Finite Group Databasebeta

← #13159949

Subgroups — #13159949

Order 1728.

900 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 186 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
1Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723399C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic
4#11722186C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic
6C32×C8×Dic3C211yesmaximal, characteristic, complemented
7#11722052C211yesmaximal, characteristic, complemented
8C4×#3744—31nomaximal
9#10590744—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88806—21no—
19#88803—21no—
20Dic3×C3⋊Dic3—21no—
21#88796C414yescomplemented
22C2×#3744—31no—
23C4×C4⋊(C3⋊D3)—31no—
24C4×#634—31no—
25Dic3×D12—34no—
26#63844—34no—
27C2×#3716—34no—
28C32×C4×C8—31no—
29#63648D311yescharacteristic
30#63644—31no—
31#63643—31no—
32#63643—34no—
33#63643—64no—
34C3×#821—34no—
35C3×C8×Dic3D314yescomplemented
36#63573—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41#6308—21no—
42#6308C2×C412yescomplemented
43C2×C3×C3⋊Dic3—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)C2×C412yes—
46C2×C32×Dic3D411yescharacteristic
47C2×C32×Dic3D411yescharacteristic
48Dic3×C3⋊D3—42no—
49C33×C8—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#4758—91no—
52C4×#204—94no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3—61no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C3×D12—34no—
59C2×D3×Dic3—64no—
60C2×C32×C8—24no—
61C2×C32×C8—31no—
62C2×C32×C8C2×D314yes—
63C2×C32×C8C2×D311yescharacteristic
64C32×C42—31no—
65#3752—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—61no—
71C4×C3⋊Dic3—61no—
72#3744—34no—
73C3×C4⋊Dic3—34no—
74C3×C4×Dic3C2×D314yes—
75C2×C3×C3⋊C8C2×D314yes—
76#3724—64no—
77(C2×D3)⋊Dic3—64no—
78Dic32—64no—
79#3716—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—22no—
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—21no—
90C32×Dic3SD1612yescomplemented
91D4×Dic3—91no—
92C2×#214—91no—
93C2×#204—94no—
94#887—91no—
95C4×D12—94no—
96C4×Dic6—94no—
97C3×#102—91no—
98C3×C4×C8—34no—
99#821—34no—
100#817—94no—
101#816—94no—
102#816—91no—
103#816—184no—
104C8×Dic3C3⋊D311yescharacteristic, complemented
105#808—91no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C2×C32×C4—24no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119C2×C3⋊Dic3—124no—
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)C4×D312yes—
122C4×C3⋊D3—122no—
123#634—32no—
124#634—38no—
125#634—68no—
126#634—61no—
127#634—64no—
128#634—124no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3D1214yes—
131C2×C3×Dic3D1214yes—
132C3×D12—38no—
133C3×D12—64no—
134D3×Dic3—128no—
135C32×C8—21no—
136C32×C8—24no—
137C32×C8—32no—
138C32×C8—44no—
139C3×C3⋊C8C4×D318yescomplemented
140C4×SD16—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C2×C33#13511yescharacteristic
144C3×C3⋊D3—42no—
145C2×C3×D4—91no—
146C22×Dic3—181no—
147C2×D12—34no—
148C2×C4×D3—184no—
149C2×Dic6—94no—
150C2×Dic6—91no—
151C2×Dic6—184no—
152C2×C3×C8—24no—
153C2×C3×C8—34no—
154C2×C3×C8D3214yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4—91no—
157C3×C42—34no—
158#217—181no—
159#214—94no—
160(C2×D3)⋊C4—184no—
161C4⋊Dic3—34no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165Dic3⋊C4—184no—
166Dic3⋊C4—181no—
167C4×Dic3—184no—
168C4×Dic3—181no—
169C4×Dic3C2×C3⋊D311yescharacteristic
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171#204—916no—
172C22×C32—24no—
173C22×C32C2×D1214yes—
174C22×C32D3×D411yescharacteristic
175C2×C3⋊D3—22no—
176C2×C3⋊D3—41no—
177C2×C3⋊D3—41no—
178C2×C3×D3—68no—
179C2×C3×D3—124no—
180C2×C3×D3—124no—
181C32×C4—24no—
182C32×C4—24no—
183C32×C4—32no—
184C32×C4—61no—
185C32×C4C2×C4×D314yes—
186C32×C4C2×C4×D311yescharacteristic
187C32×C4#23514yes—
188C32×C4#23711yescharacteristic
189C3⋊Dic3—41no—
190C3⋊Dic3—62no—
191C3⋊Dic3—68no—
