Finite Group Databasebeta

← C2×#11723363 #13176871

Subgroups — C2×#11723363 #13176871

Order 1728.

1900 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic, complemented
2C2×C32×#235C211yesmaximal, characteristic, complemented
3C22×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4C2×#88798C211yesmaximal, characteristic, complemented
5C2×#88793C211yesmaximal, characteristic, complemented
6C2×#88792C211yesmaximal, characteristic, complemented
7#11723363C218yesmaximal, direct product factor
8C2×#88789C211yesmaximal, characteristic, complemented
9C2×#64159—31nomaximal
10C2×#63822—34nomaximal
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×#3828—22no—
14C3×#3828C2214yescomplemented
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×#634C2211yescharacteristic
18C2×C32×#67—21no—
19C32×#235—22no—
20C32×#235C2214yescomplemented
21C2×C32×D12C2211yescharacteristic
22C2×C32×C4×D3—21no—
23C2×C32×Dic6C2211yescharacteristic
24C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26C2×C3⋊(C3⋊(C3⋊C8))C2214yescomplemented
27#88798C2214yescomplemented
28#88793C2214yescomplemented
29#88792C2214yescomplemented
30#88789C2214yescomplemented
31C2×C32×#41—31no—
32C2×#3828D311yescharacteristic, complemented
33C2×C3×#235—34no—
34C2×C3×#235D314yescomplemented
35#64159—38no—
36C2×#3756—31no—
37C2×#3755—31no—
38C2×#3754—31no—
39C2×#3753—31no—
40C22×C3⋊(C3⋊C8)—31no—
41C22×C3⋊(C3⋊C8)—34no—
42C22×C3⋊(C3⋊C8)—64no—
43C2×#3718—34no—
44C2×#3715—34no—
45C2×#3715—34no—
46#63822—332no—
47C2×#3713—34no—
48C23×C33D411yescharacteristic
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4D411yescharacteristic
52C2×C33×C4D412yes—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2312yes—
55C3×#638—24no—
56C2×C3×C3⋊Dic3—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)C2312yes—
59C3×C4×C3⋊D3—24no—
60C3×#634—21no—
61C3×#634C2312yes—
62C32×#67—24no—
63C2×C32×Dic3—21no—
64C32×D12—21no—
65C32×D12C2312yes—
66C32×C4×D3—24no—
67C32×Dic6—21no—
68C32×Dic6C2312yes—
69C3⋊(C3⋊(C3⋊C8))C2312yes—
70C3⋊(C3⋊(C3⋊C8))C2312yes—
71C2×#950—94no—
72C2×#950—91no—
73C32×#41—34no—
74C32×#41—62no—
75C2×C32×Q8—31no—
76C2×C32×D4—31no—
77C2×C32×D4—61no—
78C22×C32×C4—24no—
79C22×C32×C4—61no—
80C22×C32×C4C2×D314yes—
81C22×C32×C4C2×D311yescharacteristic
82C2×#638—21no—
83#3828—22no—
84#3828C2×D314yescomplemented
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
86C2×C4×C3⋊D3—21no—
87C2×#634C2×D311yescharacteristic
88C2×C3×#67—24no—
89C2×C3×#67—64no—
90C3×#235—28no—
91C3×#235—316no—
92C3×#235—68no—
93C3×#235C2×D3116yescomplemented
94C2×C3×D12—34no—
95C2×C3×D12C2×D314yes—
96C2×C3×C4×D3—24no—
97C2×C3×C4×D3—64no—
98C2×C3×Dic6—34no—
99C2×C3×Dic6C2×D314yes—
100#3756—34no—
101#3755—34no—
102#3754—34no—
