Finite Group Databasebeta

← #13160975

Subgroups — #13160975

Order 1728.

1016 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1016 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
1C2×#88861C211yesmaximal, characteristic, complemented
2C32×#938C211yesmaximal, characteristic
3#11723426C211yesmaximal, characteristic, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592740—31nomaximal
9#10591460—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×C4×D3C2211yescharacteristic
14#88861—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#217—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20#88807—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22D3×C3⋊(C3⋊C8)—22no—
23C32×#121—31no—
24C2×#3754—31no—
25C2×#3754—34no—
26C2×#3754—64no—
27#64133—31no—
28C22×C3⋊(C3⋊C8)—31no—
29C3×#938D314yes—
30#63857—34no—
31C2×D3×C3⋊C8—34no—
32#63659D311yescharacteristic, complemented
33#63637—31no—
34#63635—31no—
35C3×#832—34no—
36#63574—34no—
37#63567—34no—
38C23×C33—21no—
39C22×C32×D3D411yescharacteristic
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46C2×C3×C3⋊Dic3—21no—
47C32×#67—42no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3—21no—
51C32×C4×D3—21no—
52C3⋊(C3⋊(C3⋊C8))D412yescomplemented
53C3×C3⋊(C3⋊C8)—21no—
54#5041—94no—
55#4778—91no—
56C2×C32×D4—24no—
57C2×C32×D4—61no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C2×D311yescharacteristic
60C22×C32×C4—31no—
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×#67—24no—
65C2×C3×C4×D3C2×D314yes—
66C32×C4⋊C4—31no—
67C32×C22⋊C4—61no—
68#3757—61no—
69#3754—62no—
70#3754—68no—
71#3754—616no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3750—61no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—62no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×#217—24no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3C2×D314yes—
82C2×C3×C3⋊C8—34no—
83Dic3⋊Dic3—64no—
84(C2×D3)⋊Dic3—64no—
85D3×C3⋊C8—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3SD1612yescomplemented
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—21no—
98C32×Dic3—41no—
99C3×#121—34no—
100#938C3⋊D311yescharacteristic
101C2×#214—94no—
102C2×#214—91no—
103C2×#214—184no—
104#924—94no—
105C22×C3⋊C8—94no—
106C2×D3×C8—91no—
107#897—91no—
108C3×#102—91no—
109#832—34no—
110#816—91no—
111#810—94no—
112#808—94no—
113#808—91no—
114C23×C32—21no—
115C23×C32—24no—
116C23×C32—31no—
117C23×C32—44no—
118C22×C3×D3#6714yes—
119C32×D4—28no—
120C32×D4—21no—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—44no—
125C32×D4—122no—
126C2×C32×C4—24no—
127C2×C32×C4—31no—
128C2×C32×C4—61no—
129C2×C32×C4—61no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—61no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—124no—
137#634—68no—
138#634—61no—
139#634—64no—
140#634—61no—
141#634—64no—
142#634—124no—
143C3×#67—48no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3#6714yes—
147C3×C4×D3—24no—
148C3×C4×D3—24no—
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3×C3⋊C8—64no—
154#465—271noSylow
155C2×C33—81no—
156C2×C33#12111yescharacteristic
157C2×C33C2×SD1611yescharacteristic
158C2×C33#13511yescharacteristic
159C32×D3—22no—
160C2×C3×D4—24no—
161C2×C3×D4—64no—
162C2×C3×D4D3214yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C22×C3×C4—34no—
165C2×#67—21no—
166C2×C4×D3C2×C3⋊D311yescharacteristic
167C2×Dic6—94no—
168C2×Dic6—91no—
169C2×Dic6—184no—
170C2×C3×C8—91no—
171C3×C4⋊C4—34no—
172C3×C4⋊C4—91no—
173C3×C22⋊C4—64no—
174#217—21no—
175#217—184no—
176#214—188no—
177#214—182no—
178#214—1816no—
179(C2×D3)⋊C4—181no—
180C4⋊Dic3—34no—
181C4⋊Dic3—181no—
182C4⋊Dic3C2×C3⋊D311yescharacteristic
183Dic3⋊C4—184no—
184C2×C3⋊C8—34no—
185C2×C3⋊C8—94no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—184no—
188C2×C3⋊C8—188no—
189D3×C8—182no—
190C22×C32—24no—
191C22×C32—32no—
192C22×C32—48no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—44no—
197C22×C32—44no—
198C22×C32—44no—
199C22×C32—61no—
200C22×C32—61no—
201C22×C32—84no—
202C22×C32—84no—
203C22×C32D3×D411yescharacteristic
204C22×C32C2×#6714yes—
205C2×C3×D3—24no—
206C2×C3×D3—24no—
207C2×C3×D3#21418yescomplemented
