Finite Group Databasebeta

← #13161010

Subgroups — #13161010

Order 1728.

1092 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 142 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
1#11723981C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic, complemented
4C32×#832C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722021C211yesmaximal, characteristic, complemented
8#10592463—31nomaximal
9#10591480—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×#6312—21no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64149—31no—
22#64149—34no—
23#64149—64no—
24#64099—31no—
25#63872—34no—
26C32×#102—31no—
27#63659D311yescharacteristic, complemented
28#63648—31no—
29#63642—31no—
30#63636—31no—
31C3×#832—34no—
32C3×#832D314yescomplemented
33#63564—34no—
34#63564—34no—
35#63555—34no—
36C23×C33—21no—
37C22×C3⋊(C3⋊D3)—21no—
38C33×D4—21no—
39C33×D4—21no—
40C2×C33×C4C2311yescharacteristic
41#6312—42no—
42#6312—42no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C32×Dic3—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8—21no—
51#4764—94no—
52#4764—91no—
53C2×C32×D4—24no—
54C2×C32×D4C2×D314yes—
55C2×C32×D4C2×D311yescharacteristic
56C2×#638—61no—
57C2×#638—61no—
58C2×#638—64no—
59C2×#638—64no—
60C2×#638—68no—
61C2×#638—68no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×C32×C8—31no—
66C32×C4⋊C4—31no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8C2×D314yes—
78Dic3⋊Dic3—64no—
79Dic3⋊Dic3—64no—
80C22×C33—41no—
81C22×C33—41no—
82C22×C33—41no—
83C22×C33C2×D411yescharacteristic
84C2×C3⋊(C3⋊D3)—41no—
85C2×C3⋊(C3⋊D3)—41no—
86C2×C3⋊(C3⋊D3)—41no—
87C33×C4#4111yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—21no—
90C3⋊(C3⋊Dic3)—21no—
91C3×C3⋊Dic3—41no—
92C32×Dic3—41no—
93#940—94no—
94#940—91no—
95#940—184no—
96#890—94no—
97#890—91no—
98C3×#102—34no—
99C3×#102—91no—
100#832—34no—
101#832C3⋊D311yescharacteristic, complemented
102#821—94no—
103#821—91no—
104#815—94no—
105#815—91no—
106#809—94no—
107#809—91no—
108C23×C32—28no—
109C23×C32—24no—
110C23×C32—21no—
111C22×C3⋊D3—68no—
112C22×C3⋊D3—64no—
113C22×C3⋊D3—61no—
114C32×D4—21no—
115C32×D4—21no—
116C32×D4—24no—
117C32×D4—24no—
118C32×D4—44no—
119C32×D4—44no—
120C2×C32×C4—24no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124#638—1216no—
125#638—1216no—
126#638—128no—
127#638—122no—
128#638—128no—
129#638—122no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—31no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—34no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—64no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—124no—
142C4⋊(C3⋊D3)—124no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C32×C8—61no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—24no—
148C3×C3⋊C8—64no—
149#490—271noSylow
150C2×C33—81no—
151C2×C33#12411yescharacteristic
152C2×C33AGL(1,ℤ/8)11yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C3⋊(C3⋊D3)—81no—
155C2×C3×D4—24no—
156C2×C3×D4D3214yes—
157C2×C3×D4C2×C3⋊D311yescharacteristic
158C2×#67—184no—
159C2×#67—184no—
160C2×#67—181no—
161C2×#67—181no—
162C2×#67—188no—
163C2×#67—188no—
164C2×D12—94no—
165C2×D12—91no—
166C2×D12—184no—
167C2×C3×C8—34no—
168C2×C3×C8—91no—
169C3×C4⋊C4—34no—
170C3×C4⋊C4—91no—
171C4⋊Dic3—34no—
172C4⋊Dic3—181no—
173C4⋊Dic3—184no—
174C4⋊Dic3C2×C3⋊D311yescharacteristic
175Dic3⋊C4—181no—
176Dic3⋊C4—184no—
177C4×Dic3—94no—
178C4×Dic3—91no—
179C4×Dic3—184no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8C2×C3⋊D311yescharacteristic
182C22×C32—24no—
183C22×C32—48no—
184C22×C32—48no—
185C22×C32—48no—
186C22×C32—44no—
187C22×C32—41no—
188C22×C32—44no—
189C22×C32—41no—
190C22×C32—44no—
191C22×C32—41no—
192C22×C32D3×D414yes—
193C22×C32D3×D411yescharacteristic
194C2×C3⋊D3—128no—
195C2×C3⋊D3—128no—
196C2×C3⋊D3—128no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—124no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—121no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—121no—
