Finite Group Databasebeta

← #13159158

Subgroups — #13159158

Order 1728.

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

normal subgroups only

This group has 920 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
1C3×#64106C211yesmaximal, characteristic, complemented
2#11723526C211yesmaximal, characteristic
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88787C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C32×#807C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592585—31nomaximal
9#10590585—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×#6292—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—21no—
21#88797—22no—
22#88787—22no—
23C2×#3744—31no—
24C2×#3743—31no—
25#64106D311yescharacteristic, complemented
26#64104—31no—
27C3×#897—34no—
28#63909—34no—
29C2×#3717—34no—
30C2×#3711—34no—
31C32×#107—31no—
32#63637—31no—
33#63636—31no—
34#63636—34no—
35#63636—64no—
36C3×#807D314yescomplemented
37#63565—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C3×#634—21no—
49C3×#634—21no—
50C2×C32×Dic3—21no—
51#6292—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8D412yescomplemented
54#5049—91no—
55#4886—94no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4⋊(C3⋊D3)—61no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×C4×D3—34no—
63C2×C3×Dic6—34no—
64C2×#626—64no—
65C2×C32×C8—31no—
66C32×C4⋊C4—24no—
67C32×C4⋊C4—31no—
68C32×C4⋊C4C2×D314yes—
69C32×C4⋊C4C2×D311yescharacteristic
70#3752—21no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—21no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76#3744—62no—
77#3743—62no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4⋊Dic3—64no—
80C3×C4⋊Dic3C2×D314yes—
81C2×C3×C3⋊C8C2×D314yes—
82#3722—64no—
83#3717—68no—
84#3711—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3×C3⋊Dic3—21no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—41no—
98C3×C22⋊Q8—91no—
99C2×#215—91no—
100#930—91no—
101C2×#208—91no—
102C2×#204—94no—
103C2×ΓL(1,𝔽25)—94no—
104#897—34no—
105#895—94no—
106C3×#107—34no—
107#835—91no—
108#810—94no—
109#809—94no—
110#809—91no—
111#809—184no—
112#807C3⋊D311yescharacteristic, complemented
113C22×C3⋊D3—61no—
114C22×C3⋊D3—64no—
115C22×C3⋊D3—124no—
116C22×C3⋊D3#6711yescharacteristic
117C22×C3×D3—34no—
118C2×C32×C4—21no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—44no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3#6711yescharacteristic
127C4⋊(C3⋊D3)—68no—
128C4⋊(C3⋊D3)—61no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—64no—
132C4⋊(C3⋊D3)—124no—
133C4⋊(C3⋊D3)—122no—
134C4×C3⋊D3—21no—
135C4×C3⋊D3—21no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—64no—
141C3×C4×D3—64no—
142C3×C4×D3—64no—
143C3×Dic6—64no—
144C3×Dic6—64no—
145#626—128no—
146C32×C8—32no—
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—124no—
150C3×C3⋊C8D1218yescomplemented
151#468—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C3×C3⋊D3—41no—
157C2×C3×Q8—91no—
158C22×C3×C4—91no—
159C2×#67—181no—
160C2×D12—94no—
161C2×D12—91no—
162C2×D12—184no—
163C2×D12—184no—
164C2×C4×D3—34no—
165C2×Dic6—34no—
166C2×C3×C8—34no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—34no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4D3214yes—
171C3×C4⋊C4C2×C3⋊D311yescharacteristic
172C3×C22⋊C4—181no—
173#215—182no—
174(C2×D3)⋊C4—64no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—184no—
177C4⋊Dic3—64no—
178C4⋊Dic3C2×C3⋊D311yescharacteristic
179#208—182no—
180C2×C3⋊C8—94no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—184no—
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184#204—188no—
185ΓL(1,𝔽25)—188no—
186C22×C32—24no—
187C22×C32D3×D414yes—
188C22×C32C2×#6711yescharacteristic
189C2×C3⋊D3—21no—
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3—128no—
193C2×C3⋊D3—124no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—121no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C2×C3⋊D3—244no—
200C2×C3⋊D3—244no—
201C2×C3×D3—64no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—41no—
