Finite Group Databasebeta

← #13159220

Subgroups — #13159220

Order 1728.

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

normal subgroups only

This group has 768 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#11723429C211yesmaximal, characteristic
3#11722180C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic
5#11722042C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592464—31nomaximal
9#10590531—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634—21no—
12C2×C32×D12C2211yescharacteristic
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(C2×D3)⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#64145—31no—
22#64097D311yescharacteristic, complemented
23C3×#888—34no—
24#63860—34no—
25C32×#102—31no—
26#63659—31no—
27#63643—31no—
28#63642—31no—
29#63637—31no—
30#63637—34no—
31#63637—64no—
32C3×#809D314yes—
33#63567—34no—
34#63562—34no—
35#63554—34no—
36C22×C32×D3—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40#6308—21no—
41#6308—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×#634—42no—
46C32×D12—21no—
47C32×D12—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5089—91no—
51#4765—94no—
52C2×C32×D4—31no—
53C2×#634—21no—
54C2×#634—31no—
55C2×#634—34no—
56C2×#634—64no—
57C2×C3×D12C2×D314yes—
58C2×C3×Dic6—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63#3757—61no—
64#3757—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(C2×D3)⋊Dic3—64no—
76(C2×D3)⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—41no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C33×C4—41no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—21no—
87C3×C3⋊Dic3—41no—
88C3×C4⋊Q8—91no—
89#936—94no—
90#888—34no—
91#875—91no—
92C3×#102—34no—
93#832—94no—
94#816—94no—
95#815—94no—
96#810—94no—
97#810—91no—
98#810—91no—
99#810—184no—
100#809—91no—
101#809C3⋊D311yescharacteristic
102#803—91no—
103C23×C32—61no—
104C22×C3×D3—24no—
105C32×D4—61no—
106C32×D4—61no—
107C2×C32×C4—21no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—44no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—124no—
117C2×C3⋊Dic3#6711yescharacteristic
118C2×C3⋊Dic3#6711yescharacteristic
119#634—42no—
120#634—68no—
121#634—64no—
122#634—61no—
123#634—61no—
124#634—64no—
125#634—124no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C3×D12—24no—
130C3×D12—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
141C32×D3—81no—
142C2×C3×Q8—181no—
143C2×C3×D4—34no—
144C2×D12C2×C3⋊D311yescharacteristic
145C2×Dic6—64no—
146C2×Dic6—94no—
147C2×Dic6—91no—
148C2×Dic6—184no—
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#217—184no—
157#217—184no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
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×C32—121no—
169C22×C32—121no—
170C22×C32—121no—
171C22×C32D3×D414yes—
172C22×C32C2×#6711yescharacteristic
173C2×C3×D3—44no—
174C2×C3×D3—44no—
175C2×C3×D3—44no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—41no—
179C32×C4—44no—
180C32×C4—84no—
181C32×C4#23514yes—
182C32×C4#23714yes—
183C32×C4#23911yescharacteristic
184C32×C4#23911yescharacteristic
185C3⋊Dic3—21no—
186C3⋊Dic3—21no—
187C3⋊Dic3—41no—
188C3⋊Dic3—128no—
189C3⋊Dic3—124no—
190C3⋊Dic3—121no—
191C3×Dic3—64no—
192C3×Dic3—64no—
193C3×Dic3—124no—
194C4⋊Q8—91no—
195#124—271no—
196#103—271no—
197#103—271no—
198#102—31no—
199#102—271no—
200#97—271no—
201C33#48911yescharacteristic, complemented, Sylow
202C23×C3—64no—
203C22×D3—21no—
204C3×Q8—362no—
205C3×D4—64no—
206C3×D4—64no—
207C2×C3×C4—24no—
208C2×C3×C4—21no—
209C2×C3×C4—24no—
210C2×C3×C4—44no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—181no—
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216C2×Dic3—34no—
217C2×Dic3—34no—
218C2×Dic3—64no—
219C2×Dic3—184no—
220C2×Dic3—181no—
221C2×Dic3—364no—
222D12—21no—
223D12—21no—
224Dic6—128no—
225Dic6—188no—
226Dic6—184no—
227Dic6—184no—
228Dic6—181no—
229Dic6—181no—
230Dic6—364no—
231C3×C8—64no—
232C3⋊C8—21no—
233C3⋊C8—184no—
234C3⋊C8—181no—
235C3⋊C8—364no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—241no—
240C2×C32#88514yes—
241C2×C32#91114yes—
242C2×C32#91514yes—
243C2×C32#93211yescharacteristic
244C2×C32#94211yescharacteristic
245C2×C32#94711yescharacteristic
246C3×D3—84no—
247C2×Q8—181no—
248C2×Q8—271no—
249C2×D4—31no—
250C2×C8—31no—
251C2×C8—271no—
252C4⋊C4—91no—
253C4⋊C4—181no—
254C4⋊C4D3×C3⋊D311yescharacteristic
255C22⋊C4—541no—
256C22⋊C4—541no—
257C42—91no—
258C22×C3—24no—
259C22×C3—124no—
260C22×C3—124no—
261C22×C3—124no—
262C22×C3D3×#6714yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—41no—
270C3×C4—44no—
271C3×C4—84no—
272C3×C4—181no—
273C3×C4—181no—
274C3×C4—361no—
275C3×C4#379614yes—
276C3×C4#379914yes—
277C3×C4#382811yescharacteristic
278C3×C4#383011yescharacteristic
279Dic3—64no—
280Dic3—64no—
281Dic3—124no—
282Dic3—368no—
283Dic3—361no—
284Dic3—364no—
285C32—24no—
286C32#476514yescomplemented
287C32#508911yescharacteristic, complemented
288C23—61no—
289Q8—362no—
290Q8—541no—
291Q8—541no—
292D4—61no—
293D4—61no—
294C2×C4—21no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—181no—
298C2×C4—541no—
299C2×C4C2×D3×C3⋊D311yescharacteristic
300C8—61no—
301C8—541no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—244no—
306C2×C3#6390714yes—
307C2×C3#6393214yes—
308C2×C3#6394614yes—
309C2×C3#6409411yescharacteristic
310C2×C3#6412011yescharacteristic
311C2×C3#6412411yescharacteristic
312D3—81no—
313C22—121no—
314C22—121no—
315C22—121no—
316C22C3⋊D3×#6711yescharacteristic
317C4—41no—
318C4—181no—
319C4—181no—
320C4—361no—
321C4—1081no—
322C4#8901511yescharacteristic
323C4#8902211yescharacteristic
324C3—24no—
325C3#1059053114yescomplemented
326C3#1059246411yescharacteristic, complemented
327C2—241no—
328C2#1172352111yescharacteristic
329C2#1172356011yescharacteristic
330C2#1172358411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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
3. #11722180C2semidirect
5. #11722042C2semidirect
7. #11722020C2semidirect
22. #64097D3semidirect
201. C33#489semidirect
286. C32#4765semidirect
287. C32#5089semidirect
325. C3#10590531semidirect
326. C3#10592464semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
#117234292
#117221802
C32×#8092
#117220422
#117220312
#117220202
#105924643
#105905313