206C32×C4#23514yes—
207C32×C4#23511yescharacteristic
208C32×C4#23714yes—
209C32×C4#23711yescharacteristic
210C3⋊Dic3—41no—
211C3⋊Dic3—68no—
212C3⋊Dic3—68no—
213C3⋊Dic3—64no—
214C3⋊Dic3—61no—
215C3⋊Dic3—64no—
216C3⋊Dic3—61no—
217C3×Dic3—44no—
218C3×Dic3—124no—
219#127—271no—
220#125—271no—
221#102—31no—
222#102—91no—
223#102—271no—
224#102—271no—
225#97—271no—
226C33#49011yescharacteristic, complemented, Sylow
227C23×C3—28no—
228C23×C3—24no—
229C23×C3—21no—
230C22×D3—188no—
231C22×D3—184no—
232C22×D3—181no—
233C3×D4—24no—
234C3×D4—24no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—44no—
238C3×D4—44no—
239C2×C3×C4—24no—
240C2×C3×C4—64no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3214yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244#67—3616no—
245#67—3616no—
246#67—362no—
247#67—362no—
248#67—368no—
249#67—368no—
250C2×Dic3—21no—
251C2×Dic3—64no—
252C2×Dic3—94no—
253C2×Dic3—94no—
254C2×Dic3—91no—
255C2×Dic3—91no—
256C2×Dic3—184no—
257C2×Dic3—184no—
258D12—184no—
259D12—184no—
260D12—181no—
261D12—181no—
262D12—364no—
263D12—364no—
264C3×C8—64no—
265C3×C8—181no—
266C3⋊C8—21no—
267C3⋊C8—64no—
268C2×C32—24no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—88no—
272C2×C32—84no—
273C2×C32—81no—
274C2×C32#88514yes—
275C2×C32#88511yescharacteristic
276C2×C32#91114yes—
277C2×C32#91111yescharacteristic
278C2×C32#91414yes—
279C2×C32#91411yescharacteristic
280C3⋊D3—248no—
281C3⋊D3—244no—
282C3⋊D3—241no—
283C2×D4—271no—
284C2×D4—541no—
285C2×D4—541no—
286C2×D4D3×C3⋊D311yescharacteristic
287C2×C8—31no—
288C2×C8—91no—
289C4⋊C4—31no—
290C4⋊C4—91no—
291C4⋊C4—541no—
292C4⋊C4—541no—
293C42—271no—
294C22×C3—24no—
295C22×C3—48no—
296C22×C3—48no—
297C22×C3—48no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—44no—
304C22×C3#381114yes—
305C22×C3D4×C3⋊D311yescharacteristic
306C2×D3—368no—
307C2×D3—368no—
308C2×D3—368no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—364no—
313C2×D3—364no—
314C2×D3—364no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—124no—
318C3×C4—361no—
319C3×C4#380414yes—
320C3×C4#380414yes—
321C3×C4#382811yescharacteristic
322C3×C4#383011yescharacteristic
323Dic3—41no—
324Dic3—124no—
325Dic3—188no—
326Dic3—188no—
327Dic3—181no—
328Dic3—181no—
329Dic3—184no—
330Dic3—184no—
331C32—24no—
332C32#476414yescomplemented
333C32#476411yescharacteristic, complemented
334C23—21no—
335C23—541no—
336D4—21no—
337D4—21no—
338D4—541no—
339D4—541no—
340D4—1082no—
341D4—1082no—
342C2×C4—61no—
343C2×C4—181no—
344C2×C4—271no—
345C2×C4—271no—
346C2×C4C2×D3×C3⋊D311yescharacteristic
347C8—61no—
348C8—181no—
349C2×C3—24no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—88no—
353C2×C3—81no—
354C2×C3—84no—
355C2×C3#6393714yes—
356C2×C3#6393714yes—
357C2×C3#6395314yes—
358C2×C3#6409411yescharacteristic
359C2×C3#6412011yescharacteristic
360C2×C3#6412311yescharacteristic
361D3—728no—
362D3—721no—
363D3—724no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22#8903911yescharacteristic
371C4—121no—
372C4—361no—
373C4—541no—
374C4—541no—
375C4#8902811yescharacteristic
376C4#8903111yescharacteristic
377C3—24no—
378C3#1059148014yescomplemented
379C3#1059246311yescharacteristic, complemented
380C2—81no—
381C2—2161no—
382C2#1172356411yescharacteristic
383C2#1172356611yescharacteristic
384C2#1172359211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radical#11723981C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723450C2semidirect
3. C3×#63659C2semidirect
4. C32×#832C2semidirect
5. #11722036C2semidirect
6. #11722035C2semidirect
7. #11722021C2semidirect
27. #63659D3semidirect
32. C3×#832D3semidirect
101. #832C3⋊D3semidirect
226. C33#490semidirect
332. C32#4764semidirect
333. C32#4764semidirect
378. C3#10591480semidirect
379. C3#10592463semidirect

Maximal subgroups

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

subgroupindex
#117239812
#117234502
C3×#636592
C32×#8322
#117220362
#117220352
#117220212
#105924633
#105914803