Finite Group Databasebeta

← #13159384

Subgroups — #13159384

Order 1728.

1108 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1108 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#11723934C211yesmaximal, characteristic
2#11723410C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic, complemented
8#10592551—31nomaximal
9#10590483—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C3×#634C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88859—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21#88797C414yescomplemented
22C2×#3744—31no—
23#64102—31no—
24#64102—34no—
25#64102—64no—
26#64096—31no—
27#63843—34no—
28C2×#3717—34no—
29C32×#106—31no—
30C8×C3⋊Dic3—31no—
31#63637D311yescharacteristic, complemented
32#63636—31no—
33C3×#810—34no—
34C3×#808D314yescomplemented
35#63564—34no—
36#63553—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C412yes—
44C4×C3⋊(C3⋊D3)—42no—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634C2×C412yescomplemented
48C2×C32×Dic3—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#4852—94no—
52#4788—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C2×C4×C3⋊D3—61no—
57C2×C4×C3⋊D3—64no—
58C2×C4×C3⋊D3—68no—
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C32×C8—31no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—61no—
67#3752—64no—
68#3752—68no—
69#3750—61no—
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—61no—
73C4×C3⋊Dic3—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75#3744—34no—
76C3×C4⋊Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78C2×C3×C3⋊C8C2×D314yes—
79Dic3⋊Dic3—64no—
80Dic32—64no—
81#3717—316no—
82C22×C33C2×D411yescharacteristic
83C2×C3⋊(C3⋊D3)—22no—
84C2×C3⋊(C3⋊D3)—41no—
85C2×C3⋊(C3⋊D3)—41no—
86C33×C4—41no—
87C33×C4C22×C411yescharacteristic
88C33×C4#4111yescharacteristic
89C3⋊(C3⋊Dic3)—21no—
90C3⋊(C3⋊Dic3)SD1612yescomplemented
91C3×C3⋊Dic3—22no—
92C32×Dic3—41no—
93Q8×Dic3—91no—
94C2×#215—91no—
95C2×#204—94no—
96#893—94no—
97#893—91no—
98#893—184no—
99#887—94no—
100C3×#106—34no—
101C3×#103—91no—
102#821—91no—
103C8×Dic3—94no—
104C8×Dic3—91no—
105#810—34no—
106#809—94no—
107#808C3⋊D311yescharacteristic, complemented
108C22×C3⋊D3—68no—
109C22×C3⋊D3—64no—
110C22×C3⋊D3—61no—
111C2×C32×C4—28no—
112C2×C32×C4—24no—
113C2×C32×C4—21no—
114C2×C32×C4—24no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125C4⋊(C3⋊D3)—32no—
126C4⋊(C3⋊D3)—38no—
127C4⋊(C3⋊D3)—68no—
128C4⋊(C3⋊D3)—68no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—64no—
131C4×C3⋊D3—122no—
132C4×C3⋊D3—128no—
133C4×C3⋊D3—1216no—
134#634—21no—
135#634C4×D312yescomplemented
136C2×C3×Dic3—24no—
137C2×C3×Dic3—64no—
138C3×Dic6—38no—
139C3×Dic6—64no—
140C32×C8—32no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—64no—
143C3×C3⋊C8C4×D318yescomplemented
144C4×SD16—271noSylow
145C2×C33C4×D411yescharacteristic
146C2×C33C2×SD1611yescharacteristic
147C2×C33#13511yescharacteristic
148C3⋊(C3⋊D3)—42no—
149C2×C3×Q8—91no—
150C2×D12—94no—
151C2×D12—91no—
152C2×D12—184no—
153C2×C4×D3—184no—
154C2×C4×D3—181no—
155C2×C4×D3—188no—
156C2×Dic6—34no—
157C2×C3×C8—34no—
158C2×C3×C8—91no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—34no—
161C3×C4⋊C4D3214yes—
162C3×C4⋊C4C2×C3⋊D311yescharacteristic
163#215—94no—
164(C2×D3)⋊C4—184no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—188no—
167C4⋊Dic3—181no—
168C4⋊Dic3C2×C3⋊D311yescharacteristic
169Dic3⋊C4—184no—
170C4×Dic3—94no—
171C4×Dic3—91no—
172C4×Dic3—184no—
173C4×Dic3—181no—
174C4×Dic3—184no—
175C2×C3⋊C8—34no—
176C2×C3⋊C8C2×C3⋊D311yescharacteristic
177#204—916no—
178C22×C32—24no—
179C22×C32D3×D414yes—
180C22×C32D3×D411yescharacteristic
181C2×C3⋊D3—616no—
182C2×C3⋊D3—62no—
183C2×C3⋊D3—68no—
184C2×C3⋊D3—128no—
185C2×C3⋊D3—128no—
186C2×C3⋊D3—124no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—124no—
189C2×C3⋊D3—121no—
190C32×C4—24no—
191C32×C4—24no—
192C32×C4—48no—
