Finite Group Databasebeta

← #13161059

Subgroups — #13161059

Order 1728.

1040 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 168 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×#3753C211yesmaximal, characteristic, complemented
2C32×#940C211yesmaximal, characteristic, complemented
3Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592771—31nomaximal
9#10591428—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×#67—21no—
14C2×C32×D12C2211yescharacteristic
15C2×Dic3×C3⋊D3—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#3753—22no—
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21#88803—21no—
22#88789—22no—
23C32×#127—31no—
24C2×#3753—31no—
25C2×#3753D311yescharacteristic, complemented
26C4×C4⋊(C3⋊D3)—31no—
27C3×#940D314yescomplemented
28C2×C3×#213—34no—
29Dic3×D12—34no—
30C2×#3713—34no—
31#63659—31no—
32#63659—31no—
33#63659—34no—
34#63659—64no—
35C4⋊(C3⋊(C3⋊C8))—31no—
36Dic3⋊(C3⋊C8)—34no—
37#63563—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)D412yescomplemented
47C32×#67—42no—
48C32×#67—42no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C32×D12—21no—
52C32×D12—21no—
53Dic3×C3⋊D3—42no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C3×C3⋊(C3⋊C8)—21no—
56#5150—91no—
57#5072—94no—
58C2×C32×D4—24no—
59C2×C32×D4—31no—
60C2×C32×D4—61no—
61C2×C32×D4—61no—
62C2×C32×D4C2×D314yes—
63C2×C32×D4C2×D311yescharacteristic
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—61no—
66C2×C3×#67—24no—
67C2×C3×#67—24no—
68C2×C3×D12—34no—
69C2×C3×D12C2×D314yes—
70C2×D3×Dic3—64no—
71C32×C42—31no—
72#3753—22no—
73#3753—62no—
74#3752—61no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—34no—
77C4⋊(C3⋊Dic3)—64no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—34no—
80C2×C3⋊(C3⋊C8)—64no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#213—68no—
83C3×C4×Dic3C2×D314yes—
84C2×C3×C3⋊C8—34no—
85(C2×D3)⋊Dic3—64no—
86#3713—68no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C32×Dic3—21no—
101C32×Dic3D812yescomplemented
102C2×C3×D8—91no—
103C3×#127—34no—
104#940C3⋊D311yescharacteristic, complemented
105D4×Dic3—91no—
106C2×#213—34no—
107C2×#213—94no—
108C2×#213—91no—
109C4×D12—94no—
110#832—94no—
111#832—94no—
112#832—91no—
113#832—184no—
114#821—91no—
115Dic3⋊C8—91no—
116C4⋊(C3⋊C8)—94no—
117C23×C32—21no—
118C23×C32—24no—
119C23×C32—44no—
120C23×C32—61no—
121C22×C3⋊D3—21no—
122C22×C3×D3—24no—
123C22×C3×D3—64no—
124C32×D4—28no—
125C32×D4—24no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—24no—
129C32×D4—44no—
130C32×D4—61no—
131C32×D4—61no—
132C32×D4—122no—
133C32×D4—122no—
134C2×C32×C4—24no—
135C2×C32×C4—31no—
136C2×C32×C4—31no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—124no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)#6712yescomplemented
144C4×C3⋊D3—122no—
145C3×#67—48no—
146C3×#67—48no—
147C2×C3×Dic3#6714yes—
148C2×C3×Dic3#6714yes—
149C3×D12—24no—
150C3×D12—24no—
151C3×D12—38no—
152C3×D12—64no—
153D3×Dic3—128no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—61no—
156C3⋊(C3⋊C8)—64no—
157C3⋊(C3⋊C8)—124no—
158C3×C3⋊C8—64no—
159#459—271noSylow
160C2×C33—81no—
161C2×C33#12111yescharacteristic
162C2×C33C2×D811yescharacteristic
163C2×C33AGL(1,ℤ/8)11yescharacteristic
164C3×C3⋊D3—42no—
165C32×D3—81no—
166C2×C3×D4—24no—
167C2×C3×D4—34no—
168C2×C3×D4—64no—
169C2×C3×D4—64no—
170C2×C3×D4—91no—
171C2×C3×D4D3214yes—
172C2×C3×D4C2×C3⋊D311yescharacteristic
173C2×#67—21no—
174C2×#67—21no—
175C22×Dic3—181no—
176C2×D12—34no—
177C2×D12C2×C3⋊D311yescharacteristic
178C2×C4×D3—184no—
179C3×D8—182no—
180C2×C3×C8—91no—
181C3×C42—34no—
182#217—181no—
183#213—68no—
184#213—182no—
185#213—188no—
186(C2×D3)⋊C4—184no—
187C4⋊Dic3—94no—
188C4⋊Dic3—91no—
189C4⋊Dic3—184no—
190C4×Dic3C2×C3⋊D311yescharacteristic
191C2×C3⋊C8—34no—
192C2×C3⋊C8—94no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—184no—
195C22×C32—24no—
196C22×C32—48no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—44no—
200C22×C32—41no—
201C22×C32—44no—
202C22×C32—44no—
203C22×C32—84no—
204C22×C32—84no—
205C22×C32—121no—
206C22×C32—121no—
207C22×C32—121no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×#6714yes—
210C2×C3⋊D3—22no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3×D3—44no—
214C2×C3×D3—44no—
215C2×C3×D3—44no—
216C2×C3×D3—68no—
217C2×C3×D3—124no—
218C2×C3×D3—124no—
