Finite Group Databasebeta

← #13160962

Subgroups — #13160962

Order 1728.

2132 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 2132 conjugacy classes of subgroups — too many to draw. Only its 170 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×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3#11723532C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592472—31nomaximal
9#10591456—34nomaximal
10C23×C3⋊(C3⋊D3)—21no—
11C2×C33×D4C2211yescharacteristic
12C2×#6312—21no—
13D4×C3⋊(C3⋊D3)—22no—
14D4×C3⋊(C3⋊D3)—22no—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×#6292—21no—
19C3×#3753—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
21C32×C4⋊Dic3C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88803—21no—
24#88791—22no—
25C2×D4×C3⋊D3—31no—
26C2×D4×C3⋊D3—34no—
27C2×D4×C3⋊D3—64no—
28C2×#3753D311yescharacteristic, complemented
29C2×#3745—31no—
30#64104—31no—
31C2×C3×#213—34no—
32#63916—34no—
33C2×#3714—34no—
34C32×#102—31no—
35#63647—31no—
36#63636—31no—
37C3×#832D314yescomplemented
38#63564—34no—
39#63557—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)—24no—
42C22×C3⋊(C3⋊D3)—21no—
43C22×C3⋊(C3⋊D3)—42no—
44C22×C3⋊(C3⋊D3)—42no—
45C22×C3⋊(C3⋊D3)D411yescharacteristic
46C22×C3×C3⋊D3—21no—
47C33×D4—21no—
48C33×D4D412yescomplemented
49C2×C33×C4C2311yescharacteristic
50#6312—24no—
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C4⋊(C3⋊(C3⋊D3))—21no—
53C4⋊(C3⋊(C3⋊D3))D412yescomplemented
54C4×C3⋊(C3⋊D3)—21no—
55C4×C3⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C2×C32×Dic3—21no—
59#6292—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8—21no—
62#5156—91no—
63#4773—94no—
64C23×C3⋊D3—61no—
65C23×C3⋊D3—64no—
66C23×C3⋊D3—68no—
67C2×C32×D4—24no—
68C2×C32×D4C2×D314yes—
69C2×C32×D4C2×D311yescharacteristic
70C2×#638—61no—
71C2×#638—64no—
72C2×#638—68no—
73D4×C3⋊D3—62no—
74D4×C3⋊D3—68no—
75D4×C3⋊D3—616no—
76D4×C3⋊D3—616no—
77D4×C3⋊D3—68no—
78D4×C3⋊D3—62no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—64no—
82C2×C4⋊(C3⋊D3)—61no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—34no—
86C2×C4×C3⋊D3—64no—
87C2×C3×D12—34no—
88C2×#626—64no—
89C2×C32×C8—31no—
90C32×C4⋊C4—31no—
91#3753—22no—
92#3752—61no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
94#3745—62no—
95C3×#213—68no—
96C3×C4⋊Dic3C2×D314yescomplemented
97C2×C3×C3⋊C8—34no—
98C2×C3×C3⋊C8C2×D314yescomplemented
99(C2×D3)⋊Dic3—64no—
100#3714—68no—
101C22×C33—22no—
102C22×C33—41no—
103C22×C33—41no—
104C22×C33C2×D411yescharacteristic
105C2×C3⋊(C3⋊D3)—21no—
106C2×C3⋊(C3⋊D3)—21no—
107C2×C3⋊(C3⋊D3)—22no—
108C2×C3⋊(C3⋊D3)—48no—
109C2×C3⋊(C3⋊D3)—41no—
110C2×C3⋊(C3⋊D3)—41no—
111C2×C3⋊(C3⋊D3)D812yescomplemented
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C33×C4C2×D411yescharacteristic
116C33×C4C2×D411yescharacteristic
117C3⋊(C3⋊Dic3)—21no—
118C32×Dic3—41no—
119C2×D3×D4—94no—
120C2×D3×D4—91no—
