Finite Group Databasebeta

← #13159954

Subgroups — #13159954

Order 1728.

888 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 166 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
1D12×C3⋊Dic3C211yesmaximal, characteristic
2#11723406C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic
4#11722186C211yesmaximal, characteristic, complemented
5C3×#63642C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic
7#11722057C211yesmaximal, characteristic, complemented
8#10592456—31nomaximal
9#10590747—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88806—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21#88794C414yescomplemented
22D4×C3⋊Dic3—31no—
23C2×#3754—31no—
24#64096—31no—
25Dic3×D12—34no—
26#63844—34no—
27C2×#3716—34no—
28C32×#102—31no—
29#63643—31no—
30#63643—34no—
31#63643—64no—
32#63642D311yescharacteristic, complemented
33#63634—31no—
34C3×#821D314yes—
35C3×#815—34no—
36#63575—34no—
37C22×C32×D3—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41#6308—21no—
42#6308C2×C412yescomplemented
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C32×Dic3—21no—
46C32×D12—21no—
47C32×D12C2×C412yes—
48D3×C3⋊Dic3—42no—
49C33×C8—21no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51#4757—94no—
52#4710—91no—
53C2×C32×D4—31no—
54C22×C3⋊Dic3—61no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C3×D12C2×D314yes—
59C2×D3×Dic3—64no—
60C2×C32×C8—24no—
61C2×C32×C8C2×D314yes—
62C2×C32×C8C2×D311yescharacteristic
63C32×C4⋊C4—31no—
64#3757—61no—
65#3754—34no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—64no—
69#3750—61no—
70C4×C3⋊Dic3—61no—
71C4×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—34no—
75C2×C3×C3⋊C8—34no—
76#3724—64no—
77(C2×D3)⋊Dic3—64no—
78Dic32—64no—
79#3716—316no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—22no—
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—21no—
90C32×Dic3—41no—
91D4×Dic3—94no—
92C2×#214—94no—
93C2×#204—91no—
94#887—94no—
95C4×D12—91no—
96C4×Dic6—91no—
97C3×#102—34no—
98C3×#97—91no—
99#821C3⋊D311yescharacteristic
100#818—91no—
101#816—94no—
102#816—91no—
103#816—184no—
104#815—34no—
105#807—94no—
106C23×C32—61no—
107C22×C3×D3—24no—
108C32×D4—32no—
109C32×D4—61no—
110C2×C32×C4—24no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3—122no—
121C2×C3⋊Dic3D1211yescharacteristic
122C2×C3⋊Dic3D1211yescharacteristic
123#634—32no—
124#634—38no—
125#634—68no—
126#634—61no—
127#634—64no—
128#634—124no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C3×D12—24no—
133C3×D12C4×D318yes—
134D3×Dic3—128no—
135C32×C8—21no—
136C32×C8—24no—
137C32×C8—44no—
138C3⋊(C3⋊C8)C4×D312yescomplemented
139C3×C3⋊C8—38no—
140#440—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C32×D3—42no—
145C2×C3×D4—34no—
146C22×Dic3—184no—
147C2×D12C2×C3⋊D311yescharacteristic
148C2×C4×D3—181no—
149C2×Dic6—94no—
150C2×Dic6—91no—
151C2×Dic6—184no—
152C2×C3×C8—24no—
153C2×C3×C8—91no—
154C2×C3×C8D3214yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4—34no—
157C3×C42—91no—
158#217—184no—
159#214—916no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—94no—
162C4⋊Dic3—91no—
163C4⋊Dic3—184no—
164C4⋊Dic3C2×C3⋊D311yescharacteristic
165Dic3⋊C4—181no—
166Dic3⋊C4—184no—
167C4×Dic3—34no—
168C4×Dic3—181no—
169C4×Dic3—184no—
170C2×C3⋊C8—34no—
171#204—94no—
172C22×C32—24no—
173C22×C32—62no—
174C22×C32—121no—
175C22×C32—121no—
176C22×C32C2×D1211yescharacteristic
177C22×C32D3×D414yes—
178C2×C3×D3—28no—
179C2×C3×D3—44no—
180C2×C3×D3—44no—
181C32×C4—24no—
182C32×C4—24no—
183C32×C4—121no—
