Finite Group Databasebeta

← #13160039

Subgroups — #13160039

Order 1728.

1388 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1388 conjugacy classes of subgroups — too many to draw. Only its 160 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×#89026C211yesmaximal, characteristic, complemented
2C2×C3×#3745C211yesmaximal, characteristic, complemented
3#11723532C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592472—31nomaximal
9#10590945—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
14C2×C32×D12C2211yescharacteristic
15C2×#6292—21no—
16C2×#6290—21no—
17#89026—22no—
18#89026—22no—
19C2×C33×C8C2211yescharacteristic, complemented
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C3×#3745—22no—
22C32×C4⋊Dic3C2211yescharacteristic, complemented
23#88803—21no—
24#88789—22no—
25C2×D4×C3⋊D3—31no—
26C2×#3802—34no—
27C2×#3753—31no—
28C2×#3745D311yescharacteristic, complemented
29#64104—31no—
30C2×C3×D24—34no—
31#63916—34no—
32C2×#3713—34no—
33C32×#102—31no—
34#63647—31no—
35#63647—34no—
36#63647—64no—
37#63636—31no—
38C3×#821D314yescomplemented
39#63562—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C32×D3—21no—
44C2×D3×C3⋊D3—24no—
45C2×D3×C3⋊D3—42no—
46C2×D3×C3⋊D3—42no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C4×C3⋊(C3⋊D3)—21no—
50C4×C3⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yescomplemented
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12D412yescomplemented
58#6292—42no—
59#6290—24no—
60C33×C8—21no—
61C3⋊(C3⋊(C3⋊C8))—21no—
62#5156—91no—
63#4773—94no—
64C23×C3⋊D3—61no—
65C22×D32—64no—
66C2×C32×D4—31no—
67C2×#638—61no—
68D4×C3⋊D3—62no—
69D4×C3⋊D3—62no—
70C2×C4⋊(C3⋊D3)—61no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—64no—
76C2×C3×D12—34no—
77C2×C3×D12—34no—
78C2×C3×D12C2×D314yes—
79C2×#626—64no—
80C2×#625—64no—
81#3802—68no—
82#3802—68no—
83C2×C32×C8—24no—
84C2×C32×C8C2×D314yescomplemented
85C2×C32×C8C2×D311yescharacteristic, complemented
86C32×C4⋊C4—31no—
87#3753—62no—
88#3752—61no—
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—34no—
91C2×C3⋊(C3⋊C8)—64no—
92#3745—22no—
93C3×C4⋊Dic3C2×D314yescomplemented
94C3×D24—68no—
95(C2×D3)⋊Dic3—64no—
96#3713—68no—
97C22×C33C2×D411yescharacteristic
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)—21no—
100C2×C3⋊(C3⋊D3)D812yescomplemented
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—41no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C2×C32×D3—22no—
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110D3×C3⋊D3—48no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3⋊(C3⋊Dic3)—21no—
114C32×Dic3—41no—
115C2×D3×D4—94no—
116C2×D3×D4—91no—
117C2×C3×D8—91no—
118#938—91no—
119C2×#213—94no—
120C2×#213—91no—
121C2×D24—34no—
122#895—94no—
123C3×#102—34no—
124#832—91no—
125#821C3⋊D311yescharacteristic, complemented
126(C2×D3)⋊C8—94no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—184no—
129#809—94no—
130C23×C32—61no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—31no—
134C22×C3⋊D3—34no—
135C22×C3⋊D3—64no—
136C22×C3⋊D3—64no—
137C22×C3⋊D3—122no—
138C22×C3⋊D3—122no—
139C22×C3×D3—24no—
140C22×C3×D3—64no—
141C22×C3×D3—64no—
142C2×D32—616no—
143C2×D32—128no—
144C2×D32—128no—
145C32×D4—32no—
146C32×D4—61no—
147C2×C32×C4—24no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D314yes—
150C2×C32×C4C22×D311yescharacteristic
151#638—64no—
152C2×C3⋊Dic3—31no—
153C2×C3⋊Dic3—34no—
154C2×C3⋊Dic3—64no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—122no—
159C4⋊(C3⋊D3)#6712yescomplemented
