Finite Group Databasebeta

← C2×#11723562 #13177705

Subgroups — C2×#11723562 #13177705

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×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3C2×#88860C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11723562C218yesmaximal, direct product factor
6C2×#88793C211yesmaximal, characteristic, complemented
7C2×#88791C211yesmaximal, characteristic, complemented
8C2×#88787C211yesmaximal, characteristic, complemented
9C2×#64123—31nomaximal
10C2×#63934—34nomaximal
11C23×C3×C3⋊D3—21no—
12C2×C33×D4C2211yescharacteristic
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
14C2×C3×#638—21no—
15C3×D4×C3⋊D3—22no—
16C3×D4×C3⋊D3C2214yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic, complemented
20C2×#6292—21no—
21C2×#6289—21no—
22#89018—22no—
23#89018C2214yescomplemented
24#88860C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
26C32×#214C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic, complemented
28#88793C2214yescomplemented
29#88791C2214yescomplemented
30#88787C2214yescomplemented
31C2×#3832—31no—
32C2×D4×C3⋊D3D311yescharacteristic, complemented
33C2×C3×D3×D4—34no—
34C2×#3799—34no—
35C2×C32×SD16—31no—
36C2×#3755—31no—
37C2×#3753—31no—
38C2×#3753—34no—
39C2×#3753—64no—
40#64123—38no—
41C2×#3745—31no—
42C2×#3743—31no—
43C2×C3×#214D314yescomplemented
44#63934—332no—
45C2×#3715—34no—
46C2×#3714—34no—
47C2×#3711—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—24no—
52C22×C3×C3⋊D3—44no—
53C22×C3×C3⋊D3D411yescharacteristic
54C33×D4—21no—
55C33×D4C2312yes—
56C2×C33×C4C2311yescharacteristic
57C4⋊(C3⋊(C3⋊D3))—21no—
58C4⋊(C3⋊(C3⋊D3))C2312yescomplemented
59C3×#638—24no—
60C2×C3×C3⋊Dic3D411yescharacteristic
61C3×C4⋊(C3⋊D3)—21no—
62C3×C4⋊(C3⋊D3)C2312yes—
63C3×C4×C3⋊D3D412yes—
64C3×C4×C3⋊D3C2312yes—
65C2×C32×Dic3—21no—
66C32×Dic6—21no—
67C32×Dic6C2312yescomplemented
68#6292—24no—
69#6289—24no—
70C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
71C32×C3⋊C8C2312yescomplemented
72C2×#932—91no—
73C2×#914—94no—
74C23×C3⋊D3—21no—
75C23×C3×D3—64no—
76C2×C32×Q8—31no—
77C2×C32×D4—24no—
78C2×C32×D4C2×D314yes—
79C2×C32×D4C2×D311yescharacteristic
80C2×#638—21no—
81#3832—34no—
82#3832—62no—
83D4×C3⋊D3—22no—
84D4×C3⋊D3C2×D314yescomplemented
85C2×C4⋊(C3⋊D3)—31no—
86C2×C4⋊(C3⋊D3)—34no—
87C2×C4⋊(C3⋊D3)—61no—
88C2×C4⋊(C3⋊D3)—64no—
89C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
90C2×C4×C3⋊D3—61no—
91C2×C4×C3⋊D3C2×D311yescharacteristic
92C2×C3×#67—64no—
93C3×D3×D4—316no—
94C3×D3×D4—68no—
95C2×C3×D12—34no—
96C2×C3×C4×D3—34no—
97C2×C3×Dic6C2×D314yescomplemented
98C2×#626—64no—
99C2×#624—64no—
100#3799—316no—
101#3799—68no—
102C32×SD16—34no—
103C2×C32×C8—31no—
104#3755—34no—
105#3753—34no—
106#3753—316no—
107#3753—616no—
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—34no—
110C2×C3⋊(C3⋊C8)—64no—
111#3745—34no—
112#3743—34no—
113C3×#214C2×D3116yescomplemented
114C2×C3×C3⋊C8C2×D314yescomplemented
