Finite Group Databasebeta

← #13161186

Subgroups — #13161186

Order 1728.

1140 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1140 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×C32×#215C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592871—31nomaximal
9#10591586—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—22no—
12C3×#3832—22no—
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×#6290—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C32×#215—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88789—22no—
25C2×#3832D311yescharacteristic, complemented
26C2×C3×#239—34no—
27C2×C32×SD16—31no—
28#64147—31no—
29C2×#3753—31no—
30C2×C3×#215D314yescomplemented
31#63908—34no—
32C2×#3713—34no—
33#63662—31no—
34#63662—34no—
35#63662—64no—
36#63648—31no—
37#63647—31no—
38#63563—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C33×Q8—21no—
44C33×Q8D412yescomplemented
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—24no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yescomplemented
52C3×C4×C3⋊D3—24no—
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C32×D12—21no—
56C32×D12—21no—
57#6290—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5238—91no—
61#5172—94no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×D4—31no—
66C2×#638—61no—
67#3832—22no—
68#3832—22no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C3×#239—68no—
74C3×#239—68no—
75C2×C3×D12—34no—
76C2×C3×D12—64no—
77C2×C3×D12C2×D314yes—
78C2×C3×C4×D3—34no—
79C2×C3×C4×D3—64no—
80C2×#625—64no—
81C32×SD16—62no—
82C2×C32×C8—31no—
83#3757—61no—
84#3753—62no—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—34no—
87C4⋊(C3⋊Dic3)—64no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C3×#215—28no—
92C2×C3×C3⋊C8C2×D314yes—
93(C2×D3)⋊Dic3—64no—
94#3713—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C32×D3—41no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C33×C4—22no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3⋊(C3⋊Dic3)—41no—
109C3×C3⋊Dic3—21no—
110C2×C3×#41—91no—
111C2×#239—34no—
112C2×C3×SD16—34no—
113C2×#215C3⋊D311yescharacteristic, complemented
114#938—94no—
115C2×#213—94no—
116C2×#213—91no—
117#930—91no—
118#835—94no—
119#835—91no—
120#835—184no—
121#832—91no—
122#830—91no—
123#821—94no—
124(C2×D3)⋊C8—94no—
125C23×C32—61no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3#6711yescharacteristic
128C22×C3×D3—24no—
129C22×C3×D3—34no—
130C22×C3×D3—64no—
131C32×Q8—21no—
132C32×Q8—28no—
133C32×Q8—24no—
134C32×Q8—44no—
135C32×Q8#6718yescomplemented
136C32×Q8#6712yescomplemented
137C32×D4—61no—
138C32×D4—61no—
139C2×C32×C4—21no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—44no—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D311yescharacteristic
145#638—122no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—124no—
149C2×C3⋊Dic3#6711yescharacteristic
150C4⋊(C3⋊D3)—24no—
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)#6712yescomplemented
153C4×C3⋊D3—24no—