103#3753—34no—
104C2×C3⋊(C3⋊C8)—31no—
105C2×C3⋊(C3⋊C8)—31no—
106C2×C3⋊(C3⋊C8)—34no—
107C2×C3⋊(C3⋊C8)—34no—
108C2×C3⋊(C3⋊C8)—34no—
109C2×C3⋊(C3⋊C8)—316no—
110C2×C3⋊(C3⋊C8)—64no—
111C2×C3⋊(C3⋊C8)—64no—
112C2×C3⋊(C3⋊C8)—616no—
113#3718—316no—
114#3715—316no—
115#3715—316no—
116#3713—316no—
117C22×C33—22no—
118C22×C33C2×D412yes—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—22no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D412yes—
126C33×C4C2411yescharacteristic
127C3×C3⋊Dic3—22no—
128C32×Dic3—22no—
129C2×C3×#41—34no—
130C2×C3×#41—91no—
131C2×#235—34no—
132C2×#235C3⋊D311yescharacteristic, complemented
133#950—932no—
134#950—98no—
135C2×#216—94no—
136C2×#216—91no—
137C2×#215—94no—
138C2×#215—91no—
139C2×#214—94no—
140C2×#214—91no—
141C2×#213—94no—
142C2×#213—91no—
143C22×C3⋊C8—94no—
144C22×C3⋊C8—91no—
145C22×C3⋊C8—184no—
146C23×C32—24no—
147C23×C32—61no—
148C23×C32#6714yes—
149C23×C32#6711yescharacteristic
150C22×C3⋊D3—21no—
151C22×C3×D3—24no—
152C22×C3×D3—64no—
153C32×Q8—32no—
154C32×Q8—61no—
155C32×D4—32no—
156C32×D4—64no—
157C32×D4—61no—
158C2×C32×C4—28no—
159C2×C32×C4—24no—
160C2×C32×C4—24no—
161C2×C32×C4—28no—
162C2×C32×C4—61no—
163C2×C32×C4—64no—
164C2×C32×C4#6714yes—
165C2×C32×C4#6718yes—
166C2×C32×C4#6711yescharacteristic
167C2×C32×C4#6712yes—
168C2×C32×C4C22×D314yes—
169C2×C32×C4C22×D318yes—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D312yes—
172#638—24no—
173C2×C3⋊Dic3—21no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)C22×D312yes—
176C4×C3⋊D3—24no—
177#634—21no—
178#634C22×D312yes—
179C3×#67—216no—
180C3×#67—616no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—64no—
183C3×D12—24no—
184C3×D12—38no—
185C3×D12—64no—
186C3×D12C22×D318yes—
187C3×C4×D3—216no—
188C3×C4×D3—616no—
189C3×Dic6—24no—
190C3×Dic6—38no—
191C3×Dic6—64no—
192C3×Dic6C22×D318yes—
193C3⋊(C3⋊C8)—32no—
194C3⋊(C3⋊C8)—32no—
195C3⋊(C3⋊C8)—38no—
196C3⋊(C3⋊C8)—38no—
197C3⋊(C3⋊C8)—68no—
198C3⋊(C3⋊C8)—68no—
199C2×#135—271noSylow
200C2×C33—22no—
201C2×C33#13512yescomplemented
202C2×C33C22×D411yescharacteristic
203C3×C3⋊D3—42no—
204C32×D3—42no—
205C3×#41—316no—
206C3×#41—68no—
207C3×#41—94no—
208C3×#41—182no—
209C2×C3×Q8—34no—
210C2×C3×Q8—91no—
211C2×C3×D4—34no—
212C2×C3×D4—64no—
213C2×C3×D4—91no—
214C2×C3×D4—181no—
215C22×C3×C4—24no—
216C22×C3×C4—64no—
217C22×C3×C4—181no—
218C22×C3×C4D3214yes—
219C22×C3×C4C2×C3⋊D311yescharacteristic
220C2×#67—21no—
221C2×#67—64no—
222#235—22no—
223#235—316no—
224#235—68no—
225#235C2×C3⋊D314yescomplemented
226C2×D12—34no—
227C2×D12C2×C3⋊D311yescharacteristic
228C2×C4×D3—21no—
229C2×C4×D3—64no—
230C2×Dic6—34no—
231C2×Dic6C2×C3⋊D311yescharacteristic
232#216—916no—
233#216—94no—
234#215—916no—
235#215—94no—
236#214—916no—
