Finite Group Databasebeta

← #13159001

Subgroups — #13159001

Order 1728.

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

normal subgroups only

This group has 824 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
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×#88798C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic
5C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8#10592727—31nomaximal
9#10590267—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×#3752—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3—21no—
21#88798—22no—
22#88793—22no—
23C32×C4×Q8—31no—
24C2×#3756—31no—
25C2×#3755—31no—
26#64084D311yescharacteristic
27C3×#875—34no—
28C3×C4×Dic6D314yescomplemented
29C2×#3718—34no—
30C2×#3715—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—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×#634—21no—
46C3×#634—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6D412yes—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53#5054—91no—
54#5028—94no—
55C2×C32×Q8—31no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×C3×D12—34no—
59C2×C3×Dic6—34no—
60C2×C3×Dic6C2×D314yes—
61C32×C4⋊C4—31no—
62C32×C4⋊C4—61no—
63C32×C42—24no—
64C32×C42—61no—
65C32×C42C2×D314yes—
66C32×C42C2×D311yescharacteristic
67#3756—62no—
68#3755—62no—
69#3752—21no—
70#3752—21no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C2×C3⋊(C3⋊C8)—64no—
77C3×(C2×D3)⋊C4—64no—
78C3×(C2×D3)⋊C4—64no—
79C3×C4⋊Dic3C2×D314yes—
80C3×Dic3⋊C4—24no—
81C3×C4×Dic3—24no—
82#3718—68no—
83#3715—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—41no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C33×C4—21no—
89C33×C4—21no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—41no—
93C32×Dic3—22no—
94C32×Dic3—41no—
95C3×#124—91no—
96C3×C4×Q8—34no—
97C2×#216—94no—
98C2×#216—91no—
99C2×#215—94no—
100C2×#214—91no—
101#875—34no—
102C4×Dic6C3⋊D311yescharacteristic, complemented
103#835—91no—
104#832—91no—
105#810—94no—
106#809—94no—
107C4⋊(C3⋊C8)—94no—
108C4⋊(C3⋊C8)—91no—
109C4⋊(C3⋊C8)—184no—
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C32×Q8—32no—
113C32×Q8—61no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—61no—
118C2×C32×C4—61no—
119C2×C32×C4#6714yes—
120C2×C32×C4#6714yes—
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)—21no—
128#634—21no—
129#634—21no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133C3×D12—64no—
134C3×D12—64no—
135C3×Dic6—24no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6D1218yes—
139C3⋊(C3⋊C8)—61no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—64no—
143C3⋊(C3⋊C8)—124no—
144C3⋊(C3⋊C8)—124no—
145#464—271noSylow
146C2×C33#12111yescharacteristic
147C2×C33#13511yescharacteristic
148C2×C33#13711yescharacteristic
149C3×C3⋊D3—81no—
150C2×C3×Q8—34no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×D12—34no—
154C2×Dic6—34no—
155C2×Dic6C2×C3⋊D311yescharacteristic
156C3×C4⋊C4—34no—
157C3×C4⋊C4—64no—
158C3×C22⋊C4—181no—
159C3×C22⋊C4—181no—
160C3×C42—24no—
161C3×C42—64no—
162C3×C42D3214yes—
163C3×C42C2×C3⋊D311yescharacteristic
164#216—188no—
165#216—182no—
166#215—188no—
167#214—182no—
168(C2×D3)⋊C4—64no—
169(C2×D3)⋊C4—64no—
170C4⋊Dic3C2×C3⋊D311yescharacteristic
171Dic3⋊C4—21no—
172C4×Dic3—21no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—94no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—184no—
178C2×C3⋊C8—184no—
179C22×C32—24no—
180C22×C32C2×#6714yes—
181C22×C32C2×#6711yescharacteristic
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—124no—
186C2×C3×D3—124no—
187C2×C3×D3—124no—
188C32×C4—28no—
