Finite Group Databasebeta

← #13161065

Subgroups — #13161065

Order 1728.

1272 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 156 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×C32×#213C211yesmaximal, characteristic, complemented
3#11723499C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592849—31nomaximal
9#10591433—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×#6312—21no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×D3×C3⋊Dic3—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#213—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88790—22no—
23C2×C32×D8—31no—
24#64149—31no—
25#64149—34no—
26#64149—64no—
27D4×C3⋊Dic3—31no—
28C2×#3753—31no—
29C2×C3×#213D314yescomplemented
30#63898—34no—
31C2×#3714—34no—
32#63659D311yescharacteristic, complemented
33#63648—31no—
34#63641—31no—
35C3×#832—34no—
36#63572—34no—
37#63564—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C32×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—
50C2×C3×C3⋊Dic3—21no—
51C32×D12—21no—
52C32×D12D412yescomplemented
53D3×C3⋊Dic3—42no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#5150—94no—
57#4786—91no—
58C2×C32×D4—24no—
59C2×C32×D4—31no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—61no—
63C2×#638—61no—
64C2×#638—64no—
65C2×#638—64no—
66C2×#638—68no—
67C2×#638—68no—
68C22×C3⋊Dic3—61no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C3×D12C2×D314yes—
73C2×D3×Dic3—64no—
74C32×D8—62no—
75C2×C32×C8—31no—
76#3757—61no—
77#3753—62no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4×C3⋊Dic3—31no—
80C4×C3⋊Dic3—34no—
81C4×C3⋊Dic3—64no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C3×#213—28no—
84C3×C4⋊Dic3—34no—
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×C32×D3—22no—
97C2×C32×D3—41no—
98C2×C32×D3—41no—
99C33×C4C2×D411yescharacteristic
100C33×C4#4111yescharacteristic
101C3⋊(C3⋊Dic3)—21no—
102C3⋊(C3⋊Dic3)D812yescomplemented
103C3×C3⋊Dic3—41no—
104C2×C3×D8—34no—
105#940—94no—
106#940—91no—
107#940—184no—
108D4×Dic3—94no—
109C2×#213—94no—
110C2×#213C3⋊D311yescharacteristic, complemented
111C2×D24—91no—
112#893—91no—
113C3×#102—91no—
114#832—34no—
115#821—94no—
116Dic3⋊C8—94no—
117Dic3⋊C8—91no—
118#809—91no—
119C23×C32—28no—
120C23×C32—24no—
121C23×C32—21no—
122C23×C32—61no—
123C22×C3⋊D3—68no—
124C22×C3⋊D3—64no—
125C22×C3⋊D3—61no—
126C22×C3×D3—24no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—24no—
130C32×D4—24no—
131C32×D4—32no—
132C32×D4—44no—
133C32×D4—44no—
134C32×D4—61no—
135C2×C32×C4—24no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D311yescharacteristic
138#638—122no—
139#638—128no—
140#638—1216no—
141#638—1216no—
142#638—128no—
143#638—122no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—31no—
146C2×C3⋊Dic3—31no—
147C2×C3⋊Dic3—34no—
148C2×C3⋊Dic3—34no—
149C2×C3⋊Dic3—64no—
150C2×C3⋊Dic3—64no—
151C2×C3⋊Dic3—122no—
152C4⋊(C3⋊D3)—61no—
153C4⋊(C3⋊D3)—61no—
154C4⋊(C3⋊D3)—64no—
155C4⋊(C3⋊D3)—64no—
156C4⋊(C3⋊D3)—124no—
157C4⋊(C3⋊D3)—124no—
158C2×C3×Dic3—64no—
159C3×D12—24no—
160C3×D12#6718yescomplemented
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—
172C32×D3—42no—
173C2×C3×D4—24no—
174C2×C3×D4—34no—
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—184no—
184C2×D12—94no—
185C2×D12—91no—
186C2×D12—184no—
187C2×D12C2×C3⋊D311yescharacteristic
188C2×C4×D3—181no—
189C3×D8—68no—
190C2×C3×C8—34no—
191C2×C3×C8—91no—
192C3×C4⋊C4—91no—
193#217—184no—
194#213—22no—
195#213—188no—
196(C2×D3)⋊C4—181no—
197C4⋊Dic3—34no—
198C4×Dic3—94no—
199C4×Dic3—91no—
200C4×Dic3—184no—
201C2×C3⋊C8—34no—
202C2×C3⋊C8C2×C3⋊D311yescharacteristic
203D24—182no—
204C22×C32—24no—
205C22×C32—48no—
206C22×C32—48no—
207C22×C32—48no—
208C22×C32—41no—
209C22×C32—44no—
210C22×C32—41no—
211C22×C32—44no—
212C22×C32—41no—
213C22×C32—44no—
214C22×C32—62no—
215C22×C32—121no—
216C22×C32—121no—
217C22×C32D3×D414yes—
218C22×C32D3×D411yescharacteristic
219C2×C3⋊D3—128no—
220C2×C3⋊D3—128no—
221C2×C3⋊D3—128no—