115#3715—316no—
116#3714—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—22no—
125C2×C3×C3⋊D3—48no—
126C2×C3×C3⋊D3—42no—
127C2×C3×C3⋊D3C2×D412yes—
128C33×C4C2×D411yescharacteristic
129C33×C4C2411yescharacteristic
130C3×C3⋊Dic3C2×D412yes—
131C32×Dic3—22no—
132C22×C3×D4—91no—
133C2×#239—94no—
134C2×D3×D4—34no—
135C2×#235—91no—
136C2×C3×SD16—34no—
137C2×#215—94no—
138C2×#214—91no—
139C2×#214C3⋊D311yescharacteristic, complemented
140#932—98no—
141C2×#213—94no—
142C2×#213—91no—
143C2×#213—91no—
144C2×#213—184no—
145C2×#208—91no—
146#914—932no—
147C2×D24—94no—
148C2×ΓL(1,𝔽25)—94no—
149C23×C32—21no—
150C23×C32—24no—
151C23×C32—44no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3—24no—
154C22×C3⋊D3—44no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—64no—
157C22×C3⋊D3—124no—
158C22×C3⋊D3#6711yescharacteristic
159C22×C3×D3—34no—
160C22×C3×D3—616no—
161C22×C3×D3—64no—
162C22×C3×D3—1216no—
163C32×Q8—32no—
164C32×Q8—61no—
165C32×D4—21no—
166C32×D4—28no—
167C32×D4—24no—
168C32×D4—44no—
169C32×D4C22×D318yes—
170C32×D4C22×D312yes—
171C2×C32×C4—24no—
172C2×C32×C4—61no—
173C2×C32×C4C22×D314yes—
174C2×C32×C4C22×D311yescharacteristic
175#638—24no—
176C2×C3⋊Dic3#6711yescharacteristic
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—32no—
179C4⋊(C3⋊D3)—38no—
180C4⋊(C3⋊D3)—68no—
181C4⋊(C3⋊D3)—64no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—64no—
184C4⋊(C3⋊D3)—124no—
185C4⋊(C3⋊D3)C22×D312yes—
186C4×C3⋊D3—64no—
187C4×C3⋊D3#6712yes—
188C4×C3⋊D3C22×D312yes—
189C3×#67—616no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—34no—
192C3×D12—38no—
193C3×D12—64no—
194C3×C4×D3—38no—
195C3×C4×D3—38no—
196C3×Dic6—24no—
197C3×Dic6C22×D318yescomplemented
198#626—616no—
199#624—616no—
200C32×C8—32no—
201C3⋊(C3⋊C8)—32no—
202C3⋊(C3⋊C8)—38no—
203C3⋊(C3⋊C8)—68no—
204C3×C3⋊C8C22×D318yescomplemented
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—22no—
211C3×C3⋊D3—42no—
212C24×C3—181no—
213C23×D3—64no—
214C2×C3×Q8—34no—
215C2×C3×D4—24no—
216C2×C3×D4—94no—
217C2×C3×D4—91no—
218C2×C3×D4—182no—
219C2×C3×D4—181no—
220C2×C3×D4D3214yes—
221C2×C3×D4C2×C3⋊D311yescharacteristic
222C22×C3×C4—91no—
223C2×#67—64no—
224C2×#67—181no—
225#239—916no—
226#239—188no—
227D3×D4—316no—
228D3×D4—68no—
229#235—94no—
230#235—182no—
231C2×D12—34no—
232C2×D12—94no—
233C2×D12—91no—
234C2×D12—184no—
235C2×D12—184no—
236C2×C4×D3—34no—
237C2×C4×D3—181no—
238C2×C4×D3—184no—
239C2×Dic6C2×C3⋊D311yescharacteristic, complemented
240C3×SD16—316no—
241C2×C3×C8—34no—
242#215—916no—
243#214—94no—
244#214C2×C3⋊D314yescomplemented
245#213—916no—
246#213—94no—
247#213—94no—
248#213—1816no—
249#208—94no—
250C2×C3⋊C8—94no—
251C2×C3⋊C8—91no—
252C2×C3⋊C8—184no—
253C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
254D24—916no—
255ΓL(1,𝔽25)—916no—
256C22×C32—22no—
257C22×C32—24no—
258C22×C32—28no—
259C22×C32—48no—
260C22×C32—42no—
261C22×C32—48no—
262C22×C32—88no—
263C22×C32D3×D414yes—
