Finite Group Databasebeta

← #13159552

Subgroups — #13159552

Order 1728.

824 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 144 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
1#11723503C211yesmaximal, characteristic
2#11723449C211yesmaximal, characteristic
3#11722190C211yesmaximal, characteristic
4C3×#63643C211yesmaximal, characteristic, complemented
5C32×#817C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592561—31nomaximal
9#10590686—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×#6289—21no—
13C2×C33×C8C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88806—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20#88804—21no—
21#64106—31no—
22C4⋊C4×C3⋊D3—31no—
23#63900—34no—
24#63871—34no—
25C32×#106—31no—
26#63647—31no—
27#63647—34no—
28#63647—64no—
29#63643D311yescharacteristic, complemented
30#63637—31no—
31#63635—31no—
32C3×#817D314yescomplemented
33C3×#816—34no—
34#63574—34no—
35#63565—34no—
36C22×C3⋊(C3⋊D3)D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)D411yescharacteristic
39C4×C3⋊(C3⋊D3)—21no—
40C4×C3⋊(C3⋊D3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C3×#634—21no—
44C3×#634—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47#6289—42no—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#4862—94no—
51#4805—91no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—34no—
54C2×C4×C3⋊D3—61no—
55C2×C4×C3⋊D3—64no—
56C2×#634C2×D311yescharacteristic
57C2×C3×Dic6—34no—
58C2×#624—64no—
59C2×C32×C8—24no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C2×D311yescharacteristic
62C32×C4⋊C4—31no—
63C32×C4⋊C4—31no—
64#3752—61no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3C2×D314yes—
73C3×C4⋊Dic3C2×D314yes—
74#3724—64no—
75Dic3⋊Dic3—64no—
76#3722—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3⋊(C3⋊D3)—21no—
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)SD1612yes—
81C33×C4#4111yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—21no—
84C3×C3⋊Dic3—41no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—41no—
87C32×Dic3—41no—
88#946—91no—
89#897—94no—
90D3×C4⋊C4—94no—
91D3×C4⋊C4—91no—
92C3×#106—34no—
93C3×#103—91no—
94#835—91no—
95(C2×D3)⋊C8—94no—
96(C2×D3)⋊C8—91no—
97(C2×D3)⋊C8—184no—
98#817C3⋊D311yescharacteristic, complemented
99#816—34no—
100#810—94no—
101#808—94no—
102#808—91no—
103C22×C3⋊D3—31no—
104C22×C3⋊D3—34no—
105C22×C3⋊D3—64no—
106C2×C32×C4—24no—
107C2×C32×C4—61no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—64no—
116C4×C3⋊D3—61no—
117C4×C3⋊D3—61no—
118C4×C3⋊D3—64no—
119C4×C3⋊D3—64no—
120C4×C3⋊D3—124no—
121C4×C3⋊D3—124no—
122C4×C3⋊D3—122no—
123#634—21no—
124#634—21no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129C3×Dic6—64no—
130C3×Dic6—64no—
131#624—128no—
132C32×C8—21no—
133C32×C8—24no—
134C32×C8—44no—
135C3⋊(C3⋊C8)—61no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—124no—
138#483—271noSylow
139C2×C33#12311yescharacteristic
140C2×C33C2×SD1611yescharacteristic
141C2×C33#13711yescharacteristic
142C3⋊(C3⋊D3)—22no—
143C2×C3×Q8—91no—
144C2×C4×D3—94no—
145C2×C4×D3—91no—
146C2×C4×D3—184no—
147C2×C4×D3—181no—
148C2×C4×D3—184no—
149C2×Dic6—34no—
150C2×C3×C8—24no—
151C2×C3×C8D3214yes—
152C2×C3×C8C2×C3⋊D311yescharacteristic
153C3×C4⋊C4—34no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—91no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—184no—
158C4⋊Dic3—34no—
159C4⋊Dic3—184no—
160C4⋊Dic3C2×C3⋊D311yescharacteristic
161C4⋊Dic3C2×C3⋊D311yescharacteristic
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—184no—
165C2×C3⋊C8—94no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—184no—
168C22×C32—24no—
169C22×C32D3×D414yes—
170C22×C32D3×D411yescharacteristic