237#214—94no—
238#213—916no—
239#213—94no—
240C2×C3⋊C8—94no—
241C2×C3⋊C8—94no—
242C2×C3⋊C8—916no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—94no—
246C2×C3⋊C8—184no—
247C2×C3⋊C8—184no—
248C2×C3⋊C8—1816no—
249C22×C32—216no—
250C22×C32—28no—
251C22×C32—22no—
252C22×C32—28no—
253C22×C32—24no—
254C22×C32—62no—
255C22×C32—122no—
256C22×C32C2×#6718yes—
257C22×C32C2×#6714yes—
258C22×C32C2×#6712yes—
259C22×C32C2×#6711yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3—42no—
262C2×C3×D3—28no—
263C2×C3×D3—48no—
264C2×C3×D3—68no—
265C2×C3×D3—128no—
266C32×C4—24no—
267C32×C4—24no—
268C32×C4—28no—
269C32×C4—62no—
270C32×C4C2×#6714yes—
271C32×C4C2×#6718yes—
272C32×C4C2×#6711yescharacteristic
273C32×C4C2×#6712yes—
274C32×C4C23×D314yes—
275C32×C4C23×D311yescharacteristic
276C3⋊Dic3—22no—
277C3×Dic3—28no—
278C3×Dic3—68no—
279C2×#41—31no—
280C2×#41—91no—
281#135—278no—
282C2×Q16—271no—
283C2×SD16—271no—
284C2×SD16—271no—
285C2×D8—271no—
286C22×C8—271no—
287C33C2×#13511yescharacteristic, complemented, Sylow
288C23×C3—24no—
289C23×C3—64no—
290C23×C3—181no—
291C23×C3#62514yes—
292C23×C3#63811yescharacteristic
293C22×D3—21no—
294C22×D3—64no—
295C3×Q8—38no—
296C3×Q8—64no—
297C3×Q8—92no—
298C3×Q8—181no—
299C3×D4—38no—
300C3×D4—616no—
301C3×D4—64no—
302C3×D4—92no—
303C3×D4—184no—
304C3×D4—181no—
305C2×C3×C4—28no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—28no—
309C2×C3×C4—64no—
310C2×C3×C4—616no—
311C2×C3×C4—181no—
312C2×C3×C4—184no—
313C2×C3×C4#62518yes—
314C2×C3×C4#62514yes—
315C2×C3×C4#63812yes—
316C2×C3×C4#63811yescharacteristic
317C2×C3×C4C2×D3214yes—
318C2×C3×C4C2×D3218yes—
319C2×C3×C4C22×C3⋊D311yescharacteristic
320C2×C3×C4C22×C3⋊D312yes—
321#67—24no—
322#67—616no—
323C2×Dic3—21no—
324C2×Dic3—64no—
325D12—21no—
326D12—38no—
327D12—64no—
328D12C22×C3⋊D312yes—
329C4×D3—24no—
330C4×D3—616no—
331Dic6—21no—
332Dic6—38no—
333Dic6—64no—
334Dic6C22×C3⋊D312yes—
335C3⋊C8—98no—
336C3⋊C8—98no—
337C3⋊C8—92no—
338C3⋊C8—92no—
339C3⋊C8—188no—
340C3⋊C8—188no—
341C2×C32—216no—
342C2×C32—28no—
343C2×C32—22no—
344C2×C32—24no—
345C2×C32—28no—
346C2×C32—122no—
347C2×C32#95018yescomplemented
348C2×C32#95012yescomplemented
349C2×C32C22×#6714yes—
350C2×C32C22×#6711yescharacteristic
351C3⋊D3—42no—
352C3×D3—48no—
353C3×D3—128no—
354#41—34no—
355#41—62no—
356#41—94no—
357#41—182no—
358C2×Q8—31no—
359C2×Q8—91no—
360C2×D4—31no—
361C2×D4—61no—
362C2×D4—91no—
363C2×D4—181no—
364C22×C4—61no—
365C22×C4—181no—
366C22×C4D3×C3⋊D311yescharacteristic
367Q16—274no—
368SD16—274no—
369SD16—274no—
370D8—274no—
371C2×C8—271no—
372C2×C8—271no—
373C2×C8—274no—
374C22×C3—216no—
375C22×C3—28no—
376C22×C3—22no—
377C22×C3—28no—
378C22×C3—24no—
379C22×C3—68no—
380C22×C3—128no—
381C22×C3—182no—
