Finite Group Databasebeta

← C2×#11723566 #13177713

Subgroups — C2×#11723566 #13177713

Order 1728.

2732 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89017C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11723566C218yesmaximal, direct product factor
6C2×#88795C211yesmaximal, characteristic, complemented
7C2×#88791C211yesmaximal, characteristic, complemented
8C2×#88788C211yesmaximal, characteristic, complemented
9C2×#64123—31nomaximal
10C2×#63937—34nomaximal
11C23×C3⋊(C3⋊D3)—21no—
12C2×C33×D4C2211yescharacteristic
13C2×#6312—21no—
14D4×C3⋊(C3⋊D3)—22no—
15D4×C3⋊(C3⋊D3)C2214yescomplemented
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C2×C32×Dic6C2211yescharacteristic, complemented
20C2×#6292—21no—
21C2×Dic3×C3⋊D3—21no—
22#89017—22no—
23#89017C2214yescomplemented
24C3×#3753C2214yescomplemented
25C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
26C32×#214C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic, complemented
28#88795C2214yescomplemented
29#88791C2214yescomplemented
30#88788C2214yescomplemented
31C2×#3832—31no—
32C2×D4×C3⋊D3—31no—
33C2×D4×C3⋊D3—34no—
34C2×D4×C3⋊D3—64no—
35C2×#3796—34no—
36C2×C32×SD16—31no—
37C2×#3755—31no—
38C2×#3753D311yescharacteristic, complemented
39#64123—38no—
40C2×#3745—31no—
41C2×#3743—31no—
42C2×C3×#214D314yescomplemented
43C2×C3×#213—34no—
44#63937—332no—
45C2×#3717—34no—
46C2×#3714—34no—
47C2×#3712—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3⋊(C3⋊D3)—24no—
51C22×C3⋊(C3⋊D3)—44no—
52C22×C3⋊(C3⋊D3)D411yescharacteristic
53C22×C3×C3⋊D3—21no—
54C33×D4—21no—
55C33×D4C2312yes—
56C2×C33×C4C2311yescharacteristic
57#6312—24no—
58C2×C3⋊(C3⋊Dic3)D411yescharacteristic
59C4⋊(C3⋊(C3⋊D3))—21no—
60C4⋊(C3⋊(C3⋊D3))C2312yes—
61C4×C3⋊(C3⋊D3)D412yes—
62C4×C3⋊(C3⋊D3)C2312yes—
63C3×C4⋊(C3⋊D3)—21no—
64C3×C4⋊(C3⋊D3)C2312yescomplemented
65C2×C32×Dic3—21no—
66C32×Dic6—21no—
67C32×Dic6C2312yescomplemented
68#6292—24no—
69Dic3×C3⋊D3—24no—
70C3×C3⋊(C3⋊C8)C2312yescomplemented
71C32×C3⋊C8C2312yescomplemented
72C2×#914—94no—
73C2×#911—91no—
74C23×C3⋊D3—61no—
75C23×C3⋊D3—64no—
76C23×C3⋊D3—68no—
77C2×C32×Q8—31no—
78C2×C32×D4—24no—
79C2×C32×D4C2×D314yes—
80C2×C32×D4C2×D311yescharacteristic
81C2×#638—61no—
82C2×#638—64no—
83C2×#638—68no—
84#3832—34no—
85#3832—62no—
86D4×C3⋊D3—34no—
87D4×C3⋊D3—316no—
88D4×C3⋊D3—62no—
89D4×C3⋊D3—68no—
90D4×C3⋊D3—616no—
91D4×C3⋊D3—616no—
92C2×C4⋊(C3⋊D3)—31no—
93C2×C4⋊(C3⋊D3)—34no—
94C2×C4⋊(C3⋊D3)—64no—
95C2×C4⋊(C3⋊D3)—61no—
96C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