160C4×C3⋊D3—61no—
161C4×C3⋊D3—61no—
162C4×C3⋊D3—64no—
163C4×C3⋊D3—64no—
164C4×C3⋊D3—124no—
165C4×C3⋊D3—124no—
166C2×C3×Dic3—24no—
167C3×D12—24no—
168C3×D12—38no—
169C3×D12—64no—
170C3×D12—64no—
171C3×D12—64no—
172C3×D12D1218yescomplemented
173#626—128no—
174#625—616no—
175C32×C8—21no—
176C32×C8—24no—
177C32×C8—44no—
178C3⋊(C3⋊C8)—61no—
179C3⋊(C3⋊C8)—64no—
180C3⋊(C3⋊C8)—124no—
181#447—271noSylow
182C2×C33#12011yescharacteristic
183C2×C33C2×D811yescharacteristic
184C2×C33AGL(1,ℤ/8)11yescharacteristic
185C3⋊(C3⋊D3)—22no—
186C3×C3⋊D3—42no—
187C3×C3⋊D3—81no—
188C32×D3—42no—
189C23×D3—181no—
190C23×D3—184no—
191C2×C3×D4—34no—
192C2×C3×D4—91no—
193C2×C3×D4—91no—
194C2×#67—181no—
195C2×#67—181no—
196C2×#67—184no—
197D3×D4—182no—
198D3×D4—188no—
199D3×D4—188no—
200D3×D4—182no—
201C2×D12—34no—
202C2×D12—34no—
203C2×D12—184no—
204C2×D12C2×C3⋊D311yescharacteristic
205C2×C4×D3—94no—
206C2×C4×D3—91no—
207C2×C4×D3—184no—
208C3×D8—182no—
209C2×C3×C8—24no—
210C2×C3×C8D3214yescomplemented
211C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
212C3×C4⋊C4—34no—
213#217—181no—
214#213—188no—
215#213—182no—
216(C2×D3)⋊C4—184no—
217C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
218C2×C3⋊C8—94no—
219C2×C3⋊C8—91no—
220C2×C3⋊C8—184no—
221D24—68no—
222C22×C32—24no—
223C22×C32—62no—
224C22×C32—121no—
225C22×C32—121no—
226C22×C32D3×D414yes—
227C22×C32D3×D411yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—32no—
230C2×C3⋊D3—38no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3⋊D3—41no—
236C2×C3⋊D3—68no—
237C2×C3⋊D3—61no—
238C2×C3⋊D3—61no—
239C2×C3⋊D3—64no—
240C2×C3⋊D3—64no—
241C2×C3⋊D3—124no—
242C2×C3⋊D3—124no—
243C2×C3⋊D3—128no—
244C2×C3×D3—28no—
245C2×C3×D3—44no—
246C2×C3×D3—44no—
247C2×C3×D3—68no—
248C2×C3×D3—124no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252C2×C3×D3—124no—
253D32—1232no—
254C32×C4—24no—
255C32×C4—24no—
256C32×C4—121no—
257C32×C4C2×D1214yes—
258C32×C4D3×D414yes—
259C32×C4D3×D411yescharacteristic
260C32×C4C2×#6711yescharacteristic
261C3⋊Dic3—61no—
262C3⋊Dic3—64no—
263C3⋊Dic3—124no—
264C3×Dic3—44no—
265C22×D4—271no—
266C2×D8—91no—
267C2×D8—271no—
268#121—271no—
269#102—31no—
270#102—271no—
271C22⋊C8—271no—
272C33#44711yescharacteristic, complemented, Sylow
273C23×C3—64no—
274C23×C3—181no—
275C23×C3—181no—
276C22×D3—21no—
277C22×D3—64no—
278C22×D3—64no—
279C22×D3—94no—
280C22×D3—91no—
281C22×D3—184no—
282C22×D3—184no—
283C22×D3—1816no—
284C22×D3—362no—
285C22×D3—362no—
286C22×D3—368no—
287C22×D3—368no—
288C3×D4—38no—
289C3×D4—64no—
290C3×D4—92no—
291C3×D4—181no—
292C3×D4—181no—
293C3×D4—181no—
294C2×C3×C4—24no—
295C2×C3×C4—64no—
296C2×C3×C4C2×D3214yes—
297C2×C3×C4C22×C3⋊D311yescharacteristic
298#67—184no—
299#67—1816no—
300#67—362no—
301C2×Dic3—21no—
302C2×Dic3—94no—
303C2×Dic3—91no—
304C2×Dic3—184no—
305D12—21no—
306D12—38no—
307D12—64no—
308D12—64no—
309D12—64no—
310D12—368no—
311D12C4⋊(C3⋊D3)12yescomplemented
312C4×D3—184no—
313C4×D3—184no—
314C4×D3—181no—
315C4×D3—181no—
316C4×D3—364no—
317C4×D3—364no—
318C3×C8—24no—
319C3×C8—21no—
320C3×C8—44no—
321C3⋊C8—184no—
322C3⋊C8—181no—
323C3⋊C8—364no—
324C2×C32—24no—
325C2×C32—24no—
326C2×C32—24no—
327C2×C32—122no—
328C2×C32#88214yes—
329C2×C32D3×D814yes—
330C2×C32D3×D811yescharacteristic
331C2×C32#91111yescharacteristic
332C2×C32#91414yes—
333C2×C32#93711yescharacteristic
334C3⋊D3—42no—
335C3⋊D3—62no—
336C3⋊D3—68no—
337C3⋊D3—81no—
338C3⋊D3—128no—
339C3×D3—48no—
340C3×D3—128no—
341C3×D3—244no—
342C24—541no—
343C2×D4—31no—
344C2×D4—91no—
