Finite Group Databasebeta

← #13161067

Subgroups — #13161067

Order 1728.

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

normal subgroups only

This group has 968 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×#3754C211yesmaximal, characteristic, complemented
2C32×#940C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592774—31nomaximal
9#10591438—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×#67—21no—
14C2×C32×D12C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3754—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22#88792—22no—
23C32×#127—31no—
24C2×#3754—31no—
25C2×#3754D311yescharacteristic, complemented
26C4×#634—31no—
27C3×#940D314yes—
28C2×C3×#214—34no—
29Dic3×Dic6—34no—
30C2×#3715—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×C32×D3—21no—
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634D412yescomplemented
47C32×#67—42no—
48C32×#67—42no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C32×D12—21no—
52C32×D12—21no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C3×C3⋊(C3⋊C8)—21no—
55#5163—91no—
56#5075—94no—
57C2×C32×D4—24no—
58C2×C32×D4—31no—
59C2×C32×D4—61no—
60C2×C32×D4—61no—
61C2×C32×D4C2×D314yes—
62C2×C32×D4C2×D311yescharacteristic
63C2×#634C2×D311yescharacteristic
64C2×C3×#67—24no—
65C2×C3×#67—24no—
66C2×C3×D12C2×D314yes—
67C2×C3×Dic6—34no—
68C32×C42—31no—
69#3754—22no—
70#3754—62no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—64no—
74#3750—61no—
75C4×C3⋊Dic3—61no—
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×#214—68no—
81C3×C4×Dic3C2×D314yes—
82C2×C3×C3⋊C8—34no—
83Dic3⋊Dic3—64no—
84Dic32—64no—
85#3715—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—22no—
97C32×Dic3—21no—
98C32×Dic3SD1612yescomplemented
99C2×C3×SD16—91no—
100C3×#127—34no—
101Q8×Dic3—91no—
102C2×#215—91no—
103#940C3⋊D311yescharacteristic
104C2×#214—34no—
105C2×#214—94no—
106C4×Dic6—94no—
107#832—94no—
108#832—94no—
109#832—91no—
110#832—184no—
111#821—91no—
112Dic3⋊C8—91no—
113C4⋊(C3⋊C8)—94no—
114C23×C32—21no—
115C23×C32—24no—
116C23×C32—44no—
117C23×C32—61no—
118C22×C3×D3—24no—
119C32×D4—28no—
120C32×D4—24no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—44no—
125C32×D4—61no—
126C32×D4—61no—
127C32×D4—122no—
128C32×D4—122no—
129C2×C32×C4—24no—
130C2×C32×C4—31no—
131C2×C32×C4—31no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—124no—
138#634—21no—
139#634#6712yescomplemented
140C3×#67—48no—
141C3×#67—48no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3#6714yes—
144C2×C3×Dic3#6714yes—
145C3×D12—24no—
146C3×D12—24no—
147C3×Dic6—38no—
148C3×Dic6—64no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—61no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—124no—
153C3×C3⋊C8—64no—
154#460—271noSylow
155C2×C33—81no—
156C2×C33#12111yescharacteristic
157C2×C33C2×SD1611yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C32×D3—81no—
160C2×C3×Q8—91no—
161C2×C3×D4—24no—
162C2×C3×D4—34no—
163C2×C3×D4—64no—
164C2×C3×D4—64no—
165C2×C3×D4D3214yes—
166C2×C3×D4C2×C3⋊D311yescharacteristic
167C2×#67—21no—
168C2×#67—21no—
169C2×D12C2×C3⋊D311yescharacteristic
170C2×Dic6—34no—
171C3×SD16—182no—
172C2×C3×C8—91no—
173C3×C42—34no—
174#215—182no—
175#214—68no—
176#214—188no—
177C4⋊Dic3—94no—
178C4⋊Dic3—91no—
179C4⋊Dic3—184no—
180C4⋊Dic3—181no—
181Dic3⋊C4—184no—
182C4×Dic3—184no—
183C4×Dic3—181no—
184C4×Dic3C2×C3⋊D311yescharacteristic
185C2×C3⋊C8—34no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189C22×C32—24no—
190C22×C32—48no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—44no—
197C22×C32—84no—
198C22×C32—84no—
199C22×C32—121no—
200C22×C32—121no—
201C22×C32—121no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×#6714yes—
204C2×C3×D3—44no—
205C2×C3×D3—44no—
206C2×C3×D3—44no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—32no—
210C32×C4—61no—
211C32×C4D3×D414yes—
212C32×C4#23714yes—
213C32×C4#23711yescharacteristic
