Finite Group Databasebeta

← #13161358

Subgroups — #13161358

Order 1728.

1672 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1672 conjugacy classes of subgroups — too many to draw. Only its 144 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#11724684C211yesmaximal, characteristic, complemented
2C32×#949C211yesmaximal, characteristic, complemented
3#11723524C211yesmaximal, characteristic, complemented
4#11723378C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6#11722047C211yesmaximal, characteristic, complemented
7#11722027C211yesmaximal, characteristic, complemented
8#10592897—31nomaximal
9#10591667—34nomaximal
10C33×#41C2211yescharacteristic
11#89080—21no—
12D4×C3⋊(C3⋊D3)—22no—
13D4×C3⋊(C3⋊D3)—21no—
14#89076—21no—
15#89076C2211yescharacteristic
16C2×C4⋊(C3⋊(C3⋊D3))—21no—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C32×D12C2211yescharacteristic, complemented
19C2×#6291—22no—
20C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
21C3×#3748C2211yescharacteristic, complemented
22C32×#215—21no—
23C32×#213—21no—
24C32×#208C2211yescharacteristic, complemented
25#88794—21no—
26#88790—21no—
27#64366—31no—
28#64366—34no—
29#64366—64no—
30C32×AGL(1,ℤ/8)—31no—
31#64149—31no—
32#64141—31no—
33C3×#949D314yescomplemented
34#63911—34no—
35#63830—34no—
36#63664D311yescharacteristic, complemented
37#63652—31no—
38#63650—31no—
39C3×#837—34no—
40#63570—34no—
41#63559—34no—
42C22×C3⋊(C3⋊D3)—22no—
43C22×C3⋊(C3⋊D3)D412yes—
44C22×C32×D3—21no—
45C33×Q8D411yescharacteristic, complemented
46C33×D4—21no—
47C33×D4D411yescharacteristic, complemented
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50#6312—22no—
51#6312—21no—
52C4⋊(C3⋊(C3⋊D3))—22no—
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))—21no—
55C4⋊(C3⋊(C3⋊D3))D411yescharacteristic, complemented
56C4×C3⋊(C3⋊D3)—22no—
57C4×C3⋊(C3⋊D3)—21no—
58#6308D411yescharacteristic, complemented
59C2×C3×C3⋊Dic3—21no—
60C32×D12—21no—
61C32×D12—21no—
62#6291—44no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65#5198—94no—
66#4822—91no—
67C32×#41—24no—
68C32×#41C2×D311yescharacteristic
69C32×#41C2×D314yes—
70C2×C32×D4—31no—
71C2×#638—62no—
72#3832—68no—
73#3832—61no—
74#3832—64no—
75D4×C3⋊D3—62no—
76D4×C3⋊D3—68no—
77D4×C3⋊D3—616no—
78D4×C3⋊D3—68no—
79D4×C3⋊D3—61no—
80D4×C3⋊D3—64no—
81#3828—31no—
82#3828—34no—
83#3828—61no—
84#3828—64no—
85#3828—64no—
86#3828—68no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4⋊(C3⋊D3)—34no—
89C2×C4⋊(C3⋊D3)—61no—
90C2×C4⋊(C3⋊D3)—64no—
91C2×C4⋊(C3⋊D3)—64no—
92C2×C4⋊(C3⋊D3)—68no—
93C2×C3×D12C2×D314yescomplemented
94C2×#626—68no—
95C32×SD16—61no—
96C32×D8—61no—
97C32×M16—31no—
98#3754—61no—
99#3753—61no—
100C4×C3⋊Dic3C2×D311yescharacteristic, complemented
101#3748C2×D311yescharacteristic, complemented
102C3×#215—24no—
103C3×#213—24no—
104C3×C4×Dic3—34no—
105C3×#208—34no—
