Finite Group Databasebeta

← C2×#11723557 #13177711

Subgroups — C2×#11723557 #13177711

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×D3×D4C211yesmaximal, characteristic, complemented
2C2×#89022C211yesmaximal, characteristic, complemented
3C2×#88860C211yesmaximal, characteristic, complemented
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5#11723557C218yesmaximal, direct product factor
6C2×#88792C211yesmaximal, characteristic, complemented
7C2×#88790C211yesmaximal, characteristic, complemented
8C2×#88786C211yesmaximal, characteristic, complemented
9C2×#64141—31nomaximal
10C2×#63934—34nomaximal
11C23×C32×D3—21no—
12C2×C33×D4C2211yescharacteristic
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
14C2×C3×#634C2211yescharacteristic, complemented
15C2×C32×#67—21no—
16C32×D3×D4—22no—
17C32×D3×D4C2214yescomplemented
18C2×C32×D12C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6291—21no—
21C2×#6289—21no—
22#89022—22no—
23#89022C2214yescomplemented
24#88860C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
26C3×#3754C2214yescomplemented
27C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
28#88792C2214yescomplemented
29#88790C2214yescomplemented
30#88786C2214yescomplemented
31C22×C32×D4—31no—
32C2×#3828—31no—
33C2×C3×D3×D4D314yescomplemented
34C2×#3799—34no—
35C2×#3754—31no—
36C2×#3754D311yescharacteristic, complemented
37#64141—38no—
38C2×#3753—31no—
39C2×#3753—31no—
40C2×#3753—34no—
41C2×#3753—64no—
42C2×#3748—31no—
43C2×C3×#214—34no—
44#63934—332no—
45C2×#3715—34no—
46C2×#3714—34no—
47C2×#3711—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)—21no—
50C22×C32×D3—21no—
51C22×C32×D3—24no—
52C22×C32×D3—44no—
53C22×C32×D3D411yescharacteristic
54C33×D4—21no—
55C33×D4C2312yes—
56C2×C33×C4C2311yescharacteristic
57C4⋊(C3⋊(C3⋊D3))—21no—
58C4⋊(C3⋊(C3⋊D3))C2312yescomplemented
59C2×C3×C3⋊Dic3—21no—
60C3×#634—21no—
61C3×#634C2312yescomplemented
62C32×#67—24no—
63C2×C32×Dic3D411yescharacteristic
64C32×D12—21no—
65C32×D12C2312yes—
66C32×C4×D3D412yes—
67C32×C4×D3C2312yes—
68#6291—24no—
69#6289—24no—
70C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
71C3×C3⋊(C3⋊C8)C2312yescomplemented
72C2×#932—94no—
73C2×#914—91no—
74C24×C32—61no—
75C23×C3×D3—24no—
76C2×C32×D4—24no—
77C2×C32×D4—34no—
78C2×C32×D4—31no—
79C2×C32×D4—62no—
80C2×C32×D4—61no—
81C2×C32×D4C2×D314yes—
82C2×C32×D4C2×D311yescharacteristic
83C22×C32×C4—31no—
84C2×#638—61no—
85#3828—34no—
86#3828—62no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4⋊(C3⋊D3)—34no—
89C2×C4⋊(C3⋊D3)—64no—
90C2×C4×C3⋊D3—61no—
91C2×#634C2×D311yescharacteristic, complemented
92C2×C3×#67—24no—
93C3×D3×D4—28no—
94C3×D3×D4C2×D3116yescomplemented
95C2×C3×D12C2×D314yes—
96C2×C3×C4×D3C2×D314yes—
97C2×C3×Dic6—34no—
98C2×#626—64no—
99C2×#624—64no—
100#3799—316no—
101#3799—68no—
102#3754—34no—
103#3754C2×D314yescomplemented
104#3753—34no—
105#3753—316no—
106#3753—34no—
107#3753—616no—
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×#214—316no—
114C2×C3×C3⋊C8—34no—
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×C32×D3—22no—
124C2×C32×D3—22no—
125C2×C32×D3—48no—
126C2×C32×D3—42no—
127C2×C32×D3C2×D412yes—
128C33×C4C2×D411yescharacteristic
129C33×C4C2411yescharacteristic
130C3×C3⋊Dic3—22no—
131C32×Dic3C2×D412yes—
132C22×C3×D4—34no—
133C2×#239—91no—
134C2×D3×D4C3⋊D311yescharacteristic, complemented
135C2×#235—94no—
136C2×C3×SD16—91no—
137C2×#215—91no—
138C2×#214—34no—
139C2×#214—94no—
140#932—932no—
141C2×#213—94no—
142C2×#213—94no—
143C2×#213—91no—
144C2×#213—184no—
145C2×#208—94no—
146#914—98no—