222C2×C3⋊D3—121no—
223C2×C3⋊D3—124no—
224C2×C3⋊D3—121no—
225C2×C3⋊D3—124no—
226C2×C3⋊D3—121no—
227C2×C3⋊D3—124no—
228C2×C3×D3—28no—
229C2×C3×D3—44no—
230C2×C3×D3—44no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4#23511yescharacteristic
234C32×C4D3×D411yescharacteristic
235C32×C4#23714yes—
236C32×C4C2×#6714yes—
237C3⋊Dic3—32no—
238C3⋊Dic3—38no—
239C3⋊Dic3—41no—
240C3⋊Dic3—68no—
241C3⋊Dic3—68no—
242C3⋊Dic3—64no—
243C3⋊Dic3—61no—
244C3×Dic3—124no—
245C2×D8—31no—
246C2×D8—271no—
247#127—271no—
248C4×D4—271no—
249C4⋊C8—271no—
250#102—91no—
251#102—271no—
252C33#45911yescharacteristic, complemented, Sylow
253C23×C3—28no—
254C23×C3—24no—
255C23×C3—21no—
256C23×C3—64no—
257C22×D3—21no—
258C22×D3—188no—
259C22×D3—184no—
260C22×D3—181no—
261C3×D4—24no—
262C3×D4—24no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—38no—
266C3×D4—44no—
267C3×D4—44no—
268C3×D4—64no—
269C2×C3×C4—24no—
270C2×C3×C4—181no—
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—64no—
280C2×Dic3—94no—
281C2×Dic3—91no—
282C2×Dic3—94no—
283C2×Dic3—91no—
284C2×Dic3—184no—
285C2×Dic3—184no—
286C2×Dic3—368no—
287D12—21no—
288D12—184no—
289D12—184no—
290D12—181no—
291D12—181no—
292D12—364no—
293D12—364no—
294D12#63812yescomplemented
295C4×D3—362no—
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—84no—
305C2×C32—81no—
306C2×C32—122no—
307C2×C32#88411yescharacteristic
308C2×C32D3×D814yes—
309C2×C32D3×D811yescharacteristic
310C2×C32#91114yes—
311C2×C32#91411yescharacteristic
312C2×C32#93914yes—
313C3⋊D3—248no—
314C3⋊D3—244no—
315C3⋊D3—241no—
316C3×D3—48no—
317C2×D4—31no—
318C2×D4—271no—
319C2×D4—541no—
320C2×D4—541no—
321C2×D4D3×C3⋊D311yescharacteristic
322C22×C4—541no—
323D8—62no—
324D8—542no—
325C2×C8—31no—
326C2×C8—91no—
327C4⋊C4—91no—
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—68no—
341C22×C3—124no—
342C22×C3—124no—
343C22×C3#381114yes—
344C22×C3D4×C3⋊D311yescharacteristic
345C2×D3—22no—
346C2×D3—41no—
347C2×D3—41no—
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—361no—
360C3×C4#380414yes—
361C3×C4D3×#6714yes—
362C3×C4#383011yescharacteristic
363C3×C4C2×#63811yescharacteristic
364Dic3—92no—
365Dic3—98no—
366Dic3—124no—
367Dic3—188no—
368Dic3—188no—
369Dic3—181no—
370Dic3—184no—
371C32—24no—
372C32#478611yescharacteristic, complemented
373C32#515014yescomplemented
374C23—21no—
375C23—61no—
376C23—541no—
377D4—21no—
378D4—21no—
379D4—32no—
380D4—61no—
381D4—541no—
382D4—541no—
383D4—1082no—
384D4—1082no—
385C2×C4—181no—
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—81no—
397C2×C3—84no—
398C2×C3—128no—
399C2×C3#6392614yes—
400C2×C3#6393714yes—
401C2×C3#6397114yes—
402C2×C3D8×C3⋊D311yescharacteristic
403C2×C3#6412011yescharacteristic
404C2×C3#6414811yescharacteristic
405D3—42no—
406D3—728no—
407D3—721no—
408D3—724no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—1081no—
416C22—1081no—
417C22—1081no—
418C22#8903911yescharacteristic
419C4—272no—
420C4—361no—
421C4—541no—
422C4#8902811yescharacteristic
423C4D3×#63811yescharacteristic
424C3—24no—
425C3#1059143314yescomplemented
426C3#1059284911yescharacteristic, complemented
427C2—81no—
428C2—122no—
429C2—2161no—
430C2#1172354711yescharacteristic
431C2#1172356411yescharacteristic
432C2#1172361611yescharacteristic
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×C32×#213C2semidirect
3. #11723499C2semidirect
4. C2×#88790C2semidirect
5. C3×#63659C2semidirect
6. #11722050C2semidirect
7. #11722036C2semidirect
29. C2×C3×#213D3semidirect
32. #63659D3semidirect
52. C32×D12D4semidirect
102. C3⋊(C3⋊Dic3)D8semidirect
110. C2×#213C3⋊D3semidirect
160. C3×D12#67semidirect
252. C33#459semidirect
294. D12#638semidirect
372. C32#4786semidirect
373. C32#5150semidirect
425. C3#10591433semidirect
426. C3#10592849semidirect

Maximal subgroups

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

subgroupindex
#117239812
C2×C32×#2132
#117234992
C2×#887902
C3×#636592
#117220502
#117220362
#105928493
#105914333