Finite Group Databasebeta

← #13161060

Subgroups — #13161060

Order 1728.

1272 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 148 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
1#11723981C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3#11723501C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592485—31nomaximal
9#10591433—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×#6312—21no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×Dic3×C3⋊D3—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3753—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88803—21no—
22#88791—22no—
23#64149—31no—
24#64149—34no—
25#64149—64no—
26C2×#3753D311yescharacteristic, complemented
27C2×#3745—31no—
28#64102—31no—
29C2×C3×#213—34no—
30#63898—34no—
31C2×#3714—34no—
32C32×#102—31no—
33#63641—31no—
34#63636—31no—
35C3×#832D314yescomplemented
36#63572—34no—
37#63564—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C3×C3⋊D3—21no—
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44#6312—42no—
45#6312—42no—
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yescomplemented
52C2×C32×Dic3—21no—
53Dic3×C3⋊D3—42no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#5150—91no—
57#4786—94no—
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—61no—
62C2×#638—61no—
63C2×#638—64no—
64C2×#638—64no—
65C2×#638—68no—
66C2×#638—68no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3—61no—
72C2×C3×D12—34no—
73C2×D3×Dic3—64no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76#3753—22no—
77#3752—61no—
78C4×C3⋊Dic3—31no—
79C4×C3⋊Dic3—34no—
80C4×C3⋊Dic3—64no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82#3745—62no—
83C3×#213—68no—
84C3×C4⋊Dic3C2×D314yes—
85C2×C3×C3⋊C8—34no—
86C2×C3×C3⋊C8C2×D314yes—
87(C2×D3)⋊Dic3—64no—
88#3714—68no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33C2×D411yescharacteristic
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3⋊(C3⋊D3)—41no—
95C2×C3⋊(C3⋊D3)—41no—
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—41no—
99C33×C4C2×D411yescharacteristic
100C33×C4#4111yescharacteristic
101C3⋊(C3⋊Dic3)—21no—
102C3⋊(C3⋊Dic3)D812yescomplemented
103C32×Dic3—41no—
104C2×C3×D8—91no—
105#940—94no—
106#940—91no—
107#940—184no—
108D4×Dic3—91no—
109C2×#213—34no—
110C2×#213—91no—
111C2×D24—94no—
112#893—94no—
113C3×#102—34no—
114#832C3⋊D311yescharacteristic, complemented
115#821—91no—
116Dic3⋊C8—94no—
117Dic3⋊C8—91no—
118#809—94no—
119C23×C32—28no—
120C23×C32—24no—
121C23×C32—21no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3—68no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—61no—
126C22×C3×D3—64no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—24no—
130C32×D4—24no—
131C32×D4—44no—
132C32×D4—44no—
133C2×C32×C4—24no—
134C2×C32×C4—61no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137#638—122no—
138#638—128no—
139#638—1216no—
140#638—1216no—
141#638—122no—
142#638—128no—
143C2×C3⋊Dic3—31no—
144C2×C3⋊Dic3—31no—
145C2×C3⋊Dic3—34no—
146C2×C3⋊Dic3—34no—
147C2×C3⋊Dic3—64no—
148C2×C3⋊Dic3—64no—
