Finite Group Databasebeta

← #13159421

Subgroups — #13159421

Order 1728.

1564 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1564 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
2#11723936C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592472—31nomaximal
9#10590598—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))—21no—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C32×D12C2211yescharacteristic
16C2×#6290—21no—
17#89026—22no—
18#89026—22no—
19C33×C4⋊C4C2211yescharacteristic, complemented
20#88859—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88791—22no—
24#88790—22no—
25C2×D4×C3⋊D3—31no—
26C2×#3802—34no—
27C2×#3753—31no—
28C2×#3745—31no—
29#64104—31no—
30#64104—34no—
31#64104—64no—
32C2×#3714—34no—
33C2×#3714—34no—
34C32×#102—31no—
35#63647—31no—
36#63636D311yescharacteristic, complemented
37C3×#809—34no—
38C3×#809D314yescomplemented
39#63557—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3⋊(C3⋊D3)D411yescharacteristic
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×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))—21no—
52C4⋊(C3⋊(C3⋊D3))—42no—
53C4×C3⋊(C3⋊D3)—21no—
54C4×C3⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)D412yescomplemented
57C32×D12—21no—
58C32×D12D412yescomplemented
59#6290—24no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8—21no—
62#4773—94no—
63#4773—91no—
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)—31no—
71C2×C4⋊(C3⋊D3)—34no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—64no—
74C2×C4⋊(C3⋊D3)—64no—
75C2×C4⋊(C3⋊D3)—68no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—34no—
79C2×C4×C3⋊D3—64no—
80C2×C3×D12—34no—
81C2×C3×D12C2×D314yes—
82C2×#625—64no—
83#3802—68no—
84#3802—68no—
85C2×C32×C8—31no—
86C32×C4⋊C4—24no—
87C32×C4⋊C4C2×D314yescomplemented
88C32×C4⋊C4C2×D311yescharacteristic, complemented
89#3753—62no—
90#3752—61no—
91#3752—64no—
92#3752—68no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
94#3745—62no—
95C2×C3×C3⋊C8—34no—
96C2×C3×C3⋊C8C2×D314yescomplemented
97#3714—68no—
98#3714—68no—
99C22×C33C2×D411yescharacteristic
100C2×C3⋊(C3⋊D3)—21no—
101C2×C3⋊(C3⋊D3)—21no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3⋊(C3⋊D3)—41no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3⋊(C3⋊D3)D812yescomplemented
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C2×C32×D3—22no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112D3×C3⋊D3—48no—
113C33×C4—41no—
114C33×C4C2×D411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3⋊(C3⋊Dic3)—21no—
117C2×D3×D4—94no—
118C2×D3×D4—91no—
119C2×#213—94no—
120C2×#213—91no—
121C2×D24—94no—
122C2×D24—91no—
123#895—94no—
124#895—91no—
125#895—184no—
126C3×#102—34no—
127C3×#102—91no—