193C32×C4—44no—
194C32×C4—41no—
195C32×C4—121no—
196C32×C4C2×C4×D314yes—
197C32×C4C2×C4×D311yescharacteristic
198C32×C4#23711yescharacteristic
199C32×C4#23914yes—
200C3⋊Dic3—22no—
201C3⋊Dic3—32no—
202C3⋊Dic3—38no—
203C3⋊Dic3—68no—
204C3⋊Dic3—61no—
205C3⋊Dic3—64no—
206C3⋊Dic3—124no—
207C3×Dic3—44no—
208C3×Dic3—68no—
209C2×SD16—271no—
210C4×Q8—271no—
211C4×D4—271no—
212#106—31no—
213#103—91no—
214#102—271no—
215C4×C8—271no—
216C33C4×SD1611yescharacteristic, complemented, Sylow
217C22×D3—188no—
218C22×D3—184no—
219C22×D3—181no—
220C3×Q8—92no—
221C3×Q8—181no—
222C2×C3×C4—28no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—64no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×Dic3—21no—
231C2×Dic3—64no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—94no—
235C2×Dic3—91no—
236C2×Dic3—184no—
237C2×Dic3—184no—
238D12—98no—
239D12—92no—
240D12—188no—
241D12—188no—
242D12—184no—
243D12—181no—
244C4×D3—362no—
245C4×D3—368no—
246C4×D3—3616no—
247Dic6—38no—
248Dic6—64no—
249C3×C8—38no—
250C3×C8—181no—
251C3⋊C8—64no—
252C3⋊C8C4×C3⋊D312yescomplemented
253C2×C32—24no—
254C2×C32—24no—
255C2×C32—24no—
256C2×C32#67⋊C411yescharacteristic
257C2×C32#89314yes—
258C2×C32D3×SD1614yes—
259C2×C32D3×SD1611yescharacteristic
260C2×C32#91614yes—
261C2×C32#91911yescharacteristic
262C3⋊D3—1216no—
263C3⋊D3—122no—
264C3⋊D3—128no—
265C2×Q8—91no—
266C2×D4—271no—
267C22×C4—541no—
268SD16—274no—
269C2×C8—31no—
270C2×C8—91no—
271C4⋊C4—31no—
272C4⋊C4—541no—
273C4⋊C4D3×C3⋊D311yescharacteristic
274C22⋊C4—541no—
275C42—271no—
276C42—541no—
277C22×C3—24no—
278C22×C3#381114yes—
279C22×C3D4×C3⋊D311yescharacteristic
280C2×D3—1816no—
281C2×D3—182no—
282C2×D3—188no—
283C2×D3—368no—
284C2×D3—368no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—364no—
288C2×D3—364no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—48no—
292C3×C4—41no—
293C3×C4—44no—
294C3×C4—124no—
295C3×C4—182no—
296C3×C4#379614yes—
297C3×C4C4×D3214yes—
298C3×C4C2×C4×C3⋊D311yescharacteristic
299C3×C4#383211yescharacteristic
300Dic3—41no—
301Dic3—68no—
302Dic3—92no—
303Dic3—98no—
304Dic3—188no—
305Dic3—181no—
306Dic3—184no—
307Dic3—364no—
308C32—24no—
309C32#478811yescharacteristic, complemented
310C32#485214yescomplemented
311C23—541no—
312Q8—92no—
313Q8—181no—
314D4—272no—
315D4—541no—
316C2×C4—21no—
317C2×C4—61no—
318C2×C4—181no—
319C2×C4—271no—
320C2×C4—271no—
321C2×C4—1082no—
322C2×C4C2×D3×C3⋊D311yescharacteristic
323C8—32no—
324C8—181no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3#6388114yes—
329C2×C3#6393014yes—
330C2×C3#6395014yes—
331C2×C3#6410211yescharacteristic
332C2×C3SD16×C3⋊D311yescharacteristic
333C2×C3#6412511yescharacteristic
334D3—3616no—
335D3—368no—
336D3—362no—
337C22—542no—
338C22—1081no—
339C22—1081no—
340C22#8903911yescharacteristic
341C4—41no—
342C4—121no—
343C4—182no—
344C4—272no—
345C4—541no—
346C4#8901711yescharacteristic
347C4C4×D3×C3⋊D311yescharacteristic
348C3—24no—
349C3#1059048314yescomplemented
350C3#1059255111yescharacteristic, complemented
351C2—1082no—
352C2#1172347211yescharacteristic
353C2#1172355311yescharacteristic
354C2#1172358711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×SD16
LCSG ▷ 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. #11723410C2semidirect
4. C3×#63637C2semidirect
5. C32×#808C2semidirect
7. #11722018C2semidirect
21. #88797C4semidirect
31. #63637D3semidirect
34. C3×#808D3semidirect
47. C3×#634C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
90. C3⋊(C3⋊Dic3)SD16semidirect
107. #808C3⋊D3semidirect
135. #634C4×D3semidirect
143. C3×C3⋊C8C4×D3semidirect
216. C33C4×SD16semidirect
252. C3⋊C8C4×C3⋊D3semidirect
309. C32#4788semidirect
310. C32#4852semidirect
349. C3#10590483semidirect
350. C3#10592551semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117234102
C2×#887972
C3×#636372
C32×#8082
#117220352
#117220182
#105925513
#105904833