97C2×C4×C3⋊D3—31no—
98C2×C4×C3⋊D3—34no—
99C2×C4×C3⋊D3—61no—
100C2×C4×C3⋊D3—64no—
101C2×C3×D12—34no—
102C2×C3×Dic6C2×D314yescomplemented
103C2×#626—64no—
104C2×D3×Dic3—64no—
105#3796—316no—
106#3796—68no—
107C32×SD16—34no—
108C2×C32×C8—31no—
109#3755—34no—
110#3753C2×D314yescomplemented
111C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
112#3745—34no—
113#3743—34no—
114C3×#214C2×D3116yescomplemented
115C3×#213—316no—
116C2×C3×C3⋊C8—34no—
117C2×C3×C3⋊C8C2×D314yescomplemented
118#3717—316no—
119#3714—316no—
120#3712—316no—
121C22×C33—22no—
122C22×C33—42no—
123C22×C33C2×D411yescharacteristic
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3⋊(C3⋊D3)—22no—
126C2×C3⋊(C3⋊D3)—48no—
127C2×C3⋊(C3⋊D3)—42no—
128C2×C3⋊(C3⋊D3)C2×D412yes—
129C2×C3×C3⋊D3—22no—
130C2×C3×C3⋊D3—42no—
131C33×C4C2×D411yescharacteristic
132C33×C4C2411yescharacteristic
133C3⋊(C3⋊Dic3)C2×D412yes—
134C32×Dic3—22no—
135C2×#239—94no—
136C2×#237—91no—
137C2×D3×D4—94no—
138C2×D3×D4—91no—
139C2×D3×D4—184no—
140C2×C3×SD16—34no—
141C2×C3×D8—91no—
142C2×#215—94no—
143C2×#214C3⋊D311yescharacteristic, complemented
144C2×#213—34no—
145C2×#213—91no—
146#914—932no—
147#911—98no—
148C2×D24—94no—
149C2×#204—91no—
150C2×ΓL(1,𝔽25)—94no—
151C2×ΓL(1,𝔽25)—91no—
152C23×C32—28no—
153C23×C32—24no—
154C23×C32—21no—
155C22×C3⋊D3—21no—
156C22×C3⋊D3—31no—
157C22×C3⋊D3—34no—
158C22×C3⋊D3—64no—
159C22×C3⋊D3—64no—
160C22×C3⋊D3—616no—
161C22×C3⋊D3—68no—
162C22×C3⋊D3—632no—
163C22×C3⋊D3—64no—
164C22×C3⋊D3—61no—
165C22×C3⋊D3—1232no—
166C22×C3⋊D3—124no—
167C22×C3⋊D3—1216no—
168C22×C3×D3—64no—
169C32×Q8—32no—
170C32×Q8—61no—
171C32×D4—28no—
172C32×D4—24no—
173C32×D4—21no—
174C32×D4—28no—
175C32×D4C22×D318yes—
176C32×D4C22×D312yes—
177C2×C32×C4—24no—
178C2×C32×C4—61no—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D311yescharacteristic
181#638—64no—
182#638—616no—
183#638—632no—
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—34no—
186C2×C3⋊Dic3—64no—
187C4⋊(C3⋊D3)—21no—
188C4⋊(C3⋊D3)—32no—
189C4⋊(C3⋊D3)—38no—
190C4⋊(C3⋊D3)—68no—
191C4⋊(C3⋊D3)—68no—
192C4⋊(C3⋊D3)—64no—
193C4⋊(C3⋊D3)—64no—
194C4⋊(C3⋊D3)—61no—
195C4⋊(C3⋊D3)C22×D312yescomplemented
196C4×C3⋊D3—32no—
197C4×C3⋊D3—32no—
198C4×C3⋊D3—38no—
199C4×C3⋊D3—38no—
200C4×C3⋊D3—68no—
201C4×C3⋊D3—68no—
202C4×C3⋊D3—64no—
203C2×C3×Dic3—24no—
204C3×D12—38no—
205C3×D12—64no—
206C3×Dic6—24no—
207C3×Dic6C22×D318yescomplemented
208#626—616no—
209D3×Dic3—616no—
210C32×C8—32no—
211C3⋊(C3⋊C8)C22×D312yescomplemented
212C3×C3⋊C8—38no—
