Finite Group Databasebeta

← C2×#11723316 #13176634

Subgroups — C2×#11723316 #13176634

Order 1728.

2060 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3C2×#88851C211yesmaximal, characteristic, complemented
4C2×C3×#3745C211yesmaximal, characteristic, complemented
5C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
6C2×#88796C211yesmaximal, characteristic, complemented
7C2×#88791C211yesmaximal, characteristic, complemented
8#11723316C218yesmaximal, direct product factor
9C2×#64117—31nomaximal
10C2×#63795—34nomaximal
11C22×D3×C3⋊D3—21no—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×#6308C2211yescharacteristic, complemented
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C32×C4×D3C2211yescharacteristic
17C2×#6292—21no—
18C2×#6290—21no—
19C2×Dic3×C3⋊D3—21no—
20D3×C4⋊(C3⋊D3)—22no—
21D3×C4⋊(C3⋊D3)C2214yescomplemented
22#89021—22no—
23#89021C2214yescomplemented
24C2×C33×C8C2211yescharacteristic, complemented
25#88851C2214yescomplemented
26C3×#3745C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic, complemented
28C32×ΓL(1,𝔽25)C2214yescomplemented
29#88796C2214yescomplemented
30#88791C2214yescomplemented
31C2×#3828—31no—
32C22×C4⋊(C3⋊D3)—31no—
33C2×D3×D12—34no—
34C2×#3795—34no—
35C2×C32×M16—31no—
36#64117—38no—
37C2×#3745—31no—
38C2×#3745D311yescharacteristic, complemented
39C2×#3744—31no—
40C2×#3744—31no—
41C2×#3744—34no—
42C2×#3744—64no—
43C2×C3×D24—34no—
44C2×C3×ΓL(1,𝔽25)D314yescomplemented
45C2×#3716—34no—
46C2×#3714—34no—
47#63795—332no—
48C22×C3⋊(C3⋊D3)—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×D3×C3⋊D3—24no—
53C2×D3×C3⋊D3—44no—
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊(C3⋊D3))—21no—
57C4⋊(C3⋊(C3⋊D3))C2312yes—
58#6308—21no—
59#6308C2312yescomplemented
60C3×C4⋊(C3⋊D3)—21no—
61C3×C4⋊(C3⋊D3)—21no—
62C3×C4⋊(C3⋊D3)C2312yes—
63C3×C4⋊(C3⋊D3)C2312yescomplemented
64C2×C32×Dic3D411yescharacteristic
65C32×C4×D3D412yes—
66C32×C4×D3C2312yes—
67#6292—24no—
68#6290—24no—
69Dic3×C3⋊D3—24no—
70C33×C8C2312yescomplemented
71C32×C3⋊C8C2312yescomplemented
72C2×#911—91no—
73C2×#908—94no—
74C23×C3⋊D3—61no—
75C22×D32—64no—
76C22×C32×C4—31no—
77C2×#638—61no—
78#3828—34no—
79#3828—62no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—34no—
83C2×C4⋊(C3⋊D3)—62no—
84C2×C4⋊(C3⋊D3)—61no—
85C2×C4⋊(C3⋊D3)—64no—
86C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
87C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
88C2×C4×C3⋊D3—61no—
89C2×#634—31no—
90C2×#634—34no—
91C2×#634—64no—
92C2×C3×D12—34no—
93C2×C3×D12—34no—
94C2×C3×C4×D3C2×D314yes—
95C2×#626—64no—
96C2×#625—64no—
97C2×D3×Dic3—64no—
98D3×D12—316no—
99D3×D12—68no—
100#3795—316no—
101#3795—68no—
102C32×M16—34no—
103C2×C32×C8—24no—
104C2×C32×C8—31no—
105C2×C32×C8C2×D314yescomplemented
106C2×C32×C8C2×D311yescharacteristic, complemented
107#3745—34no—
108#3745C2×D314yescomplemented
109#3744—34no—
110#3744—34no—
111#3744—316no—
112#3744—616no—
113C2×C3×C3⋊C8C2×D314yescomplemented
114C3×D24—316no—
115C3×ΓL(1,𝔽25)C2×D3116yescomplemented
116#3716—316no—
117#3714—316no—
118C22×C33C2×D411yescharacteristic
