Finite Group Databasebeta

← #13158966

Subgroups — #13158966

Order 1728.

896 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 176 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×#64084C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic, complemented
3C2×#88792C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic
5C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592715—31nomaximal
9#10590348—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C33×C42C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3752—21no—
18C3×#3752—21no—
19C32×(C2×D3)⋊C4—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21#88792—22no—
22#88789—22no—
23C32×C4×D4—31no—
24C2×#3754—31no—
25C2×#3753—31no—
26#64084D311yescharacteristic, complemented
27C3×#875—34no—
28C3×C4×D12D314yescomplemented
29C2×#3715—34no—
30C2×#3713—34no—
31#63637—31no—
32#63636—31no—
33C4⋊(C3⋊(C3⋊C8))—31no—
34C4⋊(C3⋊(C3⋊C8))—34no—
35C4⋊(C3⋊(C3⋊C8))—64no—
36#63565—34no—
37#63562—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3—21no—
49C32×D12—21no—
50C32×D12D412yes—
51C32×C4×D3—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54#5057—91no—
55#5016—94no—
56C2×C32×D4—31no—
57C22×C32×C4—61no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—34no—
61C2×C3×D12C2×D314yes—
62C2×C3×C4×D3—24no—
63C2×C3×Dic6—34no—
64C32×C4⋊C4—31no—
65C32×C22⋊C4—61no—
66C32×C42—24no—
67C32×C42C2×D314yes—
68C32×C42C2×D311yescharacteristic
69#3754—62no—
70#3753—62no—
71#3752—21no—
72#3752—21no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)—64no—
79C3×(C2×D3)⋊C4—24no—
80C3×(C2×D3)⋊C4—64no—
81C3×(C2×D3)⋊C4—64no—
82C3×C4⋊Dic3C2×D314yes—
83#3715—68no—
84#3713—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C32×D3—22no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C33×C4—21no—
93C33×C4—21no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—41no—
97C32×Dic3—41no—
98C3×#124—91no—
99C3×C4×D4—34no—
100C2×#215—91no—
101C2×#214—94no—
102C2×#213—94no—
103C2×#213—91no—
104#875—34no—
105C4×D12C3⋊D311yescharacteristic, complemented
106#835—91no—
107#832—91no—
108#810—94no—
109#809—94no—
110C4⋊(C3⋊C8)—94no—
111C4⋊(C3⋊C8)—91no—
112C4⋊(C3⋊C8)—184no—
113C23×C32—61no—
114C22×C3⋊D3—21no—
115C22×C3×D3—24no—
116C22×C3×D3—64no—
117C32×D4—32no—
118C32×D4—61no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4—122no—
124C2×C32×C4#6714yes—
125C2×C32×C4#6714yes—
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—64no—
137C3×D12—24no—
138C3×D12—64no—
139C3×D12—64no—
140C3×D12D1218yes—
141C3×C4×D3—48no—
142C3×Dic6—64no—
143C3×Dic6—64no—
144C3⋊(C3⋊C8)—61no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—124no—
149C3⋊(C3⋊C8)—124no—
150#463—271noSylow
151C2×C33#12111yescharacteristic
152C2×C33#13511yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C3×C3⋊D3—81no—
155C32×D3—42no—
156C2×C3×Q8—91no—
157C2×C3×D4—34no—
158C2×C3×D4—91no—
159C22×C3×C4—64no—
160C2×D12—34no—
161C2×D12C2×C3⋊D311yescharacteristic
162C2×C4×D3—21no—
163C2×Dic6—34no—
164C3×C4⋊C4—34no—
165C3×C22⋊C4—64no—
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168C3×C42—24no—
169C3×C42D3214yes—
170C3×C42C2×C3⋊D311yescharacteristic
171#215—182no—
172#214—188no—
173#213—188no—
174#213—182no—
175(C2×D3)⋊C4—21no—
176(C2×D3)⋊C4—64no—
177(C2×D3)⋊C4—64no—
178C4⋊Dic3C2×C3⋊D311yescharacteristic
179C2×C3⋊C8—94no—
180C2×C3⋊C8—94no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—184no—
184C2×C3⋊C8—184no—
185C22×C32—24no—
186C22×C32—62no—
187C22×C32—121no—
188C22×C32—121no—
189C22×C32C2×#6714yes—
190C22×C32C2×#6711yescharacteristic
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3×D3—28no—
195C2×C3×D3—44no—
196C2×C3×D3—44no—
197C2×C3×D3—124no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C32×C4—28no—
201C32×C4—21no—
202C32×C4—24no—
203C32×C4—21no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—44no—
