Finite Group Databasebeta

← #13161151

Subgroups — #13161151

Order 1728.

912 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 156 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×#64156C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592864—31nomaximal
9#10591567—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×#3752—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88789—22no—
23C2×C32×SD16—31no—
24#64156D311yescharacteristic, complemented
25D4×C3⋊Dic3—31no—
26C2×#3753—31no—
27C3×#947—34no—
28C2×C3×#215D314yescomplemented
29Dic3×D12—34no—
30C2×#3713—34no—
31#63662—31no—
32#63662—34no—
33#63662—64no—
34#63648—31no—
35#63641—31no—
36Dic3⋊(C3⋊C8)—34no—
37#63563—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C32×D12—21no—
50C32×D12D412yes—
51D3×C3⋊Dic3—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8—21no—
54#5165—94no—
55#5057—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×D4—31no—
60C22×C3⋊Dic3—61no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—34no—
63C2×C3×D12C2×D314yes—
64C2×D3×Dic3—64no—
65C32×SD16—62no—
66C2×C32×C8—31no—
67#3757—61no—
68#3753—62no—
69#3752—21no—
70#3752—21no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—64no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78C3×#215—28no—
79C3×(C2×D3)⋊C4—64no—
80C3×(C2×D3)⋊C4—64no—
81C3×C4×Dic3—34no—
82C2×C3×C3⋊C8C2×D314yes—
83(C2×D3)⋊Dic3—64no—
84#3713—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C2×C3×C3⋊D3—41no—
89C2×C32×D3—22no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C33×C4—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—21no—
98C2×C3×SD16—34no—
99C3×#124—91no—
100#947—34no—
101C2×#215C3⋊D311yescharacteristic, complemented
102D4×Dic3—94no—
103C2×#213—94no—
104C2×#213—91no—
105C4×D12—91no—
106#835—94no—
107#835—91no—
108#835—184no—
109#832—91no—
110#821—94no—
111Dic3⋊C8—94no—
112C4⋊(C3⋊C8)—91no—
113C23×C32—61no—
114C22×C3⋊D3—21no—
115C22×C3×D3—24no—
116C22×C3×D3—64no—
117C32×Q8—28no—
118C32×Q8—21no—
119C32×Q8—24no—
120C32×Q8—21no—
121C32×Q8—24no—
122C32×Q8—44no—
123C32×D4—32no—
124C32×D4—61no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—44no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—124no—
134C2×C3⋊Dic3—122no—
135C2×C3⋊Dic3#6711yescharacteristic
136C2×C3⋊Dic3#6711yescharacteristic
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
141C3×D12—24no—
142C3×D12—64no—
143C3×D12—64no—
144C3×D12#6718yes—
145D3×Dic3—128no—
146C32×C8—61no—
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—124no—
150C3×C3⋊C8—24no—
151#463—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13511yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C3×C3⋊D3—81no—
156C32×D3—42no—
157C2×C3×Q8—24no—
158C2×C3×Q8D3214yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—34no—
161C2×C3×D4—91no—
162C22×Dic3—184no—
163C2×D12—34no—
164C2×D12C2×C3⋊D311yescharacteristic
165C2×C4×D3—181no—
166C3×SD16—68no—
167C2×C3×C8—34no—
168C3×C22⋊C4—181no—
169C3×C22⋊C4—181no—
170C3×C42—91no—
171#217—184no—
172#215—22no—
173#213—188no—
174#213—182no—
175(C2×D3)⋊C4—64no—
176(C2×D3)⋊C4—64no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—94no—
179C4⋊Dic3—91no—
180C4⋊Dic3—184no—
181C4×Dic3—34no—
182C2×C3⋊C8—94no—
183C2×C3⋊C8—91no—
184C2×C3⋊C8—184no—
185C2×C3⋊C8C2×C3⋊D311yescharacteristic
186C22×C32—24no—
187C22×C32—62no—
188C22×C32—121no—
189C22×C32—121no—
190C22×C32D3×D414yes—
191C22×C32C2×#6711yescharacteristic
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3×D3—28no—
196C2×C3×D3—44no—
197C2×C3×D3—44no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200C2×C3×D3—124no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—48no—
204C32×C4—41no—
205C32×C4—44no—
206C32×C4D3×D411yescharacteristic
