Finite Group Databasebeta

← #13161108

Subgroups — #13161108

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#11723979C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic, complemented
4C2×C32×#213C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7#11722024C211yesmaximal, characteristic, complemented
8#10592855—31nomaximal
9#10591516—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic, complemented
12C2×#6312—21no—
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C32×D12C2211yescharacteristic
16C2×#6290—21no—
17#89026—22no—
18#89026—22no—
19#88864—21no—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
21C3×#3753—22no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
23C32×#213—22no—
24C2×C32×C3⋊C8C2211yescharacteristic, complemented
25C2×D4×C3⋊D3—31no—
26C2×#3802—34no—
27C2×C32×D8—31no—
28#64147—31no—
29#64147—34no—
30#64147—64no—
31C2×#3753D311yescharacteristic, complemented
32C2×C3×#213—34no—
33C2×C3×#213D314yescomplemented
34#63659—31no—
35#63648—31no—
36#63647—31no—
37#63563—34no—
38#63563—34no—
39#63557—34no—
40C23×C33—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—
47C33×D4—21no—
48C33×D4—21no—
49C2×C33×C4C2311yescharacteristic
50#6312—42no—
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3⋊(C3⋊Dic3)D411yescharacteristic
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#5156—94no—
63#5156—91no—
64C23×C3⋊D3—61no—
65C22×D32—64no—
66C2×C32×D4—24no—
67C2×C32×D4—31no—
68C2×C32×D4C2×D314yescomplemented
69C2×C32×D4C2×D311yescharacteristic, complemented
70C2×#638—61no—
71C2×#638—61no—
72C2×#638—64no—
73C2×#638—68no—
74D4×C3⋊D3—62no—
75D4×C3⋊D3—62no—
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—
85C32×D8—62no—
86C2×C32×C8—31no—
87#3757—61no—
88#3757—64no—
89#3757—68no—
90#3753—22no—
91C4⋊(C3⋊Dic3)—31no—
92C4⋊(C3⋊Dic3)—34no—
93C4⋊(C3⋊Dic3)—64no—
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
95C3×#213—28no—
96C3×#213—68no—
97C2×C3×C3⋊C8—34no—
98C2×C3×C3⋊C8C2×D314yescomplemented
99C22×C33—41no—
100C22×C33—41no—
101C22×C33—41no—
102C22×C33C2×D411yescharacteristic
103C2×C3⋊(C3⋊D3)—21no—
104C2×C3⋊(C3⋊D3)—21no—
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×C4C2×D411yescharacteristic
114C33×C4C2×D411yescharacteristic
115C3⋊(C3⋊Dic3)—21no—
116C3⋊(C3⋊Dic3)—41no—
117C2×D3×D4—94no—
118C2×D3×D4—91no—
119C2×C3×D8—34no—
120C2×C3×D8—91no—
121#938—94no—
122#938—91no—
123#938—184no—
124C2×#213—34no—
125C2×#213C3⋊D311yescharacteristic, complemented
126#832—94no—
127#832—91no—
128#821—94no—
129#821—91no—
130(C2×D3)⋊C8—94no—
131(C2×D3)⋊C8—91no—
132C23×C32—28no—
133C23×C32—24no—
134C23×C32—21no—
135C23×C32—61no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3—31no—
138C22×C3⋊D3—34no—
139C22×C3⋊D3—64no—
140C22×C3⋊D3—64no—
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—21no—
149C32×D4—21no—
150C32×D4—24no—
151C32×D4—24no—
152C32×D4—32no—
153C32×D4—44no—
154C32×D4—44no—
155C32×D4—61no—
156C2×C32×C4—24no—
157C2×C32×C4C22×D314yes—
158C2×C32×C4C22×D311yescharacteristic
159#638—64no—
160#638—122no—
161#638—128no—
162#638—1216no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—34no—
165C2×C3⋊Dic3—64no—
166C2×C3⋊Dic3—68no—
167C2×C3⋊Dic3—64no—
168C2×C3⋊Dic3—61no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)#6712yescomplemented
