Finite Group Databasebeta

← #13161066

Subgroups — #13161066

Order 1728.

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

normal subgroups only

This group has 944 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
1C2×C3×#3753C211yesmaximal, characteristic
2C32×#936C211yesmaximal, characteristic, complemented
3Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722038C211yesmaximal, characteristic, complemented
8#10592756—31nomaximal
9#10591431—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3753—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×#217—21no—
20C32×C4×Dic3C2211yescharacteristic
21#88803—21no—
22#88793—22no—
23C32×#124—31no—
24C2×#3755—31no—
25C2×#3753D311yescharacteristic
26C4×C4⋊(C3⋊D3)—31no—
27C3×#936D314yescomplemented
28C2×C3×#213—34no—
29Dic3×D12—34no—
30C2×#3715—34no—
31#63662—31no—
32#63659—31no—
33#63659—34no—
34#63659—64no—
35C4⋊(C3⋊(C3⋊C8))—31no—
36Dic3⋊(C3⋊C8)—34no—
37#63566—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)D412yes—
46C2×C32×Dic3—21no—
47C2×C32×Dic3D411yescharacteristic
48C2×C32×Dic3D411yescharacteristic
49C32×Dic6—21no—
50C32×Dic6—21no—
51Dic3×C3⋊D3—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54#5153—91no—
55#5057—94no—
56C2×C32×Q8—31no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4×C3⋊D3—61no—
62C2×C3×D12—34no—
63C2×C3×Dic6C2×D314yes—
64C2×D3×Dic3—64no—
65C32×C22⋊C4—61no—
66C32×C22⋊C4—61no—
67C32×C42—31no—
68#3755—62no—
69#3753—22no—
70#3752—61no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—34no—
73C4⋊(C3⋊Dic3)—64no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×#217—24no—
79C3×#217—24no—
80C3×#213—68no—
81C3×C4×Dic3C2×D314yes—
82C2×C3×C3⋊C8—34no—
83(C2×D3)⋊Dic3—64no—
84#3715—68no—
85C22×C33—41no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C32×Dic3—21no—
96C32×Dic3—21no—
97C32×Dic3—41no—
98C2×C3×D8—91no—
99C3×#124—34no—
100C2×#215—94no—
101#936C3⋊D311yescharacteristic, complemented
102D4×Dic3—91no—
103C2×#214—91no—
104C2×#213—34no—
105C4×D12—94no—
106#835—94no—
107#832—94no—
108#832—91no—
109#832—184no—
110#816—91no—
111Dic3⋊C8—91no—
112C4⋊(C3⋊C8)—94no—
113C23×C32—21no—
114C23×C32—24no—
115C23×C32—44no—
116C22×C3⋊D3—21no—
117C22×C3×D3—64no—
118C32×Q8—61no—
119C32×Q8—61no—
120C32×D4—28no—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—24no—
125C32×D4—44no—
126C2×C32×C4—24no—
127C2×C32×C4—31no—
128C2×C32×C4—31no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—124no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)#6712yes—
137C4×C3⋊D3—122no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3#6714yes—
140C2×C3×Dic3#6714yes—
141C3×D12—38no—
142C3×D12—64no—
143C3×Dic6—24no—
144C3×Dic6—24no—
145D3×Dic3—128no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—124no—
150C3×C3⋊C8—64no—
151#463—271noSylow
152C2×C33—81no—
153C2×C33#12111yescharacteristic
154C2×C33#13511yescharacteristic
155C2×C33AGL(1,ℤ/8)11yescharacteristic
156C3×C3⋊D3—42no—
157C2×C3×Q8—34no—
158C2×C3×D4—24no—
159C2×C3×D4—91no—
160C2×C3×D4D3214yes—
161C2×C3×D4C2×C3⋊D311yescharacteristic
162C22×Dic3—181no—
163C2×D12—34no—
164C2×C4×D3—184no—
165C2×Dic6C2×C3⋊D311yescharacteristic
166C3×D8—182no—
167C2×C3×C8—91no—
168C3×C22⋊C4—64no—
169C3×C22⋊C4—64no—
170C3×C42—34no—
171#217—21no—
172#217—21no—
173#217—181no—
174#215—188no—
175#214—182no—
176#213—68no—
177(C2×D3)⋊C4—184no—
178C4⋊Dic3—94no—
179C4⋊Dic3—91no—
180C4⋊Dic3—184no—
181C4×Dic3C2×C3⋊D311yescharacteristic
182C2×C3⋊C8—34no—
183C2×C3⋊C8—94no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—184no—
186C22×C32—24no—
187C22×C32—48no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—44no—
191C22×C32—41no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—84no—
195C22×C32—84no—
196C22×C32D3×D411yescharacteristic
197C22×C32C2×#6714yes—
198C2×C3⋊D3—22no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3×D3—68no—
202C2×C3×D3—124no—