345C2×D4—91no—
346C2×D4—541no—
347C2×D4—541no—
348C2×D4—542no—
349C2×D4—542no—
350C22×C4—271no—
351D8—182no—
352D8—542no—
353C2×C8—271no—
354C2×C8D3×C3⋊D311yescharacteristic, complemented
355C4⋊C4—31no—
356C22⋊C4—541no—
357C22×C3—24no—
358C22×C3—68no—
359C22×C3—124no—
360C22×C3—124no—
361C22×C3—182no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3—361no—
366C22×C3—361no—
367C22×C3#380214yes—
368C22×C3D4×C3⋊D311yescharacteristic
369C2×D3—22no—
370C2×D3—41no—
371C2×D3—41no—
372C2×D3—68no—
373C2×D3—98no—
374C2×D3—92no—
375C2×D3—124no—
376C2×D3—124no—
377C2×D3—124no—
378C2×D3—124no—
379C2×D3—124no—
380C2×D3—188no—
381C2×D3—184no—
382C2×D3—181no—
383C2×D3—181no—
384C2×D3—184no—
385C2×D3—364no—
386C2×D3—364no—
387C2×D3—368no—
388C2×D3—3632no—
389C3×C4—24no—
390C3×C4—24no—
391C3×C4—124no—
392C3×C4D3×D1214yes—
393C3×C4D3×#6714yes—
394C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
395C3×C4D4×C3⋊D311yescharacteristic
396Dic3—41no—
397Dic3—181no—
398Dic3—184no—
399Dic3—364no—
400C32—24no—
401C32#477314yescomplemented
402C32#515611yescharacteristic, complemented
403C23—61no—
404C23—181no—
405C23—181no—
406C23—271no—
407C23—544no—
408C23—1082no—
409C23—1082no—
410D4—32no—
411D4—61no—
412D4—92no—
413D4—181no—
414D4—181no—
415D4—181no—
416D4—544no—
417D4—1082no—
418C2×C4—61no—
419C2×C4—271no—
420C2×C4—541no—
421C2×C4—541no—
422C2×C4C2×D3×C3⋊D311yescharacteristic
423C8—21no—
424C8—541no—
425C2×C3—24no—
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—128no—
429C2×C3—362no—
430C2×C3—721no—
431C2×C3#6379714yes—
432C2×C3#6379814yes—
433C2×C3#6392214yes—
434C2×C3#6409111yescharacteristic
435C2×C3D8×C3⋊D311yescharacteristic
436C2×C3#6412311yescharacteristic
437D3—42no—
438D3—128no—
439D3—182no—
440D3—188no—
441D3—244no—
442D3—368no—
443C22—62no—
444C22—121no—
445C22—121no—
446C22—182no—
447C22—272no—
448C22—361no—
449C22—361no—
450C22—361no—
451C22—361no—
452C22—361no—
453C22—541no—
454C22—541no—
455C22—1088no—
456C22#8902611yescharacteristic
457C4—121no—
458C4—541no—
459C4D3×C4⋊(C3⋊D3)11yescharacteristic
460C4C3⋊D3×#6711yescharacteristic
461C3—24no—
462C3#1059094514yescomplemented
463C3#1059247211yescharacteristic, complemented
464C2—122no—
465C2—362no—
466C2—542no—
467C2—721no—
468C2#1172332211yescharacteristic
469C2#1172332311yescharacteristic
470C2#1172354111yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#447
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#89026C2semidirect
2. C2×C3×#3745C2semidirect
3. #11723532C2semidirect
4. C2×#88789C2semidirect
5. #11722190C2semidirect
6. C32×#821C2semidirect
7. #11722033C2semidirect
13. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C33×C8C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C32×C4⋊Dic3C22semidirect
28. C2×#3745D3semidirect
38. C3×#821D3semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
57. C32×D12D4semidirect
72. C2×C4⋊(C3⋊D3)C2×D3semidirect
84. C2×C32×C8C2×D3semidirect
85. C2×C32×C8C2×D3semidirect
93. C3×C4⋊Dic3C2×D3semidirect
100. C2×C3⋊(C3⋊D3)D8semidirect
125. #821C3⋊D3semidirect
159. C4⋊(C3⋊D3)#67semidirect
172. C3×D12D12semidirect
210. C2×C3×C8D32semidirect
211. C2×C3×C8C2×C3⋊D3semidirect
217. C4⋊Dic3C2×C3⋊D3semidirect
272. C33#447semidirect
311. D12C4⋊(C3⋊D3)semidirect
354. C2×C8D3×C3⋊D3semidirect
401. C32#4773semidirect
402. C32#5156semidirect
462. C3#10590945semidirect
463. C3#10592472semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C2×C3×#37452
#117235322
C2×#887892
#117221902
C32×#8212
#117220332
#105924723
#105909453