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×D12#6718yescomplemented
181#625—616no—
182C32×C8—61no—
183C3⋊(C3⋊C8)—21no—
184C3×C3⋊C8—24no—
185C3×C3⋊C8—64no—
186#447—271noSylow
187C2×C33—81no—
188C2×C33#12011yescharacteristic
189C2×C33C2×D811yescharacteristic
190C2×C33AGL(1,ℤ/8)11yescharacteristic
191C3⋊(C3⋊D3)—22no—
192C3×C3⋊D3—42no—
193C32×D3—42no—
194C23×D3—181no—
195C23×D3—184no—
196C2×C3×D4—24no—
197C2×C3×D4—34no—
198C2×C3×D4—91no—
199C2×C3×D4D3214yescomplemented
200C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
201C2×#67—181no—
202C2×#67—184no—
203C2×#67—184no—
204C2×#67—181no—
205C2×#67—188no—
206D3×D4—182no—
207D3×D4—188no—
208D3×D4—188no—
209D3×D4—182no—
210C2×D12—34no—
211C2×D12C2×C3⋊D311yescharacteristic
212C2×C4×D3—94no—
213C2×C4×D3—91no—
214C2×C4×D3—184no—
215C3×D8—68no—
216C3×D8—182no—
217C2×C3×C8—34no—
218C2×C3×C8—91no—
219#217—184no—
220#217—181no—
221#217—188no—
222#213—22no—
223#213—68no—
224C4⋊Dic3—94no—
225C4⋊Dic3—91no—
226C4⋊Dic3—184no—
227C2×C3⋊C8—34no—
228C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
229C22×C32—24no—
230C22×C32—48no—
231C22×C32—48no—
232C22×C32—48no—
233C22×C32—44no—
234C22×C32—41no—
235C22×C32—44no—
236C22×C32—41no—
237C22×C32—44no—
238C22×C32—41no—
239C22×C32—62no—
240C22×C32—121no—
241C22×C32—121no—
242C22×C32D3×D414yes—
243C22×C32D3×D411yescharacteristic
244C2×C3⋊D3—22no—
245C2×C3⋊D3—32no—
246C2×C3⋊D3—38no—
247C2×C3⋊D3—41no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—68no—
250C2×C3⋊D3—61no—
251C2×C3⋊D3—61no—
252C2×C3⋊D3—64no—
253C2×C3⋊D3—64no—
254C2×C3⋊D3—124no—
255C2×C3⋊D3—124no—
256C2×C3⋊D3—128no—
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×C4D3×D414yes—
267C32×C4D3×D411yescharacteristic
268C32×C4C2×#6714yes—
269C32×C4C2×#6711yescharacteristic
270C3⋊Dic3—61no—
271C3⋊Dic3—64no—
272C3⋊Dic3—124no—
273C3⋊Dic3—128no—
274C3⋊Dic3—124no—
275C3⋊Dic3—121no—
276C22×D4—271no—
277C2×D8—31no—
278C2×D8—91no—
279#121—271no—
280#102—271no—
281#102—271no—
282C22⋊C8—271no—
283C33#44711yescharacteristic, complemented, Sylow
284C23×C3—28no—
285C23×C3—24no—
286C23×C3—21no—
287C23×C3—64no—
288C23×C3—181no—
289C22×D3—21no—
290C22×D3—64no—
291C22×D3—94no—
292C22×D3—91no—
293C22×D3—184no—
294C22×D3—184no—
295C22×D3—1816no—
296C22×D3—362no—
297C22×D3—362no—
298C22×D3—368no—
299C22×D3—368no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—21no—
303C3×D4—21no—
304C3×D4—38no—
305C3×D4—44no—
306C3×D4—44no—
307C3×D4—64no—
308C3×D4—92no—
309C3×D4—181no—
310C2×C3×C4—24no—
311C2×C3×C4C2×D3214yes—
312C2×C3×C4C22×C3⋊D311yescharacteristic
313#67—184no—
314#67—1816no—
315#67—362no—
316#67—368no—
317#67—3616no—
318C2×Dic3—94no—
319C2×Dic3—91no—
320C2×Dic3—184no—
321C2×Dic3—188no—
322C2×Dic3—184no—
323C2×Dic3—181no—
324D12—21no—
325D12—38no—
326D12—64no—
327D12#63812yescomplemented
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—88no—
342C2×C32—84no—
343C2×C32—81no—
344C2×C32—122no—
345C2×C32D3×D814yes—
346C2×C32D3×D811yescharacteristic
347C2×C32#91114yes—
348C2×C32#91111yescharacteristic
349C2×C32#93714yes—
350C2×C32#93711yescharacteristic
351C3⋊D3—42no—
352C3⋊D3—62no—
353C3⋊D3—68no—
354C3⋊D3—128no—
355C3×D3—48no—
356C3×D3—128no—
357C24—541no—
358C2×D4—31no—
359C2×D4—91no—
360C2×D4—541no—
361C2×D4—542no—
362C2×D4—542no—
363C2×D4—541no—
