Finite Group Databasebeta

← C2×#11723544 #13177550

Subgroups — C2×#11723544 #13177550

Order 1728.

2092 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#237C211yesmaximal, characteristic, complemented
2C2×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×#88860C211yesmaximal, characteristic, complemented
4C2×C3×#3753C211yesmaximal, characteristic, complemented
5#11723544C218yesmaximal, direct product factor
6C2×#88795C211yesmaximal, characteristic, complemented
7C2×#88793C211yesmaximal, characteristic, complemented
8C2×#88786C211yesmaximal, characteristic, complemented
9C2×#64158—31nomaximal
10C2×#63925—34nomaximal
11C22×D3×C3⋊D3—21no—
12C2×C33×D4C2211yescharacteristic, complemented
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C32×#67—21no—
16C22×C32×Dic3—21no—
17C32×#237—22no—
18C32×#237C2214yescomplemented
19C2×C32×C4×D3C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic, complemented
21C2×#6292—21no—
22D3×C4⋊(C3⋊D3)—22no—
23D3×C4⋊(C3⋊D3)C2214yescomplemented
24#88860C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
26C3×#3753C2214yescomplemented
27C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
28#88795C2214yescomplemented
29#88793C2214yescomplemented
30#88786C2214yescomplemented
31C2×C32×#41—31no—
32C22×C4⋊(C3⋊D3)—31no—
33C2×C3×#237D314yescomplemented
34C2×D3×D12—34no—
35#64158—38no—
36C2×#3755—31no—
37C2×#3755—31no—
38C2×#3753—31no—
39C2×#3753—34no—
40C2×#3753—64no—
41C2×#3753D311yescharacteristic, complemented
42C2×#3748—31no—
43C2×C3×#213—34no—
44#63925—332no—
45C2×#3717—34no—
46C2×#3715—34no—
47C2×#3711—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3×C3⋊D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×D3×C3⋊D3—24no—
53C2×D3×C3⋊D3—44no—
54C33×D4—21no—
55C33×D4C2312yescomplemented
56C2×C33×C4C2311yescharacteristic
57C4⋊(C3⋊(C3⋊D3))—21no—
58C4⋊(C3⋊(C3⋊D3))C2312yes—
59C3×C4⋊(C3⋊D3)—21no—
60C3×C4⋊(C3⋊D3)C2312yes—
61C32×#67—24no—
62C2×C32×Dic3—21no—
63C2×C32×Dic3—24no—
64C2×C32×Dic3D411yescharacteristic
65C32×C4×D3D412yes—
66C32×C4×D3C2312yes—
67C32×Dic6—21no—
68C32×Dic6C2312yescomplemented
69#6292—24no—
70C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
71C3×C3⋊(C3⋊C8)C2312yescomplemented
72C2×#949—94no—
73C2×#911—91no—
74C23×C3⋊D3—61no—
75C22×D32—64no—
76C32×#41—34no—
77C32×#41—62no—
78C2×C32×Q8—31no—
79C2×C32×D4—24no—
80C2×C32×D4—61no—
81C2×C32×D4C2×D314yescomplemented
82C2×C32×D4C2×D311yescharacteristic, complemented
83C22×C32×C4—31no—
84C22×C32×C4—61no—
85C2×C4⋊(C3⋊D3)—31no—
86C2×C4⋊(C3⋊D3)—34no—
87C2×C4⋊(C3⋊D3)—34no—
88C2×C4⋊(C3⋊D3)—62no—
89C2×C4⋊(C3⋊D3)—61no—
90C2×C4⋊(C3⋊D3)—64no—
91C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
92C2×C3×#67—24no—
93C22×C3×Dic3—24no—
94C3×#237—28no—
95C3×#237C2×D3116yescomplemented
96C2×C3×D12—34no—
97C2×C3×C4×D3C2×D314yes—
98C2×C3×Dic6C2×D314yescomplemented
99C2×#626—64no—
100D3×D12—316no—
101D3×D12—68no—
102#3755—34no—
103#3755—34no—
104#3753—34no—
105#3753—316no—
106#3753—616no—
107#3753C2×D314yescomplemented
108#3748—34no—
109C2×C3⋊(C3⋊C8)—31no—
110C2×C3⋊(C3⋊C8)—34no—
111C2×C3⋊(C3⋊C8)—64no—
112C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
