Finite Group Databasebeta

← #13161188

Subgroups — #13161188

Order 1728.

1244 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 172 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×#3832C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic, complemented
3#11723526C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C32×#835C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592509—31nomaximal
9#10591588—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×#3832—22no—
13C3×#3832—22no—
14C2×C3×C4⋊(C3⋊D3)—21no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×#6292—21no—
19#88862—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88804—21no—
24#88791—22no—
25C2×#3832D311yescharacteristic, complemented
26C2×C3×#239—34no—
27C2×#3755—31no—
28C2×#3755—34no—
29C2×#3755—64no—
30C2×#3745—31no—
31#64104—31no—
32#63909—34no—
33C2×#3714—34no—
34C32×#103—31no—
35#63647—31no—
36#63636—31no—
37C3×#835D314yescomplemented
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4⋊(C3⋊D3)—24no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)D412yescomplemented
53C3×C4×C3⋊D3—24no—
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C2×C32×Dic3—21no—
57#6292—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5238—91no—
61#4810—94no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65#3832—22no—
66#3832—22no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—34no—
70C2×C4⋊(C3⋊D3)—64no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3—21no—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C3×#239—68no—
76C3×#239—68no—
77C2×C3×D12—34no—
78C2×C3×D12—64no—
79C2×C3×C4×D3—34no—
80C2×C3×C4×D3—64no—
81C2×#626—64no—
82C2×C32×C8—31no—
83C32×C4⋊C4—31no—
84#3755—616no—
85#3755—68no—
86#3755—62no—
87#3752—61no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91#3745—62no—
92C3×C4⋊Dic3C2×D314yes—
93C2×C3×C3⋊C8C2×D314yes—
94#3722—64no—
95#3714—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—41no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C33×C4—22no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—21no—
110C32×Dic3—41no—
111C2×C3×#41—91no—
112C2×#239—34no—
113C2×#215—94no—
114C2×#215—91no—
115C2×#215—184no—
116C2×#213—91no—
117#930—91no—
118C2×D24—94no—
119#895—94no—
120C3×#103—34no—
121#835C3⋊D311yescharacteristic, complemented
122#832—91no—
123#830—91no—
124(C2×D3)⋊C8—94no—
125#809—94no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—124no—
130C22×C3⋊D3#6711yescharacteristic
131C22×C3×D3—34no—
132C22×C3×D3—64no—
133C32×Q8—21no—
134C32×Q8—28no—
135C32×Q8—24no—
136C32×Q8—44no—
137C32×Q8D1218yescomplemented
138C32×Q8#6712yescomplemented
139C2×C32×C4—21no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—44no—
143C2×C32×C4—61no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3#6711yescharacteristic
147C4⋊(C3⋊D3)—24no—
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—68no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—61no—
152C4⋊(C3⋊D3)—61no—
153C4⋊(C3⋊D3)—64no—
154C4⋊(C3⋊D3)—124no—
155C4⋊(C3⋊D3)—122no—
156C4⋊(C3⋊D3)#6712yescomplemented
157C4×C3⋊D3—24no—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—34no—
162C3×D12—38no—
163C3×D12—616no—
164C3×D12—64no—
165C3×C4×D3—616no—
166C3×C4×D3—64no—
167C3×C4×D3—64no—
168#626—128no—
169C32×C8—61no—
170C3⋊(C3⋊C8)—68no—
171C3⋊(C3⋊C8)—64no—
172C3⋊(C3⋊C8)—61no—
173C3×C3⋊C8—24no—
174#449—271noSylow
175C2×C33#12011yescharacteristic
176C2×C33#13511yescharacteristic
177C2×C33AGL(1,ℤ/8)11yescharacteristic
178C3⋊(C3⋊D3)—81no—
179C3×C3⋊D3—41no—
180C3×C3⋊D3—42no—
181C3×#41—182no—
182C3×#41—182no—
183C2×C3×Q8—24no—
184C2×C3×Q8D3214yes—
185C2×C3×Q8C2×C3⋊D311yescharacteristic
186C2×C3×D4—91no—
187C2×C3×D4—181no—
188C22×C3×C4—91no—
189C22×C3×C4—181no—
190C2×#67—181no—
191#239—68no—
192#239—68no—
193C2×D12—34no—
194C2×D12—64no—
195C2×D12—94no—
196C2×D12—91no—
197C2×D12—184no—
198C2×D12—184no—
199C2×C4×D3—34no—
200C2×C4×D3—64no—
201C2×C3×C8—34no—
202C3×C4⋊C4—34no—
203#215—1816no—
204#215—188no—
205#215—182no—
206#213—182no—
207(C2×D3)⋊C4—181no—
208(C2×D3)⋊C4—184no—
209C4⋊Dic3C2×C3⋊D311yescharacteristic
210C2×C3⋊C8—94no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—184no—
213C2×C3⋊C8C2×C3⋊D311yescharacteristic
214D24—188no—
215C22×C32—24no—
216C22×C32D3×D414yes—
217C22×C32C2×#6711yescharacteristic
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—128no—
225C2×C3⋊D3—124no—
226C2×C3⋊D3—121no—
227C2×C3⋊D3—121no—
228C2×C3⋊D3—121no—
229C2×C3⋊D3—124no—
230C2×C3⋊D3—124no—
