Finite Group Databasebeta

← #13139972

Subgroups — #13139972

Order 1728.

593 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C22×#634⋊C3C211yesmaximal, characteristic
2C2×#88538C211yesmaximal, characteristic
3#64135⋊C3C211yesmaximal, characteristic
4#11720313C214yesmaximal, complemented
5#10591107—31nomaximal
6#10591100—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×(C3⋊Dic3)⋊C3—21no—
9C2×#634⋊C3—21no—
10C2×#634⋊C3—22no—
11C2×#634⋊C3C414yescomplemented
12C2×#634⋊C3C2212yes—
13#88538—22no—
14C2×#6180C2212yes—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17C4×(C3⋊Dic3)⋊C3—21no—
18C22×#634D311yescharacteristic
19C22×C3×Dic6—31no—
20#64135D311yescharacteristic
21C3×#926—31no—
22C2×C3×#208—31no—
23C2×C3×#208—31no—
24C2×C3×#208—61no—
25#63567—34no—
26#63566—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—24no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—24no—
36#634⋊C3—22no—
37#634⋊C3C2×C414yes—
38#6180—22no—
39#5225—91no—
40#5120—91no—
41#5004—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—21no—
46C2×#634—21no—
47C2×#634—22no—
48C2×#634Dic314yescomplemented
49C2×#634C2×D312yes—
50C22×C3×Dic3—61no—
51C2×C3×Dic6—34no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—62no—
54C2×C3×Dic6—61no—
55#3757—21no—
56C4⋊(C3⋊Dic3)C2×D312yes—
57C4×C3⋊Dic3—21no—
58C3×#217—61no—
59C3×C4⋊Dic3—32no—
60C3×C4×Dic3—61no—
61C3×#208—64no—
62C3×#208—62no—
63C3×#208—62no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67C22×H(3,𝔽3)—22no—
68C22×H(3,𝔽3)C22⋊C412yes—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)C22⋊C412yes—
71C4×H(3,𝔽3)C22×C411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73(C3⋊Dic3)⋊C3—24no—
74(C3⋊Dic3)⋊C3—42no—
75C22×C3×Q8—91no—
76C22×Dic6—31no—
77C22×Dic6—91no—
78C2×C3×M16—91no—
79C3×#117—91no—
80#926—31no—
81#926—91no—
82C2×#208—31no—
83C2×#208—31no—
84C2×#208—61no—
85#835—94no—
86#816—94no—
87#810—94no—
88C23×C32—21no—
89C23×C32#6711yescharacteristic
90C23×C32#6711yescharacteristic
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4C4×D312yes—
96C2×C32×C4D1212yes—
97C2×C32×C4C2×Dic312yes—
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4#6711yescharacteristic
100C2×C32×C4#6712yes—
101C2×C32×C4C22×D311yescharacteristic
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—24no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106#634—24no—
107#634—22no—
108#634C2×Dic314yes—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C3×Dic6—34no—
113C3×Dic6—64no—
114C3×Dic6—62no—
115C3×C3⋊C8—64no—
116C3×C3⋊C8—62no—
117C3×C3⋊C8—62no—
118#419—271noSylow
119C2×H(3,𝔽3)—22no—
120C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
121C2×H(3,𝔽3)#13712yes—
122C2×C3×Q8—92no—
123C2×C3×Q8—94no—
124C2×C3×Q8—181no—
125C2×C3×Q8—182no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—61no—
129C22×C3×C4—181no—
130C22×C3×C4D3211yescharacteristic
131C22×Dic3—61no—
132C22×Dic3—181no—
133C2×Dic6—34no—
134C2×Dic6—32no—
135C2×Dic6—61no—
136C2×Dic6—62no—
137C2×Dic6—92no—
138C2×Dic6—94no—
139C2×Dic6—181no—
140C2×Dic6—182no—
141C3×M16—182no—
142C2×C3×C8—92no—
143C3×C4⋊C4—92no—
144C3×C22⋊C4—181no—
145C3×C42—181no—
146#217—61no—
147#217—181no—
148C4⋊Dic3—32no—
149C4⋊Dic3—92no—
150C4×Dic3—61no—
151C4×Dic3—181no—
152#208—62no—
153#208—62no—
154#208—64no—
155C2×C3⋊C8—32no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—62no—
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—42no—