154C4×C3⋊D3—21no—
155C4×C3⋊D3—21no—
156C2×C3×Dic3—34no—
157C3×D12—24no—
158C3×D12—24no—
159C3×D12—38no—
160C3×D12—616no—
161C3×D12—64no—
162C3×C4×D3—616no—
163C3×C4×D3—64no—
164C3×C4×D3—64no—
165#625—128no—
166C32×C8—61no—
167C3⋊(C3⋊C8)—61no—
168C3⋊(C3⋊C8)—64no—
169C3⋊(C3⋊C8)—124no—
170C3×C3⋊C8—24no—
171#449—271noSylow
172C2×C33#12011yescharacteristic
173C2×C33#13511yescharacteristic
174C2×C33AGL(1,ℤ/8)11yescharacteristic
175C3×C3⋊D3—41no—
176C3×C3⋊D3—42no—
177C32×D3—81no—
178C3×#41—182no—
179C3×#41—182no—
180C2×C3×Q8—24no—
181C2×C3×Q8D3214yes—
182C2×C3×Q8C2×C3⋊D311yescharacteristic
183C2×C3×D4—34no—
184C2×C3×D4—91no—
185C2×C3×D4—181no—
186C22×C3×C4—91no—
187C22×C3×C4—181no—
188C2×#67—181no—
189C2×#67—184no—
190#239—68no—
191#239—68no—
192C2×D12—34no—
193C2×D12—64no—
194C2×D12C2×C3⋊D311yescharacteristic
195C2×C4×D3—34no—
196C2×C4×D3—64no—
197C3×SD16—68no—
198C2×C3×C8—34no—
199#217—184no—
200#215—22no—
201#213—182no—
202#213—188no—
203(C2×D3)⋊C4—181no—
204C4⋊Dic3—94no—
205C4⋊Dic3—91no—
206C4⋊Dic3—184no—
207C2×C3⋊C8—94no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—184no—
210C2×C3⋊C8C2×C3⋊D311yescharacteristic
211C22×C32—24no—
212C22×C32—121no—
213C22×C32—121no—
214C22×C32—121no—
215C22×C32D3×D414yes—
216C22×C32C2×#6711yescharacteristic
217C2×C3⋊D3—21no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3⋊D3—22no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3×D3—44no—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C2×C3×D3—64no—
227C2×C3×D3—64no—
228C2×C3×D3—68no—
229C2×C3×D3—64no—
230C2×C3×D3—124no—
231C2×C3×D3—124no—
232C32×C4—22no—
233C32×C4—24no—
234C32×C4—24no—
235C32×C4—28no—
236C32×C4—48no—
237C32×C4D3×D414yes—
238C32×C4C2×#6714yes—
239C32×C4C2×#6711yescharacteristic
240C32×C4C2×#6711yescharacteristic
241C3⋊Dic3—21no—
242C3⋊Dic3—121no—
243C3⋊Dic3—124no—
244C3⋊Dic3—244no—
245C3×Dic3—64no—
246C2×#41—91no—
247C2×SD16—31no—
248C2×D8—271no—
249#121—271no—
250#103—271no—
251#102—271no—
252C22⋊C8—271no—
253C33#44911yescharacteristic, complemented, Sylow
254C23×C3—64no—
255C23×C3—91no—
256C23×C3—181no—
257C22×D3—21no—
258C22×D3—34no—
259C22×D3—64no—
260C3×Q8—24no—
261C3×Q8—21no—
262C3×Q8—28no—
263C3×Q8—44no—
264C3×Q8#62518yescomplemented
265C3×Q8#63812yescomplemented
266C3×D4—64no—
267C3×D4—64no—
268C3×D4—92no—
269C3×D4—184no—
270C3×D4—181no—
271C2×C3×C4—24no—
272C2×C3×C4—21no—
273C2×C3×C4—24no—
274C2×C3×C4—44no—
275C2×C3×C4—91no—
276C2×C3×C4—184no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C2×D3214yes—
280C2×C3×C4C22×C3⋊D311yescharacteristic
281#67—362no—
282#67—368no—
283C2×Dic3—34no—
284C2×Dic3—184no—
285C2×Dic3—181no—
286C2×Dic3—364no—
287D12—21no—
288D12—21no—
289D12—38no—
290D12—616no—
291D12—64no—
292C4×D3—616no—
293C4×D3—64no—
294C4×D3—64no—
295C3×C8—64no—
296C3⋊C8—21no—
297C3⋊C8—184no—
298C3⋊C8—181no—
299C3⋊C8—364no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—241no—
304C2×C32#91414yes—
305C2×C32#91614yes—
306C2×C32#93714yes—
307C2×C32#94911yescharacteristic
