Finite Group Databasebeta

← #13140731

Subgroups — #13140731

Order 1728.

554 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3701C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3#11721004C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10591588—31nomaximal
9#10588490—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C3×#3701—22no—
13C3×#3701—22no—
14C2×C3×D36—21no—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9—21no—
17C2×C3×C4×D9C2211yescharacteristic
18C2×#6192—21no—
19#88548—22no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C9×C4⋊Dic3C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23#88445—21no—
24#88417—22no—
25C2×C3×#239—31no—
26C2×#3755—31no—
27#63909—31no—
28C2×#3714—31no—
29C2×#3701D311yescharacteristic, complemented
30C3×#835D311yescharacteristic
31#63562—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3675—31no—
34C2×#3675—61no—
35C2×D72—31no—
36#63457—31no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63369—31no—
40C22×C3⋊D9—21no—
41C22×C3×D9—21no—
42C22×C3×D9D411yescharacteristic
43C3×Q8×C9—21no—
44C3×Q8×C9D412yescomplemented
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9C2311yescharacteristic
47C4⋊(C3⋊D9)—21no—
48C4⋊(C3⋊D9)—21no—
49C2×C9×Dic3—21no—
50C2×C3×Dic9D411yescharacteristic
51C3×D36—24no—
52C3×D36—21no—
53C3×D36D412yescomplemented
54C3×C4×D9—24no—
55C3×C4×D9—21no—
56C3×C4×D9—21no—
57#6192—42no—
58C3⋊(C9⋊C8)—21no—
59C9×C3⋊C8—21no—
60#5238—91no—
61#4810—91no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C4⋊(C3⋊D3)—31no—
64C3×#239—62no—
65C3×#239—62no—
66C2×C3×D12—31no—
67C2×C3×D12—61no—
68C2×C3×C4×D3—31no—
69C2×C3×C4×D3—61no—
70C2×#626—61no—
71#3755—62no—
72C2×C3⋊(C3⋊C8)—31no—
73C3×C4⋊Dic3C2×D311yescharacteristic
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3722—61no—
76#3714—62no—
77C2×Q8×C9—21no—
78C2×Q8×C9C2×D311yescharacteristic
79#3701—22no—
80#3701—22no—
81C2×D36—21no—
82C2×D36—31no—
83C2×D36—61no—
84C2×D36—61no—
85C2×D36C2×D311yescharacteristic
86C2×C4×D9—21no—
87C2×C4×D9C2×D311yescharacteristic
88C2×C8×C9—31no—
89C9×C4⋊C4—31no—
90#3675—64no—
91#3675—62no—
92(C2×D9)⋊C4—61no—
93C2×C9⋊C8—31no—
94C2×C9⋊C8—61no—
95D72—62no—
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—41no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×C3×D9—21no—
101C2×C3×D9—22no—
102C2×C3×D9—21no—
103C2×C3×D9—21no—
104C2×C3×D9—41no—
105C2×C3×D9—41no—
106C3×C4×C9—22no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C9×Dic3—41no—
110C3×Dic9—21no—
111C2×C3×#41—91no—
112C2×#239—31no—
113C2×#215—91no—
114C2×#215—91no—
115C2×#215—181no—
116C2×#213—91no—
117#930—91no—
118C2×D24—91no—
119#895—91no—
120C3×#103—31no—
121#835D911yescharacteristic, complemented
122#832—91no—
123#830—91no—
124(C2×D3)⋊C8—91no—
125#809—91no—
126C22×C3⋊D3—61no—
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C32×Q8—21no—
130C32×Q8D1212yes—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—61no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—31no—
137C3×D12—32no—
138C3×D12—64no—
139C3×D12—61no—
140C3×C4×D3—64no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143#626—122no—
144C22×D9—21no—