121C2×D3×D4—184no—
122C2×C3×D8—91no—
123#938—91no—
124C2×#213—34no—
125C2×#213—91no—
126C2×D24—94no—
127#895—94no—
128C3×#102—34no—
129#832C3⋊D311yescharacteristic, complemented
130#821—91no—
131(C2×D3)⋊C8—94no—
132(C2×D3)⋊C8—91no—
133#809—94no—
134C23×C32—28no—
135C23×C32—24no—
136C23×C32—21no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3—31no—
139C22×C3⋊D3—34no—
140C22×C3⋊D3—64no—
141C22×C3⋊D3—68no—
142C22×C3⋊D3—632no—
143C22×C3⋊D3—616no—
144C22×C3⋊D3—64no—
145C22×C3⋊D3—64no—
146C22×C3⋊D3—61no—
147C22×C3⋊D3—122no—
148C22×C3⋊D3—128no—
149C22×C3⋊D3—1216no—
150C22×C3⋊D3—1216no—
151C22×C3⋊D3—122no—
152C22×C3⋊D3—128no—
153C22×C3×D3—64no—
154C32×D4—28no—
155C32×D4—24no—
156C32×D4—21no—
157C32×D4—28no—
158C32×D4D1218yescomplemented
159C32×D4#6712yescomplemented
160C2×C32×C4—24no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D314yes—
163C2×C32×C4C22×D311yescharacteristic
164#638—632no—
165#638—616no—
166#638—64no—
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3—34no—
169C2×C3⋊Dic3—64no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)—32no—
173C4⋊(C3⋊D3)—38no—
174C4⋊(C3⋊D3)—68no—
175C4⋊(C3⋊D3)—68no—
176C4⋊(C3⋊D3)—64no—
177C4⋊(C3⋊D3)—61no—
178C4⋊(C3⋊D3)—122no—
179C4×C3⋊D3—68no—
180C4×C3⋊D3—68no—
181C4×C3⋊D3—64no—
182C4×C3⋊D3—64no—
183C4×C3⋊D3—61no—
184C4×C3⋊D3—61no—
185C2×C3×Dic3—24no—
186C3×D12—64no—
187C3×D12—64no—
188#626—128no—
189C32×C8—61no—
190C3⋊(C3⋊C8)—21no—
191C3×C3⋊C8—24no—
192C3×C3⋊C8—64no—
193#447—271noSylow
194C2×C33—42no—
195C2×C33#12011yescharacteristic
196C2×C33C2×D811yescharacteristic
197C2×C33AGL(1,ℤ/8)11yescharacteristic
198C3⋊(C3⋊D3)—22no—
199C3⋊(C3⋊D3)—42no—
200C3×C3⋊D3—81no—
201C23×D3—184no—
202C23×D3—181no—
203C23×D3—188no—
204C2×C3×D4—24no—
205C2×C3×D4—91no—
206C2×C3×D4D3214yes—
207C2×C3×D4C2×C3⋊D311yescharacteristic
208C2×#67—181no—
209C2×#67—184no—
210C2×#67—181no—
211C2×#67—188no—
212D3×D4—188no—
213D3×D4—182no—
214D3×D4—1816no—
215D3×D4—1816no—
216D3×D4—188no—
217D3×D4—182no—
218C2×D12—34no—
219C2×D12—94no—
220C2×D12—91no—
221C2×D12—184no—
222C2×D12—184no—
223C2×C4×D3—94no—
224C2×C4×D3—91no—
225C2×C4×D3—184no—
226C3×D8—182no—
227C2×C3×C8—34no—
228C2×C3×C8—91no—
229C3×C4⋊C4—34no—
230#217—181no—
231#213—68no—
232#213—182no—
233(C2×D3)⋊C4—184no—
234C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
235C2×C3⋊C8—34no—
236C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
237D24—188no—
238C22×C32—216no—
239C22×C32—28no—
240C22×C32—22no—
241C22×C32—24no—
242C22×C32—48no—
243C22×C32—48no—
244C22×C32—44no—
245C22×C32—41no—
246C22×C32—41no—
247C22×C32—44no—
248C22×C32D3×D414yes—
249C22×C32D3×D411yescharacteristic
250C2×C3⋊D3—32no—
251C2×C3⋊D3—38no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3⋊D3—68no—
256C2×C3⋊D3—68no—
257C2×C3⋊D3—68no—
258C2×C3⋊D3—64no—
259C2×C3⋊D3—61no—
260C2×C3⋊D3—64no—
261C2×C3⋊D3—61no—
262C2×C3⋊D3—616no—