214C32×C4C2×#6711yescharacteristic
215C3⋊Dic3—22no—
216C3⋊Dic3—121no—
217C3⋊Dic3—124no—
218C3⋊Dic3—244no—
219C3×Dic3—24no—
220C3×Dic3—68no—
221C3×Dic3#21418yescomplemented
222C2×SD16—91no—
223C2×SD16—271no—
224#127—31no—
225C4×Q8—271no—
226C4⋊C8—271no—
227#102—271no—
228#102—271no—
229C33#46011yescharacteristic, complemented, Sylow
230C23×C3—24no—
231C23×C3—21no—
232C23×C3—44no—
233C23×C3—64no—
234C22×D3—21no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×D4—28no—
238C3×D4—24no—
239C3×D4—24no—
240C3×D4—21no—
241C3×D4—21no—
242C3×D4—44no—
243C3×D4—64no—
244C3×D4—64no—
245C3×D4—128no—
246C3×D4—128no—
247C2×C3×C4—24no—
248C2×C3×C4—34no—
249C2×C3×C4—34no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—42no—
254#67—42no—
255C2×Dic3—64no—
256C2×Dic3—184no—
257C2×Dic3—181no—
258C2×Dic3—364no—
259C2×Dic3#63811yescharacteristic
260C2×Dic3#63811yescharacteristic
261D12—21no—
262D12—21no—
263Dic6—38no—
264Dic6—64no—
265C3×C8—181no—
266C3⋊C8—64no—
267C3⋊C8—184no—
268C3⋊C8—181no—
269C3⋊C8—364no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—88no—
274C2×C32—81no—
275C2×C32—84no—
276C2×C32—241no—
277C2×C32D3×SD1611yescharacteristic
278C2×C32#91411yescharacteristic
279C2×C32#93214yes—
280C2×C32C2×#21414yes—
281C2×C32#93814yes—
282C2×C32#93911yescharacteristic
283C3×D3—84no—
284C2×Q8—91no—
285C2×D4—31no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4D3×C3⋊D311yescharacteristic
289SD16—182no—
290SD16—542no—
291C2×C8—91no—
292C2×C8—271no—
293C4⋊C4—271no—
294C4⋊C4—541no—
295C42—31no—
296C42—541no—
297C22×C3—24no—
298C22×C3—48no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—44no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—84no—
306C22×C3—84no—
307C22×C3—124no—
308C22×C3—124no—
309C22×C3—124no—
310C22×C3D3×#6714yes—
311C22×C3C2×#63811yescharacteristic
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—38no—
318C3×C4—64no—
319C3×C4—182no—
320C3×C4#380514yes—
321C3×C4D3×#6714yes—
322C3×C4D4×C3⋊D311yescharacteristic
323C3×C4#383011yescharacteristic
324Dic3—21no—
325Dic3—68no—
326Dic3—361no—
327Dic3—364no—
328Dic3—724no—
329Dic3#375412yescomplemented
330C32—24no—
331C32#507514yescomplemented
332C32#516311yescharacteristic, complemented
333C23—21no—
334C23—61no—
335Q8—92no—
336Q8—181no—
337D4—21no—
338D4—21no—
339D4—61no—
340D4—61no—
341D4—122no—
342D4—122no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—181no—
346C2×C4—541no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—181no—
349C8—541no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—88no—
354C2×C3—81no—
355C2×C3—84no—
356C2×C3—244no—
357C2×C3D3×#21414yes—
358C2×C3#6393414yes—
359C2×C3#6397014yes—
360C2×C3#6414111yescharacteristic
361C2×C3C2×#375411yescharacteristic
362C2×C3#6414711yescharacteristic
363D3—81no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22D3×#63811yescharacteristic
371C4—32no—
372C4—61no—
373C4—182no—
374C4—1081no—
375C4#8902911yescharacteristic
376C4C3⋊D3×#6711yescharacteristic
377C3—24no—
378C3#1059143814yescomplemented
379C3#1059277411yescharacteristic, complemented
380C2—81no—
381C2—241no—
382C2D3×#375411yescharacteristic
383C2#1172355711yescharacteristic
384C2#1172363111yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#940C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#460
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×#3754C2semidirect
3. Dic3×#634C2semidirect
4. C2×#88792C2semidirect
5. #11722202C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722032C2semidirect
25. C2×#3754D3semidirect
46. C3×#634D4semidirect
98. C32×Dic3SD16semidirect
139. #634#67semidirect
221. C3×Dic3#214semidirect
229. C33#460semidirect
329. Dic3#3754semidirect
331. C32#5075semidirect
332. C32#5163semidirect
378. C3#10591438semidirect
379. C3#10592774semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×#9402
Dic3×#6342
C2×#887922
#117222022
Dic3⋊(C3⋊(C3⋊C8))2
#117220322
#105927743
#105914383