207C32×C4#23714yes—
208C32×C4#23711yescharacteristic
209C32×C4C2×#6714yes—
210C3⋊Dic3—21no—
211C3⋊Dic3—21no—
212C3⋊Dic3—121no—
213C3⋊Dic3—124no—
214C3⋊Dic3—244no—
215C3×Dic3—64no—
216C3×Dic3—64no—
217C2×SD16—31no—
218C2×D8—271no—
219#124—91no—
220C4×D4—271no—
221C4⋊C8—271no—
222#103—271no—
223#102—271no—
224C33#46311yescharacteristic, complemented, Sylow
225C23×C3—64no—
226C23×C3—181no—
227C22×D3—21no—
228C22×D3—64no—
229C3×Q8—28no—
230C3×Q8—24no—
231C3×Q8—24no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—44no—
235C3×D4—38no—
236C3×D4—64no—
237C3×D4—181no—
238C3×D4—181no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—44no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4C2×D3214yes—
246C2×C3×C4C22×C3⋊D311yescharacteristic
247C2×Dic3—34no—
248C2×Dic3—34no—
249C2×Dic3—184no—
250C2×Dic3—181no—
251C2×Dic3—364no—
252C2×Dic3—368no—
253D12—21no—
254D12—64no—
255D12—64no—
256D12#63812yes—
257C4×D3—362no—
258C3×C8—64no—
259C3⋊C8—21no—
260C3⋊C8—184no—
261C3⋊C8—181no—
262C3⋊C8—364no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—24no—
266C2×C32—122no—
267C2×C32#91414yes—
268C2×C32#91614yes—
269C2×C32#93811yescharacteristic
270C2×C32#93914yes—
271C2×C32#94911yescharacteristic
272C2×C32#95011yescharacteristic
273C3⋊D3—81no—
274C3×D3—48no—
275C3×D3—244no—
276C2×Q8D3×C3⋊D311yescharacteristic
277C2×D4—31no—
278C2×D4—91no—
279C22×C4—541no—
280SD16—62no—
281D8—542no—
282C2×C8—31no—
283C2×C8—271no—
284C4⋊C4—271no—
285C22⋊C4—181no—
286C22⋊C4—181no—
287C22⋊C4—541no—
288C42—91no—
289C22×C3—24no—
290C22×C3—68no—
291C22×C3—124no—
292C22×C3—124no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D3×#6714yes—
297C22×C3D4×C3⋊D311yescharacteristic
298C2×D3—22no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—124no—
302C2×D3—124no—
303C2×D3—124no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—48no—
307C3×C4—41no—
308C3×C4—44no—
309C3×C4—181no—
310C3×C4—181no—
311C3×C4#380514yes—
312C3×C4D3×#6714yes—
313C3×C4#383011yescharacteristic
314C3×C4C2×#63811yescharacteristic
315Dic3—64no—
316Dic3—64no—
317Dic3—361no—
318Dic3—364no—
319Dic3—724no—
320C32—24no—
321C32#505711yescharacteristic, complemented
322C32#516514yescomplemented
323C23—61no—
324C23—181no—
325Q8—21no—
326Q8—21no—
327D4—32no—
328D4—61no—
329D4—181no—
330D4—181no—
331C2×C4—21no—
332C2×C4—91no—
333C2×C4—91no—
334C2×C4—541no—
335C2×C4—1082no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—61no—
338C8—541no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3—128no—
343C2×C3—721no—
344C2×C3#6393914yes—
345C2×C3#6394714yes—
346C2×C3#6397014yes—
347C2×C3#6412311yescharacteristic
348C2×C3#6412511yescharacteristic
349C2×C3#6414811yescharacteristic
350D3—42no—
351D3—244no—
352C22—62no—
353C22—121no—
354C22—121no—
355C22—361no—
356C22—361no—
357C22—361no—
358C22C3⋊D3×#6711yescharacteristic
359C4—41no—
360C4—181no—
361C4—181no—
362C4—1081no—
363C4#8902911yescharacteristic
364C4D3×#63811yescharacteristic
365C3—24no—
366C3#1059156714yescomplemented
367C3#1059286411yescharacteristic, complemented
368C2—122no—
369C2—721no—
370C2#1172357011yescharacteristic
371C2#1172358311yescharacteristic
372C2#1172361511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#88789C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C2×C32×#215C2semidirect
3. D12×C3⋊Dic3C2semidirect
5. #11722205C2semidirect
6. #11722051C2semidirect
7. #11722034C2semidirect
24. #64156D3semidirect
28. C2×C3×#215D3semidirect
101. C2×#215C3⋊D3semidirect
224. C33#463semidirect
321. C32#5057semidirect
322. C32#5165semidirect
366. C3#10591567semidirect
367. C3#10592864semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C32×#2152
D12×C3⋊Dic32
C2×#887892
#117222052
#117220512
#117220342
#105928643
#105915673