Finite Group Databasebeta

← #13159161

Subgroups — #13159161

Order 1728.

836 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 144 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, complemented
2#11723526C211yesmaximal, characteristic
3#11722179C211yesmaximal, characteristic, complemented
4C32×#808C211yesmaximal, characteristic, complemented
5#11722056C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592565—31nomaximal
9#10590425—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6292—21no—
13C33×C4⋊C4C2211yescharacteristic
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#88804—21no—
21#64104—31no—
22#64101D311yescharacteristic, complemented
23C3×#892—34no—
24#63909—34no—
25C32×#106—31no—
26#63648—31no—
27#63647—31no—
28#63636—31no—
29#63636—34no—
30#63636—64no—
31#63634—31no—
32C3×#808D314yescomplemented
33#63576—34no—
34#63564—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3—21no—
45C3×C4×C3⋊D3—21no—
46C2×C32×Dic3—21no—
47#6292—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5003—91no—
51#4866—94no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4⋊(C3⋊D3)—34no—
54C2×C4⋊(C3⋊D3)—64no—
55C2×C4⋊(C3⋊D3)—61no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×C4×D3—34no—
58C2×#626—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4—31no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C2×D311yescharacteristic
64#3752—21no—
65#3752—61no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3—21no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—64no—
75C2×C3×C3⋊C8C2×D314yes—
76#3722—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3⋊(C3⋊D3)—41no—
79C2×C3⋊(C3⋊D3)—41no—
80C2×C3⋊(C3⋊D3)—41no—
81C2×C3×C3⋊D3—21no—
82C2×C3×C3⋊D3—21no—
83C2×C3×C3⋊D3—21no—
84C33×C4—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—41no—
90C3×#117—91no—
91#930—91no—
92#895—94no—
93#892—34no—
94C3×#106—34no—
95#830—91no—
96#821—94no—
97(C2×D3)⋊C8—94no—
98#809—94no—
99#809—91no—
100#809—91no—
101#809—184no—
102#808C3⋊D311yescharacteristic, complemented
103#807—94no—
104#807—91no—
105C22×C3⋊D3—61no—
106C22×C3⋊D3—64no—
107C22×C3⋊D3—124no—
108C22×C3⋊D3#6711yescharacteristic
109C22×C3×D3—34no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—44no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3#6711yescharacteristic
119C4⋊(C3⋊D3)—68no—
120C4⋊(C3⋊D3)—64no—
121C4⋊(C3⋊D3)—61no—
122C4⋊(C3⋊D3)—61no—
123C4⋊(C3⋊D3)—64no—
124C4⋊(C3⋊D3)—124no—
125C4⋊(C3⋊D3)—122no—
126C4×C3⋊D3—21no—
127C4×C3⋊D3—21no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—64no—
131C3×C4×D3—64no—
132C3×C4×D3—64no—
133#626—128no—
134C32×C8—61no—
135C3⋊(C3⋊C8)—61no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—124no—
138C3×C3⋊C8—24no—
139#482—271noSylow
140C2×C33#12311yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C3⋊(C3⋊D3)—81no—
144C3×C3⋊D3—41no—
145C22×C3×C4—91no—
146C2×#67—181no—
147C2×D12—94no—
148C2×D12—91no—
149C2×D12—184no—
150C2×D12—184no—
151C2×C4×D3—34no—
152C2×C3×C8—34no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4D3214yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158C3×C22⋊C4—181no—
159C3×C42—181no—
160(C2×D3)⋊C4—64no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—184no—
163C4⋊Dic3—34no—
164C4⋊Dic3C2×C3⋊D311yescharacteristic
165C4×Dic3—64no—
166C2×C3⋊C8—94no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—184no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C22×C32—24no—
171C22×C32D3×D414yes—
172C22×C32C2×#6711yescharacteristic
173C2×C3⋊D3—21no—
174C2×C3⋊D3—21no—
175C2×C3⋊D3—21no—
176C2×C3⋊D3—128no—
177C2×C3⋊D3—124no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—244no—
184C2×C3⋊D3—244no—
185C2×C3×D3—64no—
186C2×C3×D3—64no—