128(C2×D3)⋊C8—94no—
129(C2×D3)⋊C8—91no—
130#809—34no—
131#809C3⋊D311yescharacteristic, complemented
132C23×C32—61no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—31no—
135C22×C3⋊D3—34no—
136C22×C3⋊D3—64no—
137C22×C3⋊D3—64no—
138C22×C3⋊D3—68no—
139C22×C3⋊D3—64no—
140C22×C3⋊D3—61no—
141C22×C3⋊D3—122no—
142C22×C3⋊D3—122no—
143C22×C3×D3—24no—
144C22×C3×D3—64no—
145C2×D32—616no—
146C2×D32—128no—
147C2×D32—128no—
148C32×D4—32no—
149C32×D4—61no—
150C2×C32×C4—28no—
151C2×C32×C4—24no—
152C2×C32×C4—21no—
153C2×C32×C4—24no—
154C2×C32×C4C22×D314yes—
155C2×C32×C4C22×D311yescharacteristic
156#638—64no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—34no—
159C2×C3⋊Dic3—64no—
160C4⋊(C3⋊D3)—21no—
161C4⋊(C3⋊D3)—61no—
162C4⋊(C3⋊D3)—61no—
163C4⋊(C3⋊D3)—64no—
164C4⋊(C3⋊D3)—64no—
165C4⋊(C3⋊D3)—124no—
166C4⋊(C3⋊D3)—124no—
167C4⋊(C3⋊D3)—122no—
168C4⋊(C3⋊D3)—128no—
169C4⋊(C3⋊D3)—1216no—
170C4⋊(C3⋊D3)D1212yescomplemented
171C4×C3⋊D3—61no—
172C4×C3⋊D3—61no—
173C4×C3⋊D3—64no—
174C4×C3⋊D3—64no—
175C4×C3⋊D3—124no—
176C4×C3⋊D3—124no—
177C3×D12—24no—
178C3×D12—38no—
179C3×D12—64no—
180C3×D12D1218yescomplemented
181#625—616no—
182C32×C8—61no—
183C3⋊(C3⋊C8)—21no—
184C3×C3⋊C8—24no—
185C3×C3⋊C8—64no—
186#447—271noSylow
187C2×C33#12011yescharacteristic
188C2×C33C2×D811yescharacteristic
189C2×C33AGL(1,ℤ/8)11yescharacteristic
190C3⋊(C3⋊D3)—22no—
191C3⋊(C3⋊D3)—81no—
192C3×C3⋊D3—42no—
193C32×D3—42no—
194C23×D3—181no—
195C23×D3—184no—
196C2×C3×D4—34no—
197C2×C3×D4—91no—
198C2×#67—181no—
199C2×#67—184no—
200D3×D4—182no—
201D3×D4—188no—
202D3×D4—188no—
203D3×D4—182no—
204C2×D12—34no—
205C2×D12—94no—
206C2×D12—91no—
207C2×D12—184no—
208C2×D12—184no—
209C2×D12—181no—
210C2×D12—188no—
211C2×D12C2×C3⋊D311yescharacteristic
212C2×C4×D3—94no—
213C2×C4×D3—91no—
214C2×C4×D3—184no—
215C2×C3×C8—34no—
216C2×C3×C8—91no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4D3214yescomplemented
219C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
220#213—182no—
221#213—188no—
222(C2×D3)⋊C4—184no—
223(C2×D3)⋊C4—181no—
224(C2×D3)⋊C4—188no—
225C2×C3⋊C8—34no—
226C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
227D24—188no—
228D24—182no—
229C22×C32—24no—
230C22×C32—62no—
231C22×C32—121no—
232C22×C32—121no—
233C22×C32D3×D414yes—
234C22×C32D3×D411yescharacteristic
235C2×C3⋊D3—22no—
236C2×C3⋊D3—32no—
237C2×C3⋊D3—38no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—41no—
240C2×C3⋊D3—68no—
241C2×C3⋊D3—61no—
242C2×C3⋊D3—61no—
243C2×C3⋊D3—64no—
244C2×C3⋊D3—64no—
245C2×C3⋊D3—124no—
246C2×C3⋊D3—124no—
247C2×C3⋊D3—128no—
248C2×C3⋊D3—128no—
249C2×C3⋊D3—128no—
250C2×C3⋊D3—128no—
251C2×C3⋊D3—124no—
252C2×C3⋊D3—121no—
253C2×C3⋊D3—124no—
254C2×C3⋊D3—121no—
255C2×C3⋊D3—124no—
256C2×C3⋊D3—121no—