147C2×D24—91no—
148C2×ΓL(1,𝔽25)—91no—
149C23×C32—21no—
150C23×C32—24no—
151C23×C32—31no—
152C23×C32—44no—
153C23×C32—64no—
154C23×C32—61no—
155C23×C32—124no—
156C22×C3⋊D3—61no—
157C22×C3⋊D3—64no—
158C22×C3⋊D3—124no—
159C22×C3×D3—24no—
160C22×C3×D3—216no—
161C22×C3×D3—416no—
162C22×C3×D3#6714yes—
163C32×D4—21no—
164C32×D4—28no—
165C32×D4—24no—
166C32×D4—32no—
167C32×D4—44no—
168C32×D4—64no—
169C32×D4—61no—
170C32×D4C22×D318yes—
171C32×D4C22×D312yes—
172C2×C32×C4—24no—
173C2×C32×C4—31no—
174C2×C32×C4—32no—
175C2×C32×C4—32no—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D311yescharacteristic
178#638—64no—
179C2×C3⋊Dic3—21no—
180C4⋊(C3⋊D3)—32no—
181C4⋊(C3⋊D3)—38no—
182C4⋊(C3⋊D3)—68no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)—64no—
185C4⋊(C3⋊D3)—124no—
186C4×C3⋊D3—64no—
187#634—21no—
188#634C22×D312yescomplemented
189C3×#67—216no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3#6714yes—
192C3×D12—24no—
193C3×D12C22×D318yes—
194C3×C4×D3#6718yes—
195C3×C4×D3C22×D318yes—
196C3×Dic6—38no—
197C3×Dic6—64no—
198#626—616no—
199#624—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—
210C32×D3—22no—
211C32×D3—42no—
212C24×C3—64no—
213C23×D3—21no—
214C2×C3×Q8—91no—
215C2×C3×D4—24no—
216C2×C3×D4—316no—
217C2×C3×D4—34no—
218C2×C3×D4—68no—
219C2×C3×D4—64no—
220C2×C3×D4D3214yes—
221C2×C3×D4C2×C3⋊D311yescharacteristic
222C22×C3×C4—34no—
223C2×#67—21no—
224C2×#67—184no—
225#239—94no—
226#239—182no—
227D3×D4—22no—
228D3×D4C2×C3⋊D314yescomplemented
229#235—916no—
230#235—188no—
231C2×D12—94no—
232C2×D12—91no—
233C2×D12—184no—
234C2×D12—181no—
235C2×D12C2×C3⋊D311yescharacteristic
236C2×C4×D3—184no—
237C2×C4×D3—181no—
238C2×C4×D3C2×C3⋊D311yescharacteristic
239C2×Dic6—34no—
240C3×SD16—94no—
241C2×C3×C8—91no—
242#215—94no—
243#214—316no—
244#214—916no—
245#213—916no—
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—
254D24—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—62no—
265C22×C32—88no—
266C22×C32—128no—
267C22×C32—122no—
268C22×C32D3×D411yescharacteristic
269C22×C32C2×#6714yes—
270C2×C3⋊D3—62no—
271C2×C3⋊D3—68no—
272C2×C3⋊D3—128no—
273C2×C3⋊D3—122no—
274C2×C3⋊D3—128no—
275C2×C3⋊D3—248no—
276C2×C3×D3—28no—
277C2×C3×D3—28no—
278C2×C3×D3—432no—
279C2×C3×D3—48no—
280C2×C3×D3C2×#6718yes—
281C32×C4—24no—
282C32×C4—24no—
283C32×C4—32no—
284C32×C4D3×D411yescharacteristic
285C32×C4C2×#6714yes—
286C32×C4C23×D314yes—
287C32×C4C23×D311yescharacteristic
288C3⋊Dic3—22no—
289C3×Dic3—68no—
290C3×Dic3C2×#6718yes—
291C2×#41—271no—
292C22×D4—31no—
293AGL(1,ℤ/8)—278no—
294C2×SD16—91no—
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—34no—
303C23×C3—44no—
304C23×C3—616no—
305C23×C3—64no—
306C23×C3—1216no—
307C22×D3—21no—
308C22×D3—24no—
309C22×D3—44no—
310C22×D3—184no—
311C22×D3—181no—
312C22×D3—364no—
313C22×D3#63811yescharacteristic
314C3×Q8—92no—
315C3×Q8—181no—
316C3×D4—24no—
317C3×D4—21no—
318C3×D4—28no—
319C3×D4—38no—
320C3×D4—44no—
321C3×D4—616no—
322C3×D4—64no—
323C3×D4C2×D3218yes—
324C3×D4C22×C3⋊D312yes—
325C2×C3×C4—24no—
326C2×C3×C4—34no—
327C2×C3×C4—38no—
328C2×C3×C4—38no—
329C2×C3×C4—181no—
330C2×C3×C4C2×D3214yes—
331C2×C3×C4C22×C3⋊D311yescharacteristic
332#67—24no—
333#67—1816no—
334C2×Dic3—64no—
335C2×Dic3#63811yescharacteristic
336D12—21no—
337D12—98no—
338D12—92no—
339D12—188no—
340D12—184no—
341D12—184no—
342D12—181no—
343D12—364no—
344D12C22×C3⋊D312yes—
345C4×D3—1816no—
346C4×D3—184no—
347C4×D3#63812yes—
348C4×D3C22×C3⋊D312yes—
349Dic6—38no—
350Dic6—64no—
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—122no—
363C2×C32#91412yescomplemented
