Finite Group Databasebeta

← #13135153

Subgroups — #13135153

Order 1728.

356 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#64133⋊C3C211yesmaximal, characteristic
2#11720574C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#10589154—31nomaximal
5#10589153—31nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084—21no—
8C2×#634⋊C3C411yescharacteristic, complemented
9#3757⋊C3—21no—
10C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic, complemented
11#3750⋊C3—21no—
12C2×(C3⋊(C3⋊C8))⋊C3—21no—
13#64133D311yescharacteristic
14C3×#924—31no—
15#63657D311yescharacteristic
16C3×C3⋊#95—31no—
17C3×C3⋊#95—31no—
18C3×C3⋊#95—61no—
19C3×#830—31no—
20C23×H(3,𝔽3)D411yescharacteristic
21C2×C4×H(3,𝔽3)—41no—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×#1084—22no—
25C2×#1084—41no—
26C2×#1084—41no—
27C2×(C3⋊Dic3)⋊C3—21no—
28C2×(C3⋊Dic3)⋊C3—21no—
29#634⋊C3—41no—
30(C3⋊(C3⋊C8))⋊C3—41no—
31#4538—91no—
32#4467—91no—
33#4452—91no—
34C22×C32×C4—21no—
35C22×C32×C4C2×D311yescharacteristic
36C22×C32×C4C2×D311yescharacteristic
37C2×#634Dic311yescharacteristic, complemented
38C22×C3×Dic3—62no—
39C22×C3×Dic3—61no—
40C22×C3×Dic3—61no—
41C2×C3×Dic6—31no—
42#3757—21no—
43C4⋊(C3⋊Dic3)Dic311yescharacteristic, complemented
44#3750—21no—
45C2×C3⋊(C3⋊C8)—21no—
46C3×#217—61no—
47C3×C4⋊Dic3—31no—
48C3×Dic3⋊C4—61no—
49C2×C3×C3⋊C8—61no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)C22⋊C411yescharacteristic
53C22×H(3,𝔽3)SD1611yescharacteristic
54C22×H(3,𝔽3)Q1611yescharacteristic
55C4×H(3,𝔽3)—21no—
56C4×H(3,𝔽3)—21no—
57#1084—42no—
58(C3⋊Dic3)⋊C3—41no—
59(C3⋊Dic3)⋊C3—41no—
60C3×C22⋊Q8—91no—
61#924—31no—
62#924—91no—
63C3×C22⋊C8—91no—
64C3×#95—91no—
65C3⋊#95—31no—
66C3⋊#95—31no—
67C3⋊#95—61no—
68#830—31no—
69#830—91no—
70C23×C32—21no—
71C23×C32D1211yescharacteristic
72C23×C32#6711yescharacteristic
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4—41no—
76C2×C32×C4—41no—
77C2×C32×C4—42no—
78C2×C32×C4C4×D311yescharacteristic
79C2×C32×C4C2×Dic311yescharacteristic
80C2×C32×C4#6711yescharacteristic
81C2×C32×C4#6711yescharacteristic
82C2×C3⋊Dic3—21no—
83C2×C3⋊Dic3—21no—
84#634—41no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—122no—
91C2×C3×Dic3—121no—
92C2×C3×Dic3—121no—
93C2×C3×Dic3—122no—
94C2×C3×Dic3—121no—
95C2×C3×Dic3—121no—
96C3×Dic6—121no—
97C3⋊(C3⋊C8)—41no—
98C3×C3⋊C8—121no—
99#328—271noSylow
100C2×H(3,𝔽3)—21no—
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)#9911yescharacteristic
103C2×H(3,𝔽3)#10311yescharacteristic
104C2×H(3,𝔽3)#10411yescharacteristic
105C2×C3×Q8—91no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—61no—
109C22×C3×C4—181no—
110C22×C3×C4D3211yescharacteristic
111C22×Dic3—61no—
112C22×Dic3—61no—
113C22×Dic3—62no—
114C2×Dic6—31no—
115C2×Dic6—91no—
116C2×C3×C8—181no—
117C3×C4⋊C4—91no—
118C3×C4⋊C4—181no—
119C3×C22⋊C4—181no—
120#217—61no—
121#217—181no—
122C4⋊Dic3—31no—
123C4⋊Dic3—91no—
124Dic3⋊C4—61no—
125Dic3⋊C4—181no—
126C2×C3⋊C8—61no—
127C2×C3⋊C8—181no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32#20411yescharacteristic
138C22×C32Dic1211yescharacteristic
139C22×C32(C2×D3)⋊C411yescharacteristic
140C22×C32#21511yescharacteristic
141C22×C32#21611yescharacteristic
142C22×C32#21711yescharacteristic
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—41no—
148C32×C4—41no—
149C3⋊Dic3—41no—
150C3⋊Dic3—41no—
151C3×Dic3—124no—
152C3×Dic3—122no—
153C3×Dic3—122no—
154C3×Dic3—121no—
155C3×Dic3—121no—
156C22⋊Q8—91no—
157C22⋊C8—91no—
