Finite Group Databasebeta

← #13159224

Subgroups — #13159224

Order 1728.

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

normal subgroups only

This group has 800 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×#64097C211yesmaximal, characteristic, complemented
2#11723440C211yesmaximal, characteristic
3#11722179C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592493—31nomaximal
9#10590535—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634—21no—
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88804—21no—
21#64156—31no—
22#64097D311yescharacteristic, complemented
23C3×#888—34no—
24#63864—34no—
25C32×#103—31no—
26#63662—31no—
27#63648—31no—
28#63642—31no—
29#63636—31no—
30#63636—34no—
31#63636—64no—
32C3×#810D314yescomplemented
33#63565—34no—
34#63564—34no—
35#63554—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C4⋊(C3⋊(C3⋊D3))—21no—
40C4⋊(C3⋊(C3⋊D3))—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×#634—42no—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5089—91no—
51#4794—94no—
52C2×C32×Q8—31no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C2×#634—21no—
57C2×C3×Dic6—64no—
58C2×C3×Dic6C2×D314yes—
59C2×C32×C8—31no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63#3752—61no—
64#3752—61no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×C4⋊Dic3—34no—
72C3×Dic3⋊C4—64no—
73C3×C4×Dic3—34no—
74C2×C3×C3⋊C8C2×D314yes—
75#3722—64no—
76#3722—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3⋊(C3⋊D3)—41no—
79C2×C3⋊(C3⋊D3)—41no—
80C2×C3⋊(C3⋊D3)—41no—
81C33×C4—41no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C32×Dic3—41no—
88C3×C4⋊Q8—91no—
89#947—94no—
90#888—34no—
91#875—91no—
92C3×#103—34no—
93#835—94no—
94#821—94no—
95#815—94no—
96#810—91no—
97#810C3⋊D311yescharacteristic, complemented
98#809—94no—
99#809—91no—
100#809—91no—
101#809—184no—
102#803—91no—
103C22×C3⋊D3—61no—
104C22×C3⋊D3—64no—
105C22×C3⋊D3—124no—
106C32×Q8—61no—
107C32×Q8—61no—
108C2×C32×C4—21no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—44no—
112C2×C32×C4—61no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3⋊Dic3#6711yescharacteristic
118C4⋊(C3⋊D3)—68no—
119C4⋊(C3⋊D3)—64no—
120C4⋊(C3⋊D3)—61no—
121C4⋊(C3⋊D3)—61no—
122C4⋊(C3⋊D3)—64no—
123C4⋊(C3⋊D3)—124no—
124#634—42no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C3×Dic6—24no—
130C3×Dic6—24no—
131C3×Dic6—128no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—61no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—124no—
136C3×C3⋊C8—24no—
137#489—271noSylow
138C2×C33#12411yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C33#13711yescharacteristic
141C3⋊(C3⋊D3)—81no—
142C2×C3×Q8—34no—
143C2×C3×Q8—181no—
144C2×D12—94no—
145C2×D12—91no—
146C2×D12—184no—
147C2×Dic6—64no—
148C2×Dic6C2×C3⋊D311yescharacteristic
149C2×C3×C8—34no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4D3214yes—
154C3×C4⋊C4C2×C3⋊D311yescharacteristic
155C3×C42—91no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—184no—
159(C2×D3)⋊C4—184no—
160C4⋊Dic3—34no—
161Dic3⋊C4—64no—
162C4×Dic3—34no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—184no—
166C2×C3⋊C8C2×C3⋊D311yescharacteristic
167C22×C32—24no—
168C22×C32D3×D414yes—
169C22×C32C2×#6711yescharacteristic
170C2×C3⋊D3—128no—
171C2×C3⋊D3—124no—
172C2×C3⋊D3—121no—
173C2×C3⋊D3—121no—
174C2×C3⋊D3—121no—
175C2×C3⋊D3—124no—
176C2×C3⋊D3—124no—
177C2×C3⋊D3—244no—
178C2×C3⋊D3—244no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—41no—