119C2×C3⋊(C3⋊D3)—22no—
120C2×C3⋊(C3⋊D3)—42no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—42no—
124C2×C3×C3⋊D3—42no—
125C2×C32×D3—22no—
126C2×C32×D3C2×D412yes—
127D3×C3⋊D3—48no—
128C33×C4C2×D411yescharacteristic
129C33×C4C2411yescharacteristic
130C3⋊(C3⋊Dic3)—22no—
131C32×Dic3C2×D412yes—
132C2×#237—91no—
133C2×D3×D4—91no—
134C2×#235—94no—
135C22×D12—94no—
136C2×C3×D8—91no—
137C2×C3×M16—34no—
138C2×#214—91no—
139C2×#213—91no—
140#911—98no—
141#908—932no—
142C2×D24—34no—
143C2×D24—94no—
144C2×#204—94no—
145C2×#204—94no—
146C2×#204—91no—
147C2×#204—184no—
148C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
149C23×C32—31no—
150C22×C3⋊D3—21no—
151C22×C3⋊D3—21no—
152C22×C3⋊D3—64no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—64no—
155C22×C3⋊D3—124no—
156C22×C3⋊D3—124no—
157C22×C3×D3—64no—
158C22×C3×D3—64no—
159C22×C3×D3D1214yes—
160C2×D32—616no—
161C2×D32—1216no—
162C2×C32×C4—24no—
163C2×C32×C4—31no—
164C2×C32×C4—32no—
165C2×C32×C4—32no—
166C2×C32×C4C22×D314yes—
167C2×C32×C4C22×D311yescharacteristic
168#638—64no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—124no—
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)—32no—
175C4⋊(C3⋊D3)—38no—
176C4⋊(C3⋊D3)—68no—
177C4⋊(C3⋊D3)—64no—
178C4⋊(C3⋊D3)—61no—
179C4⋊(C3⋊D3)—64no—
180C4⋊(C3⋊D3)—124no—
181C4⋊(C3⋊D3)C22×D312yes—
182C4⋊(C3⋊D3)C22×D312yescomplemented
183C4×C3⋊D3—64no—
184#634—32no—
185#634—38no—
186#634—68no—
187#634—61no—
188#634—64no—
189#634—124no—
190C2×C3×Dic3D1214yes—
191C3×D12—38no—
192C3×D12—38no—
193C3×D12—64no—
194C3×D12—64no—
195C3×C4×D3D1218yes—
196C3×C4×D3C22×D318yes—
197#626—616no—
198#625—616no—
199D3×Dic3—616no—
200C32×C8—28no—
201C32×C8—32no—
202C32×C8C22×D318yescomplemented
203C32×C8C22×D312yescomplemented
204C3×C3⋊C8C22×D318yescomplemented
205C2×AGL(1,ℤ/8)—271noSylow
206C2×C33AGL(1,ℤ/8)12yescomplemented
207C2×C33C22×D411yescharacteristic
208C3⋊(C3⋊D3)—42no—
209C3×C3⋊D3—42no—
210C3×C3⋊D3—42no—
211C32×D3—22no—
212C23×D3—184no—
213C23×D3—181no—
214C2×C3×D4—91no—
215C2×C3×D4—91no—
216C22×C3×C4—34no—
217C2×#67—184no—
218C2×#67—181no—
219C2×#67—181no—
220C22×Dic3—181no—
221#237—94no—
222#237—182no—
223D3×D4—94no—
224D3×D4—182no—
225#235—916no—
226#235—188no—
227C2×D12—34no—
228C2×D12—34no—
229C2×D12—916no—
230C2×D12—94no—
231C2×D12—91no—
232C2×D12—184no—
233C2×D12—188no—
234C2×D12—184no—
235C2×C4×D3—184no—
236C2×C4×D3C2×C3⋊D311yescharacteristic
237C2×Dic6—94no—
238C2×Dic6—91no—
239C2×Dic6—184no—
240C3×D8—94no—
241C3×M16—316no—
242C2×C3×C8—24no—
243C2×C3×C8—34no—
244C2×C3×C8D3214yescomplemented
245C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
246#214—94no—
247#213—94no—
248C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
249D24—316no—
250D24—916no—
251#204—916no—
252#204—916no—
253#204—94no—
254#204—1816no—
255ΓL(1,𝔽25)C2×C3⋊D314yescomplemented
256C22×C32—24no—
257C22×C32—32no—
258C22×C32—62no—
259C22×C32C2×D1214yes—
260C22×C32D3×D411yescharacteristic
261C2×C3⋊D3—22no—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—42no—
264C2×C3⋊D3—42no—
265C2×C3⋊D3—62no—
266C2×C3⋊D3—68no—
267C2×C3⋊D3—128no—
