Finite Group Databasebeta

← #13160954

Subgroups — #13160954

Order 1728.

1276 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1276 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×#3830C211yesmaximal, characteristic, complemented
2C2×#88860C211yesmaximal, characteristic, complemented
3#11723526C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592480—31nomaximal
9#10591452—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#638—21no—
13C22×C3×C3⋊Dic3—21no—
14C3×#3830—22no—
15C3×#3830—22no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×#6292—21no—
19#88860—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88804—21no—
24#88797—22no—
25C2×#3830D311yescharacteristic, complemented
26C2×C3×#237—34no—
27C2×#3753—31no—
28C2×#3753—34no—
29C2×#3753—64no—
30C2×#3744—31no—
31#64104—31no—
32#63909—34no—
33C2×#3717—34no—
34C32×#102—31no—
35#63647—31no—
36#63637—31no—
37C3×#832D314yescomplemented
38#63565—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C33×D4—21no—
44C33×D4D412yescomplemented
45C2×C33×C4C2311yescharacteristic
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C3×#638—24no—
49C2×C3×C3⋊Dic3—24no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C3×#634—21no—
55C3×#634D412yescomplemented
56C2×C32×Dic3—21no—
57#6292—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5238—91no—
61#4781—94no—
62C2×C32×D4—24no—
63C2×C32×D4C2×D314yes—
64C2×C32×D4C2×D311yescharacteristic
65C2×#638—21no—
66C22×C3⋊Dic3—21no—
67#3830—22no—
68#3830—22no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×#634C2×D311yescharacteristic
75C2×C3×#67—64no—
76C22×C3×Dic3—64no—
77C3×#237—68no—
78C3×#237—68no—
79C2×C3×C4×D3—34no—
80C2×C3×Dic6—34no—
81C2×#626—64no—
82C2×C32×C8—31no—
83C32×C4⋊C4—31no—
84#3753—616no—
85#3753—68no—
86#3753—62no—
87#3752—61no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91#3744—62no—
92C3×C4⋊Dic3C2×D314yes—
93C2×C3×C3⋊C8C2×D314yes—
94#3722—64no—
95#3717—68no—
96C22×C33—22no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3—21no—
105C2×C3×C3⋊D3—21no—
106C33×C4C2×D411yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—22no—
109C3×C3⋊Dic3—21no—
110C32×Dic3—41no—
111C2×C3×#41—91no—
112C2×#237—34no—
113C2×#215—91no—
114C2×#213—94no—
115C2×#213—91no—
116C2×#213—184no—
117#930—91no—
118C2×#204—94no—
119#895—94no—
120C3×#102—34no—
121#835—91no—
122#832C3⋊D311yescharacteristic, complemented
123#830—91no—
124(C2×D3)⋊C8—94no—
125#810—94no—
126C23×C32—21no—
127C23×C32—24no—
128C23×C32—44no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—124no—
132C22×C3⋊D3#6711yescharacteristic
133C22×C3×D3—34no—
134C32×D4—21no—
135C32×D4—28no—
136C32×D4—24no—
137C32×D4—44no—
138C32×D4D1218yescomplemented
139C32×D4#6712yescomplemented
140C2×C32×C4—24no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144#638—24no—
145C2×C3⋊Dic3—24no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3#6711yescharacteristic
148C4⋊(C3⋊D3)—68no—
149C4⋊(C3⋊D3)—64no—
150C4⋊(C3⋊D3)—61no—
151C4⋊(C3⋊D3)—61no—
152C4⋊(C3⋊D3)—64no—
153C4⋊(C3⋊D3)—124no—
