Finite Group Databasebeta

← #13161221

Subgroups — #13161221

Order 1728.

1052 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×#88862C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3C2×C32×#216C211yesmaximal, characteristic, complemented
4#11723440C211yesmaximal, characteristic, complemented
5C2×#88793C211yesmaximal, characteristic, complemented
6C2×#88791C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592890—31nomaximal
9#10591609—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14#88862—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—22no—
21#88793—22no—
22#88791—22no—
23C2×C32×Q16—31no—
24#64156—31no—
25#64156D311yescharacteristic, complemented
26C2×#3755—31no—
27C2×#3755—31no—
28C2×#3755—34no—
29C2×#3755—64no—
30C2×#3745—31no—
31C3×#947—34no—
32C2×C3×#216D314yescomplemented
33#63864—34no—
34C2×#3715—34no—
35C2×#3714—34no—
36C8×C3⋊Dic3—31no—
37Dic3×C3⋊C8—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3D412yes—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C2×C32×Dic3—21no—
50C32×Dic6—21no—
51C32×Dic6—21no—
52C3⋊(C3⋊(C3⋊C8))D412yes—
53C32×C3⋊C8D412yes—
54#5191—94no—
55#5059—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8—31no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×Dic6C2×D314yes—
66C32×Q16—62no—
67C2×C32×C8—31no—
68#3755—62no—
69#3755—62no—
70#3755—68no—
71#3755—616no—
72#3752—22no—
73#3752—62no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3745—62no—
79C3×#216—28no—
80C3×(C2×D3)⋊C4—68no—
81C3×C4×Dic3—34no—
82C2×C3×C3⋊C8C2×D314yes—
83#3722—68no—
84#3715—68no—
85#3714—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—42no—
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—42no—
91C33×C4—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3×C3⋊Dic3—22no—
95C32×Dic3—41no—
96C2×C3×Q16—34no—
97C3×#124—91no—
98#947—34no—
99#947—94no—
100C2×#216C3⋊D311yescharacteristic, complemented
101C2×#215—94no—
102C2×#215—94no—
103C2×#215—91no—
104C2×#215—184no—
105C2×#214—91no—
106C2×#213—91no—
107C2×D24—94no—
108#875—91no—
109C8×Dic3—94no—
110C4×C3⋊C8—91no—
111C22×C3⋊D3—21no—
112C22×C3⋊D3—61no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—124no—
115C22×C3×D3—64no—
116C32×Q8—28no—
117C32×Q8—24no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—24no—
121C32×Q8—44no—
122C32×Q8—61no—
123C32×Q8—61no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—44no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3#6712yes—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—68no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—38no—
142C3×D12—64no—
143C3×D12—64no—
144C3×Dic6—24no—
145C3×Dic6—24no—
146C32×C8—32no—
147C3⋊(C3⋊C8)—32no—
148C3⋊(C3⋊C8)—38no—
149C3⋊(C3⋊C8)—68no—
150C3×C3⋊C8#6718yes—
151#495—271noSylow
152C2×C33#12711yescharacteristic
153C2×C33#13512yes—
154C3⋊(C3⋊D3)—81no—
155C3×C3⋊D3—81no—
156C2×C3×Q8—24no—
157C2×C3×Q8—34no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—91no—
161C2×D12—34no—
162C2×D12—94no—
163C2×D12—91no—
164C2×D12—184no—
165C2×Dic6C2×C3⋊D311yescharacteristic
166C3×Q16—68no—
167C2×C3×C8—34no—
168C3×C22⋊C4—182no—
169C3×C42—91no—
170#216—22no—
171#215—188no—
172#215—188no—
173#215—182no—
174#215—1816no—
175#214—182no—
176#213—182no—
177(C2×D3)⋊C4—68no—
178(C2×D3)⋊C4—182no—
179(C2×D3)⋊C4—188no—
180C4×Dic3—34no—
181C2×C3⋊C8—94no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184C2×C3⋊C8C2×C3⋊D311yescharacteristic
185D24—188no—
186C22×C32—24no—
187C22×C32D3×D414yes—
188C22×C32C2×#6711yescharacteristic
189C2×C3⋊D3—41no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—128no—