106C3×#208C2×D314yescomplemented
107#3716—64no—
108#3714—64no—
109C22×C33—21no—
110C22×C33C2×D411yescharacteristic
111C2×C3⋊(C3⋊D3)—22no—
112C2×C3⋊(C3⋊D3)—21no—
113C2×C3⋊(C3⋊D3)—42no—
114C2×C3⋊(C3⋊D3)—82no—
115C2×C32×D3—42no—
116C2×C32×D3—41no—
117C33×C4—21no—
118C33×C4C2×D411yescharacteristic
119C33×C4C2×D411yescharacteristic
120C3⋊(C3⋊Dic3)—21no—
121C3×C3⋊Dic3—22no—
122#1009—94no—
123#1009—91no—
124#1009—184no—
125C3×AGL(1,ℤ/8)—34no—
126#949C3⋊D311yescharacteristic, complemented
127#940—94no—
128#932—94no—
129#908—91no—
130#874—91no—
131C3×#104—91no—
132#837—34no—
133#825—94no—
134#823—94no—
135#823—91no—
136#805—91no—
137C23×C32—61no—
138C22×C3⋊D3—32no—
139C22×C3⋊D3—38no—
140C22×C3⋊D3—68no—
141C22×C3⋊D3—616no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—68no—
144C22×C3×D3—24no—
145C32×Q8—24no—
146C32×Q8D1211yescharacteristic, complemented
147C32×Q8#6714yescomplemented
148C32×D4—28no—
149C32×D4—24no—
150C32×D4—21no—
151C32×D4—24no—
152C32×D4—61no—
153C32×D4—61no—
154C32×D4D1211yescharacteristic, complemented
155C32×D4#6714yescomplemented
156C2×C32×C4—28no—
157C2×C32×C4—24no—
158C2×C32×C4—21no—
159C2×C32×C4—24no—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D311yescharacteristic
162#638—616no—
163#638—62no—
164#638—68no—
165#638—68no—
166#638—64no—
167#638—61no—
168#638—124no—
169C2×C3⋊Dic3—21no—
170C4⋊(C3⋊D3)—31no—
171C4⋊(C3⋊D3)—34no—
172C4⋊(C3⋊D3)—64no—
173C4⋊(C3⋊D3)—68no—
174C4⋊(C3⋊D3)—616no—
175C4⋊(C3⋊D3)—62no—
176C4⋊(C3⋊D3)—68no—
177C4⋊(C3⋊D3)—68no—
178C4⋊(C3⋊D3)—61no—
179C4⋊(C3⋊D3)—64no—
180C4⋊(C3⋊D3)—61no—
181C4⋊(C3⋊D3)—64no—
182C4×C3⋊D3—616no—
183C4×C3⋊D3—62no—
184C4×C3⋊D3—68no—
185C4×C3⋊D3—68no—
186C4×C3⋊D3—61no—
187C4×C3⋊D3—64no—
188#634—31no—
189#634—34no—
190#634—64no—
191C2×C3×Dic3—64no—
192C3×D12—24no—
193C3×D12—24no—
194#626—1216no—
195C32×C8—61no—
196C3⋊(C3⋊C8)—21no—
197C3×C3⋊C8—24no—
198C3×C3⋊C8—64no—
199#453—271noSylow
200C2×C33—21no—
201C2×C33—41no—
202C2×C33#12011yescharacteristic
203C3⋊(C3⋊D3)—42no—
204C3⋊(C3⋊D3)—41no—
205C32×D3—81no—
206C3×#41—24no—
207C3×#41D3214yes—
208C3×#41C2×C3⋊D311yescharacteristic
209C2×C3×D4—34no—
210C2×#67—188no—
211#239—188no—
212#239—184no—
213#239—181no—
214D3×D4—188no—
215D3×D4—182no—
216D3×D4—1816no—
217D3×D4—188no—
218D3×D4—184no—
219D3×D4—181no—
220#235—94no—
221#235—91no—
222#235—184no—
223#235—184no—
224#235—181no—
225#235—188no—
226C2×D12—94no—
227C2×D12—91no—
228C2×D12—184no—
229C2×D12—184no—
230C2×D12—181no—
231C2×D12—188no—
232C2×D12—182no—
233C2×D12C2×C3⋊D311yescharacteristic, complemented
234C3×SD16—64no—
235C3×D8—64no—
236C3×M16—34no—
237C3×M16—91no—
238C3×C42—91no—
239#215—21no—
240#214—184no—
241#213—21no—
242#213—184no—
243C4×Dic3—34no—
244#208—34no—
245#208C2×C3⋊D311yescharacteristic, complemented
246D24—181no—
247#204—181no—
248C22×C32—28no—
249C22×C32—21no—
250C22×C32—24no—
251C22×C32—24no—
