Finite Group Databasebeta

← C3×#10593552 #13167627

Subgroups — C3×#10593552 #13167627

Order 1728.

370 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 90 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×C4⋊#157C211yesmaximal, characteristic
2C2×C3×#3777C211yesmaximal, characteristic
3C2×C3×#3777C211yesmaximal, characteristic
4C2×C3×#3777C211yesmaximal, characteristic
5C3×#63781C211yesmaximal, characteristic
6C3×#63781C211yesmaximal, characteristic
7C3×#63747C211yesmaximal, characteristic
8#10593552C311yesmaximal, characteristic, direct product factor
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C3×C4⋊#157—22no—
17C3×#3777—22no—
18C3×#3777—22no—
19C3×#3777—22no—
20C2×C4⋊#157C2×C311yescharacteristic
21C2×#3777C2×C311yescharacteristic
22C2×#3777C2×C311yescharacteristic
23C2×#3777C2×C311yescharacteristic
24#63781C2×C311yescharacteristic
25#63781C2×C311yescharacteristic
26#63747C2×C311yescharacteristic
27C22×C3×C3⋊D3C2311yescharacteristic
28C2×C3×#157—21no—
29C2×C3×#157—21no—
30C2×C3×#157—21no—
31C2×C3×#157—21no—
32C2×C3×#157—21no—
33C2×C3×#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157—21no—
36C2×C3×#157—21no—
37C2×C3×#157—21no—
38C2×C3×#157—21no—
39C2×C3×#157—21no—
40C2×C3×#157—21no—
41C2×C3×#157—21no—
42C2×C3×#157—21no—
43C2×C3×#157D412yes—
44C2×C3×#157D412yes—
45C2×C3×#157Q812yes—
46C2×C33×C4Q811yescharacteristic
47C2×C3×C3⋊Dic3Q811yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C3×#398—91nomaximal
51C22×#157C22×C311yescharacteristic
52C22×#157C22×C311yescharacteristic
53C22×#157C22×C311yescharacteristic
54C22×#157C22×C311yescharacteristic
55C22×#157C22×C311yescharacteristic
56C22×#157C22×C311yescharacteristic
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C2×C3×C4×D3—121no—
59C4⋊#157—22no—
60#3777—22no—
61#3777—22no—
62#3777—22no—
63C22×C33C2×Q811yescharacteristic
64C2×C3×C3⋊D3C2×D411yescharacteristic
65C2×C3×C3⋊D3C2×D411yescharacteristic
66C2×C3×C3⋊D3C2×Q811yescharacteristic
67C2×C3×C3⋊D3#4111yescharacteristic
68C2×C3×C3⋊D3#4111yescharacteristic
69C2×C3×C3⋊D3#4111yescharacteristic
70C3×#157—22no—
71C3×#157—22no—
72C3×#157—22no—
73C3×#157—41no—
74C3×#157—41no—
75C3×#157—41no—
76C33×C4—21no—
77C3×C3⋊Dic3—21no—
78C2×C3×C4⋊C4—91no—
79C2×C3×C4⋊C4—91no—
80C2×C3×C4⋊C4—91no—
81C2×C3×C4⋊C4—91no—
82C3×#95—91no—
83C3×#95—91no—
84C3×#95—91no—
85C22×C3⋊D3C23×C311yescharacteristic
86C22×C3×D3—121no—
87C2×#157—21no—
88C2×#157—21no—
89C2×#157—21no—
90C2×#157—21no—
91C2×#157—21no—
92C2×#157—21no—
93C2×#157—21no—
94C2×#157—21no—
95C2×#157—21no—
96C2×#157—21no—
97C2×#157—21no—
98C2×#157—21no—
99C2×#157—21no—
100C2×#157—21no—
101C2×#157—21no—
102C2×#157C3×D412yes—
103C2×#157C3×D412yes—
104C2×#157C3×Q812yes—
105C2×C32×C4—41no—
106C2×C32×C4—81no—
107C2×C32×C4C3×Q811yescharacteristic
108C2×C3⋊Dic3C3×Q811yescharacteristic
109C4×C3⋊D3—21no—
110C4×C3⋊D3—21no—
111C2×C3×Dic3—121no—
112C3×C4×D3—122no—
113C3×C4×D3—122no—
114#398—91noSylow
115C2×C33C22⋊Q811yescharacteristic
116C2×C33#12511yescharacteristic
117C2×C33C4⋊Q811yescharacteristic
118C3×C3⋊D3#12111yescharacteristic
119C3×C3⋊D3C22⋊Q811yescharacteristic
120C3×C3⋊D3#12311yescharacteristic
121C3×C3⋊D3#12511yescharacteristic
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×C3×C4—91no—
129C2×C4×D3—121no—
130C3×C4⋊C4—182no—
131C3×C4⋊C4—182no—
132C3×C4⋊C4—182no—
133C3×C4⋊C4—182no—
134C22×C32—41no—
135C22×C32—81no—
136C22×C32C2×C3×Q811yescharacteristic
