Finite Group Databasebeta

← #13159336

Subgroups — #13159336

Order 1728.

852 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 178 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
1C3×#64101C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3C2×#88787C211yesmaximal, characteristic
4#11722177C211yesmaximal, characteristic, complemented
5C32×#807C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722052C211yesmaximal, characteristic, complemented
8#10592580—31nomaximal
9#10590436—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×Dic3×C3⋊D3—21no—
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#88787C414yescomplemented
22C2×#3743—31no—
23#64101D311yescharacteristic
24C4⋊C4×C3⋊D3—31no—
25C3×#892—34no—
26D3×C4⋊Dic3—34no—
27C2×#3711—34no—
28C32×#107—31no—
29#63644—31no—
30#63635—31no—
31#63634—31no—
32#63634—34no—
33#63634—64no—
34C3×#807D314yescomplemented
35#63574—34no—
36#63573—34no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3C2×C412yes—
44C3×C4×C3⋊D3Q812yes—
45C2×C32×Dic3—21no—
46Dic3×C3⋊D3—42no—
47C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
48C32×C3⋊C8C2×C412yescomplemented
49#4995—91no—
50#4881—94no—
51C2×C4×C3⋊D3—61no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—34no—
54C2×D3×Dic3—64no—
55C2×C32×C8—31no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4—31no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60#3752—21no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—61no—
66C4×C3⋊Dic3—21no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70#3743—34no—
71C3×(C2×D3)⋊C4—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—64no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77#3711—316no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3C4⋊C412yes—
82C33×C4—41no—
83C33×C4C22×C411yescharacteristic
84C33×C4C2×Q811yescharacteristic
85C3⋊(C3⋊Dic3)—41no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3C4⋊C412yes—
88C32×Dic3—41no—
89C3×#117—91no—
90C2×C4⋊Dic3—91no—
91C2×#208—91no—
92C2×ΓL(1,𝔽25)—94no—
93#892—34no—
94D3×C4⋊C4—94no—
95C3×#107—34no—
96#817—94no—
97#808—94no—
98#808—91no—
99#808—91no—
100#807—94no—
101#807—91no—
102#807—184no—
103#807C3⋊D311yescharacteristic, complemented
104C22×C3⋊D3#6711yescharacteristic
105C22×C3×D3—34no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—44no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3#6711yescharacteristic
118C4×C3⋊D3—122no—
119C4×C3⋊D3Dic612yes—
120C4×C3⋊D3C4×D312yes—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—64no—
124C3×C4×D3—38no—
125C3×C4×D3—38no—
126D3×Dic3—128no—
127C32×C8—32no—
128C3⋊(C3⋊C8)—32no—
129C3⋊(C3⋊C8)—38no—
130C3⋊(C3⋊C8)—68no—
131C3×C3⋊C8C4×D318yescomplemented
132#428—271noSylow
133C2×C33C2×C4⋊C411yescharacteristic
134C2×C33AGL(1,ℤ/8)11yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—22no—
137C22×C3×C4—91no—
138C22×Dic3—181no—
139C2×C4×D3—34no—
140C2×C4×D3—184no—
141C2×C3×C8—34no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—34no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4D3214yes—
146C3×C4⋊C4C2×C3⋊D311yescharacteristic
147C3×C22⋊C4—181no—
148C3×C42—181no—
149(C2×D3)⋊C4—64no—
150C4⋊Dic3—34no—
151C4⋊Dic3—94no—
152C4⋊Dic3—91no—
153C4⋊Dic3—184no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—184no—
157C4×Dic3—64no—
158#208—94no—
159C2×C3⋊C8—94no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C2×C3⋊C8C2×C3⋊D311yescharacteristic
163ΓL(1,𝔽25)—916no—
164C22×C32—24no—
165C22×C32D3×D414yes—
166C22×C32C2×#6711yescharacteristic
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3⋊D3Dic3⋊C412yes—
