Finite Group Databasebeta

← #13139956

Subgroups — #13139956

Order 1728.

463 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 121 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×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#11720573C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#11720311C212yesmaximal, complemented
5#11720310C212yesmaximal, complemented
6#10591061—31nomaximal
7#10591053—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C411yescharacteristic
10C22×(C3⋊Dic3)⋊C3C411yescharacteristic
11C2×#6180C2212yes—
12C4×(C3⋊Dic3)⋊C3C411yescharacteristic
13C4×(C3⋊Dic3)⋊C3C411yescharacteristic
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×C4×C3⋊Dic3D311yescharacteristic
18C2×C3×C4×Dic3—31no—
19#63657D311yescharacteristic
20C3×#830—31no—
21C3×#830—31no—
22C3×#830—31no—
23C3×#830—61no—
24#63572—32no—
25Dic3⋊(C3⋊C8)—32no—
26#63553—32no—
27Dic3×C3⋊C8—32no—
28C23×H(3,𝔽3)C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3C814yescomplemented
34C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
35C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
36C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
37C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
38#6180—22no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40#4925—91no—
41#4718—91no—
42#4718—91no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C2×D311yescharacteristic
46C22×C3⋊Dic3Dic311yescharacteristic
47C22×C3⋊Dic3Dic311yescharacteristic
48C22×C3×Dic3—31no—
49C22×C3×Dic3—31no—
50C4×C3⋊Dic3Dic311yescharacteristic
51C4×C3⋊Dic3Dic311yescharacteristic
52C4×C3⋊Dic3C2×D311yescharacteristic
53C4×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)C2×D312yes—
55C3×C4×Dic3—31no—
56C3×C4×Dic3—31no—
57C3×C4×Dic3—31no—
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—62no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)C2×C812yes—
66C22×H(3,𝔽3)C22×C411yescharacteristic
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)#4112yes—
69(C3⋊Dic3)⋊C3—21no—
70(C3⋊Dic3)⋊C3C2×C812yes—
71(C3⋊Dic3)⋊C3C2×C812yes—
72(C3⋊Dic3)⋊C3M1612yes—
73C2×C3×C42—91no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—91no—
76C3×C22⋊C8—91no—
77C3×C22⋊C8—91no—
78#830—31no—
79#830—31no—
80#830—31no—
81#830—61no—
82#830—91no—
83Dic3⋊C8—92no—
84Dic3⋊C8—92no—
85C8×Dic3—92no—
86C8×Dic3—92no—
87C4⋊(C3⋊C8)—92no—
88C4×C3⋊C8—92no—
89C23×C32—21no—
90C23×C32C4×D311yescharacteristic
91C23×C32C2×Dic311yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—42no—
97C2×C32×C4C4×D311yescharacteristic
98C2×C32×C4C2×Dic311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3C3⋊C814yescomplemented
103C2×C3⋊Dic3C2×Dic311yescharacteristic
104C2×C3⋊Dic3C2×Dic311yescharacteristic
105C2×C3⋊Dic3C2×Dic311yescharacteristic
106C2×C3⋊Dic3C2×Dic311yescharacteristic
107C2×C3×Dic3—34no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—31no—
112C2×C3×Dic3—62no—
113C3⋊(C3⋊C8)—22no—
114C3×C3⋊C8—62no—
115C3×C3⋊C8—62no—
116C3×C3⋊C8—62no—
117C3×C3⋊C8—122no—
118#431—271noSylow
119C2×H(3,𝔽3)—22no—
120C2×H(3,𝔽3)#11711yescharacteristic
121C2×H(3,𝔽3)C22×C811yescharacteristic
122C2×H(3,𝔽3)C2×M1611yescharacteristic
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—61no—
126C22×C3×C4—91no—
127C22×C3×C4—91no—
128C22×C3×C4D3211yescharacteristic
129C22×Dic3—31no—
130C22×Dic3—31no—
131C22×Dic3—91no—
132C22×Dic3—91no—
133C2×C3×C8—92no—
134C2×C3×C8—92no—
135C3×C42—91no—
136C3×C42—91no—
137C3×C42—91no—
138C3×C42—91no—
139C4×Dic3—31no—
140C4×Dic3—31no—
141C4×Dic3—31no—
142C4×Dic3—31no—
143C4×Dic3—91no—
144C4×Dic3—91no—
145C4×Dic3—91no—
146C4×Dic3—91no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—62no—