113C3×#213—316no—
114C2×C3×C3⋊C8—34no—
115#3717—316no—
116#3715—316no—
117#3711—316no—
118C22×C33—22no—
119C22×C33—42no—
120C22×C33C2×D411yescharacteristic
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3⋊(C3⋊D3)—42no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—42no—
125C2×C32×D3—22no—
126C2×C32×D3C2×D412yes—
127D3×C3⋊D3—48no—
128C33×C4C2×D411yescharacteristic
129C33×C4C2411yescharacteristic
130C32×Dic3—22no—
131C32×Dic3C2×D412yes—
132C2×C3×#41—34no—
133C2×#237C3⋊D311yescharacteristic, complemented
134C2×D3×D4—91no—
135C22×D12—94no—
136C2×C3×D8—91no—
137#949—932no—
138C2×#215—94no—
139C2×#215—94no—
140C2×#214—91no—
141C2×#213—34no—
142C2×#213—94no—
143C2×#213—91no—
144C2×#213—184no—
145C2×#208—94no—
146#911—98no—
147C2×#204—91no—
148C2×ΓL(1,𝔽25)—91no—
149C23×C32—21no—
150C23×C32—24no—
151C23×C32—31no—
152C23×C32—44no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3—64no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—64no—
157C22×C3⋊D3—124no—
158C22×C3⋊D3—124no—
159C22×C3×D3—64no—
160C22×C3×D3#6714yes—
161C2×D32—616no—
162C2×D32—1216no—
163C32×Q8—32no—
164C32×Q8—61no—
165C32×D4—21no—
166C32×D4—28no—
167C32×D4—24no—
168C32×D4—44no—
169C32×D4—64no—
170C32×D4C22×D318yescomplemented
171C32×D4C22×D312yescomplemented
172C2×C32×C4—24no—
173C2×C32×C4—31no—
174C2×C32×C4—32no—
175C2×C32×C4—32no—
176C2×C32×C4—64no—
177C2×C32×C4—61no—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C22×D311yescharacteristic
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—32no—
182C4⋊(C3⋊D3)—38no—
183C4⋊(C3⋊D3)—68no—
184C4⋊(C3⋊D3)—64no—
185C4⋊(C3⋊D3)—61no—
186C4⋊(C3⋊D3)—64no—
187C4⋊(C3⋊D3)—124no—
188C4⋊(C3⋊D3)C22×D312yes—
189C3×#67—216no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—216no—
192C2×C3×Dic3#6714yes—
193C3×D12—38no—
194C3×D12—64no—
195C3×C4×D3#6718yes—
196C3×C4×D3C22×D318yes—
197C3×Dic6—24no—
198C3×Dic6C22×D318yescomplemented
199#626—616no—
200C3⋊(C3⋊C8)—32no—
201C3⋊(C3⋊C8)—38no—
202C3⋊(C3⋊C8)—68no—
203C3⋊(C3⋊C8)C22×D312yescomplemented
204C3×C3⋊C8—38no—
205C2×AGL(1,ℤ/8)—271noSylow
206C2×C33—42no—
207C2×C33AGL(1,ℤ/8)12yescomplemented
208C2×C33C22×D411yescharacteristic
209C3⋊(C3⋊D3)—42no—
210C3×C3⋊D3—42no—
211C32×D3—22no—
212C23×D3—184no—
213C23×D3—181no—
214C3×#41—316no—
215C3×#41—68no—
216C2×C3×Q8—34no—
217C2×C3×D4—24no—
218C2×C3×D4—64no—
219C2×C3×D4—91no—
220C2×C3×D4D3214yescomplemented
221C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
222C22×C3×C4—34no—
223C22×C3×C4—64no—
224C2×#67—21no—
225C2×#67—181no—
226C22×Dic3—21no—
227#237—22no—
228#237C2×C3⋊D314yescomplemented
229D3×D4—94no—
230D3×D4—182no—
231C2×D12—34no—
232C2×D12—916no—
233C2×D12—94no—
234C2×D12—91no—
235C2×D12—184no—
236C2×D12—188no—
237C2×D12—184no—
238C2×C4×D3C2×C3⋊D311yescharacteristic
239C2×Dic6C2×C3⋊D311yescharacteristic, complemented
240C3×D8—94no—
241C2×C3×C8—91no—
242#215—916no—
243#215—916no—
244#214—94no—
245#213—316no—
246#213—916no—
247#213—94no—
248#213—1816no—
249#208—916no—
250C2×C3⋊C8—34no—
251C2×C3⋊C8—94no—
252C2×C3⋊C8—91no—
253C2×C3⋊C8—184no—
254#204—94no—
255ΓL(1,𝔽25)—94no—
256C22×C32—22no—
257C22×C32—24no—
258C22×C32—28no—
