Finite Group Databasebeta

← #13159385

Subgroups — #13159385

Order 1728.

888 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C32×#893C211yesmaximal, characteristic
2#11723399C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic
4#11722180C211yesmaximal, characteristic, complemented
5C3×#63634C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic
7#11722019C211yesmaximal, characteristic, complemented
8#10592717—31nomaximal
9#10590590—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#88794C414yescomplemented
22C32×C4×D4—31no—
23C2×#3754—31no—
24C4×#634—31no—
25C3×#893D314yes—
26#63844—34no—
27C2×#3716—34no—
28#63659—31no—
29#63637—31no—
30#63637—34no—
31#63637—64no—
32#63634D311yescharacteristic, complemented
33#63630—31no—
34C3×#807—34no—
35#63562—34no—
36#63554—34no—
37C22×C32×D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41#6308—21no—
42#6308C2×C412yescomplemented
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46C32×D12—21no—
47C32×D12C2×C412yes—
48C32×C4×D3—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51#5018—94no—
52#4889—91no—
53C2×C32×D4—31no—
54C22×C32×C4—61no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C3×D12C2×D314yes—
59C2×C3×C4×D3—24no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C22⋊C4—61no—
64C32×C42—31no—
65#3754—34no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68C4×C3⋊Dic3—61no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—24no—
74C3×C4⋊Dic3—34no—
75C3×C4×Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77#3724—64no—
78Dic32—64no—
79#3716—316no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—22no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—21no—
90C32×Dic3—21no—
91C3×C4×D4—34no—
92C2×#214—94no—
93C2×#204—91no—
94#893C3⋊D311yescharacteristic
95#887—91no—
96C4×Dic6—94no—
97C3×#107—91no—
98#832—94no—
99#815—91no—
100#810—94no—
101#810—91no—
102#810—184no—
103#809—91no—
104#807—34no—
105#803—94no—
106C23×C32—61no—
107C22×C3×D3—24no—
108C32×D4—32no—
109C32×D4—61no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—44no—
116C2×C32×C4—122no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123#634—32no—
124#634—38no—
125#634—68no—
126#634—61no—
127#634—64no—
128#634—124no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3#6714yes—
131C2×C3×Dic3#6714yes—
132C3×D12—24no—
133C3×D12C4×D318yes—
134C3×C4×D3—48no—
135C3⋊(C3⋊C8)—61no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—124no—
138C3⋊(C3⋊C8)C4×D312yescomplemented
139C3×C3⋊C8—38no—
140#440—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C32×D3—42no—
145C2×C3×D4—34no—
146C22×C3×C4—64no—
147C2×D12C2×C3⋊D311yescharacteristic
148C2×C4×D3—21no—
149C2×Dic6—94no—
150C2×Dic6—91no—
151C2×Dic6—184no—
152C2×C3×C8—91no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4D3214yes—
156C3×C4⋊C4C2×C3⋊D311yescharacteristic
157C3×C22⋊C4—64no—
158C3×C42—34no—
159#214—916no—
160(C2×D3)⋊C4—21no—
161C4⋊Dic3—34no—
162Dic3⋊C4—184no—
163Dic3⋊C4—181no—
164C4×Dic3—184no—
165C4×Dic3—181no—
166C4×Dic3C2×C3⋊D311yescharacteristic
167C2×C3⋊C8—34no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
171#204—94no—
172C22×C32—24no—
173C22×C32—62no—
174C22×C32—121no—
175C22×C32—121no—
176C22×C32D3×D411yescharacteristic
177C22×C32C2×#6714yes—
178C2×C3×D3—28no—
179C2×C3×D3—44no—
180C2×C3×D3—44no—
181C32×C4—24no—
182C32×C4—24no—
183C32×C4—41no—
184C32×C4—44no—
185C32×C4—61no—
186C32×C4—61no—
187C32×C4—84no—
188C32×C4C2×C4×D314yes—
189C32×C4C2×C4×D311yescharacteristic