182C32×C4—44no—
183C32×C4—84no—
184C32×C4—121no—
185C32×C4#23514yes—
186C32×C4#23714yes—
187C32×C4#23911yescharacteristic
188C32×C4#23911yescharacteristic
189C3⋊Dic3—21no—
190C3⋊Dic3—21no—
191C3⋊Dic3—41no—
192C3×Dic3—44no—
193C3×Dic3—64no—
194C3×Dic3—64no—
195C3×Dic3—124no—
196C4⋊Q8—91no—
197#124—271no—
198#103—31no—
199#103—271no—
200#102—271no—
201#102—271no—
202#97—271no—
203C33#48911yescharacteristic, complemented, Sylow
204C22×D3—184no—
205C22×D3—181no—
206C22×D3—364no—
207C3×Q8—64no—
208C3×Q8—64no—
209C3×Q8—362no—
210C2×C3×C4—24no—
211C2×C3×C4—21no—
212C2×C3×C4—24no—
213C2×C3×C4—44no—
214C2×C3×C4—64no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—34no—
222C2×Dic3—34no—
223C2×Dic3—64no—
224D12—188no—
225D12—184no—
226D12—184no—
227D12—181no—
228D12—181no—
229D12—364no—
230Dic6—21no—
231Dic6—21no—
232Dic6—128no—
233C3×C8—64no—
234C3⋊C8—21no—
235C3⋊C8—184no—
236C3⋊C8—181no—
237C3⋊C8—364no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32#88514yes—
242C2×C32#91414yes—
243C2×C32#91814yes—
244C2×C32#93211yescharacteristic
245C2×C32#94211yescharacteristic
246C2×C32#94711yescharacteristic
247C3⋊D3—248no—
248C3⋊D3—244no—
249C3⋊D3—241no—
250C2×Q8—31no—
251C2×Q8—181no—
252C2×D4—271no—
253C2×C8—31no—
254C2×C8—271no—
255C4⋊C4—91no—
256C4⋊C4—181no—
257C4⋊C4D3×C3⋊D311yescharacteristic
258C22⋊C4—541no—
259C22⋊C4—541no—
260C42—91no—
261C22×C3—24no—
262C22×C3D3×#6714yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C2×D3—368no—
265C2×D3—361no—
266C2×D3—361no—
267C2×D3—361no—
268C2×D3—364no—
269C2×D3—364no—
270C2×D3—364no—
271C2×D3—724no—
272C2×D3—724no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—41no—
276C3×C4—44no—
277C3×C4—84no—
278C3×C4—124no—
279C3×C4—181no—
280C3×C4—181no—
281C3×C4—361no—
282C3×C4#379614yes—
283C3×C4#379914yes—
284C3×C4#382811yescharacteristic
285C3×C4#383011yescharacteristic
286Dic3—41no—
287Dic3—64no—
288Dic3—64no—
289Dic3—124no—
290C32—24no—
291C32#479414yescomplemented
292C32#508911yescharacteristic, complemented
293C23—541no—
294Q8—61no—
295Q8—61no—
296Q8—362no—
297D4—541no—
298D4—541no—
299C2×C4—21no—
300C2×C4—61no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—181no—
304C2×C4C2×D3×C3⋊D311yescharacteristic
305C8—61no—
306C8—541no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3#6390714yes—
311C2×C3#6393414yes—
312C2×C3#6394814yes—
313C2×C3#6409411yescharacteristic
314C2×C3#6412311yescharacteristic
315C2×C3#6412711yescharacteristic
316D3—728no—
317D3—724no—
318D3—721no—
319C22—1081no—
320C22—1081no—
321C22—1081no—
322C22C3⋊D3×#6711yescharacteristic
323C4—41no—
324C4—121no—
325C4—181no—
326C4—181no—
327C4—361no—
328C4#8901511yescharacteristic
329C4#8902211yescharacteristic
330C3—24no—
331C3#1059053514yescomplemented
332C3#1059249311yescharacteristic, complemented
333C2—2161no—
334C2#1172352111yescharacteristic
335C2#1172356211yescharacteristic
336C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#489
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×#64097C2semidirect
4. C32×#810C2semidirect
5. #11722037C2semidirect
7. #11722020C2semidirect
22. #64097D3semidirect
32. C3×#810D3semidirect
97. #810C3⋊D3semidirect
203. C33#489semidirect
291. C32#4794semidirect
292. C32#5089semidirect
331. C3#10590535semidirect
332. C3#10592493semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
#117234402
#117221792
C32×#8102
#117220372
#117220362
#117220202
#105924933
#105905353