213C3×C3⋊C8C22×D318yescomplemented
214C2×AGL(1,ℤ/8)—271noSylow
215C2×C33—42no—
216C2×C33AGL(1,ℤ/8)12yescomplemented
217C2×C33C22×D411yescharacteristic
218C3⋊(C3⋊D3)—22no—
219C3⋊(C3⋊D3)—42no—
220C3×C3⋊D3—42no—
221C23×D3—184no—
222C23×D3—181no—
223C23×D3—188no—
224C2×C3×Q8—34no—
225C2×C3×D4—24no—
226C2×C3×D4—91no—
227C2×C3×D4D3214yes—
228C2×C3×D4C2×C3⋊D311yescharacteristic
229C2×#67—181no—
230C2×#67—184no—
231C2×#67—181no—
232C2×#67—188no—
233C22×Dic3—181no—
234#239—916no—
235#239—188no—
236#237—94no—
237#237—182no—
238D3×D4—916no—
239D3×D4—94no—
240D3×D4—1816no—
241D3×D4—188no—
242D3×D4—182no—
243D3×D4—1816no—
244C2×D12—34no—
245C2×D12—94no—
246C2×D12—91no—
247C2×D12—184no—
248C2×D12—184no—
249C2×C4×D3—94no—
250C2×C4×D3—91no—
251C2×C4×D3—184no—
252C2×C4×D3—184no—
253C2×Dic6C2×C3⋊D311yescharacteristic, complemented
254C3×SD16—316no—
255C3×D8—94no—
256C2×C3×C8—34no—
257C2×C3×C8—91no—
258#215—916no—
259#214C2×C3⋊D314yescomplemented
260#213—316no—
261#213—94no—
262C2×C3⋊C8—34no—
263C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
264D24—916no—
265#204—94no—
266ΓL(1,𝔽25)—916no—
267ΓL(1,𝔽25)—94no—
268C22×C32—216no—
269C22×C32—28no—
270C22×C32—22no—
271C22×C32—24no—
272C22×C32—416no—
273C22×C32—42no—
274C22×C32—48no—
275C22×C32D3×D414yes—
276C22×C32D3×D411yescharacteristic
277C2×C3⋊D3—22no—
278C2×C3⋊D3—32no—
279C2×C3⋊D3—38no—
280C2×C3⋊D3—42no—
281C2×C3⋊D3—68no—
282C2×C3⋊D3—616no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—616no—
286C2×C3⋊D3—68no—
287C2×C3⋊D3—62no—
288C2×C3⋊D3—1264no—
289C2×C3⋊D3—128no—
290C2×C3⋊D3—1232no—
291C2×C3⋊D3—1216no—
292C2×C3⋊D3—122no—
293C2×C3⋊D3—128no—
294C2×C3×D3—68no—
295C2×C3×D3—128no—
296C32×C4—24no—
297C32×C4—24no—
298C32×C4—62no—
299C32×C4D3×D414yes—
300C32×C4D3×D411yescharacteristic
301C32×C4C23×D314yes—
302C32×C4C23×D311yescharacteristic
303C3⋊Dic3—32no—
304C3⋊Dic3—38no—
305C3⋊Dic3—68no—
306C3×Dic3—28no—
307C2×#41—271no—
308C22×D4—271no—
309AGL(1,ℤ/8)—278no—
310C2×SD16—31no—
311C2×SD16—271no—
312C2×D8—91no—
313C2×D8—271no—
314C2×M16—271no—
315C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
316C23×C3—28no—
317C23×C3—24no—
318C23×C3—21no—
319C23×C3—181no—
320C22×D3—64no—
321C22×D3—94no—
322C22×D3—91no—
323C22×D3—184no—
324C22×D3—1816no—
325C22×D3—184no—
326C22×D3—188no—
327C22×D3—1832no—
328C22×D3—184no—
329C22×D3—181no—
330C22×D3—3632no—
331C22×D3—364no—
332C22×D3—3616no—
333C3×Q8—38no—
334C3×Q8—64no—
335C3×D4—28no—
336C3×D4—24no—
337C3×D4—21no—
338C3×D4—28no—
339C3×D4—92no—
340C3×D4—181no—