189C32×C4—24no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—44no—
196C32×C4—62no—
197C32×C4—121no—
198C32×C4C2×D1214yes—
199C32×C4#23514yes—
200C32×C4D3×D411yescharacteristic
201C32×C4#23711yescharacteristic
202C3⋊Dic3—41no—
203C3×Dic3—28no—
204C3×Dic3—44no—
205C3×Dic3—124no—
206C2×Q16—271no—
207C2×SD16—271no—
208#124—91no—
209C4×Q8—31no—
210C4⋊C8—271no—
211#103—271no—
212#102—271no—
213C33#46411yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—64no—
216C3×Q8—38no—
217C3×Q8—64no—
218C3×Q8—181no—
219C3×Q8—181no—
220C3×D4—181no—
221C3×D4—181no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—64no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4#62514yes—
229C2×C3×C4#62514yes—
230C2×C3×C4#63811yescharacteristic
231C2×C3×C4#63811yescharacteristic
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—64no—
237D12—64no—
238D12—64no—
239Dic6—21no—
240Dic6—64no—
241Dic6—64no—
242Dic6C4⋊(C3⋊D3)12yes—
243C3⋊C8—184no—
244C3⋊C8—184no—
245C3⋊C8—181no—
246C3⋊C8—181no—
247C3⋊C8—364no—
248C3⋊C8—364no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32#93014yes—
253C2×C32#93811yescharacteristic
254C2×C32#94214yes—
255C2×C32#95014yes—
256C2×C32#95011yescharacteristic
257C2×C32#95111yescharacteristic
258C3⋊D3—81no—
259C3×D3—244no—
260C2×Q8—31no—
261C2×Q8—91no—
262C2×D4—91no—
263Q16—542no—
264SD16—542no—
265C2×C8—271no—
266C2×C8—271no—
267C4⋊C4—31no—
268C4⋊C4—61no—
269C22⋊C4—181no—
270C22⋊C4—181no—
271C42—61no—
272C42D3×C3⋊D311yescharacteristic
273C22×C3—24no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3—361no—
277C22×C3C2×#62514yes—
278C22×C3C2×#63811yescharacteristic
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—124no—
282C3×C4—28no—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—24no—
288C3×C4—24no—
289C3×C4—44no—
290C3×C4—68no—
291C3×C4—124no—
292C3×C4—361no—
293C3×C4#379514yes—
294C3×C4D3×D1214yes—
295C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
296C3×C4#382811yescharacteristic
297Dic3—22no—
298Dic3—41no—
299Dic3—124no—
300C32—24no—
301C32#502814yescomplemented
302C32#505411yescharacteristic, complemented
303C23—181no—
304Q8—32no—
305Q8—61no—
306Q8—181no—
307Q8—181no—
308D4—181no—
309D4—181no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—181no—
313C2×C4#629011yescharacteristic
314C2×C4#629011yescharacteristic
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—541no—
317C8—541no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—721no—
322C2×C3#6382214yes—
323C2×C3#6382714yes—
324C2×C3#6390814yes—
325C2×C3#6413911yescharacteristic
326C2×C3#6415111yescharacteristic
327C2×C3#6415911yescharacteristic
328D3—244no—
329C22—361no—
330C22—361no—
331C22—361no—
332C22C2×#629011yescharacteristic
333C4—21no—
334C4—21no—
335C4—62no—
336C4—121no—
337C4—361no—
338C4#8902111yescharacteristic
339C4D3×C4⋊(C3⋊D3)11yescharacteristic
340C3—24no—
341C3#1059026714yescomplemented
342C3#1059272711yescharacteristic, complemented
343C2—721no—
344C2#1172336311yescharacteristic
345C2#1172337511yescharacteristic
346C2#1172351511yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42D3×C3⋊D3
hypocenterC33#464
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C32×C4×Dic6C2semidirect
3. C2×#88798C2semidirect
5. C4⋊(C3⋊(C3⋊(C3⋊C8)))C2semidirect
6. #11722039C2semidirect
28. C3×C4×Dic6D3semidirect
102. C4×Dic6C3⋊D3semidirect
213. C33#464semidirect
301. C32#5028semidirect
302. C32#5054semidirect
341. C3#10590267semidirect
342. C3#10592727semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C32×C4×Dic62
C2×#887982
C2×#887932
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
#117220392
#117220332
#105927273
#105902673