145C22×D9—61no—
146C22×D9—121no—
147C22×D9#6711yescharacteristic
148C3⋊(C3⋊C8)—61no—
149C3×C3⋊C8—21no—
150Q8×C9—21no—
151Q8×C9—22no—
152Q8×C9—41no—
153Q8×C9#6712yescomplemented
154C2×C4×C9—21no—
155C2×C4×C9—21no—
156C2×C4×C9—41no—
157C2×C4×C9—61no—
158C2×C4×C9C22×D311yescharacteristic
159C2×Dic9#6711yescharacteristic
160D36—24no—
161D36—21no—
162D36—62no—
163D36—61no—
164D36—61no—
165D36—121no—
166D36—122no—
167D36#6712yescomplemented
168C4×D9—24no—
169C4×D9—21no—
170C4×D9—21no—
171C8×C9—61no—
172C9⋊C8—62no—
173C9⋊C8—61no—
174#449—271noSylow
175C2×C3×C9#12011yescharacteristic
176C2×C3×C9#13511yescharacteristic
177C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
178C3⋊D9—81no—
179C3×D9—41no—
180C3×D9—42no—
181C3×#41—182no—
182C3×#41—182no—
183C2×C3×Q8—21no—
184C2×C3×Q8C2×D911yescharacteristic
185C2×C3×Q8D3211yescharacteristic
186C2×C3×D4—91no—
187C2×C3×D4—181no—
188C22×C3×C4—91no—
189C22×C3×C4—181no—
190C2×#67—181no—
191#239—62no—
192#239—62no—
193C2×D12—31no—
194C2×D12—61no—
195C2×D12—91no—
196C2×D12—91no—
197C2×D12—181no—
198C2×D12—181no—
199C2×C4×D3—31no—
200C2×C4×D3—61no—
201C2×C3×C8—31no—
202C3×C4⋊C4—31no—
203#215—184no—
204#215—182no—
205#215—182no—
206#213—182no—
207(C2×D3)⋊C4—181no—
208(C2×D3)⋊C4—181no—
209C4⋊Dic3C2×D911yescharacteristic
210C2×C3⋊C8—91no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—181no—
213C2×C3⋊C8C2×D911yescharacteristic
214D24—182no—
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—121no—
217C2×C3⋊D3—121no—
218C2×C3⋊D3—121no—
219C2×C3×D3—62no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C32×C4—22no—
226C32×C4C2×D1211yescharacteristic
227C32×C4D3×D411yescharacteristic
228C3×Dic3—41no—
229C3×Dic3—61no—
230C22×C9—21no—
231C22×C9C2×#6711yescharacteristic
232C2×D9—21no—
233C2×D9—21no—
234C2×D9—21no—
235C2×D9—22no—
236C2×D9—41no—
237C2×D9—41no—
238C2×D9—122no—
239C2×D9—121no—
240C2×D9—121no—
241C2×D9—121no—
242C2×D9—241no—
243C2×D9—241no—
244C4×C9—22no—
245C4×C9—21no—
246C4×C9—21no—
247C4×C9—42no—
248C4×C9—121no—
249C4×C9C2×#6711yescharacteristic
250C4×C9C2×#6711yescharacteristic
251Dic9—21no—
252C2×#41—91no—
253C2×SD16—271no—
254C2×D8—271no—
255#121—271no—
256#103—31no—
257#102—271no—
258C22⋊C8—271no—
259C3×C9#44911yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C23×C3—181no—
262C22×D3—31no—
263C22×D3—61no—
264C22×D3—181no—
265C22×D3—181no—
266C22×D3—361no—
267C3×Q8—21no—
268C3×Q8—21no—
269C3×Q8—22no—
270C3×Q8—41no—
271C3×Q8D3612yescomplemented
272C3×Q8#62612yes—
273C3×D4—92no—
274C3×D4—184no—
275C3×D4—181no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—41no—
280C2×C3×C4—61no—
281C2×C3×C4—91no—
282C2×C3×C4—184no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C22×D911yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287#67—362no—
288C2×Dic3—21no—
289C2×Dic3—31no—
290D12—32no—
291D12—64no—
292D12—61no—
293D12—182no—
294D12—181no—
295D12—181no—
296D12—181no—
297D12—181no—
298D12—361no—
299D12—362no—
300C4×D3—64no—
301C4×D3—61no—
302C4×D3—61no—
303C3×C8—61no—
304C3⋊C8—21no—
305C3⋊C8—182no—
306C3⋊C8—181no—
307C3⋊C8—181no—
308C2×C32#88211yescharacteristic
309C2×C32#91411yescharacteristic