154C4⋊(C3⋊D3)—122no—
155C4×C3⋊D3—21no—
156C4×C3⋊D3—21no—
157#634—21no—
158#634#6712yescomplemented
159C3×#67—616no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—616no—
163C2×C3×Dic3—64no—
164C3×C4×D3—64no—
165C3×C4×D3—64no—
166C3×Dic6—38no—
167C3×Dic6—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—42no—
176C2×C33#12011yescharacteristic
177C2×C33#13511yescharacteristic
178C2×C33AGL(1,ℤ/8)11yescharacteristic
179C3⋊(C3⋊D3)—81no—
180C3×C3⋊D3—41no—
181C3×#41—182no—
182C3×#41—182no—
183C2×C3×Q8—91no—
184C2×C3×D4—24no—
185C2×C3×D4—181no—
186C2×C3×D4D3214yes—
187C2×C3×D4C2×C3⋊D311yescharacteristic
188C22×C3×C4—91no—
189C22×C3×C4—181no—
190C2×#67—64no—
191C2×#67—181no—
192C22×Dic3—64no—
193#237—68no—
194#237—68no—
195C2×D12—94no—
196C2×D12—91no—
197C2×D12—184no—
198C2×D12—184no—
199C2×C4×D3—34no—
200C2×Dic6—34no—
201C2×C3×C8—34no—
202C3×C4⋊C4—34no—
203#215—182no—
204#213—1816no—
205#213—188no—
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
214#204—188no—
215C22×C32—22no—
216C22×C32—24no—
217C22×C32—28no—
218C22×C32—48no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—44no—
222C22×C32—44no—
223C22×C32—84no—
224C22×C32—84no—
225C22×C32D3×D414yes—
226C22×C32C2×#6711yescharacteristic
227C2×C3⋊D3—21no—
228C2×C3⋊D3—21no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3—128no—
231C2×C3⋊D3—124no—
232C2×C3⋊D3—121no—
233C2×C3⋊D3—121no—
234C2×C3⋊D3—121no—
235C2×C3⋊D3—124no—
236C2×C3⋊D3—124no—
237C2×C3⋊D3—244no—
238C2×C3⋊D3—244no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—64no—
242C32×C4—24no—
243C32×C4—24no—
244C32×C4—121no—
245C32×C4C2×D1214yes—
246C32×C4D3×D414yes—
247C32×C4C2×#6711yescharacteristic
248C32×C4C2×#6711yescharacteristic
249C3⋊Dic3—22no—
250C3⋊Dic3—21no—
251C3×Dic3—44no—
252C3×Dic3—68no—
253C3×Dic3—64no—
254C2×#41—91no—
255C2×SD16—271no—
256C2×D8—271no—
257#121—271no—
258#103—271no—
259#102—31no—
260C22⋊C8—271no—
261C33#44911yescharacteristic, complemented, Sylow
262C23×C3—24no—
263C23×C3—21no—
264C23×C3—44no—
265C23×C3—91no—
266C22×D3—34no—
267C22×D3—184no—
268C22×D3—181no—
269C22×D3—364no—
270C3×Q8—92no—
271C3×Q8—181no—
272C3×D4—24no—
273C3×D4—21no—
274C3×D4—28no—
275C3×D4—44no—
276C3×D4—184no—
277C3×D4#62618yescomplemented
278C3×D4C4⋊(C3⋊D3)12yescomplemented
279C2×C3×C4—24no—
280C2×C3×C4—64no—
281C2×C3×C4—91no—
282C2×C3×C4—184no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4C2×D3214yes—
287C2×C3×C4C22×C3⋊D311yescharacteristic
288#67—616no—
289#67—362no—
290C2×Dic3—21no—
291C2×Dic3—34no—
292C2×Dic3—616no—
293C2×Dic3—64no—
294D12—188no—
295D12—184no—
296D12—184no—
297D12—181no—
298D12—181no—
299D12—364no—
300D12—368no—
301C4×D3—64no—
302C4×D3—64no—
303Dic6—38no—
304Dic6—64no—
305C3×C8—64no—
306C3⋊C8—21no—
307C3⋊C8—188no—
308C3⋊C8—184no—
309C3⋊C8—181no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—42no—
314C2×C32—48no—
315C2×C32—88no—
316C2×C32#88214yes—
317C2×C32#91114yes—
318C2×C32#91614yes—
319C2×C32#94911yescharacteristic
320C2×C32#95011yescharacteristic
321C2×C32#95311yescharacteristic
322C3⋊D3—41no—
323C3⋊D3—248no—