364C2×C32#93218yescomplemented
365C2×C32C2×D3×D411yescharacteristic
366C2×C32C22×#6714yes—
367C3⋊D3—122no—
368C3⋊D3—128no—
369C3⋊D3—248no—
370C3×D3—28no—
371C3×D3—48no—
372C24—61no—
373#41—274no—
374#41—542no—
375C2×Q8—91no—
376C2×D4—34no—
377C2×D4—31no—
378C2×D4—62no—
379C2×D4—61no—
380C2×D4—271no—
381C2×D4—541no—
382C2×D4D3×C3⋊D311yescharacteristic
383C22×C4—31no—
384C22×C4—541no—
385SD16—94no—
386SD16—274no—
387D8—274no—
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—68no—
401C22×C3—88no—
402C22×C3—1232no—
403C22×C3—128no—
404C22×C3D3×#6714yes—
405C22×C3C2×#63811yescharacteristic
406C2×D3—22no—
407C2×D3—22no—
408C2×D3—48no—
409C2×D3—42no—
410C2×D3—188no—
411C2×D3—182no—
412C2×D3—368no—
413C2×D3—362no—
414C2×D3—368no—
415C2×D3—728no—
416C2×D3C2×#63812yes—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—38no—
420C3×C4—182no—
421C3×C4D3×#6714yes—
422C3×C4C2×#63811yescharacteristic
423C3×C4C22×D3214yes—
424C3×C4C23×C3⋊D311yescharacteristic
425Dic3—68no—
426Dic3C2×#63812yes—
427C32—24no—
428C32C2×#91411yescharacteristic, complemented
429C32C2×#93214yescomplemented
430C23—21no—
431C23—31no—
432C23—64no—
433C23—61no—
434C23—124no—
435C23—541no—
436Q8—92no—
437Q8—181no—
438D4—21no—
439D4—32no—
440D4—64no—
441D4—61no—
442D4—272no—
443D4—544no—
444D4—541no—
445D4C2×D3×C3⋊D312yes—
446C2×C4—31no—
447C2×C4—32no—
448C2×C4—32no—
449C2×C4—181no—
450C2×C4—544no—
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—128no—
461C2×C3#6393418yescomplemented
462C2×C3#6414112yescomplemented
463C2×C3C2×D3×#6714yes—
464C2×C3C22×#63811yescharacteristic
465D3—22no—
466D3—42no—
467D3—368no—
468D3—362no—
469D3—728no—
470C22—22no—
471C22—32no—
472C22—42no—
473C22—62no—
474C22—62no—
475C22—128no—
476C22—122no—
477C22—542no—
478C22—1082no—
479C22D3×#63811yescharacteristic
480C4—32no—
481C4—182no—
482C4D3×#63811yescharacteristic
483C4C22×D3×C3⋊D311yescharacteristic
484C3—24no—
485C3C2×#6393414yescomplemented
486C3C2×#6414111yescharacteristic, complemented
487C2—42no—
488C2—62no—
489C2—122no—
490C2—1082no—
491C2#1172355712yesdirect 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×D3×D4C2semidirect
2. C2×#89022C2semidirect
3. C2×#88860C2semidirect
4. C2×C3×#3754C2semidirect
5. #11723557C2direct (N × H)
6. C2×#88792C2semidirect
7. C2×#88790C2semidirect
8. C2×#88786C2semidirect
13. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
14. C2×C3×#634C22semidirect
17. C32×D3×D4C22semidirect
23. #89022C22semidirect
24. #88860C22semidirect
25. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C3×#3754C22semidirect
27. C2×C3×C3⋊(C3⋊C8)C22semidirect
28. #88792C22semidirect
29. #88790C22semidirect
30. #88786C22semidirect
33. C2×C3×D3×D4D3semidirect
36. C2×#3754D3semidirect
58. C4⋊(C3⋊(C3⋊D3))C23semidirect
61. C3×#634C23semidirect
70. C3⋊(C3⋊(C3⋊C8))C23semidirect
71. C3×C3⋊(C3⋊C8)C23semidirect
91. C2×#634C2×D3semidirect
94. C3×D3×D4C2×D3semidirect
103. #3754C2×D3semidirect
112. C2×C3⋊(C3⋊C8)C2×D3semidirect
134. C2×D3×D4C3⋊D3semidirect
188. #634C22×D3semidirect
203. C3⋊(C3⋊C8)C22×D3semidirect
207. C2×C33AGL(1,ℤ/8)semidirect
228. D3×D4C2×C3⋊D3semidirect
299. C33C2×AGL(1,ℤ/8)semidirect
363. C2×C32#914semidirect
364. C2×C32#932semidirect
428. C32C2×#914semidirect
429. C32C2×#932semidirect
461. C2×C3#63934semidirect
462. C2×C3#64141semidirect
485. C3C2×#63934semidirect
486. C3C2×#64141semidirect
491. C2#11723557direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×D3×D42
C2×#890222
C2×#888602
C2×C3×#37542
#117235572
C2×#887922
C2×#887902
C2×#887862
C2×#641413
C2×#639343