184C32×C4C2×C4×D314yes—
185C32×C4C2×C4×D311yescharacteristic
186C32×C4#23511yescharacteristic
187C32×C4#23714yes—
188C3⋊Dic3—21no—
189C3⋊Dic3—21no—
190C3⋊Dic3—62no—
191C3⋊Dic3—68no—
192C3⋊Dic3—128no—
193C3⋊Dic3—121no—
194C3⋊Dic3—124no—
195C3⋊Dic3—244no—
196C3×Dic3—44no—
197C3×Dic3—64no—
198C3×Dic3—64no—
199C2×SD16—271no—
200C4×Q8—271no—
201C4×D4—271no—
202#107—271no—
203#103—271no—
204#102—31no—
205#97—91no—
206C33#44011yescharacteristic, complemented, Sylow
207C23×C3—64no—
208C22×D3—21no—
209C3×D4—38no—
210C3×D4—64no—
211C2×C3×C4—24no—
212C2×C3×C4—64no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—34no—
219C2×Dic3—34no—
220C2×Dic3—184no—
221C2×Dic3—181no—
222C2×Dic3—184no—
223C2×Dic3—181no—
224C2×Dic3—364no—
225C2×Dic3—364no—
226C2×Dic3—368no—
227D12—21no—
228D12C4×C3⋊D312yes—
229C4×D3—362no—
230Dic6—98no—
231Dic6—92no—
232Dic6—188no—
233Dic6—184no—
234Dic6—181no—
235Dic6—364no—
236C3×C8—24no—
237C3×C8—21no—
238C3×C8—44no—
239C3×C8—92no—
240C3⋊C8—38no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—122no—
245C2×C32C4×D1211yescharacteristic
246C2×C32#67⋊C414yes—
247C2×C32#90811yescharacteristic
248C2×C32#90911yescharacteristic
249C2×C32#91114yes—
250C2×C32#91514yes—
251C3×D3—48no—
252C2×Q8—271no—
253C2×D4—31no—
254C22×C4—541no—
255SD16—274no—
256C2×C8—91no—
257C2×C8D3×C3⋊D311yescharacteristic
258C4⋊C4—31no—
259C4⋊C4—271no—
260C4⋊C4—541no—
261C22⋊C4—541no—
262C42—91no—
263C42—541no—
264C22×C3—24no—
265C22×C3—68no—
266C22×C3—124no—
267C22×C3—124no—
268C22×C3D3×D1214yes—
269C22×C3D4×C3⋊D311yescharacteristic
270C2×D3—22no—
271C2×D3—41no—
272C2×D3—41no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—124no—
276C3×C4—181no—
277C3×C4—181no—
278C3×C4C4×D3214yes—
279C3×C4#380414yes—
280C3×C4C2×C4×C3⋊D311yescharacteristic
281C3×C4#383011yescharacteristic
282Dic3—41no—
283Dic3—64no—
284Dic3—64no—
285Dic3—182no—
286Dic3—188no—
287Dic3—368no—
288Dic3—364no—
289Dic3—361no—
290Dic3—724no—
291C32—24no—
292C32#471011yescharacteristic, complemented
293C32#475714yescomplemented
294C23—61no—
295Q8—272no—
296Q8—541no—
297D4—32no—
298D4—61no—
299C2×C4—61no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—541no—
303C2×C4—541no—
304C2×C4—1082no—
305C2×C4C2×D3×C3⋊D311yescharacteristic
306C8—21no—
307C8—92no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—24no—
311C2×C3—128no—
312C2×C3#6379514yes—
313C2×C3#6380014yes—
314C2×C3#6388014yes—
315C2×C3#6409011yescharacteristic
316C2×C3#6412011yescharacteristic
317C2×C3#6412411yescharacteristic
318D3—42no—
319C22—62no—
320C22—121no—
321C22—121no—
322C22C3⋊D3×D1211yescharacteristic
323C4—121no—
324C4—181no—
325C4—181no—
326C4—542no—
327C4—1081no—
328C4C4×D3×C3⋊D311yescharacteristic
329C4#8902811yescharacteristic
330C3—24no—
331C3#1059074714yescomplemented
332C3#1059245611yescharacteristic, complemented
333C2—122no—
334C2#1172332011yescharacteristic
335C2#1172334011yescharacteristic
336C2#1172346111yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11723406C2semidirect
4. #11722186C2semidirect
5. C3×#63642C2semidirect
7. #11722057C2semidirect
21. #88794C4semidirect
32. #63642D3semidirect
42. #6308C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
138. C3⋊(C3⋊C8)C4×D3semidirect
206. C33#440semidirect
292. C32#4710semidirect
293. C32#4757semidirect
331. C3#10590747semidirect
332. C3#10592456semidirect

Maximal subgroups

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

subgroupindex
D12×C3⋊Dic32
#117234062
C2×#887942
#117221862
C3×#636422
C32×#8212
#117220572
#105924563
#105907473