310C2×C32#91911yescharacteristic
311C3⋊D3—241no—
312C3×D3—121no—
313C3×D3—122no—
314C2×C9—21no—
315C2×C9—21no—
316C2×C9—21no—
317C2×C9#94911yescharacteristic
318C2×C9#95011yescharacteristic
319C2×C9#95311yescharacteristic
320D9—41no—
321D9—42no—
322D9—242no—
323D9—241no—
324#41—182no—
325#41—182no—
326C2×Q8D3×D911yescharacteristic
327C2×D4—91no—
328C2×D4—181no—
329C2×D4—271no—
330C2×D4—541no—
331C22×C4—91no—
332C22×C4—181no—
333SD16—542no—
334D8—542no—
335C2×C8—31no—
336C2×C8—271no—
337C4⋊C4—31no—
338C22⋊C4—541no—
339C22×C3—21no—
340C22×C3—182no—
341C22×C3—181no—
342C22×C3—181no—
343C22×C3—181no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3D4×D911yescharacteristic
347C22×C3D3×#6711yescharacteristic
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—62no—
351C2×D3—61no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—362no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C2×D3—361no—
361C2×D3—721no—
362C2×D3—721no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—42no—
368C3×C4—121no—
369C3×C4—181no—
370C3×C4C2×D3611yescharacteristic
371C3×C4D4×D911yescharacteristic
372C3×C4C2×#62611yescharacteristic
373C3×C4D3×#6711yescharacteristic
374Dic3—41no—
375Dic3—61no—
376C32#481011yescharacteristic
377C9—21no—
378C9#523811yescharacteristic, complemented
379C23—91no—
380C23—181no—
381C23—541no—
382Q8—21no—
383Q8#619212yescomplemented
384D4—92no—
385D4—184no—
386D4—181no—
387D4—541no—
388D4—541no—
389D4—1082no—
390C2×C4—21no—
391C2×C4—61no—
392C2×C4—91no—
393C2×C4—184no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4C2×D3×D911yescharacteristic
397C8—61no—
398C8—541no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—361no—
403C2×C3—362no—
404C2×C3#6344411yescharacteristic
405C2×C3#6347611yescharacteristic
406C2×C3#6348111yescharacteristic
407C2×C3#6393911yescharacteristic
408C2×C3#6394911yescharacteristic
409C2×C3#6397711yescharacteristic
410D3—121no—
411D3—122no—
412D3—722no—
413D3—721no—
414D3—721no—
415C22—181no—
416C22—182no—
417C22—181no—
418C22—181no—
419C22—361no—
420C22—361no—
421C22—1081no—
422C22—1081no—
423C22—1081no—
424C22D9×#6711yescharacteristic
425C4—22no—
426C4—121no—
427C4—181no—
428C4C2×#619211yescharacteristic
429C4D9×#6711yescharacteristic
430C3—21no—
431C3#1058849011yescharacteristic, complemented
432C3#1059158811yescharacteristic
433C2—361no—
434C2—362no—
435C2—2161no—
436C2#1172121111yescharacteristic
437C2#1172122311yescharacteristic
438C2#1172127611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#449
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3701C2semidirect
2. C2×#88548C2semidirect
3. #11721004C2semidirect
4. C2×#88417C2semidirect
5. C9×#835C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720266C2semidirect
29. C2×#3701D3semidirect
44. C3×Q8×C9D4semidirect
53. C3×D36D4semidirect
121. #835D9semidirect
153. Q8×C9#67semidirect
167. D36#67semidirect
259. C3×C9#449semidirect
271. C3×Q8D36semidirect
378. C9#5238semidirect
383. Q8#6192semidirect
431. C3#10588490semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37012
C2×#885482
#117210042
C2×#884172
C9×#8352
(C2×D9)⋊(C3⋊C8)2
#117202662
#105915883
#105884903