257C2×C3×D3—28no—
258C2×C3×D3—44no—
259C2×C3×D3—44no—
260C2×C3×D3—68no—
261C2×C3×D3—124no—
262C2×C3×D3—124no—
263D32—1232no—
264C32×C4—24no—
265C32×C4—24no—
266C32×C4—48no—
267C32×C4—44no—
268C32×C4—41no—
269C32×C4C2×D1214yes—
270C32×C4C2×D1211yescharacteristic
271C32×C4D3×D414yes—
272C32×C4D3×D411yescharacteristic
273C3⋊Dic3—61no—
274C3⋊Dic3—64no—
275C3⋊Dic3—124no—
276C22×D4—271no—
277C2×D8—271no—
278C2×D8—271no—
279#121—271no—
280#102—31no—
281#102—91no—
282C22⋊C8—271no—
283C33#44711yescharacteristic, complemented, Sylow
284C23×C3—64no—
285C23×C3—181no—
286C22×D3—21no—
287C22×D3—64no—
288C22×D3—94no—
289C22×D3—91no—
290C22×D3—184no—
291C22×D3—184no—
292C22×D3—1816no—
293C22×D3—188no—
294C22×D3—184no—
295C22×D3—181no—
296C22×D3—362no—
297C22×D3—362no—
298C22×D3—368no—
299C22×D3—368no—
300C3×D4—38no—
301C3×D4—64no—
302C3×D4—92no—
303C3×D4—181no—
304C2×C3×C4—28no—
305C2×C3×C4—24no—
306C2×C3×C4—21no—
307C2×C3×C4—24no—
308C2×C3×C4C2×D3214yes—
309C2×C3×C4C22×C3⋊D311yescharacteristic
310#67—184no—
311#67—1816no—
312C2×Dic3—94no—
313C2×Dic3—91no—
314C2×Dic3—184no—
315D12—21no—
316D12—38no—
317D12—64no—
318D12—184no—
319D12—184no—
320D12—181no—
321D12—181no—
322D12—364no—
323D12—364no—
324D12—362no—
325D12—368no—
326D12—3616no—
327D12C4⋊(C3⋊D3)12yescomplemented
328C4×D3—184no—
329C4×D3—184no—
330C4×D3—181no—
331C4×D3—181no—
332C4×D3—364no—
333C4×D3—364no—
334C3×C8—64no—
335C3×C8—181no—
336C3⋊C8—21no—
337C3⋊C8—64no—
338C2×C32—24no—
339C2×C32—24no—
340C2×C32—24no—
341C2×C32—122no—
342C2×C32#88214yes—
343C2×C32#88211yescharacteristic
344C2×C32D3×D814yes—
345C2×C32D3×D811yescharacteristic
346C2×C32#91414yes—
347C2×C32#91411yescharacteristic
348C3⋊D3—42no—
349C3⋊D3—62no—
350C3⋊D3—68no—
351C3⋊D3—128no—
352C3⋊D3—248no—
353C3⋊D3—244no—
354C3⋊D3—241no—
355C3×D3—48no—
356C3×D3—128no—
357C24—541no—
358C2×D4—31no—
359C2×D4—91no—
360C2×D4—271no—
361C2×D4—541no—
362C2×D4—541no—
363C2×D4—542no—
364C2×D4—542no—
365C22×C4—271no—
366D8—542no—
367D8—542no—
368C2×C8—31no—
369C2×C8—91no—
370C4⋊C4D3×C3⋊D311yescharacteristic, complemented
371C22⋊C4—541no—
372C22×C3—24no—
373C22×C3—68no—
374C22×C3—124no—
375C22×C3—124no—
376C22×C3—182no—
377C22×C3—361no—
378C22×C3—361no—
379C22×C3#381114yes—
380C22×C3D4×C3⋊D311yescharacteristic
381C2×D3—22no—
382C2×D3—41no—
383C2×D3—41no—
384C2×D3—68no—
385C2×D3—92no—
386C2×D3—98no—
387C2×D3—124no—
388C2×D3—124no—
389C2×D3—188no—
390C2×D3—181no—
391C2×D3—181no—
392C2×D3—184no—
393C2×D3—184no—
394C2×D3—364no—
395C2×D3—364no—
396C2×D3—368no—
397C2×D3—3632no—
398C2×D3—368no—
399C2×D3—368no—
400C2×D3—368no—
401C2×D3—361no—
402C2×D3—361no—
403C2×D3—361no—
404C2×D3—364no—
405C2×D3—364no—
406C2×D3—364no—
407C3×C4—24no—
408C3×C4—24no—