151C2×C3⋊C8—92no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32D3×C812yes—
161C22×C32C2×C3⋊C812yes—
162C22×C32C2×C4×D311yescharacteristic
163C22×C32C22×Dic311yescharacteristic
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—22no—
167C32×C4—41no—
168C32×C4#23512yes—
169C32×C4#23712yes—
170C3⋊Dic3—21no—
171C3⋊Dic3C2×C3⋊C812yes—
172C3⋊Dic3C2×C3⋊C812yes—
173C3⋊Dic3#20812yes—
174C3×Dic3—32no—
175C3×Dic3—32no—
176C3×Dic3—32no—
177C3×Dic3—61no—
178C2×C42—91no—
179C4⋊C8—272no—
180C22⋊C8—91no—
181C22⋊C8—91no—
182C4×C8—272no—
183H(3,𝔽3)#43111yescharacteristic, complemented, Sylow
184C23×C3—31no—
185C23×C3—31no—
186C23×C3—61no—
187C23×C3D3×Dic311yescharacteristic
188C2×C3×C4—22no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—91no—
200C2×C3×C4—94no—
201C2×C3×C4—91no—
202C2×C3×C4—122no—
203C2×C3×C4—182no—
204C2×C3×C4D3×Dic311yescharacteristic
205C2×C3×C4C2×D3211yescharacteristic
206C2×Dic3—34no—
207C2×Dic3—31no—
208C2×Dic3—31no—
209C2×Dic3—31no—
210C2×Dic3—31no—
211C2×Dic3—62no—
212C2×Dic3—94no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3—91no—
216C2×Dic3—91no—
217C2×Dic3—182no—
218C3×C8—182no—
219C3×C8—182no—
220C3⋊C8—62no—
221C3⋊C8—62no—
222C3⋊C8—62no—
223C3⋊C8—122no—
224C3⋊C8—182no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—42no—
231C2×C32#87711yescharacteristic
232C2×C32C2×D3×C811yescharacteristic
233C2×C32D3×M1611yescharacteristic
234C2×C32C22×C3⋊C811yescharacteristic
235C2×C32C2×#20811yescharacteristic
236C2×C32#92611yescharacteristic
237C22×C4—91no—
238C22×C4—91no—
239C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
240C2×C8—92no—
241C2×C8—92no—
242C42—91no—
243C42—91no—
244C42—91no—
245C42—91no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—62no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3D3×C3⋊C812yes—
261C22×C3C2×D3×Dic311yescharacteristic
262C3×C4—21no—
263C3×C4—32no—
264C3×C4—32no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—62no—
268C3×C4—92no—
269C3×C4—92no—
270C3×C4—92no—
271C3×C4—121no—
272C3×C4—181no—
273C3×C4#380412yes—
274Dic3—32no—
275Dic3—32no—
276Dic3—32no—
277Dic3—61no—
278Dic3—92no—
279Dic3—92no—
280Dic3—92no—
281Dic3—181no—
282C32—21no—
283C32#471811yescharacteristic, complemented
284C32#492511yescharacteristic, complemented
285C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
286C2×C4—22no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—94no—
291C2×C4—91no—
292C2×C4—182no—
293C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
294C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C8—182no—
296C8—182no—
297C2×C3—22no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—62no—
305C2×C3—62no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—122no—
310C2×C3C2×D3×C3⋊C811yescharacteristic
311C2×C3D3×#20811yescharacteristic
312C2×C3#6395511yescharacteristic
313C22—21no—
314C22—21no—
315C22((C3⋊D3)⋊C3)⋊C812yes—
316C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
317C4—21no—
318C4—92no—
319C4—92no—
320C4—92no—
321C4—181no—
322C4#8867112yes—
323C3—31no—
324C3—31no—
325C3—61no—
326C3#1059105311yescharacteristic
327C2—22no—
328C2C2×((C3⋊D3)⋊C3)⋊C811yescharacteristic
329C2#1172105311yescharacteristic
330C2#1172129611yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#431
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720311C2semidirect
5. #11720310C2semidirect
33. C2×(C3⋊Dic3)⋊C3C8semidirect
102. C2×C3⋊Dic3C3⋊C8semidirect
183. H(3,𝔽3)#431semidirect
283. C32#4718semidirect
284. C32#4925semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊Dic3)⋊C32
#117205732
#63657⋊C32
#117203112
#117203102
#105910613
#105910533