382C22×C3—362no—
383C22×C3C2×#62518yes—
384C22×C3C2×#62514yes—
385C22×C3C2×#63812yes—
386C22×C3C2×#63811yescharacteristic
387C2×D3—22no—
388C2×D3—42no—
389C2×D3—68no—
390C2×D3—128no—
391C3×C4—24no—
392C3×C4—28no—
393C3×C4—24no—
394C3×C4—68no—
395C3×C4—182no—
396C3×C4C2×#62514yes—
397C3×C4C2×#62518yes—
398C3×C4C2×#63811yescharacteristic
399C3×C4C2×#63812yes—
400C3×C4C22×D3214yes—
401C3×C4C23×C3⋊D311yescharacteristic
402Dic3—22no—
403Dic3—68no—
404C32—24no—
405C32C2×#95014yescomplemented
406C32C2×#95011yescharacteristic, complemented
407C23—61no—
408C23—181no—
409C23#629011yescharacteristic
410Q8—32no—
411Q8—61no—
412Q8—92no—
413Q8—181no—
414D4—32no—
415D4—64no—
416D4—61no—
417D4—92no—
418D4—184no—
419D4—181no—
420C2×C4—61no—
421C2×C4—64no—
422C2×C4—181no—
423C2×C4—184no—
424C2×C4#629012yes—
425C2×C4#629011yescharacteristic
426C2×C4C2×D3×C3⋊D311yescharacteristic
427C2×C4C2×D3×C3⋊D312yes—
428C8—272no—
429C8—272no—
430C2×C3—216no—
431C2×C3—22no—
432C2×C3—28no—
433C2×C3—24no—
434C2×C3—28no—
435C2×C3—128no—
436C2×C3—362no—
437C2×C3#6382218yescomplemented
438C2×C3#6415912yescomplemented
439C2×C3C22×#62514yes—
440C2×C3C22×#63811yescharacteristic
441D3—42no—
442D3—128no—
443C22—22no—
444C22—62no—
445C22—122no—
446C22—182no—
447C22—362no—
448C22C2×#629012yes—
449C22C2×#629011yescharacteristic
450C4—62no—
451C4—182no—
452C4C2×#629012yes—
453C4C2×#629011yescharacteristic
454C4C22×D3×C3⋊D311yescharacteristic
455C3—24no—
456C3C2×#6382214yescomplemented
457C3C2×#6415911yescharacteristic, complemented
458C2—22no—
459C2—122no—
460C2—362no—
461C2#1172336312yesdirect product factor
462C2C22×#629011yescharacteristic
subgroupG/H
centerC2×C4#6290
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C2×C32×#235C2semidirect
3. C22×C3⋊(C3⋊(C3⋊C8))C2semidirect
4. C2×#88798C2semidirect
5. C2×#88793C2semidirect
6. C2×#88792C2semidirect
7. #11723363C2direct (N × H)
8. C2×#88789C2semidirect
14. C3×#3828C22semidirect
20. C32×#235C22semidirect
26. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
27. #88798C22semidirect
28. #88793C22semidirect
29. #88792C22semidirect
30. #88789C22semidirect
32. C2×#3828D3semidirect
34. C2×C3×#235D3semidirect
84. #3828C2×D3semidirect
93. C3×#235C2×D3semidirect
132. C2×#235C3⋊D3semidirect
201. C2×C33#135semidirect
225. #235C2×C3⋊D3semidirect
287. C33C2×#135semidirect
347. C2×C32#950semidirect
348. C2×C32#950semidirect
405. C32C2×#950semidirect
406. C32C2×#950semidirect
437. C2×C3#63822semidirect
438. C2×C3#64159semidirect
456. C3C2×#63822semidirect
457. C3C2×#64159semidirect
461. C2#11723363direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C32×#2352
C22×C3⋊(C3⋊(C3⋊C8))2
C2×#887982
C2×#887932
C2×#887922
#117233632
C2×#887892
C2×#641593
C2×#638223