308C2×C32#95011yescharacteristic
309C2×C32#95311yescharacteristic
310C3⋊D3—41no—
311C3⋊D3—42no—
312C3×D3—84no—
313C3×D3—124no—
314C3×D3—128no—
315#41—182no—
316#41—182no—
317C2×Q8D3×C3⋊D311yescharacteristic
318C2×D4—31no—
319C2×D4—91no—
320C2×D4—181no—
321C2×D4—541no—
322C22×C4—91no—
323C22×C4—181no—
324SD16—62no—
325D8—542no—
326C2×C8—31no—
327C2×C8—271no—
328C4⋊C4—271no—
329C22⋊C4—541no—
330C22×C3—24no—
331C22×C3—124no—
332C22×C3—124no—
333C22×C3—124no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3—182no—
337C22×C3—181no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3D3×#6714yes—
341C22×C3D4×C3⋊D311yescharacteristic
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—68no—
348C2×D3—64no—
349C2×D3—124no—
350C2×D3—124no—
351C3×C4—22no—
352C3×C4—28no—
353C3×C4—24no—
354C3×C4—24no—
355C3×C4—48no—
356C3×C4—181no—
357C3×C4C2×#62514yes—
358C3×C4D3×#6714yes—
359C3×C4D4×C3⋊D311yescharacteristic
360C3×C4C2×#63811yescharacteristic
361Dic3—64no—
362Dic3—361no—
363Dic3—364no—
364Dic3—724no—
365C32—24no—
366C32#517214yescomplemented
367C32#523811yescharacteristic, complemented
368C23—61no—
369C23—91no—
370C23—181no—
371Q8—21no—
372Q8#629012yescomplemented
373D4—61no—
374D4—61no—
375D4—92no—
376D4—184no—
377D4—181no—
378D4—1082no—
379C2×C4—21no—
380C2×C4—91no—
381C2×C4—184no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—541no—
385C2×C4C2×D3×C3⋊D311yescharacteristic
386C8—61no—
387C8—541no—
388C2×C3—24no—
389C2×C3—24no—
390C2×C3—24no—
391C2×C3—244no—
392C2×C3—361no—
393C2×C3—362no—
394C2×C3#6393914yes—
395C2×C3#6394714yes—
396C2×C3#6397614yes—
397C2×C3#6412311yescharacteristic
398C2×C3#6412511yescharacteristic
399C2×C3#6414611yescharacteristic
400D3—81no—
401D3—124no—
402D3—128no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—181no—
407C22—181no—
408C22—182no—
409C22—181no—
410C22—361no—
411C22—361no—
412C22C3⋊D3×#6711yescharacteristic
413C4—22no—
414C4—181no—
415C4—1081no—
416C4C2×#629011yescharacteristic
417C4C3⋊D3×#6711yescharacteristic
418C3—24no—
419C3#1059158614yescomplemented
420C3#1059287111yescharacteristic, complemented
421C2—241no—
422C2—361no—
423C2—362no—
424C2#1172357011yescharacteristic
425C2#1172358311yescharacteristic
426C2#1172363411yescharacteristic
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×C32×#215C2semidirect
3. #11723522C2semidirect
4. C2×#88789C2semidirect
5. #11722205C2semidirect
6. #11722034C2semidirect
7. #11722023C2semidirect
25. C2×#3832D3semidirect
30. C2×C3×#215D3semidirect
44. C33×Q8D4semidirect
51. C3×C4⋊(C3⋊D3)D4semidirect
113. C2×#215C3⋊D3semidirect
135. C32×Q8#67semidirect
136. C32×Q8#67semidirect
152. C4⋊(C3⋊D3)#67semidirect
253. C33#449semidirect
264. C3×Q8#625semidirect
265. C3×Q8#638semidirect
366. C32#5172semidirect
367. C32#5238semidirect
372. Q8#6290semidirect
419. C3#10591586semidirect
420. C3#10592871semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C32×#2152
#117235222
C2×#887892
#117222052
#117220342
#117220232
#105928713
#105915863