263C2×C3⋊D3—68no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—1264no—
266C2×C3⋊D3—128no—
267C2×C3⋊D3—1232no—
268C2×C3⋊D3—128no—
269C2×C3⋊D3—128no—
270C2×C3⋊D3—124no—
271C2×C3⋊D3—121no—
272C2×C3⋊D3—121no—
273C2×C3⋊D3—124no—
274C2×C3×D3—124no—
275C2×C3×D3—124no—
276C2×C3×D3—124no—
277C32×C4—24no—
278C32×C4—24no—
279C32×C4—121no—
280C32×C4C2×D1214yes—
281C32×C4D3×D414yes—
282C32×C4D3×D411yescharacteristic
283C32×C4C2×#6711yescharacteristic
284C3⋊Dic3—68no—
285C3⋊Dic3—64no—
286C3⋊Dic3—61no—
287C3×Dic3—44no—
288C22×D4—271no—
289C2×D8—91no—
290C2×D8—271no—
291#121—271no—
292#102—31no—
293#102—271no—
294C22⋊C8—271no—
295C33#44711yescharacteristic, complemented, Sylow
296C23×C3—28no—
297C23×C3—24no—
298C23×C3—21no—
299C23×C3—181no—
300C22×D3—64no—
301C22×D3—94no—
302C22×D3—91no—
303C22×D3—184no—
304C22×D3—188no—
305C22×D3—1832no—
306C22×D3—184no—
307C22×D3—1816no—
308C22×D3—184no—
309C22×D3—181no—
310C22×D3—362no—
311C22×D3—368no—
312C22×D3—3616no—
313C22×D3—3616no—
314C22×D3—362no—
315C22×D3—368no—
316C3×D4—28no—
317C3×D4—24no—
318C3×D4—21no—
319C3×D4—28no—
320C3×D4—181no—
321C3×D4—181no—
322C3×D4#62618yescomplemented
323C3×D4C4⋊(C3⋊D3)12yescomplemented
324C2×C3×C4—24no—
325C2×C3×C4—64no—
326C2×C3×C4C2×D3214yes—
327C2×C3×C4C22×C3⋊D311yescharacteristic
328#67—1832no—
329#67—184no—
330#67—1816no—
331#67—362no—
332C2×Dic3—21no—
333C2×Dic3—94no—
334C2×Dic3—91no—
335C2×Dic3—184no—
336D12—64no—
337D12—64no—
338D12—98no—
339D12—92no—
340D12—188no—
341D12—188no—
342D12—184no—
343D12—181no—
344D12—368no—
345C4×D3—188no—
346C4×D3—188no—
347C4×D3—184no—
348C4×D3—184no—
349C4×D3—181no—
350C4×D3—181no—
351C3×C8—64no—
352C3×C8—181no—
353C3⋊C8—21no—
354C3⋊C8—64no—
355C2×C32—24no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—416no—
359C2×C32—42no—
360C2×C32—48no—
361C2×C32#88214yes—
362C2×C32D3×D814yes—
363C2×C32D3×D811yescharacteristic
364C2×C32#91111yescharacteristic
365C2×C32#91414yes—
366C2×C32#93711yescharacteristic
367C3⋊D3—616no—
368C3⋊D3—62no—
369C3⋊D3—68no—
370C3⋊D3—81no—
371C3⋊D3—1216no—
372C3⋊D3—122no—
373C3⋊D3—128no—
374C3×D3—244no—
375C24—541no—
376C2×D4—91no—
377C2×D4—271no—
378C2×D4—541no—
379C2×D4—541no—
380C2×D4—542no—
381C2×D4—542no—
382C2×D4D3×C3⋊D311yescharacteristic
383C22×C4—271no—
384D8—182no—
385D8—542no—
386C2×C8—31no—
387C2×C8—91no—
388C4⋊C4—31no—
389C22⋊C4—541no—
390C22×C3—216no—
391C22×C3—22no—
392C22×C3—28no—
393C22×C3—24no—
394C22×C3—48no—
395C22×C3—48no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—361no—
401C22×C3—361no—
402C22×C3—361no—
403C22×C3#381114yes—
404C22×C3D4×C3⋊D311yescharacteristic
405C2×D3—92no—
406C2×D3—98no—
407C2×D3—124no—
408C2×D3—124no—
409C2×D3—124no—
410C2×D3—188no—
411C2×D3—188no—
412C2×D3—188no—
413C2×D3—181no—
414C2×D3—181no—
415C2×D3—184no—
416C2×D3—184no—
417C2×D3—1816no—
418C2×D3—182no—
419C2×D3—188no—
420C2×D3—3664no—