341C3×D4C2×D3218yes—
342C3×D4C22×C3⋊D312yes—
343C2×C3×C4—24no—
344C2×C3×C4—64no—
345C2×C3×C4C2×D3214yes—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347#67—184no—
348#67—1816no—
349#67—184no—
350#67—1832no—
351C2×Dic3—21no—
352C2×Dic3—94no—
353C2×Dic3—91no—
354C2×Dic3—184no—
355C2×Dic3—184no—
356D12—38no—
357D12—64no—
358D12—98no—
359D12—92no—
360D12—188no—
361D12—188no—
362D12—1816no—
363D12—184no—
364D12—181no—
365C4×D3—98no—
366C4×D3—92no—
367C4×D3—98no—
368C4×D3—92no—
369C4×D3—188no—
370C4×D3—188no—
371C4×D3—1816no—
372Dic6—21no—
373Dic6C22×C3⋊D312yescomplemented
374C3×C8—38no—
375C3×C8—92no—
376C3⋊C8—38no—
377C3⋊C8C22×C3⋊D312yescomplemented
378C2×C32—24no—
379C2×C32—28no—
380C2×C32—416no—
381C2×C32—42no—
382C2×C32—48no—
383C2×C32#91112yescomplemented
384C2×C32#91418yescomplemented
385C2×C32C2×D3×D414yes—
386C2×C32C2×D3×D411yescharacteristic
387C3⋊D3—42no—
388C3⋊D3—616no—
389C3⋊D3—62no—
390C3⋊D3—68no—
391C3⋊D3—1216no—
392C3⋊D3—122no—
393C3⋊D3—128no—
394C3×D3—128no—
395C24—541no—
396#41—274no—
397#41—542no—
398C2×Q8—31no—
399C2×D4—91no—
400C2×D4—274no—
401C2×D4—271no—
402C2×D4—541no—
403C2×D4—542no—
404C2×D4—541no—
405C2×D4D3×C3⋊D311yescharacteristic
406C22×C4—271no—
407C22×C4—541no—
408SD16—34no—
409SD16—274no—
410D8—94no—
411D8—274no—
412M16—274no—
413C2×C8—31no—
414C2×C8—91no—
415C22×C3—216no—
416C22×C3—28no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3—416no—
420C22×C3—42no—
421C22×C3—48no—
422C22×C3—182no—
423C22×C3—362no—
424C22×C3#381114yes—
425C22×C3D4×C3⋊D311yescharacteristic
426C2×D3—68no—
427C2×D3—98no—
428C2×D3—92no—
429C2×D3—128no—
430C2×D3—188no—
431C2×D3—1816no—
432C2×D3—182no—
433C2×D3—188no—
434C2×D3—1816no—
435C2×D3—188no—
436C2×D3—182no—
437C2×D3—3664no—
438C2×D3—3632no—
439C2×D3—368no—
440C2×D3—3616no—
441C2×D3—362no—
442C2×D3—368no—
443C3×C4—24no—
444C3×C4—24no—
445C3×C4—68no—
446C3×C4#381114yes—
447C3×C4D4×C3⋊D311yescharacteristic
448C3×C4C22×D3214yes—
449C3×C4C23×C3⋊D311yescharacteristic
450Dic3—22no—
451Dic3—98no—
452Dic3—92no—
453Dic3—188no—
454C32—24no—
455C32C2×#91111yescharacteristic, complemented
456C32C2×#91414yescomplemented
457C23—21no—
458C23—181no—
459C23—271no—
460C23—544no—
461C23—541no—
462C23—1084no—
463Q8—32no—
464Q8—61no—
465D4—21no—
466D4—92no—
467D4—181no—
468D4—272no—
469D4—544no—
470D4—544no—
471D4—541no—
472D4C2×D3×C3⋊D312yes—
473C2×C4—61no—
474C2×C4—271no—
475C2×C4—272no—
476C2×C4—272no—
477C2×C4—544no—
478C2×C4C2×D3×C3⋊D311yescharacteristic
479C8—32no—
480C8—92no—