192C3⋊Dic3—128no—
193C3⋊Dic3—121no—
194C3⋊Dic3—124no—
195C3⋊Dic3—244no—
196C3×Dic3—24no—
197C3×Dic3—124no—
198C3×Dic3#20418yescomplemented
199C2×SD16—271no—
200C4×Q8—271no—
201C4×D4—271no—
202#106—271no—
203#103—271no—
204#102—91no—
205C4×C8—31no—
206C33C4×SD1611yescharacteristic, complemented, Sylow
207C23×C3—181no—
208C22×D3—64no—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—24no—
212C2×C3×C4—34no—
213C2×C3×C4—34no—
214C2×C3×C4—181no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—64no—
218C2×Dic3—184no—
219C2×Dic3—181no—
220C2×Dic3—184no—
221C2×Dic3—181no—
222C2×Dic3—364no—
223C2×Dic3—364no—
224C2×Dic3—362no—
225C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
226C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
227D12—38no—
228D12—64no—
229C4×D3—368no—
230Dic6—98no—
231Dic6—92no—
232Dic6—188no—
233Dic6—184no—
234Dic6—181no—
235Dic6—364no—
236C3×C8—24no—
237C3×C8—21no—
238C3×C8—38no—
239C3×C8—44no—
240C3⋊C8C4×C3⋊D312yescomplemented
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32C4×D1214yes—
245C2×C32#67⋊C411yescharacteristic
246C2×C32C2×#20414yes—
247C2×C32#90414yes—
248C2×C32#91211yescharacteristic
249C2×C32D3×SD1611yescharacteristic
250C3⋊D3—42no—
251C3×D3—128no—
252C2×Q8—271no—
253C2×D4—91no—
254C22×C4—541no—
255SD16—274no—
256C2×C8—31no—
257C2×C8D3×C3⋊D311yescharacteristic
258C4⋊C4—91no—
259C4⋊C4—271no—
260C4⋊C4—541no—
261C22⋊C4—541no—
262C42—31no—
263C42—541no—
264C22×C3—24no—
265C22×C3—182no—
266C22×C3—361no—
267C22×C3—361no—
268C22×C3D3×D1214yes—
269C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
270C2×D3—68no—
271C2×D3—124no—
272C2×D3—124no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—38no—
276C3×C4—64no—
277C3×C4—361no—
278C3×C4C4×D3214yes—
279C3×C4#380414yes—
280C3×C4C2×C4×C3⋊D311yescharacteristic
281C3×C4#382811yescharacteristic
282Dic3—21no—
283Dic3—124no—
284Dic3—182no—
285Dic3—188no—
286Dic3—368no—
287Dic3—364no—
288Dic3—361no—
289Dic3—724no—
290Dic3#374412yescomplemented
291C32—24no—
292C32C4×#20414yescomplemented
293C32#475811yescharacteristic, complemented
294C23—181no—
295Q8—272no—
296Q8—541no—
297D4—92no—
298D4—181no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—181no—
302C2×C4—541no—
303C2×C4—541no—
304C2×C4—1082no—
305C2×C4C2×D3×C3⋊D311yescharacteristic
306C8—21no—
307C8—32no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—24no—
311C2×C3—362no—
312C2×C3D3×#20414yes—
313C2×C3#6380714yes—
314C2×C3#6388014yes—
315C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
316C2×C3C2×#374411yescharacteristic
317C2×C3#6411311yescharacteristic
318D3—128no—
319C22—182no—
320C22—361no—
321C22—361no—
322C22D3×C4⋊(C3⋊D3)11yescharacteristic
323C4—32no—
324C4—61no—
325C4—361no—
326C4—542no—
327C4—1081no—
328C4C4×D3×C3⋊D311yescharacteristic
329C4#8903111yescharacteristic
330C3—24no—
331C3#1059074414yescomplemented
332C3C4×#374411yescharacteristic, complemented
333C2—362no—
334C2D3×#374411yescharacteristic
335C2#1172333611yescharacteristic
336C2#1172347111yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C4×SD16
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. #11723399C2semidirect
4. #11722186C2semidirect
6. C32×C8×Dic3C2semidirect
7. #11722052C2semidirect
21. #88796C4semidirect
35. C3×C8×Dic3D3semidirect
42. #6308C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
90. C32×Dic3SD16semidirect
104. C8×Dic3C3⋊D3semidirect
139. C3×C3⋊C8C4×D3semidirect
198. C3×Dic3#204semidirect
206. C33C4×SD16semidirect
240. C3⋊C8C4×C3⋊D3semidirect
290. Dic3#3744semidirect
292. C32C4×#204semidirect
293. C32#4758semidirect
331. C3#10590744semidirect
332. C3C4×#3744semidirect

Maximal subgroups

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

subgroupindex
Dic3×C4⋊(C3⋊D3)2
#117233992
C2×#887962
#117221862
C3×#636482
C32×C8×Dic32
#117220522
C4×#37443
#105907443