324C3⋊D3—241no—
325C3⋊D3—244no—
326C3×D3—124no—
327#41—182no—
328#41—182no—
329C2×Q8—91no—
330C2×D4—181no—
331C2×D4—271no—
332C2×D4—541no—
333C2×D4D3×C3⋊D311yescharacteristic
334C22×C4—91no—
335C22×C4—181no—
336SD16—542no—
337D8—542no—
338C2×C8—31no—
339C2×C8—271no—
340C4⋊C4—31no—
341C22⋊C4—541no—
342C22×C3—22no—
343C22×C3—28no—
344C22×C3—24no—
345C22×C3—48no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—84no—
351C22×C3—84no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3D3×#6714yes—
356C22×C3D4×C3⋊D311yescharacteristic
357C2×D3—64no—
358C2×D3—64no—
359C2×D3—64no—
360C2×D3—368no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C2×D3—364no—
365C2×D3—364no—
366C2×D3—364no—
367C2×D3—724no—
368C2×D3—724no—
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—124no—
372C3×C4—181no—
373C3×C4—182no—
374C3×C4C2×#62614yes—
375C3×C4D3×#6714yes—
376C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
377C3×C4D4×C3⋊D311yescharacteristic
378Dic3—41no—
379Dic3—68no—
380Dic3—64no—
381C32—24no—
382C32#478114yescomplemented
383C32#523811yescharacteristic, complemented
384C23—21no—
385C23—91no—
386C23—541no—
387Q8—92no—
388Q8—181no—
389D4—21no—
390D4—184no—
391D4—541no—
392D4—541no—
393D4—1082no—
394D4#629212yescomplemented
395C2×C4—61no—
396C2×C4—91no—
397C2×C4—184no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4C2×D3×C3⋊D311yescharacteristic
402C8—61no—
403C8—541no—
404C2×C3—24no—
405C2×C3—24no—
406C2×C3—24no—
407C2×C3—48no—
408C2×C3—42no—
409C2×C3—88no—
410C2×C3—361no—
411C2×C3#6392514yes—
412C2×C3#6393514yes—
413C2×C3#6397714yes—
414C2×C3#6409111yescharacteristic
415C2×C3#6412011yescharacteristic
416C2×C3#6412511yescharacteristic
417D3—124no—
418D3—728no—
419D3—724no—
420D3—721no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—181no—
425C22—181no—
426C22—181no—
427C22—1081no—
428C22—1081no—
429C22—1081no—
430C22C3⋊D3×#6711yescharacteristic
431C4—121no—
432C4—181no—
433C4—182no—
434C4C2×#629211yescharacteristic
435C4C3⋊D3×#6711yescharacteristic
436C3—24no—
437C3#1059145214yescomplemented
438C3#1059248011yescharacteristic, complemented
439C2—42no—
440C2—361no—
441C2—2161no—
442C2#1172354611yescharacteristic
443C2#1172356111yescharacteristic
444C2#1172363611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
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×C3×#3830C2semidirect
2. C2×#88860C2semidirect
3. #11723526C2semidirect
4. C2×#88797C2semidirect
5. C32×#832C2semidirect
6. #11722039C2semidirect
7. #11722023C2semidirect
25. C2×#3830D3semidirect
37. C3×#832D3semidirect
44. C33×D4D4semidirect
55. C3×#634D4semidirect
122. #832C3⋊D3semidirect
138. C32×D4D12semidirect
139. C32×D4#67semidirect
158. #634#67semidirect
261. C33#449semidirect
277. C3×D4#626semidirect
278. C3×D4C4⋊(C3⋊D3)semidirect
382. C32#4781semidirect
383. C32#5238semidirect
394. D4#6292semidirect
437. C3#10591452semidirect
438. C3#10592480semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#888602
#117235262
C2×#887972
C32×#8322
#117220392
#117220232
#105924803
#105914523