Finite Group Databasebeta

← C2×#11724563 #13181588

Subgroups — C2×#11724563 #13181588

Order 1728.

5388 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6308C211yesmaximal, characteristic, complemented
2C2×C3×#3828C211yesmaximal, characteristic, complemented
3C2×C32×#235C211yesmaximal, characteristic, complemented
4C2×#89029C214yesmaximal, complemented
5C2×#89021C212yesmaximal, complemented
6C2×C3⋊D3×Dic6C212yesmaximal, complemented
7#11724563C2116yesmaximal, direct product factor
8C2×#89014C212yesmaximal, complemented
9C2×D3×#634C212yesmaximal, complemented
10C2×#64367—31nomaximal
11C2×#64298—34nomaximal
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2212yes—
14C2×#6308C2212yes—
15C2×#6308C2212yes—
16C2×#6308C2218yescomplemented
17C2×C3×#638C2212yes—
18C3×#3828C2218yescomplemented
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2212yes—
21C2×C3×#634C2211yescharacteristic
22C2×C32×#67C2212yes—
23C32×#235C2218yescomplemented
24C2×C32×D12C2211yescharacteristic
25C2×C32×C4×D3C2212yes—
26C2×C32×Dic6C2211yescharacteristic
27C2×#6293C2214yes—
28C2×#6290C2214yes—
29C2×Dic3×C3⋊D3C2214yes—
30#89029C22132yescomplemented
31C2×D3×C3⋊Dic3C2214yes—
32#89021C22116yescomplemented
33C3⋊D3×Dic6C22116yescomplemented
34#89014C22116yescomplemented
35D3×#634C22116yescomplemented
36C2×C32×#41—31no—
37#64367—316no—
38C2×Q8×C3⋊D3—32no—
39C2×#3830—34no—
40C2×#3830—32no—
41C2×#3828—32no—
42C2×#3828D311yescharacteristic, complemented
43C22×#634—31no—
44C22×#634—32no—
45C22×#634—34no—
46C22×#634—64no—
47C2×C3×#235—34no—
48C2×C3×#235D314yescomplemented
49C2×#3805—316no—
50#64298—364no—
51C2×#3795—38no—
52C2×#3795—38no—
53C2×D3×Dic6—38no—
54C2×D3×Dic6—38no—
55C23×C33C2311yescharacteristic
56C22×C3×C3⋊D3C2312yes—
57C22×C32×D3C2312yes—
58C2×C33×C4C2311yescharacteristic
59C2×C33×C4C2311yescharacteristic
60C2×C33×C4C2314yes—
61C2×C3⋊(C3⋊Dic3)C2314yes—
62C2×C3⋊(C3⋊Dic3)C2318yes—
63#6308C2318yes—
64#6308C2318yes—
65C3×#638C2318yes—
66C2×C3×C3⋊Dic3C2312yes—
67C3×C4⋊(C3⋊D3)C2314yes—
68C3×C4×C3⋊D3C2318yes—
69C3×#634C2314yes—
70C32×#67C2318yes—
71C2×C32×Dic3C2312yes—
72C32×D12C2314yes—
73C32×C4×D3C2318yes—
74C32×Dic6C2314yes—
75#6293C23116yes—
76#6290C23116yes—
77Dic3×C3⋊D3C23116yes—
78D3×C3⋊Dic3C23116yes—
79C2×#1010—94no—
80C2×#1010—91no—
81C32×#41—38no—
82C2×C32×Q8—31no—
83C2×C32×D4—32no—
84C2×C32×D4—31no—
85C22×C32×C4—24no—
86C22×C32×C4—32no—
87C22×C32×C4C2×D314yes—
88C22×C32×C4C2×D311yescharacteristic
89C2×#638—34no—
90C2×#638C2×D312yes—
91C22×C3⋊Dic3—32no—
92C22×C3⋊Dic3—38no—
93C22×C3⋊Dic3—34no—
94C22×C3⋊Dic3—68no—
95Q8×C3⋊D3—316no—
96#3830—332no—
97#3830—316no—
98#3828—316no—
99#3828C2×D318yescomplemented
100C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
101C2×C4×C3⋊D3—34no—
102C2×C4×C3⋊D3C2×D312yes—
103C2×#634—32no—
104C2×#634—32no—