264C22×C32C2×#6711yescharacteristic
265C2×C3⋊D3—22no—
266C2×C3⋊D3—22no—
267C2×C3⋊D3—48no—
268C2×C3⋊D3—42no—
269C2×C3⋊D3—62no—
270C2×C3⋊D3—68no—
271C2×C3⋊D3—128no—
272C2×C3⋊D3—122no—
273C2×C3⋊D3—128no—
274C2×C3⋊D3—248no—
275C2×C3⋊D3C2×#6712yes—
276C2×C3×D3—38no—
277C2×C3×D3—68no—
278C2×C3×D3—68no—
279C2×C3×D3—1232no—
280C2×C3×D3—128no—
281C32×C4—24no—
282C32×C4—24no—
283C32×C4—62no—
284C32×C4D3×D414yes—
285C32×C4C2×#6711yescharacteristic
286C32×C4C23×D314yes—
287C32×C4C23×D311yescharacteristic
288C3⋊Dic3C2×#6712yes—
289C3×Dic3—28no—
290C3×Dic3—38no—
291C2×#41—271no—
292C22×D4—91no—
293AGL(1,ℤ/8)—278no—
294C2×SD16—31no—
295C2×SD16—271no—
296C2×D8—271no—
297C2×D8—271no—
298C2×M16—271no—
299C33C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
300C23×C3—24no—
301C23×C3—21no—
302C23×C3—44no—
303C23×C3—91no—
304C23×C3—184no—
305C23×C3—181no—
306C23×C3—364no—
307C22×D3—34no—
308C22×D3—616no—
309C22×D3—64no—
310C22×D3—1216no—
311C22×D3—184no—
312C22×D3—181no—
313C22×D3—364no—
314C3×Q8—38no—
315C3×Q8—64no—
316C3×D4—24no—
317C3×D4—21no—
318C3×D4—28no—
319C3×D4—44no—
320C3×D4—92no—
321C3×D4—184no—
322C3×D4—181no—
323C3×D4C2×D3218yes—
324C3×D4C22×C3⋊D312yes—
325C2×C3×C4—24no—
326C2×C3×C4—64no—
327C2×C3×C4—91no—
328C2×C3×C4—92no—
329C2×C3×C4—92no—
330C2×C3×C4C2×D3214yes—
331C2×C3×C4C22×C3⋊D311yescharacteristic
332#67—616no—
333#67—184no—
334C2×Dic3—21no—
335C2×Dic3—34no—
336D12—38no—
337D12—64no—
338D12—98no—
339D12—92no—
340D12—188no—
341D12—1816no—
342D12—184no—
343D12—181no—
344D12—364no—
345C4×D3—38no—
346C4×D3—38no—
347C4×D3—184no—
348C4×D3—1816no—
349Dic6—21no—
350Dic6C22×C3⋊D312yescomplemented
351C3×C8—38no—
352C3⋊C8—98no—
353C3⋊C8—92no—
354C3⋊C8—188no—
355C3⋊C8C22×C3⋊D312yescomplemented
356C2×C32—24no—
357C2×C32—28no—
358C2×C32—42no—
359C2×C32—48no—
360C2×C32—88no—
361C2×C32#91418yescomplemented
362C2×C32#93212yescomplemented
363C2×C32C2×D3×D414yes—
364C2×C32C22×#6711yescharacteristic
365C3⋊D3—22no—
366C3⋊D3—42no—
367C3⋊D3—122no—
368C3⋊D3—128no—
369C3⋊D3—248no—
370C3×D3—68no—
371C3×D3—128no—
372C24—181no—
373#41—274no—
374#41—542no—
375C2×Q8—31no—
376C2×D4—94no—
377C2×D4—91no—
378C2×D4—182no—
379C2×D4—181no—
380C2×D4—271no—
381C2×D4—541no—
382C2×D4D3×C3⋊D311yescharacteristic
383C22×C4—91no—
384C22×C4—541no—
385SD16—34no—
386SD16—274no—
387D8—274no—
388D8—274no—
389M16—274no—
390C2×C8—31no—
391C2×C8—271no—
392C22×C3—28no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—48no—
396C22×C3—42no—
397C22×C3—48no—
398C22×C3—88no—
399C22×C3—92no—
400C22×C3—182no—
401C22×C3—182no—
402C22×C3—368no—
403C22×C3—362no—
404C22×C3D3×#6714yes—
405C22×C3D4×C3⋊D311yescharacteristic
406C2×D3—38no—
407C2×D3—68no—
408C2×D3—68no—
409C2×D3—1232no—
410C2×D3—128no—
411C2×D3—188no—
412C2×D3—182no—
413C2×D3—368no—
414C2×D3—362no—
415C2×D3—368no—
416C2×D3—728no—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—68no—
420C3×C4—92no—
421C3×C4D3×#6714yes—
422C3×C4D4×C3⋊D311yescharacteristic