137C2×C3⋊D3C2×C3×D411yescharacteristic
138C2×C3⋊D3C2×C3×D411yescharacteristic
139C2×C3⋊D3C2×C3×Q811yescharacteristic
140C2×C3⋊D3C3×#4111yescharacteristic
141C2×C3⋊D3C3×#4111yescharacteristic
142C2×C3⋊D3C3×#4111yescharacteristic
143C2×C3×D3—121no—
144C2×C3×D3—121no—
145C2×C3×D3—121no—
146C2×C3×D3—121no—
147C2×C3×D3—121no—
148C2×C3×D3—121no—
149#157—22no—
150#157—22no—
151#157—22no—
152#157—41no—
153#157—41no—
154#157—41no—
155C32×C4—21no—
156C32×C4—42no—
157C32×C4—82no—
158C3⋊Dic3—21no—
159C3×Dic3—122no—
160C2×C4⋊C4—91no—
161C2×C4⋊C4—91no—
162C2×C4⋊C4—91no—
163C2×C4⋊C4—91no—
164#95—91no—
165#95—91no—
166#95—91no—
167C33#39811yescharacteristic, complemented, Sylow
168C23×C3—91no—
169C22×D3—121no—
170C2×C3×C4—41no—
171C2×C3×C4—81no—
172C2×C3×C4—92no—
173C2×C3×C4—92no—
174C2×C3×C4—92no—
175C2×C3×C4—91no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4PSU(3,𝔽2)11yescharacteristic
194C2×Dic3—121no—
195C4×D3—122no—
196C4×D3—122no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—81no—
201C2×C32—81no—
202C2×C32—81no—
203C2×C32C3×C22⋊Q811yescharacteristic
204C2×C32C3×#12511yescharacteristic
205C2×C32C3×C4⋊Q811yescharacteristic
206C3⋊D3C3×#12111yescharacteristic
207C3⋊D3C3×C22⋊Q811yescharacteristic
208C3⋊D3C3×#12311yescharacteristic
209C3⋊D3C3×#12511yescharacteristic
210C3×D3—121no—
211C3×D3—121no—
212C3×D3—121no—
213C3×D3—121no—
214C22×C4—91no—
215C22×C4—91no—
216C22×C4—91no—
217C22×C4—91no—
218C22×C4—91no—
219C22×C4—91no—
220C22×C4—91no—
221C4⋊C4—182no—
222C4⋊C4—182no—
223C4⋊C4—182no—
224C4⋊C4—182no—
225C22×C3—41no—
226C22×C3—81no—
227C22×C3—91no—
228C22×C3—91no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—91no—
232C22×C3—91no—
233C22×C3C2×PSU(3,𝔽2)11yescharacteristic
234C2×D3—121no—
235C2×D3—121no—
236C2×D3—121no—
237C2×D3—121no—
238C2×D3—121no—
239C2×D3—121no—
240C3×C4—21no—
241C3×C4—42no—
242C3×C4—82no—
243C3×C4—182no—
244C3×C4—182no—
245C3×C4—182no—
246C3×C4—181no—
247C3×C4—361no—
248C3×C4—361no—
249C3×C4—361no—
250Dic3—122no—
251C32—41no—
252C32—81no—
253C32C3×#39811yescharacteristic, complemented
254C23—91no—
255C2×C4—92no—
256C2×C4—92no—
257C2×C4—92no—
258C2×C4—91no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—181no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C4—181no—
276C2×C4C3×PSU(3,𝔽2)11yescharacteristic
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—81no—
281C2×C3—81no—
282C2×C3—81no—
283C2×C3—91no—
284C2×C3—91no—
285C2×C3—91no—
286C2×C3—91no—
287C2×C3#6424311yescharacteristic
288C2×C3C4⋊PSU(3,𝔽2)11yescharacteristic
289C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
290D3—121no—
291D3—121no—
292D3—121no—
293D3—121no—
294C22—91no—
295C22—91no—
296C22—91no—
297C22—91no—
298C22—91no—
299C22—91no—
300C22C2×C3×PSU(3,𝔽2)11yescharacteristic
301C4—21no—
302C4—182no—
303C4—182no—
304C4—182no—
305C4—181no—
306C4—361no—
307C4—361no—
308C4—361no—
309C3—41no—
310C3—81no—
311C3#1059355211yescharacteristic, direct product factor
312C2—91no—
313C2—91no—
314C2—91no—
315C2—91no—
316C2C3×#6424311yescharacteristic
317C2C3×C4⋊PSU(3,𝔽2)11yescharacteristic
318C2C3×C22⋊PSU(3,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#398
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10593552C3direct (N × H)
167. C33#398semidirect
253. C32C3×#398semidirect
311. C3#10593552direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊#1572
C2×C3×#37772
C3×#637812
C3×#637472
#105935523
C3×#3989