421C2×D3—368no—
422C2×D3—3632no—
423C2×D3—368no—
424C2×D3—368no—
425C2×D3—361no—
426C2×D3—361no—
427C2×D3—364no—
428C2×D3—364no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—124no—
432C3×C4C2×#62614yes—
433C3×C4#381114yes—
434C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
435C3×C4D4×C3⋊D311yescharacteristic
436Dic3—41no—
437Dic3—188no—
438Dic3—181no—
439Dic3—184no—
440C32—24no—
441C32#477314yescomplemented
442C32#515611yescharacteristic, complemented
443C23—21no—
444C23—181no—
445C23—271no—
446C23—544no—
447C23—541no—
448C23—1082no—
449C23—1082no—
450D4—21no—
451D4—181no—
452D4—181no—
453D4—272no—
454D4—544no—
455D4—541no—
456D4—1082no—
457D4#629212yescomplemented
458C2×C4—61no—
459C2×C4—271no—
460C2×C4—541no—
461C2×C4—541no—
462C2×C4C2×D3×C3⋊D311yescharacteristic
463C8—61no—
464C8—181no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—416no—
469C2×C3—42no—
470C2×C3—48no—
471C2×C3—721no—
472C2×C3#6392614yes—
473C2×C3#6393714yes—
474C2×C3#6398114yes—
475C2×C3#6409111yescharacteristic
476C2×C3D8×C3⋊D311yescharacteristic
477C2×C3#6412311yescharacteristic
478D3—1816no—
479D3—182no—
480D3—188no—
481D3—244no—
482D3—3616no—
483D3—362no—
484D3—368no—
485C22—22no—
486C22—41no—
487C22—41no—
488C22—272no—
489C22—361no—
490C22—361no—
491C22—361no—
492C22—541no—
493C22—541no—
494C22—542no—
495C22—1088no—
496C22—1081no—
497C22—1081no—
498C22#8903911yescharacteristic
499C4—121no—
500C4—541no—
501C4C2×#629211yescharacteristic
502C4#8903911yescharacteristic
503C3—24no—
504C3#1059145614yescomplemented
505C3#1059247211yescharacteristic, complemented
506C2—42no—
507C2—542no—
508C2—721no—
509C2—1082no—
510C2#1172354711yescharacteristic
511C2#1172356611yescharacteristic
512C2#1172363911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#447
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×D4×C3⋊(C3⋊D3)C2semidirect
2. C2×C3×#3753C2semidirect
3. #11723532C2semidirect
4. C2×#88791C2semidirect
5. C32×#832C2semidirect
6. #11722035C2semidirect
7. #11722024C2semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. C32×C4⋊Dic3C22semidirect
22. C2×C32×C3⋊C8C22semidirect
28. C2×#3753D3semidirect
37. C3×#832D3semidirect
48. C33×D4D4semidirect
53. C4⋊(C3⋊(C3⋊D3))D4semidirect
83. C2×C4⋊(C3⋊D3)C2×D3semidirect
93. C2×C3⋊(C3⋊C8)C2×D3semidirect
96. C3×C4⋊Dic3C2×D3semidirect
98. C2×C3×C3⋊C8C2×D3semidirect
111. C2×C3⋊(C3⋊D3)D8semidirect
129. #832C3⋊D3semidirect
158. C32×D4D12semidirect
159. C32×D4#67semidirect
234. C4⋊Dic3C2×C3⋊D3semidirect
236. C2×C3⋊C8C2×C3⋊D3semidirect
295. C33#447semidirect
322. C3×D4#626semidirect
323. C3×D4C4⋊(C3⋊D3)semidirect
441. C32#4773semidirect
442. C32#5156semidirect
457. D4#6292semidirect
504. C3#10591456semidirect
505. C3#10592472semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×C3×#37532
#117235322
C2×#887912
C32×#8322
#117220352
#117220242
#105924723
#105914563