409C3×C4—48no—
410C3×C4—41no—
411C3×C4—44no—
412C3×C4D3×D1214yes—
413C3×C4D3×D1214yes—
414C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
415C3×C4D4×C3⋊D311yescharacteristic
416Dic3—181no—
417Dic3—184no—
418Dic3—364no—
419C32—24no—
420C32#477314yescomplemented
421C32#477311yescharacteristic, complemented
422C23—61no—
423C23—181no—
424C23—271no—
425C23—544no—
426C23—541no—
427C23—1082no—
428C23—1082no—
429D4—32no—
430D4—61no—
431D4—92no—
432D4—181no—
433D4—544no—
434D4—541no—
435D4—541no—
436D4—1082no—
437C2×C4—21no—
438C2×C4—271no—
439C2×C4—541no—
440C2×C4—541no—
441C2×C4C2×D3×C3⋊D311yescharacteristic
442C8—61no—
443C8—181no—
444C2×C3—24no—
445C2×C3—24no—
446C2×C3—24no—
447C2×C3—128no—
448C2×C3—362no—
449C2×C3#6392314yes—
450C2×C3#6392614yes—
451C2×C3#6394114yes—
452C2×C3#6409111yescharacteristic
453C2×C3D8×C3⋊D311yescharacteristic
454C2×C3#6412311yescharacteristic
455D3—42no—
456D3—128no—
457D3—188no—
458D3—182no—
459D3—368no—
460D3—728no—
461D3—724no—
462D3—721no—
463C22—62no—
464C22—121no—
465C22—121no—
466C22—182no—
467C22—272no—
468C22—361no—
469C22—361no—
470C22—541no—
471C22—541no—
472C22—1088no—
473C22—1081no—
474C22—1081no—
475C22—1081no—
476C22#8903911yescharacteristic
477C4—41no—
478C4—541no—
479C4D3×C4⋊(C3⋊D3)11yescharacteristic
480C4C3⋊D3×D1211yescharacteristic
481C3—24no—
482C3#1059059814yescomplemented
483C3#1059247211yescharacteristic, complemented
484C2—122no—
485C2—362no—
486C2—542no—
487C2—2161no—
488C2#1172354211yescharacteristic
489C2#1172354711yescharacteristic
490C2#1172357211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#447
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89026C2semidirect
2. #11723936C2semidirect
3. C2×#88791C2semidirect
4. C2×#88790C2semidirect
5. C3×#63636C2semidirect
6. C32×#809C2semidirect
7. #11722024C2semidirect
12. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C33×C4⋊C4C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. C2×C32×C3⋊C8C22semidirect
36. #63636D3semidirect
38. C3×#809D3semidirect
56. C3×C4⋊(C3⋊D3)D4semidirect
58. C32×D12D4semidirect
87. C32×C4⋊C4C2×D3semidirect
88. C32×C4⋊C4C2×D3semidirect
93. C2×C3⋊(C3⋊C8)C2×D3semidirect
96. C2×C3×C3⋊C8C2×D3semidirect
105. C2×C3⋊(C3⋊D3)D8semidirect
131. #809C3⋊D3semidirect
170. C4⋊(C3⋊D3)D12semidirect
180. C3×D12D12semidirect
218. C3×C4⋊C4D32semidirect
219. C3×C4⋊C4C2×C3⋊D3semidirect
226. C2×C3⋊C8C2×C3⋊D3semidirect
283. C33#447semidirect
327. D12C4⋊(C3⋊D3)semidirect
370. C4⋊C4D3×C3⋊D3semidirect
420. C32#4773semidirect
421. C32#4773semidirect
482. C3#10590598semidirect
483. C3#10592472semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
#117239362
C2×#887912
C2×#887902
C3×#636362
C32×#8092
#117220242
#105924723
#105905983