Finite Group Databasebeta

← #13159159

Subgroups — #13159159

Order 1728.

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

normal subgroups only

This group has 780 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#11723522C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic, complemented
5#11722055C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592483—31nomaximal
9#10590505—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6290—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#64147—31no—
22#64101D311yescharacteristic, complemented
23C3×#892—34no—
24#63908—34no—
25C32×#102—31no—
26#63659—31no—
27#63647—31no—
28#63645—31no—
29#63635—31no—
30#63635—34no—
31#63635—64no—
32C3×#809D314yescomplemented
33#63575—34no—
34#63562—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3—21no—
44C3×C4×C3⋊D3—21no—
45C32×D12—21no—
46C32×D12—21no—
47#6290—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5003—91no—
51#4784—94no—
52C2×C32×D4—31no—
53C2×#638—61no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×D12C2×D314yes—
56C2×C3×C4×D3—34no—
57C2×#625—64no—
58C2×C32×C8—31no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D311yescharacteristic
62#3757—61no—
63#3752—21no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4×C3⋊Dic3—21no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×(C2×D3)⋊C4—64no—
73C3×C4⋊Dic3—34no—
74C3×C4×Dic3—64no—
75C2×C3×C3⋊C8C2×D314yes—
76(C2×D3)⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C32×D3—41no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—41no—
90C3×#117—91no—
91#938—94no—
92#930—91no—
93#892—34no—
94C3×#102—34no—
95#832—94no—
96#830—91no—
97(C2×D3)⋊C8—94no—
98#818—94no—
99#809—91no—
100#809C3⋊D311yescharacteristic, complemented
101#808—94no—
102#808—91no—
103#808—184no—
104#807—91no—
105C23×C32—61no—
106C22×C3⋊D3#6711yescharacteristic
107C22×C3×D3—24no—
108C22×C3×D3—34no—
109C32×D4—61no—
110C32×D4—61no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—44no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117#638—122no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—124no—
122C2×C3⋊Dic3#6711yescharacteristic
123C4×C3⋊D3—21no—
124C4×C3⋊D3—21no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—64no—
127C3×D12—24no—
128C3×D12—24no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131#625—128no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—68no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—61no—
136C3×C3⋊C8—24no—
137#482—271noSylow
138C2×C33#12311yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C3×C3⋊D3—41no—
142C32×D3—81no—
143C2×C3×D4—34no—
144C22×C3×C4—91no—
145C2×#67—181no—
146C2×#67—184no—
147C2×D12C2×C3⋊D311yescharacteristic
148C2×C4×D3—34no—
149C2×C3×C8—34no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4D3214yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154C3×C22⋊C4—181no—
155C3×C42—181no—
156#217—184no—
157(C2×D3)⋊C4—64no—
158(C2×D3)⋊C4—181no—
159C4⋊Dic3—34no—
160C4⋊Dic3—94no—
161C4⋊Dic3—91no—
162C4⋊Dic3—184no—
163C4×Dic3—64no—
164C2×C3⋊C8—94no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—184no—
167C2×C3⋊C8C2×C3⋊D311yescharacteristic
168C22×C32—24no—
169C22×C32—121no—
170C22×C32—121no—
171C22×C32—121no—
172C22×C32D3×D414yes—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—21no—
175C2×C3⋊D3—21no—
176C2×C3⋊D3—21no—
177C2×C3×D3—44no—
178C2×C3×D3—44no—
179C2×C3×D3—44no—
180C2×C3×D3—64no—
181C2×C3×D3—64no—
182C2×C3×D3—64no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—41no—
186C32×C4—44no—
187C32×C4—84no—
