Finite Group Databasebeta

← #13159299

Subgroups — #13159299

Order 1728.

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

normal subgroups only

This group has 888 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
1C32×#893C211yesmaximal, characteristic
2#11723443C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic
4C2×#88792C211yesmaximal, characteristic
5#11722180C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592718—31nomaximal
9#10590559—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#88794—22no—
22#88792—22no—
23C32×C4×D4—31no—
24C2×#3754—31no—
25C2×#3754—31no—
26#64078—31no—
27C3×#893D314yes—
28#63866—34no—
29C2×#3716—34no—
30C2×#3715—34no—
31#63637—31no—
32#63637—34no—
33#63637—64no—
34#63637D311yescharacteristic, complemented
35C4⋊(C3⋊(C3⋊C8))—31no—
36C3×#810—34no—
37Dic3⋊(C3⋊C8)—34no—
38C22×C32×D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×D12—21no—
50C32×D12D412yes—
51C32×C4×D3—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#5019—94no—
55#4818—91no—
56C2×C32×D4—31no—
57C22×C32×C4—61no—
58C2×#634—31no—
59C2×#634—34no—
60C2×#634—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×D12C2×D314yes—
63C2×C3×C4×D3—24no—
64C2×C3×Dic6—34no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C22⋊C4—61no—
69C32×C42—31no—
70#3754—62no—
71#3754—62no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)—61no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×(C2×D3)⋊C4—24no—
79C3×C4×Dic3C2×D314yes—
80C2×C3×C3⋊C8—34no—
81Dic3⋊Dic3—64no—
82Dic3⋊Dic3—64no—
83#3716—68no—
84#3715—68no—
85C22×C33C2×D411yescharacteristic
86C2×C32×D3—22no—
87C2×C32×D3—41no—
88C2×C32×D3—41no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—21no—
95C32×Dic3SD1612yescomplemented
96C3×C4×D4—34no—
97#944—91no—
98C2×#215—91no—
99C2×#214—94no—
100C2×#214—94no—
101C2×#204—91no—
102#893C3⋊D311yescharacteristic
103#869—94no—
104C3×#103—91no—
105Dic3⋊C8—91no—
106#810—34no—
107#810—94no—
108#810—91no—
109#810—184no—
110C4⋊(C3⋊C8)—94no—
111C23×C32—61no—
112C22×C3×D3—24no—
113C32×D4—32no—
114C32×D4—61no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4—44no—
121C2×C32×C4—122no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—124no—
128#634—21no—
129#634—21no—
130#634—68no—
131#634—64no—
132#634—61no—
133#634—61no—
134#634—64no—
135#634—124no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3#6714yes—
138C2×C3×Dic3#6714yes—
139C3×D12—24no—
140C3×D12D1218yes—
141C3×C4×D3—48no—
142C3×Dic6—64no—
143C3×Dic6—64no—
144C3⋊(C3⋊C8)—21no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—124no—
148C3×C3⋊C8—64no—
149#461—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33C2×SD1611yescharacteristic
152C2×C33#13711yescharacteristic
153C32×D3—42no—
154C2×C3×Q8—91no—
155C2×C3×D4—34no—
156C22×C3×C4—64no—
157C2×D12C2×C3⋊D311yescharacteristic
158C2×C4×D3—21no—
159C2×Dic6—34no—
160C2×Dic6—94no—
161C2×Dic6—91no—
162C2×Dic6—184no—
163C2×C3×C8—91no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4D3214yes—
166C3×C4⋊C4C2×C3⋊D311yescharacteristic
167C3×C22⋊C4—64no—
168C3×C42—34no—
169#215—182no—
170#214—188no—
171#214—188no—
172(C2×D3)⋊C4—21no—
173C4⋊Dic3—184no—
174C4⋊Dic3—184no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177C4×Dic3C2×C3⋊D311yescharacteristic
178C2×C3⋊C8—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182#204—182no—
183C22×C32—24no—
184C22×C32—62no—
185C22×C32—121no—
186C22×C32—121no—
187C22×C32D3×D411yescharacteristic
188C22×C32C2×#6714yes—
189C2×C3×D3—28no—
190C2×C3×D3—44no—
191C2×C3×D3—44no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—32no—
195C32×C4—41no—
196C32×C4—44no—
197C32×C4—61no—
198C32×C4—84no—
199C32×C4C2×D1214yes—