171C2×C3⋊D3—32no—
172C2×C3⋊D3—38no—
173C2×C3⋊D3—68no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—64no—
178C2×C3⋊D3—124no—
179C2×C3⋊D3—124no—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—121no—
183C32×C4—121no—
184C32×C4#23514yes—
185C32×C4#23511yescharacteristic
186C32×C4#23711yescharacteristic
187C32×C4#23914yes—
188C3⋊Dic3—41no—
189C3⋊Dic3—41no—
190C3⋊Dic3—61no—
191C3⋊Dic3—64no—
192C3⋊Dic3—124no—
193C3×Dic3—44no—
194C3×Dic3—44no—
195C3×Dic3—124no—
196C3×Dic3—124no—
197C22⋊Q8—271no—
198C2×C4⋊C4—271no—
199#106—31no—
200#106—271no—
201#103—91no—
202#103—271no—
203C22⋊C8—271no—
204C33#48311yescharacteristic, complemented, Sylow
205C22×D3—94no—
206C22×D3—91no—
207C22×D3—184no—
208C3×Q8—181no—
209C3×Q8—181no—
210C2×C3×C4—24no—
211C2×C3×C4—64no—
212C2×C3×C4—64no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—64no—
220C2×Dic3—64no—
221C2×Dic3—94no—
222C2×Dic3—91no—
223C2×Dic3—184no—
224C4×D3—184no—
225C4×D3—184no—
226C4×D3—181no—
227C4×D3—181no—
228C4×D3—364no—
229C4×D3—364no—
230C4×D3—362no—
231C4×D3—368no—
232Dic6—64no—
233Dic6—64no—
234C3×C8—24no—
235C3×C8—21no—
236C3×C8—44no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C3⋊C8—364no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32#88311yescharacteristic
244C2×C32#89414yes—
245C2×C32D3×SD1614yes—
246C2×C32D3×SD1611yescharacteristic
247C2×C32#91514yes—
248C2×C32#91811yescharacteristic
249C3⋊D3—62no—
250C3⋊D3—68no—
251C3⋊D3—128no—
252C2×Q8—91no—
253C22×C4—271no—
254C22×C4—541no—
255C2×C8—271no—
256C2×C8D3×C3⋊D311yescharacteristic
257C4⋊C4—31no—
258C4⋊C4—31no—
259C4⋊C4—91no—
260C4⋊C4—541no—
261C4⋊C4—541no—
262C22⋊C4—541no—
263C22×C3—24no—
264C22×C3#380214yes—
265C22×C3D4×C3⋊D311yescharacteristic
266C2×D3—98no—
267C2×D3—92no—
268C2×D3—188no—
269C2×D3—184no—
270C2×D3—181no—
271C2×D3—184no—
272C2×D3—181no—
273C2×D3—364no—
274C2×D3—364no—
275C3×C4—24no—
276C3×C4—24no—
277C3×C4—124no—
278C3×C4—124no—
279C3×C4—361no—
280C3×C4—361no—
281C3×C4#379914yes—
282C3×C4#380414yes—
283C3×C4#382811yescharacteristic
284C3×C4#383211yescharacteristic
285Dic3—41no—
286Dic3—41no—
287Dic3—124no—
288Dic3—124no—
289Dic3—181no—
290Dic3—184no—
291Dic3—364no—
292C32—24no—
293C32#480511yescharacteristic, complemented
294C32#486214yescomplemented
295C23—271no—
296Q8—181no—
297Q8—181no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—271no—
303C2×C4—541no—
304C2×C4—541no—
305C2×C4—1082no—
306C2×C4C2×D3×C3⋊D311yescharacteristic
307C8—21no—
308C8—541no—
309C2×C3—24no—
310C2×C3—24no—
311C2×C3—24no—
312C2×C3#6379614yes—
313C2×C3#6381114yes—
314C2×C3#6385414yes—
315C2×C3#6410311yescharacteristic
316C2×C3SD16×C3⋊D311yescharacteristic
317C2×C3#6412411yescharacteristic
318D3—182no—
319D3—188no—
320D3—368no—
321C22—272no—
322C22—541no—
323C22—541no—
324C22#8902611yescharacteristic
325C4—121no—
326C4—121no—
327C4—361no—
328C4—361no—
329C4—541no—
330C4#8901811yescharacteristic
331C4#8903111yescharacteristic
332C3—24no—
333C3#1059068614yescomplemented
334C3#1059256111yescharacteristic, complemented
335C2—542no—
336C2#1172332111yescharacteristic
337C2#1172334411yescharacteristic
338C2#1172350411yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63643C2semidirect
5. C32×#817C2semidirect
6. #11722053C2semidirect
7. #11722039C2semidirect
29. #63643D3semidirect
32. C3×#817D3semidirect
98. #817C3⋊D3semidirect
204. C33#483semidirect
293. C32#4805semidirect
294. C32#4862semidirect
333. C3#10590686semidirect
334. C3#10592561semidirect

Maximal subgroups

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

subgroupindex
#117235032
#117234492
#117221902
C3×#636432
C32×#8172
#117220532
#117220392
#105925613
#105906863