268C2×C3⋊D3—128no—
269C2×C3⋊D3—122no—
270C2×C3⋊D3—128no—
271C2×C3⋊D3—248no—
272C2×C3×D3—28no—
273C2×C3×D3—68no—
274C2×C3×D3—68no—
275C2×C3×D3—128no—
276C2×C3×D3—128no—
277C2×C3×D3C2×D1218yes—
278D32—1232no—
279C32×C4—24no—
280C32×C4—24no—
281C32×C4—32no—
282C32×C4C2×D1214yes—
283C32×C4D3×D411yescharacteristic
284C32×C4C23×D314yes—
285C32×C4C23×D311yescharacteristic
286C3⋊Dic3—62no—
287C3⋊Dic3—68no—
288C3⋊Dic3—128no—
289C3×Dic3C2×D1218yes—
290C2×#41—271no—
291C22×D4—271no—
292AGL(1,ℤ/8)—278no—
293C2×SD16—271no—
294C2×SD16—271no—
295C2×D8—91no—
296C2×D8—271no—
297C2×M16—31no—
298C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
299C23×C3—34no—
300C23×C3—181no—
301C23×C3—181no—
302C22×D3—64no—
303C22×D3—64no—
304C22×D3—1816no—
305C22×D3—184no—
306C22×D3—181no—
307C22×D3—184no—
308C22×D3—364no—
309C22×D3—364no—
310C22×D3—3616no—
311C22×D3C4⋊(C3⋊D3)11yescharacteristic
312C3×D4—92no—
313C3×D4—92no—
314C3×D4—181no—
315C3×D4—181no—
316C2×C3×C4—24no—
317C2×C3×C4—34no—
318C2×C3×C4—38no—
319C2×C3×C4—38no—
320C2×C3×C4C2×D3214yes—
321C2×C3×C4C22×C3⋊D311yescharacteristic
322#67—1816no—
323#67—184no—
324#67—184no—
325C2×Dic3—184no—
326C2×Dic3—181no—
327C2×Dic3—184no—
328C2×Dic3—364no—
329C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
330D12—38no—
331D12—38no—
332D12—64no—
333D12—64no—
334D12—98no—
335D12—92no—
336D12—188no—
337D12—1816no—
338D12—184no—
339D12—181no—
340D12—364no—
341C4×D3—1816no—
342C4×D3C4⋊(C3⋊D3)12yes—
343C4×D3C22×C3⋊D312yes—
344Dic6—98no—
345Dic6—92no—
346Dic6—188no—
347Dic6—184no—
348Dic6—181no—
349Dic6—364no—
350C3×C8—28no—
351C3×C8—38no—
352C3×C8C2×D3218yescomplemented
353C3×C8C22×C3⋊D312yescomplemented
354C3⋊C8C22×C3⋊D312yescomplemented
355C2×C32—24no—
356C2×C32—28no—
357C2×C32—62no—
358C2×C32#90818yescomplemented
359C2×C32#91112yescomplemented
360C2×C32C22×D1214yes—
361C2×C32C2×D3×D411yescharacteristic
362C3⋊D3—42no—
363C3⋊D3—42no—
364C3⋊D3—122no—
365C3⋊D3—128no—
366C3⋊D3—248no—
367C3×D3—28no—
368C3×D3—128no—
369C3×D3—128no—
370C24—541no—
371#41—274no—
372#41—542no—
373C2×Q8—271no—
374C2×D4—91no—
375C2×D4—91no—
376C2×D4—274no—
377C2×D4—271no—
378C2×D4—542no—
379C2×D4—541no—
380C2×D4—541no—
381C22×C4—31no—
382C22×C4—541no—
383SD16—274no—
384SD16—274no—
385D8—94no—
386D8—274no—
387M16—34no—
388C2×C8—31no—
389C2×C8D3×C3⋊D311yescharacteristic, complemented
390C22×C3—24no—
391C22×C3—38no—
392C22×C3—68no—
393C22×C3—182no—
394C22×C3—182no—
395C22×C3—362no—
396C22×C3—362no—
397C22×C3D3×D1214yes—
398C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
399C2×D3—22no—
400C2×D3—68no—
401C2×D3—68no—
402C2×D3—128no—
403C2×D3—128no—
404C2×D3—188no—
405C2×D3—182no—
406C2×D3—368no—
407C2×D3—368no—
408C2×D3—3632no—
409C2×D3—368no—
410C2×D3—362no—
411C2×D3—728no—
412C2×D3C2×C4⋊(C3⋊D3)12yes—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—38no—
416C3×C4D3×D1214yes—
417C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
418C3×C4C22×D3214yes—
419C3×C4C23×C3⋊D311yescharacteristic
420Dic3—188no—
421Dic3—182no—
422Dic3—368no—
423Dic3C2×C4⋊(C3⋊D3)12yes—
424C32—24no—
425C32C2×#90814yescomplemented