252C22×C32—122no—
253C22×C32—121no—
254C22×C32D3×D414yes—
255C22×C32D3×D411yescharacteristic
256C2×C3⋊D3—68no—
257C2×C3⋊D3—616no—
258C2×C3⋊D3—61no—
259C2×C3⋊D3—62no—
260C2×C3⋊D3—64no—
261C2×C3⋊D3—68no—
262C2×C3⋊D3—122no—
263C2×C3⋊D3—128no—
264C2×C3⋊D3—248no—
265C2×C3⋊D3—2416no—
266C2×C3⋊D3—248no—
267C2×C3⋊D3—242no—
268C2×C3×D3—48no—
269C2×C3×D3—44no—
270C32×C4—28no—
271C32×C4—21no—
272C32×C4—24no—
273C32×C4—24no—
274C32×C4—24no—
275C32×C4C2×D1211yescharacteristic
276C32×C4D3×D414yes—
277C32×C4D3×D411yescharacteristic
278C32×C4C2×#6714yes—
279C3⋊Dic3—22no—
280C3⋊Dic3—68no—
281C3⋊Dic3—61no—
282C3⋊Dic3—64no—
283C3×Dic3—68no—
284H(5,𝔽2)—271no—
285AGL(1,ℤ/8)—31no—
286AGL(1,ℤ/8)—271no—
287#127—271no—
288#104—91no—
289#104—271no—
290#100—271no—
291C33#45311yescharacteristic, complemented, Sylow
292C23×C3—64no—
293C22×D3—21no—
294C22×D3—98no—
295C22×D3—92no—
296C22×D3—188no—
297C22×D3—1816no—
298C22×D3—182no—
299C22×D3—188no—
300C3×Q8—24no—
301C3×Q8#62614yescomplemented
302C3×Q8#63811yescharacteristic, complemented
303C3×D4—28no—
304C3×D4—24no—
305C3×D4—21no—
306C3×D4—24no—
307C3×D4—64no—
308C3×D4—64no—
309C3×D4#62614yescomplemented
310C3×D4#63811yescharacteristic, complemented
311C2×C3×C4—28no—
312C2×C3×C4—24no—
313C2×C3×C4—21no—
314C2×C3×C4—24no—
315C2×C3×C4—181no—
316C2×C3×C4C2×D3214yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318#67—1816no—
319#67—182no—
320#67—188no—
321#67—188no—
322#67—184no—
323#67—181no—
324#67—3616no—
325C2×Dic3—64no—
326D12—21no—
327D12—21no—
328D12—94no—
329D12—91no—
330D12—184no—
331D12—188no—
332D12—1816no—
333D12—182no—
334D12—188no—
335D12—188no—
336D12—184no—
337D12—184no—
338D12—181no—
339D12—181no—
340D12—364no—
341C4×D3—1816no—
342C4×D3—182no—
343C4×D3—188no—
344C4×D3—188no—
345C4×D3—184no—
346C4×D3—181no—
347Dic6—94no—
348Dic6—91no—
349Dic6—184no—
350C3×C8—64no—
351C3×C8—181no—
352C3⋊C8—21no—
353C3⋊C8—64no—
354C2×C32—21no—
355C2×C32—24no—
356C2×C32—24no—
357C2×C32—44no—
358C2×C32—48no—
359C2×C32—44no—
360C2×C32—41no—
361C2×C32—241no—
362C2×C32#88211yescharacteristic
363C2×C32#93714yes—
364C3⋊D3—1216no—
365C3⋊D3—128no—
366C3⋊D3—128no—
367C3⋊D3—124no—
368C3⋊D3—122no—
369C3⋊D3—121no—
370C3×D3—84no—
371#41—271no—
372#41—541no—
373#41—541no—
374#41D3×C3⋊D311yescharacteristic
375C2×D4—31no—
376C2×D4—271no—
377C2×D4—541no—
378C2×D4—542no—
379C2×D4—542no—
380C2×D4—541no—
381SD16—61no—
382SD16—541no—
383D8—61no—
384D8—541no—
385M16—31no—
386M16—91no—
387C42—91no—
388C22×C3—28no—
389C22×C3—21no—
390C22×C3—24no—
391C22×C3—24no—
392C22×C3—128no—
393C22×C3—124no—
394C22×C3#381114yes—
395C22×C3D4×C3⋊D311yescharacteristic
396C2×D3—42no—
397C2×D3—41no—
398C2×D3—188no—
399C2×D3—1816no—
400C2×D3—181no—
401C2×D3—182no—
402C2×D3—184no—
403C2×D3—188no—
404C2×D3—362no—
405C2×D3—368no—
406C2×D3—728no—
407C2×D3—7216no—
408C2×D3—722no—
409C2×D3—728no—