158#95—91no—
159H(3,𝔽3)#32811yescharacteristic, complemented, Sylow
160C23×C3—31no—
161C23×C3—31no—
162C23×C3—61no—
163C23×C3#62611yescharacteristic
164C3×Q8—361no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—41no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—121no—
173C2×C3×C4—121no—
174C2×C3×C4—122no—
175C2×C3×C4—182no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4—361no—
179C2×C3×C4—361no—
180C2×C3×C4D3×Dic311yescharacteristic
181C2×C3×C4#62511yescharacteristic
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—64no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187C2×Dic3—121no—
188C2×Dic3—121no—
189C2×Dic3—121no—
190C2×Dic3—121no—
191C2×Dic3—122no—
192C2×Dic3—122no—
193C2×Dic3—181no—
194C2×Dic3—181no—
195Dic6—121no—
196Dic6—361no—
197C3×C8—361no—
198C3⋊C8—121no—
199C3⋊C8—361no—
200C2×C32—22no—
201C2×C32—22no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32#80611yescharacteristic
210C2×C32#81611yescharacteristic
211C2×C32#82511yescharacteristic
212C2×C32#83411yescharacteristic
213C2×C32#83511yescharacteristic
214C2×C32#83711yescharacteristic
215C2×Q8—91no—
216C22×C4—181no—
217C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C2×C8—181no—
219C4⋊C4—91no—
220C4⋊C4—181no—
221C22⋊C4—181no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—31no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—62no—
235C22×C3—62no—
236C22×C3—61no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3#371711yescharacteristic
240C22×C3#371911yescharacteristic
241C22×C3(C2×D3)⋊Dic311yescharacteristic
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—61no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—121no—
249C3×C4—121no—
250C3×C4—362no—
251C3×C4—361no—
252C3×C4—361no—
253Dic3—122no—
254Dic3—122no—
255Dic3—124no—
256Dic3—121no—
257Dic3—121no—
258Dic3—361no—
259Dic3—361no—
260C32—21no—
261C32#446711yescharacteristic, complemented
262C32#453811yescharacteristic, complemented
263C23#620011yescharacteristic
264Q8—361no—
265C2×C4—41no—
266C2×C4—182no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—361no—
270C2×C4—361no—
271C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
272C2×C4#619811yescharacteristic
273C8—361no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—61no—
283C2×C3—61no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3—62no—
287C2×C3—62no—
288C2×C3—61no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3#6356611yescharacteristic
292C2×C3#6356811yescharacteristic
293C2×C3#6358011yescharacteristic
294C22—21no—
295C22—21no—
296C22#8843711yescharacteristic
297C22#8843911yescharacteristic
298C22#8845011yescharacteristic
299C4—21no—
300C4—21no—
301C4—362no—
302C4—361no—
303C4—361no—
304C3—31no—
305C3—31no—
306C3—61no—
307C3#1058915211yescharacteristic
308C2—21no—
309C2—21no—
310C2#1172030511yescharacteristic
311C2#1172030911yescharacteristic
312C2#1172033311yescharacteristic
subgroupG/H
centerC22#88450
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC22×C3(C2×D3)⋊Dic3
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#328
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
8. C2×#634⋊C3C4semidirect
10. C4⋊((C3⋊Dic3)⋊C3)C4semidirect
37. C2×#634Dic3semidirect
43. C4⋊(C3⋊Dic3)Dic3semidirect
159. H(3,𝔽3)#328semidirect
261. C32#4467semidirect
262. C32#4538semidirect

Maximal subgroups

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

subgroupindex
#64133⋊C32
#117205742
#63657⋊C32
#105891543
#105891533