163C22×C32(C2×D3)⋊C412yes—
164C22×C32#21712yes—
165C22×C32C2×#6711yescharacteristic
166C22×C32C2×#6711yescharacteristic
167C32×C4—22no—
168C32×C4—21no—
169C32×C4—21no—
170C32×C4(C2×D3)⋊C412yes—
171C32×C4#21712yes—
172C32×C4C2×C4×D311yescharacteristic
173C32×C4C2×D1211yescharacteristic
174C32×C4C22×Dic311yescharacteristic
175C32×C4C2×#6711yescharacteristic
176C3⋊Dic3—24no—
177C3⋊Dic3—42no—
178C3×Dic3—64no—
179C3×Dic3—122no—
180C22×Q8—91no—
181C2×M16—91no—
182#117—91no—
183#103—274no—
184H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3#62511yescharacteristic
189C3×Q8—94no—
190C3×Q8—182no—
191C3×Q8—184no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—182no—
205C2×C3×C4—184no—
206C2×C3×C4—182no—
207C2×C3×C4D3×Dic312yes—
208C2×C3×C4#62511yescharacteristic
209C2×C3×C4#62612yes—
210C2×C3×C4C2×D3211yescharacteristic
211C2×Dic3—64no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—182no—
215C2×Dic3—184no—
216C2×Dic3—182no—
217Dic6—34no—
218Dic6—64no—
219Dic6—62no—
220Dic6—94no—
221Dic6—182no—
222Dic6—184no—
223C3×C8—182no—
224C3⋊C8—62no—
225C3⋊C8—62no—
226C3⋊C8—64no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—42no—
232C2×C32C2×(C2×D3)⋊C411yescharacteristic
233C2×C32#94212yes—
234C2×C32#95112yes—
235C2×C32C2×#21711yescharacteristic
236C2×Q8—94no—
237C2×Q8—92no—
238C2×Q8—181no—
239C2×Q8—182no—
240C22×C4—181no—
241C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
242M16—182no—
243C2×C8—92no—
244C4⋊C4—92no—
245C22⋊C4—181no—
246C42—181no—
247C22×C3—22no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—62no—
253C22×C3—62no—
254C22×C3—61no—
255C22×C3—62no—
256C22×C3—122no—
257C22×C3(C2×D3)⋊Dic312yes—
258C22×C3C2×#62511yescharacteristic
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—32no—
264C3×C4—32no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—62no—
268C3×C4—184no—
269C3×C4—362no—
270C3×C4(C2×D3)⋊Dic312yes—
271C3×C4C2×D3×Dic311yescharacteristic
272C3×C4C2×#62611yescharacteristic
273Dic3—64no—
274Dic3—122no—
275Dic3—184no—
276Dic3—362no—
277C32—21no—
278C32#500411yescharacteristic, complemented
279C32#522511yescharacteristic, complemented
280C23#619811yescharacteristic
281Q8—94no—
282Q8—182no—
283Q8—184no—
284C2×C4—182no—
285C2×C4—184no—
286C2×C4—182no—
287C2×C4((C3⋊D3)⋊C3)⋊C412yes—
288C2×C4#619811yescharacteristic
289C2×C4#620012yes—
290C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
291C8—182no—
292C2×C3—22no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—32no—
296C2×C3—32no—
297C2×C3—62no—
298C2×C3—62no—
299C2×C3—61no—
300C2×C3—62no—
301C2×C3—122no—
302C2×C3#6382712yes—
303C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
304C22—22no—
305C22#8845012yes—
306C22C2×#619811yescharacteristic
307C4—184no—
308C4—362no—
309C4#8845012yes—
310C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
311C4C2×#620011yescharacteristic
312C3—31no—
313C3—31no—
314C3—61no—
315C3#1059109911yescharacteristic
316C2—22no—
317C2#1172105812yes—
318C2C2×#8845011yescharacteristic
subgroupG/H
centerC22C2×#6198
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#419
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720313C2semidirect
11. C2×#634⋊C3C4semidirect
48. C2×#634Dic3semidirect
184. H(3,𝔽3)#419semidirect
278. C32#5004semidirect
279. C32#5225semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×#885382
#64135⋊C32
#117203132
#105911073
#105911003