Finite Group Databasebeta

← #13161271

Subgroups — #13161271

Order 1728.

888 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 156 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×C3×#3755C211yesmaximal, characteristic, complemented
2C32×#946C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic
4C2×#88786C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592748—31nomaximal
9#10591587—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6290—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3755—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4—21no—
21#88803—21no—
22#88786—22no—
23C32×C22⋊Q8—31no—
24C2×#3755D311yescharacteristic, complemented
25#64139—31no—
26C2×#3748—31no—
27C3×#946D314yescomplemented
28C2×C3×#215—34no—
29#63908—34no—
30C2×#3711—34no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63636—31no—
35#63634—31no—
36#63575—34no—
37#63562—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3D411yescharacteristic
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3D411yescharacteristic
49C32×C4×D3—21no—
50C32×C4×D3—21no—
51#6290—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54#5174—91no—
55#5049—94no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C22×C32×C4—31no—
60C2×#638—61no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—34no—
63C2×C3×C4×D3C2×D314yes—
64C2×#625—64no—
65C32×C4⋊C4—31no—
66C32×C4⋊C4—61no—
67C32×C22⋊C4—61no—
68#3755—22no—
69#3752—61no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—34no—
72C4⋊(C3⋊Dic3)—64no—
73#3748—62no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×#215—68no—
79C3×(C2×D3)⋊C4—24no—
80C3×C4⋊Dic3C2×D314yes—
81C3×Dic3⋊C4—24no—
82C2×C3×C3⋊C8—34no—
83(C2×D3)⋊Dic3—64no—
84#3711—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C32×D3—21no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C32×Dic3—21no—
97C32×Dic3—41no—
98C2×C3×SD16—91no—
99C3×C22⋊Q8—34no—
100#946C3⋊D311yescharacteristic, complemented
101C2×#215—34no—
102#938—91no—
103#930—94no—
104C2×#208—94no—
105C2×ΓL(1,𝔽25)—91no—
106#835—94no—
107#835—91no—
108#835—184no—
109#832—91no—
110#818—91no—
111#809—94no—
112#807—94no—
113C23×C32—31no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C22×C3×D3#6714yes—
117C32×Q8—28no—
118C32×Q8—24no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—24no—
122C32×Q8—44no—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—24no—
126C2×C32×C4—31no—
127C2×C32×C4—44no—
128C2×C32×C4—61no—
129C2×C32×C4—61no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133#638—122no—
134C2×C3⋊Dic3—61no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—124no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3#6714yes—
141C3×D12—64no—
142C3×D12—64no—
143C3×C4×D3—24no—
144C3×C4×D3—24no—
145#625—128no—
146C3⋊(C3⋊C8)—61no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—124no—
149C3⋊(C3⋊C8)#6712yescomplemented
150C3×C3⋊C8—38no—
151#468—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C2×C33#13711yescharacteristic
155C3×C3⋊D3—81no—
156C32×D3—41no—
157C2×C3×Q8—24no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—91no—
161C22×C3×C4—34no—
162C2×#67—184no—
163C2×#67—181no—
164C2×D12—34no—
165C2×C4×D3C2×C3⋊D311yescharacteristic
166C3×SD16—182no—
167C2×C3×C8—91no—
168C3×C4⋊C4—34no—
169C3×C4⋊C4—64no—
170C3×C22⋊C4—64no—
171#217—181no—
172#215—68no—
173(C2×D3)⋊C4—21no—
174(C2×D3)⋊C4—184no—
175C4⋊Dic3—94no—
176C4⋊Dic3—91no—
177C4⋊Dic3—184no—
178C4⋊Dic3C2×C3⋊D311yescharacteristic
179Dic3⋊C4—21no—
180#208—188no—
181C2×C3⋊C8—34no—
182C2×C3⋊C8—94no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—184no—
185ΓL(1,𝔽25)—182no—
186C22×C32—24no—
187C22×C32—61no—
188C22×C32—61no—
189C22×C32—61no—
190C22×C32D3×D411yescharacteristic
191C22×C32C2×#6714yes—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3×D3—24no—
196C2×C3×D3—24no—
197C2×C3×D3—24no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C2×C3×D3—124no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—48no—
