋊C4C3×C4semidirect
49. #3757C3×C4semidirect
126. C3×C22⋊C4#157semidirect
171. C33#342semidirect
230. C22⋊C4C3×#157semidirect
272. C32C3×#342semidirect
328. C3#10590155direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊#1572
C3×C22⋊C4×C3⋊D32
#105901553
C3×#3429