207C32×C4—44no—
208C32×C4—84no—
209C32×C4—121no—
210C32×C4C2×D1214yes—
211C32×C4D3×D411yescharacteristic
212C32×C4#23711yescharacteristic
213C32×C4#23914yes—
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3×Dic3—44no—
217C3×Dic3—64no—
218C3×Dic3—124no—
219C2×SD16—271no—
220C2×M16—271no—
221C22⋊Q8—91no—
222#121—271no—
223#107—31no—
224#103—271no—
225#102—271no—
226C33#46811yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C22×D3—34no—
229C22×D3—184no—
230C22×D3—181no—
231C22×D3—364no—
232C3×Q8—181no—
233C3×Q8—181no—
234C2×C3×C4—24no—
235C2×C3×C4—21no—
236C2×C3×C4—24no—
237C2×C3×C4—44no—
238C2×C3×C4—64no—
239C2×C3×C4—91no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245#67—362no—
246C2×Dic3—21no—
247C2×Dic3—34no—
248C2×Dic3—64no—
249D12—188no—
250D12—184no—
251D12—184no—
252D12—181no—
253D12—181no—
254D12—364no—
255D12—368no—
256C4×D3—64no—
257C4×D3—64no—
258Dic6—64no—
259Dic6—64no—
260C3×C8—38no—
261C3⋊C8—184no—
262C3⋊C8—181no—
263C3⋊C8—364no—
264C3⋊C8C4⋊(C3⋊D3)12yescomplemented
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#89514yes—
269C2×C32#91114yes—
270C2×C32#91814yes—
271C2×C32#93911yescharacteristic
272C2×C32#94211yescharacteristic
273C2×C32#94911yescharacteristic
274C3⋊D3—41no—
275C3⋊D3—248no—
276C3⋊D3—244no—
277C3⋊D3—241no—
278C3×D3—124no—
279C2×Q8—91no—
280C2×D4—271no—
281C2×D4—541no—
282C22×C4—91no—
283SD16—542no—
284M16—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4—31no—
288C4⋊C4—181no—
289C4⋊C4D3×C3⋊D311yescharacteristic
290C22⋊C4—181no—
291C22⋊C4—541no—
292C22×C3—24no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—181no—
296C22×C3D3×#6714yes—
297C22×C3D4×C3⋊D311yescharacteristic
298C2×D3—64no—
299C2×D3—64no—
300C2×D3—64no—
301C2×D3—368no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—364no—
306C2×D3—364no—
307C2×D3—364no—
308C2×D3—724no—
309C2×D3—724no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—41no—
313C3×C4—44no—
314C3×C4—84no—
315C3×C4—124no—
316C3×C4—181no—
317C3×C4—361no—
318C3×C4#379614yes—
319C3×C4D3×D1214yes—
320C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
321C3×C4#383211yescharacteristic
322Dic3—41no—
323Dic3—64no—
324Dic3—124no—
325C32—24no—
326C32#488614yescomplemented
327C32#504911yescharacteristic, complemented
328C23—91no—
329C23—541no—
330Q8—181no—
331Q8—181no—
332D4—541no—
333D4—541no—
334D4—1082no—
335C2×C4—21no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—32no—
343C8—541no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—361no—
348C2×C3#6388614yes—
349C2×C3#6392514yes—
350C2×C3#6394814yes—
351C2×C3#6410411yescharacteristic
352C2×C3#6412011yescharacteristic
353C2×C3#6412711yescharacteristic
354D3—124no—
355D3—728no—
356D3—721no—
357D3—724no—
358C22—181no—
359C22—181no—
360C22—181no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22C3⋊D3×#6711yescharacteristic
365C4—41no—
366C4—121no—
367C4—181no—
368C4—361no—
369C4#8901711yescharacteristic
370C4D3×C4⋊(C3⋊D3)11yescharacteristic
371C3—24no—
372C3#1059058514yescomplemented
373C3#1059258511yescharacteristic, complemented
374C2—361no—
375C2—2161no—
376C2#1172348211yescharacteristic
377C2#1172354611yescharacteristic
378C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723526C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#468
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×#64106C2semidirect
3. C2×#88797C2semidirect
4. C2×#88787C2semidirect
5. #11722179C2semidirect
6. C32×#807C2semidirect
7. #11722039C2semidirect
25. #64106D3semidirect
36. C3×#807D3semidirect
53. C32×C3⋊C8D4semidirect
112. #807C3⋊D3semidirect
150. C3×C3⋊C8D12semidirect
226. C33#468semidirect
264. C3⋊C8C4⋊(C3⋊D3)semidirect
326. C32#4886semidirect
327. C32#5049semidirect
372. C3#10590585semidirect
373. C3#10592585semidirect

Maximal subgroups

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

subgroupindex
C3×#641062
#117235262
C2×#887972
C2×#887872
#117221792
C32×#8072
#117220392
#105925853
#105905853