259C22×C32—32no—
260C22×C32—48no—
261C22×C32—42no—
262C22×C32—48no—
263C22×C32—62no—
264C22×C32—88no—
265C22×C32D3×D411yescharacteristic
266C22×C32C2×#6714yes—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—42no—
269C2×C3⋊D3—62no—
270C2×C3⋊D3—68no—
271C2×C3⋊D3—128no—
272C2×C3⋊D3—128no—
273C2×C3⋊D3—122no—
274C2×C3⋊D3—128no—
275C2×C3⋊D3—248no—
276C2×C3×D3—28no—
277C2×C3×D3—68no—
278C2×C3×D3—128no—
279C2×C3×D3C2×#6718yes—
280D32—1232no—
281C32×C4—24no—
282C32×C4—24no—
283C32×C4—32no—
284C32×C4—62no—
285C32×C4D3×D411yescharacteristic
286C32×C4C2×#6714yes—
287C32×C4C23×D314yes—
288C32×C4C23×D311yescharacteristic
289C3×Dic3—28no—
290C3×Dic3C2×#6718yes—
291C2×#41—31no—
292C22×D4—271no—
293AGL(1,ℤ/8)—278no—
294C2×SD16—271no—
295C2×SD16—271no—
296C2×D8—91no—
297C2×D8—271no—
298C2×M16—271no—
299C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
300C23×C3—24no—
301C23×C3—21no—
302C23×C3—34no—
303C23×C3—44no—
304C23×C3—181no—
305C22×D3—64no—
306C22×D3—1816no—
307C22×D3—184no—
308C22×D3—181no—
309C22×D3—184no—
310C22×D3—364no—
311C22×D3—364no—
312C22×D3—3616no—
313C22×D3#63811yescharacteristic
314C3×Q8—38no—
315C3×Q8—64no—
316C3×D4—24no—
317C3×D4—21no—
318C3×D4—28no—
319C3×D4—44no—
320C3×D4—616no—
321C3×D4—92no—
322C3×D4—181no—
323C3×D4C2×D3218yescomplemented
324C3×D4C22×C3⋊D312yescomplemented
325C2×C3×C4—24no—
326C2×C3×C4—34no—
327C2×C3×C4—38no—
328C2×C3×C4—38no—
329C2×C3×C4—616no—
330C2×C3×C4—64no—
331C2×C3×C4C2×D3214yes—
332C2×C3×C4C22×C3⋊D311yescharacteristic
333#67—24no—
334#67—184no—
335C2×Dic3—21no—
336C2×Dic3—24no—
337C2×Dic3#63811yescharacteristic
338D12—38no—
339D12—64no—
340D12—98no—
341D12—92no—
342D12—188no—
343D12—1816no—
344D12—184no—
345D12—181no—
346D12—364no—
347C4×D3#63812yes—
348C4×D3C22×C3⋊D312yes—
349Dic6—21no—
350Dic6C22×C3⋊D312yescomplemented
351C3×C8—92no—
352C3⋊C8—38no—
353C3⋊C8—98no—
354C3⋊C8—92no—
355C3⋊C8—188no—
356C2×C32—24no—
357C2×C32—28no—
358C2×C32—42no—
359C2×C32—48no—
360C2×C32—62no—
361C2×C32—88no—
362C2×C32#91112yescomplemented
363C2×C32#94918yescomplemented
364C2×C32C2×D3×D411yescharacteristic
365C2×C32C22×#6714yes—
366C3⋊D3—42no—
367C3⋊D3—122no—
368C3⋊D3—128no—
369C3⋊D3—248no—
370C3×D3—28no—
371C3×D3—128no—
372C24—541no—
373#41—34no—
374#41—62no—
375C2×Q8—31no—
376C2×D4—61no—
377C2×D4—91no—
378C2×D4—274no—
379C2×D4—271no—
380C2×D4—542no—
381C2×D4—541no—
382C2×D4D3×C3⋊D311yescharacteristic, complemented
383C22×C4—31no—
384C22×C4—61no—
385SD16—274no—
386SD16—274no—
387D8—94no—
388D8—274no—
389M16—274no—
390C2×C8—91no—
391C2×C8—271no—
392C22×C3—28no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—38no—
396C22×C3—48no—
397C22×C3—42no—
398C22×C3—48no—
399C22×C3—68no—
400C22×C3—88no—
401C22×C3—182no—
402C22×C3—362no—
403C22×C3D3×#6714yes—
404C22×C3C2×#63811yescharacteristic
405C2×D3—22no—
406C2×D3—68no—
407C2×D3—128no—
408C2×D3—188no—
409C2×D3—182no—
410C2×D3—368no—
411C2×D3—368no—
412C2×D3—3632no—
413C2×D3—362no—
414C2×D3—368no—
415C2×D3—728no—
416C2×D3C2×#63812yes—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—38no—
420C3×C4—68no—
421C3×C4D3×#6714yes—
422C3×C4C2×#63811yescharacteristic
423C3×C4C22×D3214yes—