105C2×#634—38no—
106C2×#634—38no—
107C2×#634—38no—
108C2×#634—332no—
109C2×#634—34no—
110C2×#634—316no—
111C2×#634—68no—
112C2×#634—68no—
113C2×#634—632no—
114C2×#634C2×D311yescharacteristic
115C2×C3×#67—38no—
116C2×C3×#67C2×D318yes—
117C3×#235—332no—
118C3×#235C2×D3132yescomplemented
119C2×C3×D12—34no—
120C2×C3×D12C2×D314yes—
121C2×C3×C4×D3—38no—
122C2×C3×C4×D3C2×D318yes—
123C2×C3×Dic6—34no—
124C2×C3×Dic6C2×D314yes—
125C2×#627—316no—
126C2×#625—316no—
127#3805—3128no—
128C2×D3×Dic3—316no—
129C2×D3×Dic3—316no—
130#3795—364no—
131#3795—364no—
132D3×Dic6—364no—
133D3×Dic6—364no—
134C22×C33—22no—
135C22×C33C2412yes—
136C22×C33C2411yescharacteristic
137C2×C3×C3⋊D3—24no—
138C2×C3×C3⋊D3C2414yes—
139C2×C32×D3—24no—
140C2×C32×D3C2414yes—
141C33×C4C2412yes—
142C33×C4C2412yes—
143C3⋊(C3⋊Dic3)C2418yes—
144C3×C3⋊Dic3C2414yes—
145C32×Dic3C2414yes—
146C2×C3×#41—34no—
147C2×C3×#41—91no—
148#1010—964no—
149#1010—916no—
150C2×D3×Q8—98no—
151C2×D3×Q8—92no—
152C2×#237—98no—
153C2×#237—916no—
154C2×#237—94no—
155C2×#237—92no—
156C2×#235—34no—
157C2×#235—98no—
158C2×#235—92no—
159C2×#235C3⋊D311yescharacteristic, complemented
160C22×Dic6—98no—
161C22×Dic6—94no—
162C22×Dic6—91no—
163C22×Dic6—92no—
164C22×Dic6—184no—
165C23×C32—24no—
166C23×C32—32no—
167C23×C32C22×D314yes—
168C23×C32C22×D311yescharacteristic
169C22×C3⋊D3C22×D312yes—
170C22×C3×D3—38no—
171C22×C3×D3C22×D318yes—
172C32×Q8—34no—
173C32×D4—38no—
174C32×D4—34no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—216no—
178C2×C32×C4—32no—
179C2×C32×C4—38no—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D3116yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D314yes—
186#638—316no—
187#638C22×D318yes—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—316no—
191C2×C3⋊Dic3—316no—
192C2×C3⋊Dic3—332no—
193C2×C3⋊Dic3—616no—
194C2×C3⋊Dic3—632no—
195C2×C3⋊Dic3C22×D312yes—
196C4⋊(C3⋊D3)C22×D314yes—
197C4×C3⋊D3—316no—
198C4×C3⋊D3C22×D318yes—
199#634—38no—
200#634—38no—
201#634—316no—
202#634—332no—
203#634—332no—
204#634—632no—
205#634—632no—
206#634C22×D314yes—
207C3×#67—332no—
208C3×#67C22×D3132yes—
209C2×C3×Dic3—38no—
210C2×C3×Dic3C22×D318yes—
211C3×D12—316no—
212C3×D12C22×D3116yes—
213C3×C4×D3—332no—
214C3×C4×D3C22×D3132yes—
215C3×Dic6—316no—
216C3×Dic6C22×D3116yes—
217#627—364no—
218#625—364no—
219D3×Dic3—364no—
220D3×Dic3—364no—
221C2×2−1+4—271noSylow
222C2×C33—22no—
223C2×C332−1+412yescomplemented
224C2×C33C2511yescharacteristic
225C3×C3⋊D3—24no—
226C32×D3—24no—
227C3×#41—332no—
228C3×#41—98no—
229C2×C3×Q8—34no—
230C2×C3×Q8—91no—
231C2×C3×D4—38no—
232C2×C3×D4—34no—
233C2×C3×D4—92no—
234C2×C3×D4—91no—
235C22×C3×C4—24no—
236C22×C3×C4—38no—
237C22×C3×C4—92no—
238C22×C3×C4D3214yes—
239C22×C3×C4C2×C3⋊D311yescharacteristic
240C2×#67—38no—
241C2×#67—916no—
242C2×#67—94no—
243C2×#67C2×C3⋊D312yes—
