Finite Group Databasebeta

← #13159932

Subgroups — #13159932

Order 1728.

992 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 166 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#11723494C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic
4#11722191C211yesmaximal, characteristic, complemented
5C3×#63642C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722054C211yesmaximal, characteristic, complemented
8#10592484—31nomaximal
9#10590746—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×#6289—21no—
13C2×D3×C3⋊Dic3—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88804—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88790C414yescomplemented
22D4×C3⋊Dic3—31no—
23C2×#3753—31no—
24#64102—31no—
25#63894—34no—
26Dic3×D12—34no—
27C2×#3714—34no—
28C32×#102—31no—
29#63648—31no—
30#63648—34no—
31#63648—64no—
32#63642D311yescharacteristic, complemented
33#63635—31no—
34C3×#821D314yescomplemented
35C3×#815—34no—
36#63573—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C22×C32×D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C32×Dic3—21no—
46C32×D12—21no—
47C32×D12C2×C412yescomplemented
48#6289—42no—
49D3×C3⋊Dic3—42no—
50C33×C8—21no—
51C3×C3⋊(C3⋊C8)C2×C412yescomplemented
52#4785—94no—
53#4711—91no—
54C2×C32×D4—31no—
55C22×C3⋊Dic3—61no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4×C3⋊D3—61no—
60C2×C3×D12C2×D314yes—
61C2×#624—64no—
62C2×D3×Dic3—64no—
63C2×C32×C8—24no—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C2×D311yescharacteristic
66C32×C4⋊C4—31no—
67#3757—61no—
68#3753—34no—
69#3752—61no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—34no—
72C4⋊(C3⋊Dic3)—64no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×C4⋊Dic3C2×D314yes—
76C3×C4×Dic3—34no—
77C2×C3×C3⋊C8—34no—
78#3722—64no—
79(C2×D3)⋊Dic3—64no—
80#3714—316no—
81C22×C33C2×D411yescharacteristic
82C2×C3⋊(C3⋊D3)—22no—
83C2×C3⋊(C3⋊D3)—41no—
84C2×C3⋊(C3⋊D3)—41no—
85C2×C32×D3—22no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4C22×C411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3×C3⋊Dic3—21no—
92C3×C3⋊Dic3—21no—
93C32×Dic3—41no—
94D4×Dic3—94no—
95C2×#213—94no—
96C2×D24—91no—
97#893—94no—
98C4×D12—91no—
99C4×D12—91no—
100C3×#102—34no—
101C3×#97—91no—
102#821—94no—
103#821—91no—
104#821—184no—
105#821C3⋊D311yescharacteristic, complemented
106#817—91no—
107#815—34no—
108#808—94no—
109C23×C32—61no—
110C22×C3⋊D3—61no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—124no—
113C22×C3×D3—24no—
114C32×D4—32no—
115C32×D4—61no—
116C2×C32×C4—24no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3D1211yescharacteristic
125C2×C3⋊Dic3D1211yescharacteristic
126C4⋊(C3⋊D3)—32no—
127C4⋊(C3⋊D3)—38no—
128C4⋊(C3⋊D3)—68no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—124no—
132C4×C3⋊D3—122no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C3×D12—24no—
137C3×D12C4×D318yescomplemented
138#624—128no—
139D3×Dic3—128no—
140C32×C8—21no—
141C32×C8—24no—
142C32×C8—44no—
143C3⋊(C3⋊C8)C4×D312yescomplemented
144C3×C3⋊C8—38no—
145#442—271noSylow
146C2×C33C4×D411yescharacteristic
147C2×C33AGL(1,ℤ/8)11yescharacteristic
148C2×C33AGL(1,ℤ/8)11yescharacteristic
149C3⋊(C3⋊D3)—42no—
150C32×D3—42no—
151C2×C3×D4—34no—
152C22×Dic3—184no—
153C2×D12—94no—
154C2×D12—91no—
155C2×D12—184no—
156C2×D12C2×C3⋊D311yescharacteristic
157C2×C4×D3—181no—
158C2×C4×D3—184no—
159C2×C4×D3—181no—
160C2×C3×C8—24no—
161C2×C3×C8—91no—
162C2×C3×C8D3214yes—
163C2×C3×C8C2×C3⋊D311yescharacteristic
164C3×C4⋊C4—34no—
165C3×C42—91no—
166#217—184no—
167#213—916no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—184no—
170(C2×D3)⋊C4—181no—
171C4⋊Dic3—94no—
172C4⋊Dic3—91no—
173C4⋊Dic3—184no—
174C4⋊Dic3C2×C3⋊D311yescharacteristic
175C4×Dic3—34no—
176C2×C3⋊C8—34no—
177D24—94no—
178C22×C32—24no—
179C22×C32—62no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32C2×D1211yescharacteristic
183C22×C32D3×D414yes—
184C2×C3⋊D3—62no—
185C2×C3⋊D3—68no—
186C2×C3⋊D3—128no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—124no—
190C2×C3⋊D3—124no—
191C2×C3⋊D3—244no—
192C2×C3⋊D3—244no—
193C2×C3×D3—28no—
194C2×C3×D3—44no—
195C2×C3×D3—44no—
196C32×C4—24no—
197C32×C4—24no—
198C32×C4—121no—
199C32×C4C2×C4×D314yes—