424C3×C4C23×C3⋊D311yescharacteristic
425Dic3—22no—
426Dic3C2×#63812yes—
427C32—24no—
428C32C2×#91111yescharacteristic, complemented
429C32C2×#94914yescomplemented
430C23—21no—
431C23—31no—
432C23—181no—
433C23—544no—
434C23—541no—
435C23—1084no—
436Q8—32no—
437Q8—61no—
438D4—21no—
439D4—64no—
440D4—92no—
441D4—181no—
442D4—272no—
443D4—544no—
444D4—541no—
445D4C2×D3×C3⋊D312yescomplemented
446C2×C4—31no—
447C2×C4—32no—
448C2×C4—32no—
449C2×C4—64no—
450C2×C4—61no—
451C2×C4C2×D3×C3⋊D311yescharacteristic
452C8—92no—
453C8—272no—
454C2×C3—28no—
455C2×C3—24no—
456C2×C3—48no—
457C2×C3—42no—
458C2×C3—68no—
459C2×C3—88no—
460C2×C3—362no—
461C2×C3#6392518yescomplemented
462C2×C3#6415812yescomplemented
463C2×C3C2×D3×#6714yes—
464C2×C3C22×#63811yescharacteristic
465D3—22no—
466D3—128no—
467D3—368no—
468D3—362no—
469D3—728no—
470C22—22no—
471C22—32no—
472C22—42no—
473C22—62no—
474C22—182no—
475C22—362no—
476C22—542no—
477C22—1088no—
478C22—1082no—
479C22D3×#63811yescharacteristic
480C4—32no—
481C4—62no—
482C4D3×#63811yescharacteristic
483C4C22×D3×C3⋊D311yescharacteristic
484C3—24no—
485C3C2×#6392514yescomplemented
486C3C2×#6415811yescharacteristic, complemented
487C2—42no—
488C2—62no—
489C2—362no—
490C2—1082no—
491C2#1172354412yesdirect product factor
492C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ 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×C32×#237C2semidirect
2. C2×D3×C4⋊(C3⋊D3)C2semidirect
3. C2×#88860C2semidirect
4. C2×C3×#3753C2semidirect
5. #11723544C2direct (N × H)
6. C2×#88795C2semidirect
7. C2×#88793C2semidirect
8. C2×#88786C2semidirect
12. C2×C33×D4C22semidirect
18. C32×#237C22semidirect
20. C2×C32×Dic6C22semidirect
23. D3×C4⋊(C3⋊D3)C22semidirect
24. #88860C22semidirect
25. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C3×#3753C22semidirect
27. C2×C3×C3⋊(C3⋊C8)C22semidirect
28. #88795C22semidirect
29. #88793C22semidirect
30. #88786C22semidirect
33. C2×C3×#237D3semidirect
41. C2×#3753D3semidirect
55. C33×D4C23semidirect
68. C32×Dic6C23semidirect
70. C3⋊(C3⋊(C3⋊C8))C23semidirect
71. C3×C3⋊(C3⋊C8)C23semidirect
81. C2×C32×D4C2×D3semidirect
82. C2×C32×D4C2×D3semidirect
95. C3×#237C2×D3semidirect
98. C2×C3×Dic6C2×D3semidirect
107. #3753C2×D3semidirect
112. C2×C3⋊(C3⋊C8)C2×D3semidirect
133. C2×#237C3⋊D3semidirect
170. C32×D4C22×D3semidirect
171. C32×D4C22×D3semidirect
198. C3×Dic6C22×D3semidirect
203. C3⋊(C3⋊C8)C22×D3semidirect
207. C2×C33AGL(1,ℤ/8)semidirect
220. C2×C3×D4D32semidirect
221. C2×C3×D4C2×C3⋊D3semidirect
228. #237C2×C3⋊D3semidirect
239. C2×Dic6C2×C3⋊D3semidirect
299. C33C2×AGL(1,ℤ/8)semidirect
323. C3×D4C2×D32semidirect
324. C3×D4C22×C3⋊D3semidirect
350. Dic6C22×C3⋊D3semidirect
362. C2×C32#911semidirect
363. C2×C32#949semidirect
382. C2×D4D3×C3⋊D3semidirect
428. C32C2×#911semidirect
429. C32C2×#949semidirect
445. D4C2×D3×C3⋊D3semidirect
461. C2×C3#63925semidirect
462. C2×C3#64158semidirect
485. C3C2×#63925semidirect
486. C3C2×#64158semidirect
491. C2#11723544direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C2×D3×C4⋊(C3⋊D3)2
C2×#888602
C2×C3×#37532
#117235442
C2×#887952
C2×#887932
C2×#887862
C2×#641583
C2×#639253