203C2×C3×D3—124no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—61no—
207C32×C4—61no—
208C32×C4—121no—
209C32×C4D3×D414yes—
210C32×C4#23714yes—
211C32×C4#23711yescharacteristic
212C32×C4C2×#6711yescharacteristic
213C3⋊Dic3—121no—
214C3⋊Dic3—124no—
215C3⋊Dic3—244no—
216C3×Dic3—24no—
217C3×Dic3—24no—
218C3×Dic3—44no—
219C2×SD16—271no—
220C2×D8—91no—
221#124—31no—
222C4×D4—271no—
223C4⋊C8—271no—
224#103—271no—
225#102—271no—
226C33#46311yescharacteristic, complemented, Sylow
227C23×C3—24no—
228C23×C3—21no—
229C23×C3—44no—
230C23×C3—181no—
231C22×D3—64no—
232C3×Q8—64no—
233C3×Q8—64no—
234C3×D4—28no—
235C3×D4—24no—
236C3×D4—24no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—44no—
240C3×D4—92no—
241C3×D4—181no—
242C2×C3×C4—24no—
243C2×C3×C4—34no—
244C2×C3×C4—34no—
245C2×C3×C4—64no—
246C2×C3×C4C2×D3214yes—
247C2×C3×C4C22×C3⋊D311yescharacteristic
248C2×Dic3—21no—
249C2×Dic3—184no—
250C2×Dic3—181no—
251C2×Dic3—364no—
252C2×Dic3—362no—
253C2×Dic3#63811yescharacteristic
254C2×Dic3#63811yescharacteristic
255D12—38no—
256D12—64no—
257C4×D3—368no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—181no—
261C3⋊C8—64no—
262C3⋊C8—184no—
263C3⋊C8—181no—
264C3⋊C8—364no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—88no—
269C2×C32—81no—
270C2×C32—84no—
271C2×C32#91111yescharacteristic
272C2×C32#91211yescharacteristic
273C2×C32#93814yes—
274C2×C32#93911yescharacteristic
275C2×C32#94914yes—
276C2×C32#95014yes—
277C3⋊D3—42no—
278C3×D3—128no—
279C2×Q8—31no—
280C2×D4—91no—
281C2×D4D3×C3⋊D311yescharacteristic
282C22×C4—541no—
283SD16—542no—
284D8—182no—
285C2×C8—91no—
286C2×C8—271no—
287C4⋊C4—271no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—541no—
291C42—31no—
292C22×C3—24no—
293C22×C3—48no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—84no—
301C22×C3—84no—
302C22×C3—182no—
303C22×C3—361no—
304C22×C3—361no—
305C22×C3D3×#6714yes—
306C22×C3C2×#63811yescharacteristic
307C2×D3—68no—
308C2×D3—124no—
309C2×D3—124no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—64no—
313C3×C4—64no—
314C3×C4—124no—
315C3×C4#380514yes—
316C3×C4D3×#6714yes—
317C3×C4D4×C3⋊D311yescharacteristic
318C3×C4#383011yescharacteristic
319Dic3—21no—
320Dic3—21no—
321Dic3—41no—
322Dic3—361no—
323Dic3—364no—
324Dic3—724no—
325C32—24no—
326C32#505714yescomplemented
327C32#515311yescharacteristic, complemented
328C23—21no—
329C23—181no—
330Q8—61no—
331Q8—61no—
332D4—21no—
333D4—21no—
334D4—92no—
335D4—181no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—61no—
339C2×C4—541no—
340C2×C4—1082no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—181no—
343C8—541no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—88no—
348C2×C3—84no—
349C2×C3—81no—
350C2×C3—362no—
351C2×C3#6392514yes—
352C2×C3#6393314yes—
353C2×C3#6397014yes—
354C2×C3#6414711yescharacteristic
355C2×C3#6415811yescharacteristic
356C2×C3#6415911yescharacteristic
357D3—128no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—182no—
362C22—361no—
363C22—361no—
364C22D3×#63811yescharacteristic
365C4—61no—
366C4—61no—
367C4—121no—
368C4—1081no—
369C4#8902911yescharacteristic
370C4C3⋊D3×#6711yescharacteristic
371C3—24no—
372C3#1059143114yescomplemented
373C3#1059275611yescharacteristic, complemented
374C2—81no—
375C2—362no—
376C2#1172354411yescharacteristic
377C2#1172355611yescharacteristic
378C2#1172363111yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C32×#936C2semidirect
3. Dic3×C4⋊(C3⋊D3)C2semidirect
4. C2×#88793C2semidirect
5. #11722202C2semidirect
6. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722038C2semidirect
27. C3×#936D3semidirect
101. #936C3⋊D3semidirect
226. C33#463semidirect
326. C32#5057semidirect
327. C32#5153semidirect
372. C3#10591431semidirect
373. C3#10592756semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×#9362
Dic3×C4⋊(C3⋊D3)2
C2×#887932
#117222022
Dic3⋊(C3⋊(C3⋊C8))2
#117220382
#105927563
#105914313