426C32C2×#91111yescharacteristic, complemented
427C23—31no—
428C23—181no—
429C23—181no—
430C23—544no—
431C23—541no—
432C23—1084no—
433Q8—272no—
434Q8—541no—
435D4—92no—
436D4—92no—
437D4—181no—
438D4—181no—
439D4—272no—
440D4—544no—
441D4—544no—
442D4—541no—
443C2×C4—31no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—544no—
447C2×C4—541no—
448C2×C4C2×D3×C3⋊D311yescharacteristic
449C8—32no—
450C8C2×D3×C3⋊D312yescomplemented
451C2×C3—24no—
452C2×C3—28no—
453C2×C3—68no—
454C2×C3—362no—
455C2×C3—362no—
456C2×C3#6379518yescomplemented
457C2×C3#6411712yescomplemented
458C2×C3C2×D3×D1214yes—
459C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
460D3—22no—
461D3—128no—
462D3—128no—
463D3—368no—
464D3—362no—
465D3—728no—
466C22—32no—
467C22—62no—
468C22—182no—
469C22—182no—
470C22—362no—
471C22—362no—
472C22—542no—
473C22—1088no—
474C22—1082no—
475C22D3×C4⋊(C3⋊D3)11yescharacteristic
476C4—32no—
477C4—542no—
478C4D3×C4⋊(C3⋊D3)11yescharacteristic
479C4C22×D3×C3⋊D311yescharacteristic
480C3—24no—
481C3C2×#6379514yescomplemented
482C3C2×#6411711yescharacteristic, complemented
483C2—62no—
484C2—362no—
485C2—362no—
486C2—1082no—
487C2#1172331612yesdirect product factor
488C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ 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×D3×C4⋊(C3⋊D3)C2semidirect
2. C2×#89021C2semidirect
3. C2×#88851C2semidirect
4. C2×C3×#3745C2semidirect
5. C2×C32×ΓL(1,𝔽25)C2semidirect
6. C2×#88796C2semidirect
7. C2×#88791C2semidirect
8. #11723316C2direct (N × H)
13. C2×#6308C22semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
21. D3×C4⋊(C3⋊D3)C22semidirect
23. #89021C22semidirect
24. C2×C33×C8C22semidirect
25. #88851C22semidirect
26. C3×#3745C22semidirect
27. C2×C32×C3⋊C8C22semidirect
28. C32×ΓL(1,𝔽25)C22semidirect
29. #88796C22semidirect
30. #88791C22semidirect
38. C2×#3745D3semidirect
44. C2×C3×ΓL(1,𝔽25)D3semidirect
59. #6308C23semidirect
63. C3×C4⋊(C3⋊D3)C23semidirect
70. C33×C8C23semidirect
71. C32×C3⋊C8C23semidirect
87. C2×C4⋊(C3⋊D3)C2×D3semidirect
105. C2×C32×C8C2×D3semidirect
106. C2×C32×C8C2×D3semidirect
108. #3745C2×D3semidirect
113. C2×C3×C3⋊C8C2×D3semidirect
115. C3×ΓL(1,𝔽25)C2×D3semidirect
148. C2×ΓL(1,𝔽25)C3⋊D3semidirect
182. C4⋊(C3⋊D3)C22×D3semidirect
202. C32×C8C22×D3semidirect
203. C32×C8C22×D3semidirect
204. C3×C3⋊C8C22×D3semidirect
206. C2×C33AGL(1,ℤ/8)semidirect
244. C2×C3×C8D32semidirect
245. C2×C3×C8C2×C3⋊D3semidirect
248. C2×C3⋊C8C2×C3⋊D3semidirect
255. ΓL(1,𝔽25)C2×C3⋊D3semidirect
298. C33C2×AGL(1,ℤ/8)semidirect
352. C3×C8C2×D32semidirect
353. C3×C8C22×C3⋊D3semidirect
354. C3⋊C8C22×C3⋊D3semidirect
358. C2×C32#908semidirect
359. C2×C32#911semidirect
389. C2×C8D3×C3⋊D3semidirect
425. C32C2×#908semidirect
426. C32C2×#911semidirect
450. C8C2×D3×C3⋊D3semidirect
456. C2×C3#63795semidirect
457. C2×C3#64117semidirect
481. C3C2×#63795semidirect
482. C3C2×#64117semidirect
487. C2#11723316direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
C2×#890212
C2×#888512
C2×C3×#37452
C2×C32×ΓL(1,𝔽25)2
C2×#887962
C2×#887912
#117233162
C2×#641173
C2×#637953