188C32×C4#23514yes—
189C32×C4#23511yescharacteristic
190C32×C4#23511yescharacteristic
191C32×C4#23714yes—
192C3⋊Dic3—21no—
193C3⋊Dic3—41no—
194C3⋊Dic3—121no—
195C3⋊Dic3—124no—
196C3⋊Dic3—244no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199#121—271no—
200#117—91no—
201#107—271no—
202#106—271no—
203#102—31no—
204#102—271no—
205C22⋊C8—271no—
206C33#48211yescharacteristic, complemented, Sylow
207C23×C3—64no—
208C23×C3—91no—
209C22×D3—21no—
210C22×D3—34no—
211C3×D4—64no—
212C3×D4—64no—
213C2×C3×C4—24no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—44no—
217C2×C3×C4—91no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4C2×D3214yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223#67—362no—
224#67—368no—
225C2×Dic3—34no—
226C2×Dic3—64no—
227C2×Dic3—184no—
228C2×Dic3—181no—
229C2×Dic3—364no—
230D12—21no—
231D12—21no—
232C4×D3—64no—
233C4×D3—64no—
234C3×C8—64no—
235C3⋊C8—21no—
236C3⋊C8—188no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—241no—
243C2×C32#88314yes—
244C2×C32#91214yes—
245C2×C32#91414yes—
246C2×C32#92911yescharacteristic
247C2×C32#94911yescharacteristic
248C2×C32#95011yescharacteristic
249C3⋊D3—41no—
250C3×D3—84no—
251C3×D3—124no—
252C2×D4—31no—
253C2×D4—541no—
254C22×C4—91no—
255C2×C8—31no—
256C2×C8—271no—
257C4⋊C4—91no—
258C4⋊C4—271no—
259C4⋊C4D3×C3⋊D311yescharacteristic
260C22⋊C4—181no—
261C22⋊C4—541no—
262C42—181no—
263C22×C3—24no—
264C22×C3—124no—
265C22×C3—124no—
266C22×C3—124no—
267C22×C3—181no—
268C22×C3—181no—
269C22×C3—181no—
270C22×C3D3×#6714yes—
271C22×C3D4×C3⋊D311yescharacteristic
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—64no—
276C2×D3—64no—
277C2×D3—64no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—41no—
281C3×C4—44no—
282C3×C4—84no—
283C3×C4—181no—
284C3×C4—361no—
285C3×C4#379514yes—
286C3×C4#379814yes—
287C3×C4#382811yescharacteristic
288C3×C4#383011yescharacteristic
289Dic3—64no—
290Dic3—124no—
291Dic3—361no—
292Dic3—364no—
293Dic3—724no—
294C32—24no—
295C32#478414yescomplemented
296C32#500311yescharacteristic, complemented
297C23—61no—
298C23—91no—
299D4—61no—
300D4—61no—
301D4—1082no—
302C2×C4—21no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4—541no—
308C2×C4C2×D3×C3⋊D311yescharacteristic
309C8—61no—
310C8—541no—
311C2×C3—24no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3—244no—
315C2×C3—361no—
316C2×C3#6390514yes—
317C2×C3#6393314yes—
318C2×C3#6393914yes—
319C2×C3#6409211yescharacteristic
320C2×C3#6412111yescharacteristic
321C2×C3#6412311yescharacteristic
322D3—81no—
323D3—124no—
324C22—121no—
325C22—121no—
326C22—121no—
327C22—181no—
328C22—181no—
329C22—181no—
330C22C3⋊D3×#6711yescharacteristic
331C4—41no—
332C4—181no—
333C4—361no—
334C4—1081no—
335C4#8901411yescharacteristic
336C4#8902011yescharacteristic
337C3—24no—
338C3#1059050514yescomplemented
339C3#1059248311yescharacteristic, complemented
340C2—241no—
341C2—361no—
342C2#1172351811yescharacteristic
343C2#1172355911yescharacteristic
344C2#1172357011yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723522C2
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. #11722178C2semidirect
4. C32×#809C2semidirect
5. #11722055C2semidirect
6. #11722031C2semidirect
7. #11722023C2semidirect
22. #64101D3semidirect
32. C3×#809D3semidirect
100. #809C3⋊D3semidirect
206. C33#482semidirect
295. C32#4784semidirect
296. C32#5003semidirect
338. C3#10590505semidirect
339. C3#10592483semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
#117235222
#117221782
C32×#8092
#117220552
#117220312
#117220232
#105924833
#105905053