Finite Group Databasebeta

← #13160959

Subgroups — #13160959

Order 1728.

1172 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 188 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×C32×#237C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592937—31nomaximal
9#10591450—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×#67—21no—
13C22×C32×Dic3—21no—
14C32×#237—22no—
15C32×#237—22no—
16C2×C32×C4×D3C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×#6290—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3753—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23#88803—21no—
24#88793—22no—
25C2×C32×#41—31no—
26C2×C3×#237D314yescomplemented
27C2×#3755—31no—
28C2×#3753D311yescharacteristic, complemented
29#64139—31no—
30C2×C3×#213—34no—
31#63908—34no—
32C2×#3715—34no—
33#63662—31no—
34#63659—31no—
35#63659—34no—
36#63659—64no—
37#63657—31no—
38#63566—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—21no—
42C22×C32×D3D411yescharacteristic
43C33×D4—21no—
44C33×D4D412yescomplemented
45C2×C33×C4C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C32×#67—24no—
50C2×C32×Dic3—24no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55C32×Dic6—21no—
56C32×Dic6D412yescomplemented
57#6290—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5238—94no—
61#5158—91no—
62C32×#41—62no—
63C32×#41—62no—
64C2×C32×Q8—31no—
65C2×C32×D4—24no—
66C2×C32×D4—61no—
67C2×C32×D4C2×D314yes—
68C2×C32×D4C2×D311yescharacteristic
69C22×C32×C4—31no—
70C22×C32×C4—61no—
71C2×#638—61no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C3×#67—24no—
74C22×C3×Dic3—24no—
75C3×#237—28no—
76C3×#237—28no—
77C2×C3×D12—34no—
78C2×C3×C4×D3C2×D314yes—
79C2×C3×Dic6C2×D314yes—
80C2×#625—64no—
81#3755—62no—
82#3753—22no—
83#3752—61no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—34no—
86C4⋊(C3⋊Dic3)—64no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C3×#213—68no—
92C2×C3×C3⋊C8—34no—
93(C2×D3)⋊Dic3—64no—
94#3715—68no—
95C22×C33—22no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33C2×D411yescharacteristic
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C32×D3—21no—
103C2×C32×D3—21no—
104C2×C32×D3—21no—
105C33×C4C2×D411yescharacteristic
106C33×C4C2×D411yescharacteristic
107C3⋊(C3⋊Dic3)—41no—
108C32×Dic3—22no—
109C32×Dic3—21no—
110C2×C3×#41—34no—
111C2×#237C3⋊D311yescharacteristic, complemented
112C2×C3×D8—91no—
113C2×#215—94no—
114#938—91no—
115C2×#214—91no—
116C2×#213—34no—
117#930—94no—
118#835—94no—
119#832—94no—
120#832—91no—
121#832—184no—
122#830—94no—
123(C2×D3)⋊C8—91no—
124#816—91no—
125C23×C32—21no—
126C23×C32—24no—
127C23×C32—31no—
128C23×C32—44no—
129C22×C3⋊D3—21no—
130C22×C3×D3—64no—
131C22×C3×D3#6714yes—
132C32×Q8—32no—
133C32×Q8—61no—
134C32×D4—21no—
135C32×D4—28no—
136C32×D4—24no—
137C32×D4—44no—
138C32×D4—64no—
139C32×D4#6718yescomplemented
140C32×D4#6712yescomplemented
141C2×C32×C4—24no—
142C2×C32×C4—31no—
143C2×C32×C4—64no—
144C2×C32×C4—61no—
145C2×C32×C4—61no—
146C2×C32×C4—61no—
147C2×C32×C4C22×D314yes—
148C2×C32×C4C22×D311yescharacteristic
149#638—122no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—64no—