200C32×C4C2×C4×D311yescharacteristic
201C32×C4#23511yescharacteristic
202C32×C4#23714yes—
203C3⋊Dic3—21no—
204C3⋊Dic3—21no—
205C3⋊Dic3—121no—
206C3⋊Dic3—124no—
207C3⋊Dic3—244no—
208C3×Dic3—44no—
209C3×Dic3—64no—
210C3×Dic3—64no—
211C2×D8—271no—
212C4×D4—271no—
213C4×D4—271no—
214#106—271no—
215#102—31no—
216#102—271no—
217#97—91no—
218C33#44211yescharacteristic, complemented, Sylow
219C23×C3—64no—
220C22×D3—21no—
221C22×D3—184no—
222C22×D3—181no—
223C22×D3—364no—
224C3×D4—38no—
225C3×D4—64no—
226C2×C3×C4—24no—
227C2×C3×C4—64no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4C2×D3214yes—
231C2×C3×C4C22×C3⋊D311yescharacteristic
232C2×Dic3—21no—
233C2×Dic3—34no—
234C2×Dic3—34no—
235C2×Dic3—184no—
236C2×Dic3—181no—
237C2×Dic3—364no—
238C2×Dic3—368no—
239D12—21no—
240D12—98no—
241D12—92no—
242D12—188no—
243D12—184no—
244D12—181no—
245D12—364no—
246D12C4×C3⋊D312yescomplemented
247C4×D3—362no—
248C4×D3—368no—
249C4×D3—362no—
250C3×C8—24no—
251C3×C8—21no—
252C3×C8—44no—
253C3×C8—92no—
254C3⋊C8—38no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—122no—
259C2×C32C4×D1211yescharacteristic
260C2×C32#67⋊C414yes—
261C2×C32#90811yescharacteristic
262C2×C32#90811yescharacteristic
263C2×C32#91114yes—
264C2×C32#91414yes—
265C3⋊D3—122no—
266C3⋊D3—128no—
267C3⋊D3—248no—
268C3×D3—48no—
269C2×D4—31no—
270C2×D4—271no—
271C22×C4—541no—
272C22×C4—541no—
273D8—274no—
274C2×C8—91no—
275C2×C8D3×C3⋊D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—271no—
278C22⋊C4—541no—
279C22⋊C4—541no—
280C42—91no—
281C22×C3—24no—
282C22×C3—68no—
283C22×C3—124no—
284C22×C3—124no—
285C22×C3D3×D1214yes—
286C22×C3D4×C3⋊D311yescharacteristic
287C2×D3—22no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—182no—
291C2×D3—188no—
292C2×D3—368no—
293C2×D3—364no—
294C2×D3—364no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—724no—
298C2×D3—724no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—124no—
302C3×C4—181no—
303C3×C4—181no—
304C3×C4C4×D3214yes—
305C3×C4#380414yes—
306C3×C4C2×C4×C3⋊D311yescharacteristic
307C3×C4#383011yescharacteristic
308Dic3—41no—
309Dic3—64no—
310Dic3—64no—
311Dic3—364no—
312Dic3—361no—
313Dic3—724no—
314C32—24no—
315C32#471111yescharacteristic, complemented
316C32#478514yescomplemented
317C23—61no—
318C23—541no—
319D4—32no—
320D4—61no—
321D4—272no—
322D4—541no—
323C2×C4—61no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—541no—
327C2×C4—1082no—
328C2×C4—1082no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—21no—
331C8—92no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—128no—
336C2×C3#6379414yes—
337C2×C3#6379514yes—
338C2×C3#6388014yes—
339C2×C3#6409011yescharacteristic
340C2×C3#6412011yescharacteristic
341C2×C3#6412311yescharacteristic
342D3—42no—
343D3—368no—
344D3—362no—
345D3—728no—
346C22—62no—
347C22—121no—
348C22—121no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22C3⋊D3×D1211yescharacteristic
353C4—121no—
354C4—181no—
355C4—181no—
356C4—1081no—
357C4C4×D3×C3⋊D311yescharacteristic
358C4#8902811yescharacteristic
359C3—24no—
360C3#1059074614yescomplemented
361C3#1059248411yescharacteristic, complemented
362C2—122no—
363C2—1082no—
364C2#1172331911yescharacteristic
365C2#1172332011yescharacteristic
366C2#1172346111yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#88790C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#442
LCSG ▷ 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
1. #11723494C2semidirect
2. D12×C3⋊Dic3C2semidirect
4. #11722191C2semidirect
5. C3×#63642C2semidirect
6. C32×#821C2semidirect
7. #11722054C2semidirect
21. #88790C4semidirect
32. #63642D3semidirect
34. C3×#821D3semidirect
42. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
47. C32×D12C2×C4semidirect
51. C3×C3⋊(C3⋊C8)C2×C4semidirect
105. #821C3⋊D3semidirect
137. C3×D12C4×D3semidirect
143. C3⋊(C3⋊C8)C4×D3semidirect
218. C33#442semidirect
246. D12C4×C3⋊D3semidirect
315. C32#4711semidirect
316. C32#4785semidirect
360. C3#10590746semidirect
361. C3#10592484semidirect

Maximal subgroups

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

subgroupindex
#117234942
D12×C3⋊Dic32
C2×#887902
#117221912
C3×#636422
C32×#8212
#117220542
#105924843
#105907463