187C2×C3×D3—64no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—41no—
191C32×C4—44no—
192C32×C4—84no—
193C32×C4—121no—
194C32×C4#23514yes—
195C32×C4#23511yescharacteristic
196C32×C4#23511yescharacteristic
197C32×C4#23914yes—
198C3⋊Dic3—21no—
199C3⋊Dic3—41no—
200C3×Dic3—44no—
201C3×Dic3—64no—
202C3×Dic3—124no—
203#121—271no—
204#117—91no—
205#107—271no—
206#106—31no—
207#102—271no—
208#102—271no—
209C22⋊C8—271no—
210C33#48211yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—34no—
213C22×D3—184no—
214C22×D3—181no—
215C22×D3—364no—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—44no—
220C2×C3×C4—64no—
221C2×C3×C4—91no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227#67—362no—
228C2×Dic3—21no—
229C2×Dic3—34no—
230C2×Dic3—64no—
231D12—188no—
232D12—184no—
233D12—184no—
234D12—181no—
235D12—181no—
236D12—364no—
237D12—368no—
238C4×D3—64no—
239C4×D3—64no—
240C3×C8—64no—
241C3⋊C8—21no—
242C3⋊C8—184no—
243C3⋊C8—181no—
244C3⋊C8—364no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32#89414yes—
249C2×C32#91414yes—
250C2×C32#91614yes—
251C2×C32#92911yescharacteristic
252C2×C32#94911yescharacteristic
253C2×C32#95011yescharacteristic
254C3⋊D3—41no—
255C3⋊D3—248no—
256C3⋊D3—244no—
257C3⋊D3—241no—
258C3×D3—124no—
259C2×D4—271no—
260C2×D4—541no—
261C22×C4—91no—
262C2×C8—31no—
263C2×C8—271no—
264C4⋊C4—31no—
265C4⋊C4—91no—
266C4⋊C4D3×C3⋊D311yescharacteristic
267C22⋊C4—181no—
268C22⋊C4—541no—
269C42—181no—
270C22×C3—24no—
271C22×C3—181no—
272C22×C3—181no—
273C22×C3—181no—
274C22×C3D3×#6714yes—
275C22×C3D4×C3⋊D311yescharacteristic
276C2×D3—64no—
277C2×D3—64no—
278C2×D3—64no—
279C2×D3—368no—
280C2×D3—361no—
281C2×D3—364no—
282C2×D3—361no—
283C2×D3—361no—
284C2×D3—364no—
285C2×D3—364no—
286C2×D3—724no—
287C2×D3—724no—
288C3×C4—24no—
289C3×C4—24no—
290C3×C4—41no—
291C3×C4—44no—
292C3×C4—84no—
293C3×C4—124no—
294C3×C4—181no—
295C3×C4—361no—
296C3×C4#379814yes—
297C3×C4#379914yes—
298C3×C4#382811yescharacteristic
299C3×C4#383211yescharacteristic
300Dic3—41no—
301Dic3—64no—
302Dic3—124no—
303C32—24no—
304C32#486614yescomplemented
305C32#500311yescharacteristic, complemented
306C23—91no—
307C23—541no—
308D4—541no—
309D4—541no—
310D4—1082no—
311C2×C4—21no—
312C2×C4—61no—
313C2×C4—91no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—61no—
319C8—541no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—361no—
324C2×C3#6385014yes—
325C2×C3#6393514yes—
326C2×C3#6393914yes—
327C2×C3#6410311yescharacteristic
328C2×C3#6412311yescharacteristic
329C2×C3#6412511yescharacteristic
330D3—124no—
331D3—728no—
332D3—721no—
333D3—724no—
334C22—181no—
335C22—181no—
336C22—181no—
337C22—1081no—
338C22—1081no—
339C22—1081no—
340C22C3⋊D3×#6711yescharacteristic
341C4—41no—
342C4—121no—
343C4—181no—
344C4—361no—
345C4#8901811yescharacteristic
346C4#8902011yescharacteristic
347C3—24no—
348C3#1059042514yescomplemented
349C3#1059256511yescharacteristic, complemented
350C2—361no—
351C2—2161no—
352C2#1172341711yescharacteristic
353C2#1172356111yescharacteristic
354C2#1172357011yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723526C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#482
LCSG ▷ C2×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
1. C3×#64101C2semidirect
3. #11722179C2semidirect
4. C32×#808C2semidirect
5. #11722056C2semidirect
6. #11722036C2semidirect
7. #11722023C2semidirect
22. #64101D3semidirect
32. C3×#808D3semidirect
102. #808C3⋊D3semidirect
210. C33#482semidirect
304. C32#4866semidirect
305. C32#5003semidirect
348. C3#10590425semidirect
349. C3#10592565semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
#117235262
#117221792
C32×#8082
#117220562
#117220362
#117220232
#105925653
#105904253