423C3×C4C22×D3214yes—
424C3×C4C23×C3⋊D311yescharacteristic
425Dic3—22no—
426Dic3—38no—
427C32—24no—
428C32C2×#91414yescomplemented
429C32C2×#93211yescharacteristic, complemented
430C23—21no—
431C23—91no—
432C23—184no—
433C23—181no—
434C23—364no—
435C23—541no—
436Q8—32no—
437Q8—61no—
438D4—21no—
439D4—92no—
440D4—184no—
441D4—181no—
442D4—272no—
443D4—544no—
444D4—541no—
445D4C2×D3×C3⋊D312yes—
446C2×C4—61no—
447C2×C4—91no—
448C2×C4—92no—
449C2×C4—92no—
450C2×C4—544no—
451C2×C4C2×D3×C3⋊D311yescharacteristic
452C8—32no—
453C8—272no—
454C2×C3—28no—
455C2×C3—24no—
456C2×C3—48no—
457C2×C3—42no—
458C2×C3—88no—
459C2×C3—182no—
460C2×C3—362no—
461C2×C3#6393418yescomplemented
462C2×C3#6412312yescomplemented
463C2×C3C2×D3×#6714yes—
464C2×C3C2×D4×C3⋊D311yescharacteristic
465D3—68no—
466D3—128no—
467D3—368no—
468D3—362no—
469D3—728no—
470C22—22no—
471C22—42no—
472C22—92no—
473C22—182no—
474C22—182no—
475C22—368no—
476C22—362no—
477C22—542no—
478C22—1082no—
479C22C3⋊D3×#6711yescharacteristic
480C4—62no—
481C4—92no—
482C4C3⋊D3×#6711yescharacteristic
483C4C22×D3×C3⋊D311yescharacteristic
484C3—24no—
485C3C2×#6393414yescomplemented
486C3C2×#6412311yescharacteristic, complemented
487C2—42no—
488C2—182no—
489C2—362no—
490C2—1082no—
491C2#1172356212yesdirect product factor
492C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
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×C3×D4×C3⋊D3C2semidirect
2. C2×#89018C2semidirect
3. C2×#88860C2semidirect
4. C2×C32×#214C2semidirect
5. #11723562C2direct (N × H)
6. C2×#88793C2semidirect
7. C2×#88791C2semidirect
8. C2×#88787C2semidirect
13. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C3×D4×C3⋊D3C22semidirect
19. C2×C32×Dic6C22semidirect
23. #89018C22semidirect
24. #88860C22semidirect
25. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C32×#214C22semidirect
27. C2×C32×C3⋊C8C22semidirect
28. #88793C22semidirect
29. #88791C22semidirect
30. #88787C22semidirect
32. C2×D4×C3⋊D3D3semidirect
43. C2×C3×#214D3semidirect
58. C4⋊(C3⋊(C3⋊D3))C23semidirect
67. C32×Dic6C23semidirect
70. C3⋊(C3⋊(C3⋊C8))C23semidirect
71. C32×C3⋊C8C23semidirect
84. D4×C3⋊D3C2×D3semidirect
97. C2×C3×Dic6C2×D3semidirect
113. C3×#214C2×D3semidirect
114. C2×C3×C3⋊C8C2×D3semidirect
139. C2×#214C3⋊D3semidirect
197. C3×Dic6C22×D3semidirect
204. C3×C3⋊C8C22×D3semidirect
207. C2×C33AGL(1,ℤ/8)semidirect
239. C2×Dic6C2×C3⋊D3semidirect
244. #214C2×C3⋊D3semidirect
253. C2×C3⋊C8C2×C3⋊D3semidirect
299. C33C2×AGL(1,ℤ/8)semidirect
350. Dic6C22×C3⋊D3semidirect
355. C3⋊C8C22×C3⋊D3semidirect
361. C2×C32#914semidirect
362. C2×C32#932semidirect
428. C32C2×#914semidirect
429. C32C2×#932semidirect
461. C2×C3#63934semidirect
462. C2×C3#64123semidirect
485. C3C2×#63934semidirect
486. C3C2×#64123semidirect
491. C2#11723562direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#890182
C2×#888602
C2×C32×#2142
#117235622
C2×#887932
C2×#887912
C2×#887872
C2×#641233
C2×#639343