152C2×C3⋊Dic3—124no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—21no—
155C3×#67—216no—
156C2×C3×Dic3—216no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3#6714yes—
159C3×D12—64no—
160C3×D12—64no—
161C3×C4×D3—24no—
162C3×C4×D3—24no—
163C3×Dic6—24no—
164C3×Dic6#6718yescomplemented
165#625—128no—
166C3⋊(C3⋊C8)—21no—
167C3⋊(C3⋊C8)—61no—
168C3⋊(C3⋊C8)—64no—
169C3⋊(C3⋊C8)—124no—
170C3×C3⋊C8—64no—
171#449—271noSylow
172C2×C33—42no—
173C2×C33#12011yescharacteristic
174C2×C33#13511yescharacteristic
175C2×C33AGL(1,ℤ/8)11yescharacteristic
176C3×C3⋊D3—81no—
177C32×D3—41no—
178C3×#41—68no—
179C3×#41—68no—
180C2×C3×Q8—34no—
181C2×C3×D4—24no—
182C2×C3×D4—64no—
183C2×C3×D4—91no—
184C2×C3×D4D3214yes—
185C2×C3×D4C2×C3⋊D311yescharacteristic
186C22×C3×C4—34no—
187C22×C3×C4—64no—
188C2×#67—21no—
189C2×#67—184no—
190C2×#67—181no—
191C22×Dic3—21no—
192#237—22no—
193#237—22no—
194C2×D12—34no—
195C2×C4×D3C2×C3⋊D311yescharacteristic
196C2×Dic6C2×C3⋊D311yescharacteristic
197C3×D8—182no—
198C2×C3×C8—91no—
199#217—181no—
200#215—188no—
201#214—182no—
202#213—68no—
203(C2×D3)⋊C4—184no—
204C4⋊Dic3—94no—
205C4⋊Dic3—91no—
206C4⋊Dic3—184no—
207C2×C3⋊C8—34no—
208C2×C3⋊C8—94no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—184no—
211C22×C32—22no—
212C22×C32—24no—
213C22×C32—28no—
214C22×C32—48no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—44no—
218C22×C32—44no—
219C22×C32—61no—
220C22×C32—61no—
221C22×C32—61no—
222C22×C32—84no—
223C22×C32—84no—
224C22×C32D3×D411yescharacteristic
225C22×C32C2×#6714yes—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3×D3—24no—
230C2×C3×D3—24no—
231C2×C3×D3—24no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234C2×C3×D3—124no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—61no—
238C32×C4—62no—
239C32×C4D3×D411yescharacteristic
240C32×C4C2×#6714yes—
241C32×C4C2×#6714yes—
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—121no—
244C3⋊Dic3—124no—
245C3⋊Dic3—244no—
246C3×Dic3—28no—
247C3×Dic3—24no—
248C2×#41—31no—
249C2×SD16—271no—
250C2×D8—91no—
251#121—271no—
252#103—271no—
253#102—271no—
254C22⋊C8—271no—
255C33#44911yescharacteristic, complemented, Sylow
256C23×C3—24no—
257C23×C3—21no—
258C23×C3—34no—
259C23×C3—44no—
260C23×C3—181no—
261C22×D3—64no—
262C22×D3#63811yescharacteristic
263C3×Q8—38no—
264C3×Q8—64no—
265C3×D4—24no—
266C3×D4—21no—
267C3×D4—28no—
268C3×D4—44no—
269C3×D4—616no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4#62518yescomplemented
273C3×D4#63812yescomplemented
274C2×C3×C4—24no—
275C2×C3×C4—34no—
276C2×C3×C4—616no—
277C2×C3×C4—64no—
278C2×C3×C4—64no—
279C2×C3×C4—64no—
280C2×C3×C4C2×D3214yes—
281C2×C3×C4C22×C3⋊D311yescharacteristic
282#67—24no—
283#67—362no—
284#67—368no—
285C2×Dic3—24no—
286C2×Dic3—21no—
287C2×Dic3—184no—
288C2×Dic3—181no—
289C2×Dic3—364no—
290C2×Dic3#63811yescharacteristic
291D12—64no—
292D12—64no—
293C4×D3—21no—
294C4×D3—21no—
295Dic6—21no—
296Dic6#63812yescomplemented
297C3×C8—181no—
298C3⋊C8—64no—
299C3⋊C8—184no—
300C3⋊C8—181no—
301C3⋊C8—364no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—24no—
305C2×C32—42no—
306C2×C32—48no—
307C2×C32—88no—
308C2×C32—121no—
309C2×C32#91111yescharacteristic