208C32×C4—121no—
209C32×C4C2×D1214yes—
210C32×C4#23514yes—
211C32×C4D3×D411yescharacteristic
212C32×C4#23711yescharacteristic
213C3⋊Dic3—41no—
214C3×Dic3—44no—
215C3×Dic3—124no—
216C2×SD16—271no—
217C2×D8—271no—
218#124—91no—
219C4×D4—31no—
220C4⋊C8—271no—
221#103—271no—
222#102—271no—
223C33#46311yescharacteristic, complemented, Sylow
224C23×C3—64no—
225C23×C3—181no—
226C22×D3—21no—
227C22×D3—64no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×D4—38no—
231C3×D4—64no—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—64no—
238C2×C3×C4—128no—
239C2×C3×C4—181no—
240C2×C3×C4#62514yes—
241C2×C3×C4#62514yes—
242C2×C3×C4#63811yescharacteristic
243C2×C3×C4#63811yescharacteristic
244C2×C3×C4C2×D3214yes—
245C2×C3×C4C22×C3⋊D311yescharacteristic
246C2×Dic3—21no—
247C2×Dic3—64no—
248D12—21no—
249D12—64no—
250D12—64no—
251D12C4⋊(C3⋊D3)12yes—
252C4×D3—42no—
253Dic6—64no—
254Dic6—64no—
255C3⋊C8—184no—
256C3⋊C8—184no—
257C3⋊C8—181no—
258C3⋊C8—181no—
259C3⋊C8—364no—
260C3⋊C8—364no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—122no—
265C2×C32#93014yes—
266C2×C32#93214yes—
267C2×C32#93811yescharacteristic
268C2×C32#94911yescharacteristic
269C2×C32#95014yes—
270C2×C32#95011yescharacteristic
271C3⋊D3—81no—
272C3×D3—48no—
273C3×D3—244no—
274C2×Q8—91no—
275C2×D4—31no—
276C2×D4—91no—
277C22×C4—61no—
278SD16—542no—
279D8—542no—
280C2×C8—271no—
281C2×C8—271no—
282C4⋊C4—31no—
283C22⋊C4—61no—
284C22⋊C4—181no—
285C22⋊C4—181no—
286C42D3×C3⋊D311yescharacteristic
287C22×C3—24no—
288C22×C3—68no—
289C22×C3—124no—
290C22×C3—124no—
291C22×C3—361no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3C2×#62514yes—
295C22×C3C2×#63811yescharacteristic
296C2×D3—22no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—124no—
300C2×D3—124no—
301C2×D3—124no—
302C3×C4—28no—
303C3×C4—24no—
304C3×C4—24no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—24no—
308C3×C4—24no—
309C3×C4—44no—
310C3×C4—124no—
311C3×C4—361no—
312C3×C4#379514yes—
313C3×C4D3×D1214yes—
314C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
315C3×C4#382811yescharacteristic
316Dic3—41no—
317Dic3—124no—
318C32—24no—
319C32#501614yescomplemented
320C32#505711yescharacteristic, complemented
321C23—61no—
322C23—181no—
323Q8—181no—
324Q8—181no—
325D4—32no—
326D4—61no—
327D4—181no—
328D4—181no—
329C2×C4—61no—
330C2×C4—122no—
331C2×C4—181no—
332C2×C4#629011yescharacteristic
333C2×C4#629011yescharacteristic
334C2×C4C2×D3×C3⋊D311yescharacteristic
335C8—541no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—128no—
341C2×C3—721no—
342C2×C3#6382214yes—
343C2×C3#6382314yes—
344C2×C3#6390814yes—
345C2×C3#6413911yescharacteristic
346C2×C3#6414111yescharacteristic
347C2×C3#6415911yescharacteristic
348D3—42no—
349D3—244no—
350C22—62no—
351C22—121no—
352C22—121no—
353C22—361no—
354C22—361no—
355C22—361no—
356C22C2×#629011yescharacteristic
357C4—21no—
358C4—21no—
359C4—121no—
360C4—361no—
361C4#8902111yescharacteristic
362C4D3×C4⋊(C3⋊D3)11yescharacteristic
363C3—24no—
364C3#1059034814yescomplemented
365C3#1059271511yescharacteristic, complemented
366C2—122no—
367C2—721no—
368C2#1172336311yescharacteristic
369C2#1172336411yescharacteristic
370C2#1172351511yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×#88789C2
hypercenterC42D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×#64084C2semidirect
2. C32×C4×D12C2semidirect
3. C2×#88792C2semidirect
5. C4⋊(C3⋊(C3⋊(C3⋊C8)))C2semidirect
6. #11722039C2semidirect
7. #11722033C2semidirect
26. #64084D3semidirect
28. C3×C4×D12D3semidirect
105. C4×D12C3⋊D3semidirect
223. C33#463semidirect
319. C32#5016semidirect
320. C32#5057semidirect
364. C3#10590348semidirect
365. C3#10592715semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C32×C4×D122
C2×#887922
C2×#887892
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
#117220392
#117220332
#105927153
#105903483