192C2×C3⋊D3—124no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—122no—
195C2×C3⋊D3—128no—
196C2×C3⋊D3—248no—
197C2×C3×D3—128no—
198C2×C3×D3—124no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—44no—
203C32×C4—41no—
204C32×C4—121no—
205C32×C4D3×D414yes—
206C32×C4D3×D411yescharacteristic
207C32×C4D3×D411yescharacteristic
208C32×C4C2×#6714yes—
209C3⋊Dic3—22no—
210C3×Dic3—44no—
211C3×Dic3—68no—
212C2×Q16—31no—
213C2×SD16—271no—
214C2×SD16—271no—
215C2×D8—271no—
216#124—91no—
217#124—271no—
218C4×C8—271no—
219C33#49511yescharacteristic, complemented, Sylow
220C23×C3—181no—
221C22×D3—64no—
222C22×D3—184no—
223C22×D3—181no—
224C22×D3—364no—
225C3×Q8—28no—
226C3×Q8—24no—
227C3×Q8—24no—
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—44no—
231C3×Q8—64no—
232C3×Q8—64no—
233C3×D4—181no—
234C3×D4—181no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—44no—
239C2×C3×C4—64no—
240C2×C3×C4—92no—
241C2×C3×C4C2×D3214yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×Dic3—21no—
244C2×Dic3—38no—
245D12—64no—
246D12—64no—
247D12—188no—
248D12—184no—
249D12—184no—
250D12—181no—
251D12—181no—
252D12—364no—
253Dic6—21no—
254Dic6—21no—
255C3×C8—38no—
256C3⋊C8—98no—
257C3⋊C8—92no—
258C3⋊C8—188no—
259C3⋊C8#63812yes—
260C2×C32—28no—
261C2×C32—24no—
262C2×C32#91918yes—
263C2×C32#94014yes—
264C2×C32#94011yescharacteristic
265C2×C32#95012yes—
266C3⋊D3—81no—
267C3⋊D3—248no—
268C3⋊D3—244no—
269C3⋊D3—241no—
270C3×D3—244no—
271C2×Q8—31no—
272C2×Q8D3×C3⋊D311yescharacteristic
273C2×D4—91no—
274C2×D4—271no—
275Q16—62no—
276SD16—542no—
277SD16—542no—
278D8—542no—
279C2×C8—31no—
280C2×C8—271no—
281C22⋊C4—182no—
282C22⋊C4—542no—
283C42—91no—
284C22×C3—24no—
285C22×C3—362no—
286C22×C3—361no—
287C22×C3D3×#6714yes—
288C22×C3D4×C3⋊D311yescharacteristic
289C2×D3—128no—
290C2×D3—124no—
291C2×D3—368no—
292C2×D3—362no—
293C2×D3—361no—
294C2×D3—368no—
295C2×D3—364no—
296C2×D3—728no—
297C3×C4—24no—
298C3×C4—24no—
299C3×C4—48no—
300C3×C4—41no—
301C3×C4—44no—
302C3×C4—124no—
303C3×C4—182no—
304C3×C4D3×#6714yes—
305C3×C4#381114yes—
306C3×C4D4×C3⋊D311yescharacteristic
307C3×C4C2×#63811yescharacteristic
308Dic3—41no—
309Dic3—68no—
310C32—24no—
311C32#505911yescharacteristic, complemented
312C32#519114yescomplemented
313C23—181no—
314C23—541no—
315Q8—21no—
316Q8—21no—
317Q8—61no—
318Q8—61no—
319D4—181no—
320D4—181no—
321D4—541no—
322D4—541no—
323C2×C4—21no—
324C2×C4—61no—
325C2×C4—92no—
326C2×C4C2×D3×C3⋊D311yescharacteristic
327C8—32no—
328C8—272no—
329C2×C3—28no—
330C2×C3—24no—
331C2×C3—721no—
332C2×C3#6394918yes—
333C2×C3#6397914yes—
334C2×C3#6412812yes—
335C2×C3#6414911yescharacteristic
336D3—244no—
337D3—728no—
338D3—724no—
339D3—721no—
340C22—362no—
341C22—361no—
342C22—1082no—
343C22—1081no—
344C22C3⋊D3×#6711yescharacteristic
345C4—41no—
346C4—121no—
347C4—182no—
348C4D3×#63811yescharacteristic
349C4#8903911yescharacteristic
350C3—24no—
351C3#1059160914yescomplemented
352C3#1059289011yescharacteristic, complemented
353C2—721no—
354C2—2161no—
355C2#1172358512yes—
356C2#1172363311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#495
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88862C2semidirect
2. C3×#64156C2semidirect
3. C2×C32×#216C2semidirect
4. #11723440C2semidirect
5. C2×#88793C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
25. #64156D3semidirect
32. C2×C3×#216D3semidirect
100. C2×#216C3⋊D3semidirect
219. C33#495semidirect
311. C32#5059semidirect
312. C32#5191semidirect
351. C3#10591609semidirect
352. C3#10592890semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×#641562
C2×C32×#2162
#117234402
C2×#887932
C2×#887912
C3⋊C8×C3⋊Dic32
#105928903
#105916093