310C2×C32#91211yescharacteristic
311C2×C32#93711yescharacteristic
312C2×C32#94914yes—
313C2×C32#95014yes—
314C2×C32#95314yes—
315C3⋊D3—81no—
316C3×D3—44no—
317C3×D3—244no—
318#41—62no—
319#41—62no—
320C2×Q8—31no—
321C2×D4—61no—
322C2×D4—91no—
323C2×D4—541no—
324C2×D4D3×C3⋊D311yescharacteristic
325C22×C4—31no—
326C22×C4—61no—
327SD16—542no—
328D8—182no—
329C2×C8—91no—
330C2×C8—271no—
331C4⋊C4—271no—
332C22⋊C4—541no—
333C22×C3—22no—
334C22×C3—28no—
335C22×C3—24no—
336C22×C3—48no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—44no—
340C22×C3—44no—
341C22×C3—64no—
342C22×C3—64no—
343C22×C3—64no—
344C22×C3—84no—
345C22×C3—84no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3D3×#6714yes—
350C22×C3C2×#63811yescharacteristic
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—21no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—124no—
357C3×C4—24no—
358C3×C4—24no—
359C3×C4—64no—
360C3×C4—68no—
361C3×C4C2×#62514yes—
362C3×C4D3×#6714yes—
363C3×C4C2×#63811yescharacteristic
364C3×C4C2×#63811yescharacteristic
365Dic3—22no—
366Dic3—21no—
367Dic3—361no—
368Dic3—364no—
369Dic3—724no—
370C32—24no—
371C32#515811yescharacteristic, complemented
372C32#523814yescomplemented
373C23—21no—
374C23—31no—
375C23—181no—
376Q8—32no—
377Q8—61no—
378D4—21no—
379D4—64no—
380D4—181no—
381D4—181no—
382D4—1082no—
383D4#629012yescomplemented
384C2×C4—31no—
385C2×C4—64no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—541no—
390C2×C4C2×D3×C3⋊D311yescharacteristic
391C8—181no—
392C8—541no—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—42no—
397C2×C3—48no—
398C2×C3—88no—
399C2×C3—124no—
400C2×C3—721no—
401C2×C3#6392514yes—
402C2×C3#6393314yes—
403C2×C3#6397614yes—
404C2×C3#6415811yescharacteristic
405C2×C3#6415911yescharacteristic
406C2×C3#6416211yescharacteristic
407D3—41no—
408D3—244no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22D3×#63811yescharacteristic
419C4—61no—
420C4—62no—
421C4—1081no—
422C4C2×#629011yescharacteristic
423C4D3×#63811yescharacteristic
424C3—24no—
425C3#1059145014yescomplemented
426C3#1059293711yescharacteristic, complemented
427C2—42no—
428C2—121no—
429C2—721no—
430C2#1172354411yescharacteristic
431C2#1172355611yescharacteristic
432C2#1172362911yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#449
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×C32×#237C2semidirect
2. C2×C3×#3753C2semidirect
3. #11723515C2semidirect
4. C2×#88793C2semidirect
5. #11722202C2semidirect
6. #11722038C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
26. C2×C3×#237D3semidirect
28. C2×#3753D3semidirect
44. C33×D4D4semidirect
56. C32×Dic6D4semidirect
111. C2×#237C3⋊D3semidirect
139. C32×D4#67semidirect
140. C32×D4#67semidirect
164. C3×Dic6#67semidirect
255. C33#449semidirect
272. C3×D4#625semidirect
273. C3×D4#638semidirect
296. Dic6#638semidirect
371. C32#5158semidirect
372. C32#5238semidirect
383. D4#6290semidirect
425. C3#10591450semidirect
426. C3#10592937semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C2×C3×#37532
#117235152
C2×#887932
#117222022
#117220382
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105929373
#105914503