219C32×C4—24no—
220C32×C4—24no—
221C32×C4—32no—
222C32×C4—61no—
223C32×C4D3×D414yes—
224C32×C4#23714yes—
225C32×C4#23711yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C3⋊Dic3—121no—
228C3⋊Dic3—124no—
229C3⋊Dic3—244no—
230C3×Dic3—24no—
231C3×Dic3#21318yescomplemented
232C2×D8—91no—
233C2×D8—271no—
234#127—31no—
235C4×D4—271no—
236C4⋊C8—271no—
237#102—271no—
238#102—271no—
239C33#45911yescharacteristic, complemented, Sylow
240C23×C3—24no—
241C23×C3—21no—
242C23×C3—44no—
243C23×C3—64no—
244C23×C3—181no—
245C22×D3—21no—
246C22×D3—64no—
247C3×D4—28no—
248C3×D4—24no—
249C3×D4—24no—
250C3×D4—21no—
251C3×D4—21no—
252C3×D4—44no—
253C3×D4—64no—
254C3×D4—64no—
255C3×D4—92no—
256C3×D4—128no—
257C3×D4—128no—
258C3×D4—181no—
259C2×C3×C4—24no—
260C2×C3×C4—34no—
261C2×C3×C4—34no—
262C2×C3×C4C2×D3214yes—
263C2×C3×C4C22×C3⋊D311yescharacteristic
264#67—42no—
265#67—42no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—364no—
269C2×Dic3—362no—
270C2×Dic3#63811yescharacteristic
271C2×Dic3#63811yescharacteristic
272D12—21no—
273D12—21no—
274D12—38no—
275D12—64no—
276C4×D3—368no—
277C3×C8—181no—
278C3⋊C8—64no—
279C3⋊C8—184no—
280C3⋊C8—181no—
281C3⋊C8—364no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—88no—
286C2×C32—81no—
287C2×C32—84no—
288C2×C32—241no—
289C2×C32D3×D811yescharacteristic
290C2×C32#91111yescharacteristic
291C2×C32C2×#21314yes—
292C2×C32#93214yes—
293C2×C32#93814yes—
294C2×C32#93911yescharacteristic
295C3⋊D3—42no—
296C3×D3—84no—
297C3×D3—128no—
298C2×D4—31no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—91no—
302C2×D4D3×C3⋊D311yescharacteristic
303C22×C4—541no—
304D8—182no—
305D8—542no—
306C2×C8—91no—
307C2×C8—271no—
308C4⋊C4—271no—
309C22⋊C4—541no—
310C42—31no—
311C22×C3—24no—
312C22×C3—48no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—44no—
317C22×C3—44no—
318C22×C3—44no—
319C22×C3—84no—
320C22×C3—84no—
321C22×C3—124no—
322C22×C3—124no—
323C22×C3—124no—
324C22×C3—182no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3D3×#6714yes—
328C22×C3C2×#63811yescharacteristic
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—68no—
333C2×D3—124no—
334C2×D3—124no—
335C3×C4—24no—
336C3×C4—24no—
337C3×C4—38no—
338C3×C4—64no—
339C3×C4#380514yes—
340C3×C4D3×#6714yes—
341C3×C4D4×C3⋊D311yescharacteristic
342C3×C4#383011yescharacteristic
343Dic3—21no—
344Dic3—361no—
345Dic3—364no—
346Dic3—724no—
347Dic3#375312yescomplemented
348C32—24no—
349C32#507214yescomplemented
350C32#515011yescharacteristic, complemented
351C23—21no—
352C23—61no—
353C23—181no—
354D4—21no—
355D4—21no—
356D4—61no—
357D4—61no—
358D4—92no—
359D4—122no—
360D4—122no—
361D4—181no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—541no—
365C2×C4—1082no—
366C2×C4C2×D3×C3⋊D311yescharacteristic
367C8—181no—
368C8—541no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—88no—
373C2×C3—81no—
374C2×C3—84no—
375C2×C3—244no—
376C2×C3—362no—
377C2×C3D3×#21314yes—
378C2×C3#6393214yes—
379C2×C3#6397014yes—
380C2×C3C2×#375311yescharacteristic
381C2×C3#6414111yescharacteristic
382C2×C3#6414711yescharacteristic
383D3—81no—
384D3—128no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—182no—
392C22—361no—
393C22—361no—
394C22D3×#63811yescharacteristic
395C4—32no—
396C4—61no—
397C4—1081no—
398C4#8902911yescharacteristic
399C4C3⋊D3×#6711yescharacteristic
400C3—24no—
401C3#1059142814yescomplemented
402C3#1059277111yescharacteristic, complemented
403C2—81no—
404C2—241no—
405C2—362no—
406C2D3×#375311yescharacteristic
407C2#1172355511yescharacteristic
408C2#1172363111yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#459
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
1. C2×C3×#3753C2semidirect
2. C32×#940C2semidirect
3. Dic3×C4⋊(C3⋊D3)C2semidirect
4. C2×#88789C2semidirect
5. #11722202C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722032C2semidirect
25. C2×#3753D3semidirect
27. C3×#940D3semidirect
46. C3×C4⋊(C3⋊D3)D4semidirect
101. C32×Dic3D8semidirect
104. #940C3⋊D3semidirect
143. C4⋊(C3⋊D3)#67semidirect
231. C3×Dic3#213semidirect
239. C33#459semidirect
347. Dic3#3753semidirect
349. C32#5072semidirect
350. C32#5150semidirect
401. C3#10591428semidirect
402. C3#10592771semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×#9402
Dic3×C4⋊(C3⋊D3)2
C2×#887892
#117222022
Dic3⋊(C3⋊(C3⋊C8))2
#117220322
#105927713
#105914283