410C3×C4—28no—
411C3×C4—21no—
412C3×C4—24no—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—182no—
416C3×C4C2×#62614yes—
417C3×C4#381114yes—
418C3×C4D4×C3⋊D311yescharacteristic
419C3×C4C2×#63811yescharacteristic
420Dic3—68no—
421Dic3—188no—
422Dic3—181no—
423Dic3—184no—
424C32—24no—
425C32#482211yescharacteristic, complemented
426C32#519814yescomplemented
427C23—61no—
428C23—272no—
429C23—542no—
430Q8—271no—
431Q8#629111yescharacteristic, complemented
432D4—21no—
433D4—61no—
434D4—61no—
435D4—271no—
436D4—542no—
437D4—542no—
438D4—541no—
439D4—541no—
440D4—541no—
441D4—1084no—
442D4#629111yescharacteristic, complemented
443C2×C4—21no—
444C2×C4—181no—
445C2×C4—542no—
446C2×C4—541no—
447C2×C4C2×D3×C3⋊D311yescharacteristic
448C8—61no—
449C8—181no—
450C2×C3—21no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—44no—
454C2×C3—48no—
455C2×C3—41no—
456C2×C3—44no—
457C2×C3—244no—
458C2×C3#6398114yes—
459C2×C3#6414611yescharacteristic
460D3—81no—
461D3—3616no—
462D3—368no—
463D3—362no—
464D3—361no—
465D3—368no—
466D3—364no—
467C22—21no—
468C22—122no—
469C22—121no—
470C22—541no—
471C22—542no—
472C22—1082no—
473C22—2162no—
474C22#8903911yescharacteristic
475C4—21no—
476C4—182no—
477C4—541no—
478C4C2×#629111yescharacteristic
479C4#8903911yescharacteristic
480C3—24no—
481C3#1059166714yescomplemented
482C3#1059289711yescharacteristic, complemented
483C2—21no—
484C2—41no—
485C2—241no—
486C2—1082no—
487C2—1081no—
488C2#1172363811yescharacteristic
subgroupG/H
centerC2#11723638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33#120
hypercenter#41D3×C3⋊D3
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11724684C2semidirect
2. C32×#949C2semidirect
3. #11723524C2semidirect
4. #11723378C2semidirect
5. C3×#63664C2semidirect
6. #11722047C2semidirect
7. #11722027C2semidirect
18. C2×C32×D12C22semidirect
20. C3×C4×C3⋊Dic3C22semidirect
21. C3×#3748C22semidirect
24. C32×#208C22semidirect
33. C3×#949D3semidirect
36. #63664D3semidirect
45. C33×Q8D4semidirect
47. C33×D4D4semidirect
55. C4⋊(C3⋊(C3⋊D3))D4semidirect
58. #6308D4semidirect
93. C2×C3×D12C2×D3semidirect
100. C4×C3⋊Dic3C2×D3semidirect
101. #3748C2×D3semidirect
106. C3×#208C2×D3semidirect
126. #949C3⋊D3semidirect
146. C32×Q8D12semidirect
147. C32×Q8#67semidirect
154. C32×D4D12semidirect
155. C32×D4#67semidirect
233. C2×D12C2×C3⋊D3semidirect
245. #208C2×C3⋊D3semidirect
291. C33#453semidirect
301. C3×Q8#626semidirect
302. C3×Q8#638semidirect
309. C3×D4#626semidirect
310. C3×D4#638semidirect
425. C32#4822semidirect
426. C32#5198semidirect
431. Q8#6291semidirect
442. D4#6291semidirect
481. C3#10591667semidirect
482. C3#10592897semidirect

Maximal subgroups

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

subgroupindex
#117246842
C32×#9492
#117235242
#117233782
C3×#636642
#117220472
#117220272
#105928973
#105916673