204C32×C4—41no—
205C32×C4—44no—
206C32×C4—61no—
207C32×C4—121no—
208C32×C4D3×D414yes—
209C32×C4#23714yes—
210C32×C4#23711yescharacteristic
211C32×C4C2×#6711yescharacteristic
212C3⋊Dic3—121no—
213C3⋊Dic3—124no—
214C3⋊Dic3—244no—
215C3×Dic3—24no—
216C3×Dic3—44no—
217C2×SD16—91no—
218C2×M16—271no—
219C22⋊Q8—31no—
220#121—271no—
221#107—271no—
222#103—271no—
223#102—271no—
224C33#46811yescharacteristic, complemented, Sylow
225C23×C3—34no—
226C23×C3—181no—
227C22×D3—64no—
228C22×D3#63811yescharacteristic
229C3×Q8—28no—
230C3×Q8—24no—
231C3×Q8—24no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—44no—
235C3×D4—181no—
236C3×D4—181no—
237C2×C3×C4—24no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—34no—
241C2×C3×C4—44no—
242C2×C3×C4—64no—
243C2×C3×C4—64no—
244C2×C3×C4—64no—
245C2×C3×C4C2×D3214yes—
246C2×C3×C4C22×C3⋊D311yescharacteristic
247#67—362no—
248#67—368no—
249C2×Dic3—21no—
250C2×Dic3—184no—
251C2×Dic3—181no—
252C2×Dic3—364no—
253C2×Dic3#63811yescharacteristic
254D12—64no—
255D12—64no—
256C4×D3—21no—
257C4×D3—21no—
258C3×C8—92no—
259C3⋊C8—38no—
260C3⋊C8—184no—
261C3⋊C8—181no—
262C3⋊C8—364no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—121no—
267C2×C32#91411yescharacteristic
268C2×C32#91511yescharacteristic
269C2×C32#93811yescharacteristic
270C2×C32#93914yes—
271C2×C32#94214yes—
272C2×C32#94914yes—
273C3⋊D3—81no—
274C3×D3—44no—
275C3×D3—244no—
276C2×Q8D3×C3⋊D311yescharacteristic
277C2×D4—91no—
278C2×D4—541no—
279C22×C4—31no—
280SD16—182no—
281M16—542no—
282C2×C8—91no—
283C2×C8—271no—
284C4⋊C4—31no—
285C4⋊C4—61no—
286C4⋊C4—271no—
287C22⋊C4—61no—
288C22⋊C4—541no—
289C22×C3—24no—
290C22×C3—64no—
291C22×C3—64no—
292C22×C3—64no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D3×#6714yes—
297C22×C3C2×#63811yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—124no—
302C2×D3—124no—
303C2×D3—124no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—48no—
307C3×C4—41no—
308C3×C4—44no—
309C3×C4—64no—
310C3×C4—124no—
311C3×C4#380514yes—
312C3×C4D3×#6714yes—
313C3×C4D4×C3⋊D311yescharacteristic
314C3×C4#383011yescharacteristic
315Dic3—21no—
316Dic3—41no—
317Dic3—361no—
318Dic3—364no—
319Dic3—724no—
320C32—24no—
321C32#504914yescomplemented
322C32#517411yescharacteristic, complemented
323C23—31no—
324C23—181no—
325Q8—21no—
326Q8—21no—
327D4—181no—
328D4—181no—
329D4—1082no—
330C2×C4—21no—
331C2×C4—31no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—541no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—92no—
338C8—541no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—124no—
343C2×C3—721no—
344C2×C3#6393914yes—
345C2×C3#6394614yes—
346C2×C3#6397014yes—
347C2×C3#6414811yescharacteristic
348C2×C3#6415111yescharacteristic
349C2×C3#6415811yescharacteristic
350D3—41no—
351D3—244no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—361no—
356C22—361no—
357C22—361no—
358C22D3×#63811yescharacteristic
359C4—41no—
360C4—61no—
361C4—121no—
362C4—1081no—
363C4#8902911yescharacteristic
364C4C3⋊D3×#6711yescharacteristic
365C3—24no—
366C3#1059158714yescomplemented
367C3#1059274811yescharacteristic, complemented
368C2—121no—
369C2—721no—
370C2#1172356811yescharacteristic
371C2#1172357911yescharacteristic
372C2#1172361511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723515C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#468
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×C3×#3755C2semidirect
2. C32×#946C2semidirect
4. C2×#88786C2semidirect
5. #11722205C2semidirect
6. #11722057C2semidirect
7. #11722033C2semidirect
24. C2×#3755D3semidirect
27. C3×#946D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
100. #946C3⋊D3semidirect
149. C3⋊(C3⋊C8)#67semidirect
224. C33#468semidirect
321. C32#5049semidirect
322. C32#5174semidirect
366. C3#10591587semidirect
367. C3#10592748semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37552
C32×#9462
#117235152
C2×#887862
#117222052
#117220572
#117220332
#105927483
#105915873