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—61no—
151C4⋊(C3⋊D3)—61no—
152C4⋊(C3⋊D3)—64no—
153C4⋊(C3⋊D3)—64no—
154C4⋊(C3⋊D3)—124no—
155C4⋊(C3⋊D3)—124no—
156C4⋊(C3⋊D3)#6712yescomplemented
157C4×C3⋊D3—122no—
158C2×C3×Dic3—24no—
159C3×D12—38no—
160C3×D12—64no—
161D3×Dic3—128no—
162C32×C8—61no—
163C3⋊(C3⋊C8)—21no—
164C3×C3⋊C8—24no—
165C3×C3⋊C8—64no—
166#459—271noSylow
167C2×C33—81no—
168C2×C33#12111yescharacteristic
169C2×C33C2×D811yescharacteristic
170C2×C33AGL(1,ℤ/8)11yescharacteristic
171C3⋊(C3⋊D3)—81no—
172C3×C3⋊D3—42no—
173C2×C3×D4—24no—
174C2×C3×D4—91no—
175C2×C3×D4D3214yes—
176C2×C3×D4C2×C3⋊D311yescharacteristic
177C2×#67—184no—
178C2×#67—181no—
179C2×#67—184no—
180C2×#67—181no—
181C2×#67—188no—
182C2×#67—188no—
183C22×Dic3—181no—
184C2×D12—34no—
185C2×D12—94no—
186C2×D12—91no—
187C2×D12—184no—
188C2×C4×D3—184no—
189C3×D8—182no—
190C2×C3×C8—34no—
191C2×C3×C8—91no—
192C3×C4⋊C4—34no—
193#217—181no—
194#213—68no—
195#213—182no—
196(C2×D3)⋊C4—184no—
197C4⋊Dic3C2×C3⋊D311yescharacteristic
198C4×Dic3—94no—
199C4×Dic3—91no—
200C4×Dic3—184no—
201C2×C3⋊C8—34no—
202C2×C3⋊C8C2×C3⋊D311yescharacteristic
203D24—188no—
204C22×C32—24no—
205C22×C32—48no—
206C22×C32—48no—
207C22×C32—48no—
208C22×C32—44no—
209C22×C32—44no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—44no—
213C22×C32—41no—
214C22×C32D3×D414yes—
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—128no—
220C2×C3⋊D3—128no—
221C2×C3⋊D3—128no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—124no—
224C2×C3⋊D3—121no—
225C2×C3⋊D3—121no—
226C2×C3⋊D3—124no—
227C2×C3⋊D3—121no—
228C2×C3×D3—68no—
229C2×C3×D3—124no—
230C2×C3×D3—124no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4—121no—
234C32×C4#23514yes—
235C32×C4D3×D414yes—
236C32×C4#23711yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—32no—
239C3⋊Dic3—38no—
240C3⋊Dic3—68no—
241C3⋊Dic3—68no—
242C3⋊Dic3—61no—
243C3⋊Dic3—64no—
244C3×Dic3—44no—
245C2×D8—91no—
246C2×D8—271no—
247#127—271no—
248C4×D4—271no—
249C4⋊C8—271no—
250#102—31no—
251#102—271no—
252C33#45911yescharacteristic, complemented, Sylow
253C23×C3—28no—
254C23×C3—24no—
255C23×C3—21no—
256C23×C3—181no—
257C22×D3—64no—
258C22×D3—188no—
259C22×D3—184no—
260C22×D3—181no—
261C3×D4—24no—
262C3×D4—24no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—44no—
266C3×D4—44no—
267C3×D4—92no—
268C3×D4—181no—
269C2×C3×C4—24no—
270C2×C3×C4—64no—
271C2×C3×C4C2×D3214yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273#67—362no—
274#67—368no—
275#67—3616no—
276#67—3616no—
277#67—362no—
278#67—368no—
279C2×Dic3—21no—
280C2×Dic3—94no—
281C2×Dic3—91no—
282C2×Dic3—94no—
283C2×Dic3—91no—
284C2×Dic3—184no—
285C2×Dic3—184no—
286C2×Dic3—362no—
287D12—38no—
288D12—64no—
289D12—184no—
290D12—184no—
291D12—181no—
292D12—181no—
293D12—364no—
294D12—364no—
295C4×D3—368no—
296C3×C8—64no—
297C3×C8—181no—
298C3⋊C8—21no—
299C3⋊C8—64no—
300C2×C32—24no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—88no—
304C2×C32—81no—
305C2×C32—84no—
306C2×C32#88414yes—
307C2×C32D3×D814yes—
308C2×C32D3×D811yescharacteristic
309C2×C32#91111yescharacteristic
