Finite Group Databasebeta

← #13159966

Subgroups — #13159966

Order 1728.

888 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 148 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×#3743C211yesmaximal, characteristic, complemented
2#11723522C211yesmaximal, characteristic
3#11723431C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722057C211yesmaximal, characteristic, complemented
8#10592479—31nomaximal
9#10590828—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#3743—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88803—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88794—22no—
23#64147—31no—
24C2×#3754—31no—
25C2×#3743D311yescharacteristic, complemented
26#64106—31no—
27C2×C3×ΓL(1,𝔽25)—34no—
28#63908—34no—
29#63857—34no—
30C2×#3716—34no—
31C32×#102—31no—
32#63643—31no—
33#63643—34no—
34#63643—64no—
35#63634—31no—
36C3×#821D314yescomplemented
37#63575—34no—
38C22×C3×C3⋊D3D411yescharacteristic
39C22×C32×D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)—21no—
43#6308—21no—
44#6308—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C2×C32×Dic3—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6290—42no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54#5134—91no—
55#4780—94no—
56C2×C32×D4—31no—
57C2×#638—61no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×D12C2×D314yes—
63C2×C3×C4×D3—34no—
64C2×#625—64no—
65C2×C32×C8—24no—
66C2×C32×C8C2×D314yes—
67C2×C32×C8C2×D311yescharacteristic
68C32×C4⋊C4—31no—
69#3757—61no—
70#3754—62no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77#3743—22no—
78C3×C4⋊Dic3C2×D314yes—
79C2×C3×C3⋊C8—34no—
80C3×ΓL(1,𝔽25)—68no—
81Dic3⋊Dic3—64no—
82(C2×D3)⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84#3716—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C32×D3—41no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—21no—
97C32×Dic3—41no—
98C2×C3×M16—91no—
99#938—94no—
100C2×#214—94no—
101#930—91no—
102#924—91no—
103C2×#204—91no—
104C2×ΓL(1,𝔽25)—34no—
105#897—94no—
106C3×#102—34no—
107#821C3⋊D311yescharacteristic, complemented
108#818—91no—
109#816—94no—
110#816—91no—
111#816—184no—
112#807—94no—
113C23×C32—61no—
114C22×C3⋊D3D1211yescharacteristic
115C22×C3×D3—24no—
116C22×C3×D3—34no—
117C32×D4—61no—
118C32×D4—61no—
119C2×C32×C4—24no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123#638—122no—
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—124no—
129C2×C3⋊Dic3—124no—
130C2×C3⋊Dic3D1211yescharacteristic
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133#634—68no—
134#634—64no—
135#634—61no—
136#634—61no—
137#634—64no—
138#634—124no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—34no—
141C3×D12—24no—
142C3×D12—24no—
143C3×C4×D3—64no—
144C3×C4×D3—64no—
145#625—128no—
146C32×C8—21no—
147C32×C8—24no—
148C32×C8—44no—
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—41no—
156C32×D3—81no—
157C2×C3×D4—34no—
158C22×C3×C4—91no—
159C2×#67—181no—
160C2×#67—184no—
161C2×D12C2×C3⋊D311yescharacteristic
162C2×C4×D3—34no—
163C2×Dic6—94no—
164C2×Dic6—91no—
165C2×Dic6—184no—
166C3×M16—182no—
167C2×C3×C8—24no—
168C2×C3×C8—91no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C4⋊C4—34no—
172#217—181no—
173#217—184no—
174#214—188no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—94no—
178C4⋊Dic3—91no—
179C4⋊Dic3—184no—
180C4⋊Dic3—184no—
181C4⋊Dic3C2×C3⋊D311yescharacteristic
182Dic3⋊C4—181no—
183C2×C3⋊C8—34no—
184#204—182no—
185ΓL(1,𝔽25)—68no—
186C22×C32—24no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32—121no—
190C22×C32C2×D1211yescharacteristic
191C22×C32D3×D414yes—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3×D3—44no—
196C2×C3×D3—44no—
197C2×C3×D3—44no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—64no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—121no—