170C2×C3×D3—38no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—41no—
176C32×C4—44no—
177C32×C4—84no—
178C32×C4—121no—
179C32×C4C2×Dic611yescharacteristic
180C32×C4C2×C4×D314yes—
181C32×C4C2×C4×D311yescharacteristic
182C32×C4D3×Q814yes—
183C3⋊Dic3—41no—
184C3⋊Dic3—121no—
185C3⋊Dic3—124no—
186C3⋊Dic3—244no—
187C3⋊Dic3Dic3⋊C412yes—
188C3×Dic3—38no—
189C3×Dic3—44no—
190C3×Dic3—124no—
191C2×M16—271no—
192#117—91no—
193C2×C4⋊C4—271no—
194#107—31no—
195#107—271no—
196#106—271no—
197#106—271no—
198C33#42811yescharacteristic, complemented, Sylow
199C23×C3—91no—
200C22×D3—34no—
201C2×C3×C4—24no—
202C2×C3×C4—21no—
203C2×C3×C4—24no—
204C2×C3×C4—44no—
205C2×C3×C4—64no—
206C2×C3×C4—91no—
207C2×C3×C4—92no—
208C2×C3×C4—92no—
209C2×C3×C4—181no—
210C2×C3×C4C2×D3214yes—
211C2×C3×C4C22×C3⋊D311yescharacteristic
212C2×Dic3—21no—
213C2×Dic3—34no—
214C2×Dic3—64no—
215C2×Dic3—184no—
216C2×Dic3—181no—
217C2×Dic3—364no—
218C2×Dic3—362no—
219C4×D3—38no—
220C4×D3—38no—
221C4×D3—368no—
222C3×C8—38no—
223C3⋊C8—98no—
224C3⋊C8—92no—
225C3⋊C8—188no—
226C3⋊C8C4×C3⋊D312yescomplemented
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32D3×C4⋊C414yes—
231C2×C32#91114yes—
232C2×C32#91814yes—
233C2×C32C2×Dic3⋊C411yescharacteristic
234C2×C32#94911yescharacteristic
235C2×C32#95111yescharacteristic
236C3⋊D3—22no—
237C3×D3—68no—
238C22×C4—91no—
239C22×C4—541no—
240M16—274no—
241C2×C8—31no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—91no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C4⋊C4D3×C3⋊D311yescharacteristic
248C22⋊C4—181no—
249C42—181no—
250C22×C3—24no—
251C22×C3—92no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3D3×#6714yes—
255C22×C3D4×C3⋊D311yescharacteristic
256C2×D3—38no—
257C2×D3—64no—
258C2×D3—64no—
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—41no—
262C3×C4—44no—
263C3×C4—84no—
264C3×C4—92no—
265C3×C4—124no—
266C3×C4—361no—
267C3×C4D3×Dic614yes—
268C3×C4C4×D3214yes—
269C3×C4C2×C4×C3⋊D311yescharacteristic
270C3×C4Q8×C3⋊D311yescharacteristic
271Dic3—38no—
272Dic3—41no—
273Dic3—124no—
274Dic3—361no—
275Dic3—364no—
276Dic3—724no—
277C32—24no—
278C32#488114yescomplemented
279C32#499511yescharacteristic, complemented
280C23—91no—
281C2×C4—21no—
282C2×C4—61no—
283C2×C4—91no—
284C2×C4—92no—
285C2×C4—92no—
286C2×C4—181no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—32no—
291C8—272no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—182no—
296C2×C3D3×Dic3⋊C414yes—
297C2×C3#6392514yes—
298C2×C3#6394314yes—
299C2×C3C4⋊C4×C3⋊D311yescharacteristic
300C2×C3#6412011yescharacteristic
301C2×C3#6412711yescharacteristic
302D3—68no—
303C22—92no—
304C22—181no—
305C22—181no—
306C22C3⋊D3×#6711yescharacteristic
307C4—41no—
308C4—92no—
309C4—121no—
310C4—361no—
311C4—1081no—
312C4C3⋊D3×Dic611yescharacteristic
313C4C4×D3×C3⋊D311yescharacteristic
314C3—24no—
315C3#1059043614yescomplemented
316C3#1059258011yescharacteristic, complemented
317C2—182no—
318C2C3⋊D3×Dic3⋊C411yescharacteristic
319C2#1172354611yescharacteristic
320C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #11722177C2semidirect
5. C32×#807C2semidirect
6. #11722053C2semidirect
7. #11722052C2semidirect
21. #88787C4semidirect
34. C3×#807D3semidirect
47. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
48. C32×C3⋊C8C2×C4semidirect
103. #807C3⋊D3semidirect
131. C3×C3⋊C8C4×D3semidirect
198. C33#428semidirect
226. C3⋊C8C4×C3⋊D3semidirect
278. C32#4881semidirect
279. C32#4995semidirect
315. C3#10590436semidirect
316. C3#10592580semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
C3⋊D3×C4⋊Dic32
C2×#887872
#117221772
C32×#8072
#117220532
#117220522
#105925803
#105904363