310C2×C32#91414yes—
311C2×C32#93911yescharacteristic
312C3⋊D3—42no—
313C3⋊D3—248no—
314C3⋊D3—241no—
315C3⋊D3—244no—
316C3×D3—128no—
317C2×D4—91no—
318C2×D4—271no—
319C2×D4—541no—
320C2×D4—541no—
321C2×D4D3×C3⋊D311yescharacteristic
322C22×C4—541no—
323D8—182no—
324D8—542no—
325C2×C8—31no—
326C2×C8—91no—
327C4⋊C4—31no—
328C22⋊C4—541no—
329C42—271no—
330C22×C3—24no—
331C22×C3—48no—
332C22×C3—48no—
333C22×C3—48no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—44no—
338C22×C3—44no—
339C22×C3—44no—
340C22×C3—182no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3#381114yes—
344C22×C3D4×C3⋊D311yescharacteristic
345C2×D3—68no—
346C2×D3—124no—
347C2×D3—124no—
348C2×D3—368no—
349C2×D3—368no—
350C2×D3—368no—
351C2×D3—361no—
352C2×D3—361no—
353C2×D3—361no—
354C2×D3—364no—
355C2×D3—364no—
356C2×D3—364no—
357C3×C4—24no—
358C3×C4—24no—
359C3×C4—124no—
360C3×C4#380414yes—
361C3×C4D3×#6714yes—
362C3×C4#382811yescharacteristic
363C3×C4D4×C3⋊D311yescharacteristic
364Dic3—41no—
365Dic3—92no—
366Dic3—98no—
367Dic3—188no—
368Dic3—188no—
369Dic3—181no—
370Dic3—184no—
371C32—24no—
372C32#478614yescomplemented
373C32#515011yescharacteristic, complemented
374C23—21no—
375C23—181no—
376C23—541no—
377D4—21no—
378D4—21no—
379D4—92no—
380D4—181no—
381D4—541no—
382D4—541no—
383D4—1082no—
384D4—1082no—
385C2×C4—61no—
386C2×C4—271no—
387C2×C4—271no—
388C2×C4—1082no—
389C2×C4C2×D3×C3⋊D311yescharacteristic
390C8—61no—
391C8—181no—
392C2×C3—24no—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—88no—
396C2×C3—84no—
397C2×C3—81no—
398C2×C3—362no—
399C2×C3#6392614yes—
400C2×C3#6393714yes—
401C2×C3#6397114yes—
402C2×C3#6409311yescharacteristic
403C2×C3D8×C3⋊D311yescharacteristic
404C2×C3#6412311yescharacteristic
405D3—128no—
406D3—728no—
407D3—724no—
408D3—721no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—182no—
413C22—361no—
414C22—361no—
415C22—1081no—
416C22—1081no—
417C22—1081no—
418C22#8903911yescharacteristic
419C4—121no—
420C4—272no—
421C4—541no—
422C4#8903111yescharacteristic
423C4C3⋊D3×#6711yescharacteristic
424C3—24no—
425C3#1059143314yescomplemented
426C3#1059248511yescharacteristic, complemented
427C2—81no—
428C2—362no—
429C2—2161no—
430C2#1172354711yescharacteristic
431C2#1172356611yescharacteristic
432C2#1172362411yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#459
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. #11723981C2semidirect
2. C2×C3×#3753C2semidirect
3. #11723501C2semidirect
4. C2×#88791C2semidirect
5. C32×#832C2semidirect
6. #11722050C2semidirect
7. #11722035C2semidirect
26. C2×#3753D3semidirect
35. C3×#832D3semidirect
51. C3×C4⋊(C3⋊D3)D4semidirect
102. C3⋊(C3⋊Dic3)D8semidirect
114. #832C3⋊D3semidirect
156. C4⋊(C3⋊D3)#67semidirect
252. C33#459semidirect
372. C32#4786semidirect
373. C32#5150semidirect
425. C3#10591433semidirect
426. C3#10592485semidirect

Maximal subgroups

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

subgroupindex
#117239812
C2×C3×#37532
#117235012
C2×#887912
C32×#8322
#117220502
#117220352
#105924853
#105914333