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—61no—
211C32×C4—121no—
212C32×C4#23511yescharacteristic
213C32×C4D3×D414yes—
214C32×C4D3×D411yescharacteristic
215C32×C4#23714yes—
216C3⋊Dic3—41no—
217C3⋊Dic3—128no—
218C3⋊Dic3—121no—
219C3⋊Dic3—124no—
220C3×Dic3—24no—
221C3×Dic3—44no—
222C3×Dic3—124no—
223C2×SD16—271no—
224C22×C8—271no—
225C22⋊Q8—271no—
226#121—31no—
227#106—271no—
228#103—271no—
229#102—91no—
230C33#46511yescharacteristic, complemented, Sylow
231C23×C3—24no—
232C23×C3—21no—
233C23×C3—34no—
234C23×C3—44no—
235C22×D3#63811yescharacteristic
236C3×D4—28no—
237C3×D4—24no—
238C3×D4—24no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—44no—
242C3×D4—128no—
243C2×C3×C4—24no—
244C2×C3×C4—34no—
245C2×C3×C4—64no—
246C2×C3×C4—64no—
247C2×C3×C4—64no—
248C2×C3×C4—181no—
249C2×C3×C4C2×D3214yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251#67—42no—
252C2×Dic3—21no—
253C2×Dic3—64no—
254C2×Dic3—184no—
255C2×Dic3—181no—
256C2×Dic3—364no—
257C2×Dic3#63811yescharacteristic
258C4×D3—21no—
259C4×D3—21no—
260Dic6—188no—
261Dic6—184no—
262Dic6—184no—
263Dic6—181no—
264Dic6—181no—
265Dic6—364no—
266C3×C8—181no—
267C3⋊C8—64no—
268C3⋊C8—98no—
269C3⋊C8—92no—
270C3⋊C8—188no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—62no—
275C2×C32—88no—
276C2×C32—81no—
277C2×C32—84no—
278C2×C32#88411yescharacteristic
279C2×C32#91211yescharacteristic
280C2×C32D3×SD1611yescharacteristic
281C2×C32C2×#21414yes—
282C2×C32#93914yes—
283C2×C32#95014yes—
284C3×D3—28no—
285C2×Q8—271no—
286C2×D4—61no—
287C2×D4D3×C3⋊D311yescharacteristic
288C22×C4—31no—
289SD16—542no—
290C2×C8—91no—
291C2×C8—271no—
292C2×C8—542no—
293C4⋊C4—31no—
294C4⋊C4—91no—
295C4⋊C4—541no—
296C22⋊C4—61no—
297C22⋊C4—541no—
298C22×C3—24no—
299C22×C3—38no—
300C22×C3—48no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—44no—
307C22×C3—64no—
308C22×C3—64no—
309C22×C3—84no—
310C22×C3—84no—
311C22×C3D3×#6714yes—
312C22×C3C2×#63811yescharacteristic
313C2×D3—21no—
314C2×D3—21no—
315C2×D3#375412yescomplemented
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—64no—
319C3×C4—124no—
320C3×C4—361no—
321C3×C4#380414yes—
322C3×C4#381114yes—
323C3×C4D4×C3⋊D311yescharacteristic
324C3×C4#383011yescharacteristic
325Dic3—21no—
326Dic3—41no—
327Dic3—124no—
328Dic3—368no—
329Dic3—361no—
330Dic3—364no—
331C32—24no—
332C32#477811yescharacteristic, complemented
333C32#504114yescomplemented
334C23—21no—
335C23—31no—
336Q8—541no—
337Q8—541no—
338D4—21no—
339D4—21no—
340D4—122no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—181no—
346C2×C4—541no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—181no—
349C8—272no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—68no—
354C2×C3—88no—
355C2×C3—84no—
356C2×C3—81no—
357C2×C3D3×#21414yes—
358C2×C3#6393314yes—
359C2×C3#6397114yes—
360C2×C3C2×#375411yescharacteristic
361C2×C3#6414811yescharacteristic
362C2×C3#6415911yescharacteristic
363D3—22no—
364C22—32no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—61no—
369C22—61no—
370C22D3×#63811yescharacteristic
371C4—61no—
372C4—121no—
373C4—361no—
374C4—1081no—
375C4#8902811yescharacteristic
376C4#8903911yescharacteristic
377C3—24no—
378C3#1059146014yescomplemented
379C3#1059274011yescharacteristic, complemented
380C2—62no—
381C2—81no—
382C2D3×#375411yescharacteristic
383C2#1172355611yescharacteristic
384C2#1172361611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#938C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#465
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
1. C2×#88861C2semidirect
3. #11723426C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63659C2semidirect
6. #11722053C2semidirect
7. #11722041C2semidirect
32. #63659D3semidirect
52. C3⋊(C3⋊(C3⋊C8))D4semidirect
92. C2×C32×D3SD16semidirect
207. C2×C3×D3#214semidirect
230. C33#465semidirect
315. C2×D3#3754semidirect
332. C32#4778semidirect
333. C32#5041semidirect
378. C3#10591460semidirect
379. C3#10592740semidirect

Maximal subgroups

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

subgroupindex
C2×#888612
C32×#9382
#117234262
C2×D3×C3⋊(C3⋊C8)2
C3×#636592
#117220532
#117220412
#105927403
#105914603