204C32×C4#23514yes—
205C32×C4#23511yescharacteristic
206C32×C4D3×D414yes—
207C32×C4C2×#6711yescharacteristic
208C3⋊Dic3—21no—
209C3⋊Dic3—128no—
210C3⋊Dic3—124no—
211C3⋊Dic3—121no—
212C3⋊Dic3—121no—
213C3⋊Dic3—124no—
214C3⋊Dic3—244no—
215C3×Dic3—44no—
216C3×Dic3—64no—
217C2×SD16—271no—
218C2×M16—91no—
219C22⋊Q8—271no—
220#121—271no—
221#107—271no—
222#103—271no—
223#102—31no—
224C33#46811yescharacteristic, complemented, Sylow
225C23×C3—64no—
226C23×C3—91no—
227C22×D3—21no—
228C22×D3—34no—
229C3×D4—64no—
230C3×D4—64no—
231C2×C3×C4—24no—
232C2×C3×C4—64no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238#67—362no—
239#67—368no—
240C2×Dic3—21no—
241C2×Dic3—34no—
242C2×Dic3—184no—
243C2×Dic3—184no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C2×Dic3—364no—
247C2×Dic3—364no—
248D12—21no—
249D12—21no—
250C4×D3—64no—
251C4×D3—64no—
252Dic6—188no—
253Dic6—184no—
254Dic6—184no—
255Dic6—181no—
256Dic6—181no—
257Dic6—364no—
258C3×C8—24no—
259C3×C8—21no—
260C3×C8—44no—
261C3×C8—92no—
262C3⋊C8—38no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—241no—
267C2×C32#88414yes—
268C2×C32#90811yescharacteristic
269C2×C32#90911yescharacteristic
270C2×C32#91114yes—
271C2×C32#91514yes—
272C2×C32#93011yescharacteristic
273C3⋊D3—41no—
274C3×D3—84no—
275C3×D3—124no—
276C2×Q8—271no—
277C2×D4—31no—
278C2×D4—541no—
279C22×C4—91no—
280SD16—542no—
281M16—182no—
282C2×C8—91no—
283C2×C8D3×C3⋊D311yescharacteristic
284C4⋊C4—31no—
285C4⋊C4—271no—
286C4⋊C4—541no—
287C22⋊C4—541no—
288C22⋊C4—541no—
289C22×C3—24no—
290C22×C3—124no—
291C22×C3—124no—
292C22×C3—124no—
293C22×C3—181no—
294C22×C3—181no—
295C22×C3—181no—
296C22×C3D3×D1214yes—
297C22×C3D4×C3⋊D311yescharacteristic
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—64no—
302C2×D3—64no—
303C2×D3—64no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—124no—
307C3×C4—181no—
308C3×C4#379814yes—
309C3×C4D3×#6714yes—
310C3×C4#382811yescharacteristic
311C3×C4D4×C3⋊D311yescharacteristic
312Dic3—41no—
313Dic3—64no—
314Dic3—368no—
315Dic3—364no—
316Dic3—361no—
317Dic3—361no—
318Dic3—364no—
319Dic3—724no—
320C32—24no—
321C32#478014yescomplemented
322C32#513411yescharacteristic, complemented
323C23—61no—
324C23—91no—
325Q8—541no—
326Q8—541no—
327D4—61no—
328D4—61no—
329D4—1082no—
330C2×C4—61no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—541no—
335C2×C4—541no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—21no—
338C8—92no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—244no—
343C2×C3—361no—
344C2×C3#6379514yes—
345C2×C3#6380014yes—
346C2×C3#6390614yes—
347C2×C3#6409311yescharacteristic
348C2×C3#6412011yescharacteristic
349C2×C3#6412411yescharacteristic
350D3—81no—
351D3—124no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—181no—
356C22—181no—
357C22—181no—
358C22C3⋊D3×D1211yescharacteristic
359C4—121no—
360C4—181no—
361C4—1081no—
362C4—1081no—
363C4#8902011yescharacteristic
364C4C3⋊D3×#6711yescharacteristic
365C3—24no—
366C3#1059082814yescomplemented
367C3#1059247911yescharacteristic, complemented
368C2—241no—
369C2—361no—
370C2#1172332011yescharacteristic
371C2#1172334011yescharacteristic
372C2#1172352011yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723522C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3743C2semidirect
3. #11723431C2semidirect
4. C2×#88794C2semidirect
5. #11722186C2semidirect
6. C32×#821C2semidirect
7. #11722057C2semidirect
25. C2×#3743D3semidirect
36. C3×#821D3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
107. #821C3⋊D3semidirect
149. C3⋊(C3⋊C8)#67semidirect
224. C33#468semidirect
321. C32#4780semidirect
322. C32#5134semidirect
366. C3#10590828semidirect
367. C3#10592479semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
#117235222
#117234312
C2×#887942
#117221862
C32×#8212
#117220572
#105924793
#105908283