364C2×D4D3×C3⋊D311yescharacteristic, complemented
365C22×C4—271no—
366D8—62no—
367D8—182no—
368C2×C8—31no—
369C2×C8—91no—
370C4⋊C4—271no—
371C22⋊C4—541no—
372C22×C3—24no—
373C22×C3—48no—
374C22×C3—48no—
375C22×C3—48no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—44no—
382C22×C3—68no—
383C22×C3—124no—
384C22×C3—124no—
385C22×C3—182no—
386C22×C3—361no—
387C22×C3—361no—
388C22×C3#381114yes—
389C22×C3D4×C3⋊D311yescharacteristic
390C2×D3—22no—
391C2×D3—41no—
392C2×D3—41no—
393C2×D3—68no—
394C2×D3—92no—
395C2×D3—98no—
396C2×D3—124no—
397C2×D3—124no—
398C2×D3—188no—
399C2×D3—181no—
400C2×D3—181no—
401C2×D3—184no—
402C2×D3—184no—
403C2×D3—364no—
404C2×D3—364no—
405C2×D3—368no—
406C2×D3—3632no—
407C3×C4—24no—
408C3×C4—24no—
409C3×C4D3×#6714yes—
410C3×C4D3×#6714yes—
411C3×C4D4×C3⋊D311yescharacteristic
412C3×C4C2×#63811yescharacteristic
413Dic3—181no—
414Dic3—184no—
415Dic3—364no—
416Dic3—368no—
417Dic3—361no—
418Dic3—364no—
419C32—24no—
420C32#515614yescomplemented
421C32#515611yescharacteristic, complemented
422C23—21no—
423C23—61no—
424C23—181no—
425C23—271no—
426C23—544no—
427C23—1082no—
428C23—1082no—
429D4—21no—
430D4—21no—
431D4—32no—
432D4—61no—
433D4—92no—
434D4—181no—
435D4—544no—
436D4—1082no—
437C2×C4—271no—
438C2×C4—541no—
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—88no—
448C2×C3—84no—
449C2×C3—81no—
450C2×C3—128no—
451C2×C3—362no—
452C2×C3#6392614yes—
453C2×C3#6392714yes—
454C2×C3#6398314yes—
455C2×C3D8×C3⋊D311yescharacteristic
456C2×C3#6412011yescharacteristic
457C2×C3#6414611yescharacteristic
458D3—42no—
459D3—128no—
460D3—188no—
461D3—182no—
462D3—368no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—62no—
467C22—121no—
468C22—121no—
469C22—182no—
470C22—272no—
471C22—361no—
472C22—361no—
473C22—541no—
474C22—541no—
475C22—1088no—
476C22#8903911yescharacteristic
477C4—541no—
478C4—1081no—
479C4D3×#63811yescharacteristic
480C4C3⋊D3×#6711yescharacteristic
481C3—24no—
482C3#1059151614yescomplemented
483C3#1059285511yescharacteristic, complemented
484C2—81no—
485C2—122no—
486C2—362no—
487C2—542no—
488C2#1172354711yescharacteristic
489C2#1172354811yescharacteristic
490C2#1172364311yescharacteristic
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×#89026C2semidirect
2. #11723979C2semidirect
3. C2×C3×#3753C2semidirect
4. C2×C32×#213C2semidirect
5. #11722034C2semidirect
6. #11722032C2semidirect
7. #11722024C2semidirect
11. C2×C33×D4C22semidirect
20. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
22. C2×C3×C3⋊(C3⋊C8)C22semidirect
24. C2×C32×C3⋊C8C22semidirect
31. C2×#3753D3semidirect
33. C2×C3×#213D3semidirect
56. C3×C4⋊(C3⋊D3)D4semidirect
58. C32×D12D4semidirect
68. C2×C32×D4C2×D3semidirect
69. C2×C32×D4C2×D3semidirect
94. C2×C3⋊(C3⋊C8)C2×D3semidirect
98. C2×C3×C3⋊C8C2×D3semidirect
105. C2×C3⋊(C3⋊D3)D8semidirect
125. C2×#213C3⋊D3semidirect
170. C4⋊(C3⋊D3)#67semidirect
180. C3×D12#67semidirect
199. C2×C3×D4D32semidirect
200. C2×C3×D4C2×C3⋊D3semidirect
228. C2×C3⋊C8C2×C3⋊D3semidirect
283. C33#447semidirect
327. D12#638semidirect
364. C2×D4D3×C3⋊D3semidirect
420. C32#5156semidirect
421. C32#5156semidirect
482. C3#10591516semidirect
483. C3#10592855semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
#117239792
C2×C3×#37532
C2×C32×#2132
#117220342
#117220322
#117220242
#105928553
#105915163