190C32×C4#23514yes—
191C32×C4#23911yescharacteristic
192C3⋊Dic3—41no—
193C3⋊Dic3—62no—
194C3⋊Dic3—68no—
195C3⋊Dic3—128no—
196C3×Dic3—24no—
197C3×Dic3—24no—
198C3×Dic3—124no—
199C2×SD16—271no—
200C4×Q8—271no—
201C4×D4—31no—
202#107—91no—
203#103—271no—
204#102—271no—
205#97—271no—
206C33#44011yescharacteristic, complemented, Sylow
207C23×C3—64no—
208C22×D3—21no—
209C3×D4—38no—
210C3×D4—64no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—24no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—44no—
217C2×C3×C4—128no—
218C2×C3×C4—181no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—64no—
222C2×Dic3—184no—
223C2×Dic3—181no—
224C2×Dic3—364no—
225C2×Dic3#63811yescharacteristic
226C2×Dic3#63811yescharacteristic
227D12—21no—
228D12C4×C3⋊D312yes—
229C4×D3—42no—
230Dic6—98no—
231Dic6—92no—
232Dic6—188no—
233Dic6—184no—
234Dic6—181no—
235Dic6—364no—
236C3×C8—92no—
237C3⋊C8—38no—
238C3⋊C8—184no—
239C3⋊C8—181no—
240C3⋊C8—364no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—122no—
245C2×C32#89311yescharacteristic
246C2×C32#91111yescharacteristic
247C2×C32#91811yescharacteristic
248C2×C32C4×#6714yes—
249C2×C32#93214yes—
250C2×C32#95114yes—
251C3×D3—48no—
252C2×Q8—271no—
253C2×D4—31no—
254C22×C4—61no—
255SD16—274no—
256C2×C8—91no—
257C2×C8—271no—
258C4⋊C4—91no—
259C4⋊C4—541no—
260C4⋊C4D3×C3⋊D311yescharacteristic
261C22⋊C4—61no—
262C42—31no—
263C42—541no—
264C22×C3—24no—
265C22×C3—68no—
266C22×C3—124no—
267C22×C3—124no—
268C22×C3D3×#6714yes—
269C22×C3C2×#63811yescharacteristic
270C2×D3—22no—
271C2×D3—41no—
272C2×D3—41no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—41no—
276C3×C4—44no—
277C3×C4—64no—
278C3×C4—64no—
279C3×C4—84no—
280C3×C4—361no—
281C3×C4#379914yes—
282C3×C4C4×D3214yes—
283C3×C4C2×C4×C3⋊D311yescharacteristic
284C3×C4#382811yescharacteristic
285Dic3—21no—
286Dic3—21no—
287Dic3—124no—
288Dic3—182no—
289Dic3—188no—
290Dic3—368no—
291C32—24no—
292C32#488911yescharacteristic, complemented
293C32#501814yescomplemented
294C23—61no—
295Q8—272no—
296Q8—541no—
297D4—32no—
298D4—61no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—122no—
303C2×C4—181no—
304C2×C4—541no—
305C2×C4C2×D3×C3⋊D311yescharacteristic
306C8—92no—
307C8—541no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—24no—
311C2×C3—128no—
312C2×C3#6390414yes—
313C2×C3#6393214yes—
314C2×C3#6394314yes—
315C2×C3C4×#63811yescharacteristic
316C2×C3#6414111yescharacteristic
317C2×C3#6416011yescharacteristic
318D3—42no—
319C22—62no—
320C22—121no—
321C22—121no—
322C22D3×#63811yescharacteristic
323C4—41no—
324C4—61no—
325C4—61no—
326C4—361no—
327C4—542no—
328C4#8902211yescharacteristic
329C4C4×D3×C3⋊D311yescharacteristic
330C3—24no—
331C3#1059059014yescomplemented
332C3#1059271711yescharacteristic, complemented
333C2—122no—
334C2#1172351011yescharacteristic
335C2#1172355511yescharacteristic
336C2#1172357411yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#440
LCSG ▷ 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
2. #11723399C2semidirect
4. #11722180C2semidirect
5. C3×#63634C2semidirect
7. #11722019C2semidirect
21. #88794C4semidirect
32. #63634D3semidirect
42. #6308C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
138. C3⋊(C3⋊C8)C4×D3semidirect
206. C33#440semidirect
292. C32#4889semidirect
293. C32#5018semidirect
331. C3#10590590semidirect
332. C3#10592717semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
#117233992
C2×#887942
#117221802
C3×#636342
#117220312
#117220192
#105927173
#105905903