481C2×C3—28no—
482C2×C3—24no—
483C2×C3—416no—
484C2×C3—48no—
485C2×C3—42no—
486C2×C3—362no—
487C2×C3#6393718yescomplemented
488C2×C3#6412312yescomplemented
489C2×C3C2×#381114yes—
490C2×C3C2×D4×C3⋊D311yescharacteristic
491D3—128no—
492D3—1816no—
493D3—188no—
494D3—182no—
495D3—3616no—
496D3—368no—
497D3—362no—
498C22—22no—
499C22—42no—
500C22—182no—
501C22—272no—
502C22—362no—
503C22—542no—
504C22—542no—
505C22—1088no—
506C22—1082no—
507C22#8903911yescharacteristic
508C4—62no—
509C4—272no—
510C4#8903911yescharacteristic
511C4C22×D3×C3⋊D311yescharacteristic
512C3—24no—
513C3C2×#6393714yescomplemented
514C3C2×#6412311yescharacteristic, complemented
515C2—42no—
516C2—362no—
517C2—542no—
518C2—1082no—
519C2#1172356612yesdirect product factor
520C2C2×#8903911yescharacteristic
subgroupG/H
centerC22#89039
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×D4×C3⋊(C3⋊D3)C2semidirect
2. C2×#89017C2semidirect
3. C2×C3×#3753C2semidirect
4. C2×C32×#214C2semidirect
5. #11723566C2direct (N × H)
6. C2×#88795C2semidirect
7. C2×#88791C2semidirect
8. C2×#88788C2semidirect
15. D4×C3⋊(C3⋊D3)C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C32×Dic6C22semidirect
23. #89017C22semidirect
24. C3×#3753C22semidirect
25. C2×C3×C3⋊(C3⋊C8)C22semidirect
26. C32×#214C22semidirect
27. C2×C32×C3⋊C8C22semidirect
28. #88795C22semidirect
29. #88791C22semidirect
30. #88788C22semidirect
38. C2×#3753D3semidirect
42. C2×C3×#214D3semidirect
64. C3×C4⋊(C3⋊D3)C23semidirect
67. C32×Dic6C23semidirect
70. C3×C3⋊(C3⋊C8)C23semidirect
71. C32×C3⋊C8C23semidirect
96. C2×C4⋊(C3⋊D3)C2×D3semidirect
102. C2×C3×Dic6C2×D3semidirect
110. #3753C2×D3semidirect
111. C2×C3⋊(C3⋊C8)C2×D3semidirect
114. C3×#214C2×D3semidirect
117. C2×C3×C3⋊C8C2×D3semidirect
143. C2×#214C3⋊D3semidirect
195. C4⋊(C3⋊D3)C22×D3semidirect
207. C3×Dic6C22×D3semidirect
211. C3⋊(C3⋊C8)C22×D3semidirect
213. C3×C3⋊C8C22×D3semidirect
216. C2×C33AGL(1,ℤ/8)semidirect
253. C2×Dic6C2×C3⋊D3semidirect
259. #214C2×C3⋊D3semidirect
263. C2×C3⋊C8C2×C3⋊D3semidirect
315. C33C2×AGL(1,ℤ/8)semidirect
373. Dic6C22×C3⋊D3semidirect
377. C3⋊C8C22×C3⋊D3semidirect
383. C2×C32#911semidirect
384. C2×C32#914semidirect
455. C32C2×#911semidirect
456. C32C2×#914semidirect
487. C2×C3#63937semidirect
488. C2×C3#64123semidirect
513. C3C2×#63937semidirect
514. C3C2×#64123semidirect
519. C2#11723566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×#890172
C2×C3×#37532
C2×C32×#2142
#117235662
C2×#887952
C2×#887912
C2×#887882
C2×#641233
C2×#639373