200C32×C4#23514yes—
201C32×C4#23511yescharacteristic
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—41no—
204C3⋊Dic3—128no—
205C3⋊Dic3—121no—
206C3⋊Dic3—124no—
207C3×Dic3—24no—
208C3×Dic3—124no—
209C3×Dic3#21418yescomplemented
210C2×SD16—271no—
211C2×SD16—271no—
212C4⋊Q8—271no—
213C4×D4—31no—
214C4⋊C8—271no—
215#103—91no—
216#103—271no—
217C33#46111yescharacteristic, complemented, Sylow
218C23×C3—64no—
219C22×D3—21no—
220C3×Q8—181no—
221C3×Q8—181no—
222C3×D4—38no—
223C3×D4—64no—
224C2×C3×C4—24no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—34no—
228C2×C3×C4—34no—
229C2×C3×C4—44no—
230C2×C3×C4—128no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×Dic3—64no—
235C2×Dic3—184no—
236C2×Dic3—181no—
237C2×Dic3—364no—
238C2×Dic3#63811yescharacteristic
239C2×Dic3#63811yescharacteristic
240D12—21no—
241D12C4⋊(C3⋊D3)12yes—
242C4×D3—42no—
243Dic6—64no—
244Dic6—64no—
245Dic6—188no—
246Dic6—184no—
247Dic6—184no—
248Dic6—181no—
249Dic6—181no—
250Dic6—364no—
251C3×C8—181no—
252C3⋊C8—64no—
253C3⋊C8—184no—
254C3⋊C8—181no—
255C3⋊C8—364no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—122no—
260C2×C32#88411yescharacteristic
261C2×C32D3×SD1611yescharacteristic
262C2×C32#91811yescharacteristic
263C2×C32#93014yes—
264C2×C32C2×#21414yes—
265C2×C32#95114yes—
266C3×D3—48no—
267C2×Q8—91no—
268C2×Q8—271no—
269C2×D4—31no—
270C22×C4—61no—
271SD16—542no—
272SD16—542no—
273C2×C8—91no—
274C2×C8—271no—
275C4⋊C4—541no—
276C4⋊C4—541no—
277C4⋊C4D3×C3⋊D311yescharacteristic
278C22⋊C4—61no—
279C42—31no—
280C22×C3—24no—
281C22×C3—68no—
282C22×C3—124no—
283C22×C3—124no—
284C22×C3D3×#6714yes—
285C22×C3C2×#63811yescharacteristic
286C2×D3—22no—
287C2×D3—41no—
288C2×D3—41no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—38no—
292C3×C4—41no—
293C3×C4—44no—
294C3×C4—64no—
295C3×C4—84no—
296C3×C4—361no—
297C3×C4#379814yes—
298C3×C4D3×D1214yes—
299C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
300C3×C4#382811yescharacteristic
301Dic3—21no—
302Dic3—124no—
303Dic3—368no—
304Dic3—361no—
305Dic3—364no—
306Dic3#375412yescomplemented
307C32—24no—
308C32#481811yescharacteristic, complemented
309C32#501914yescomplemented
310C23—61no—
311Q8—181no—
312Q8—181no—
313Q8—541no—
314Q8—541no—
315D4—32no—
316D4—61no—
317C2×C4—21no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—122no—
321C2×C4—181no—
322C2×C4—541no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—181no—
325C8—541no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—128no—
330C2×C3#6390614yes—
331C2×C3D3×#21414yes—
332C2×C3#6394314yes—
333C2×C3#6413911yescharacteristic
334C2×C3C2×#375411yescharacteristic
335C2×C3#6416011yescharacteristic
336D3—42no—
337C22—62no—
338C22—121no—
339C22—121no—
340C22D3×#63811yescharacteristic
341C4—32no—
342C4—41no—
343C4—61no—
344C4—361no—
345C4—1081no—
346C4#8902011yescharacteristic
347C4D3×C4⋊(C3⋊D3)11yescharacteristic
348C3—24no—
349C3#1059055914yescomplemented
350C3#1059271811yescharacteristic, complemented
351C2—122no—
352C2#1172351411yescharacteristic
353C2D3×#375411yescharacteristic
354C2#1172357411yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#461
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723443C2semidirect
5. #11722180C2semidirect
6. C3×#63637C2semidirect
7. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
34. #63637D3semidirect
95. C32×Dic3SD16semidirect
209. C3×Dic3#214semidirect
217. C33#461semidirect
306. Dic3#3754semidirect
308. C32#4818semidirect
309. C32#5019semidirect
349. C3#10590559semidirect
350. C3#10592718semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
#117234432
C2×#887942
C2×#887922
#117221802
C3×#636372
Dic3⋊(C3⋊(C3⋊C8))2
#105927183
#105905593