231C2×C3⋊D3—244no—
232C2×C3⋊D3—244no—
233C2×C3×D3—68no—
234C2×C3×D3—64no—
235C2×C3×D3—64no—
236C2×C3×D3—64no—
237C2×C3×D3—124no—
238C2×C3×D3—124no—
239C32×C4—22no—
240C32×C4—24no—
241C32×C4—24no—
242C32×C4—28no—
243C32×C4—48no—
244C32×C4—121no—
245C32×C4C2×D1214yes—
246C32×C4D3×D414yes—
247C32×C4C2×#6711yescharacteristic
248C32×C4C2×#6711yescharacteristic
249C3⋊Dic3—21no—
250C3×Dic3—44no—
251C3×Dic3—64no—
252C2×#41—91no—
253C2×SD16—271no—
254C2×D8—271no—
255#121—271no—
256#103—31no—
257#102—271no—
258C22⋊C8—271no—
259C33#44911yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C23×C3—181no—
262C22×D3—34no—
263C22×D3—64no—
264C22×D3—184no—
265C22×D3—181no—
266C22×D3—364no—
267C3×Q8—24no—
268C3×Q8—21no—
269C3×Q8—28no—
270C3×Q8—44no—
271C3×Q8#62618yescomplemented
272C3×Q8C4⋊(C3⋊D3)12yescomplemented
273C3×D4—92no—
274C3×D4—184no—
275C3×D4—181no—
276C2×C3×C4—24no—
277C2×C3×C4—21no—
278C2×C3×C4—24no—
279C2×C3×C4—44no—
280C2×C3×C4—64no—
281C2×C3×C4—91no—
282C2×C3×C4—184no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3214yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—362no—
288C2×Dic3—21no—
289C2×Dic3—34no—
290D12—38no—
291D12—616no—
292D12—64no—
293D12—188no—
294D12—184no—
295D12—184no—
296D12—181no—
297D12—181no—
298D12—364no—
299D12—368no—
300C4×D3—616no—
301C4×D3—64no—
302C4×D3—64no—
303C3×C8—64no—
304C3⋊C8—21no—
305C3⋊C8—188no—
306C3⋊C8—184no—
307C3⋊C8—181no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32#88214yes—
312C2×C32#91414yes—
313C2×C32#91914yes—
314C2×C32#94911yescharacteristic
315C2×C32#95011yescharacteristic
316C2×C32#95311yescharacteristic
317C3⋊D3—41no—
318C3⋊D3—42no—
319C3⋊D3—248no—
320C3⋊D3—241no—
321C3⋊D3—244no—
322C3×D3—124no—
323C3×D3—128no—
324#41—182no—
325#41—182no—
326C2×Q8D3×C3⋊D311yescharacteristic
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—24no—
340C22×C3—182no—
341C22×C3—181no—
342C22×C3—181no—
343C22×C3—181no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3D3×#6714yes—
347C22×C3D4×C3⋊D311yescharacteristic
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—68no—
351C2×D3—64no—
352C2×D3—124no—
353C2×D3—124no—
354C2×D3—368no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—364no—
359C2×D3—364no—
360C2×D3—364no—
361C2×D3—724no—
362C2×D3—724no—
363C3×C4—22no—
364C3×C4—28no—
365C3×C4—24no—
366C3×C4—24no—
367C3×C4—48no—
368C3×C4—124no—
369C3×C4—181no—
370C3×C4C2×#62614yes—
371C3×C4D3×#6714yes—
372C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
373C3×C4D4×C3⋊D311yescharacteristic
374Dic3—41no—
375Dic3—64no—
376C32—24no—
377C32#481014yescomplemented
378C32#523811yescharacteristic, complemented
379C23—91no—
380C23—181no—
381C23—541no—
382Q8—21no—
383Q8#629212yescomplemented
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×C3⋊D311yescharacteristic
397C8—61no—
398C8—541no—
399C2×C3—24no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—361no—
403C2×C3—362no—
404C2×C3#6393914yes—
405C2×C3#6394914yes—
406C2×C3#6397714yes—
407C2×C3#6409111yescharacteristic
408C2×C3#6412311yescharacteristic
409C2×C3#6412811yescharacteristic
410D3—124no—
411D3—128no—
412D3—728no—
413D3—724no—
414D3—721no—
415C22—181no—
416C22—181no—
417C22—182no—
418C22—181no—
419C22—361no—
420C22—361no—
421C22—1081no—
422C22—1081no—
423C22—1081no—
424C22C3⋊D3×#6711yescharacteristic
425C4—22no—
426C4—121no—
427C4—181no—
428C4C2×#629211yescharacteristic
429C4C3⋊D3×#6711yescharacteristic
430C3—24no—
431C3#1059158814yescomplemented
432C3#1059250911yescharacteristic, complemented
433C2—361no—
434C2—362no—
435C2—2161no—
436C2#1172357011yescharacteristic
437C2#1172358511yescharacteristic
438C2#1172363611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#449
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. C2×C3×#3832C2semidirect
2. C2×#88862C2semidirect
3. #11723526C2semidirect
4. C2×#88791C2semidirect
5. C32×#835C2semidirect
6. #11722033C2semidirect
7. #11722023C2semidirect
25. C2×#3832D3semidirect
37. C3×#835D3semidirect
44. C33×Q8D4semidirect
52. C3×C4⋊(C3⋊D3)D4semidirect
121. #835C3⋊D3semidirect
137. C32×Q8D12semidirect
138. C32×Q8#67semidirect
156. C4⋊(C3⋊D3)#67semidirect
259. C33#449semidirect
271. C3×Q8#626semidirect
272. C3×Q8C4⋊(C3⋊D3)semidirect
377. C32#4810semidirect
378. C32#5238semidirect
383. Q8#6292semidirect
431. C3#10591588semidirect
432. C3#10592509semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×#888622
#117235262
C2×#887912
C32×#8352
#117220332
#117220232
#105925093
#105915883