244C22×Dic3—916no—
245C22×Dic3—98no—
246C22×Dic3—92no—
247C22×Dic3—94no—
248C22×Dic3—188no—
249D3×Q8—964no—
250D3×Q8—916no—
251#237—964no—
252#237—9128no—
253#237—932no—
254#237—916no—
255#235—332no—
256#235—964no—
257#235—916no—
258#235C2×C3⋊D318yescomplemented
259C2×D12—34no—
260C2×D12C2×C3⋊D311yescharacteristic
261C2×C4×D3—38no—
262C2×C4×D3—916no—
263C2×C4×D3—94no—
264C2×C4×D3C2×C3⋊D312yes—
265C2×Dic6—34no—
266C2×Dic6—964no—
267C2×Dic6—916no—
268C2×Dic6—98no—
269C2×Dic6—98no—
270C2×Dic6—932no—
271C2×Dic6—92no—
272C2×Dic6—92no—
273C2×Dic6—98no—
274C2×Dic6—94no—
275C2×Dic6—916no—
276C2×Dic6—188no—
277C2×Dic6—188no—
278C2×Dic6—1832no—
279C2×Dic6C2×C3⋊D311yescharacteristic
280C22×C32—216no—
281C22×C32—28no—
282C22×C32—22no—
283C22×C32—28no—
284C22×C32—24no—
285C22×C32—34no—
286C22×C32—64no—
287C22×C32C23×D318yes—
288C22×C32C23×D314yes—
289C22×C32C23×D312yes—
290C22×C32C23×D311yescharacteristic
291C2×C3⋊D3—24no—
292C2×C3⋊D3C23×D314yes—
293C2×C3×D3—216no—
294C2×C3×D3—316no—
295C2×C3×D3—616no—
296C2×C3×D3C23×D3116yes—
297C32×C4—28no—
298C32×C4—28no—
299C32×C4—34no—
300C32×C4C23×D318yes—
301C32×C4C23×D318yes—
302C32×C4C23×D312yes—
303C32×C4C23×D312yes—
304C3⋊Dic3—38no—
305C3⋊Dic3—332no—
306C3⋊Dic3—632no—
307C3⋊Dic3C23×D314yes—
308C3×Dic3—316no—
309C3×Dic3C23×D3116yes—
3102−1+4—2716no—
311C2×#41—31no—
312C2×#41—91no—
313C2×#41—272no—
314C2×#41—272no—
315C2×#41—274no—
316C22×Q8—272no—
317C22×Q8—272no—
318C22×Q8—271no—
319C33C2×2−1+411yescharacteristic, complemented, Sylow
320C23×C3—24no—
321C23×C3—38no—
322C23×C3—92no—
323C23×C3C2×D3214yes—
324C23×C3C22×C3⋊D311yescharacteristic
325C22×D3—38no—
326C22×D3C22×C3⋊D312yes—
327C3×Q8—316no—
328C3×Q8—94no—
329C3×D4—332no—
330C3×D4—316no—
331C3×D4—98no—
332C3×D4—94no—
333C2×C3×C4—216no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—38no—
337C2×C3×C4—332no—
338C2×C3×C4—92no—
339C2×C3×C4—98no—
340C2×C3×C4C2×D3214yes—
341C2×C3×C4C2×D3214yes—
342C2×C3×C4C2×D32116yes—
343C2×C3×C4C22×C3⋊D311yescharacteristic
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D314yes—
346#67—332no—
347#67—964no—
348#67—916no—
349#67C22×C3⋊D318yes—
350C2×Dic3—38no—
351C2×Dic3—964no—
352C2×Dic3—916no—
353C2×Dic3—932no—
354C2×Dic3—94no—
355C2×Dic3—98no—
356C2×Dic3—916no—
357C2×Dic3—1832no—
358C2×Dic3—1816no—
359C2×Dic3C22×C3⋊D312yes—
360D12—316no—
361D12C22×C3⋊D314yes—
362C4×D3—332no—
363C4×D3—964no—
364C4×D3—916no—
365C4×D3C22×C3⋊D318yes—
366Dic6—316no—
367Dic6—964no—
368Dic6—932no—
369Dic6—932no—
370Dic6—98no—
371Dic6—98no—
372Dic6—916no—
373Dic6—1832no—
374Dic6—1832no—
375Dic6C22×C3⋊D314yes—
376C2×C32—216no—
377C2×C32—28no—
378C2×C32—22no—
379C2×C32—28no—
380C2×C32—24no—
381C2×C32—64no—
382C2×C32#101018yescomplemented
383C2×C32#101012yescomplemented
384C2×C32C24×D314yes—
385C2×C32C24×D311yescharacteristic
386C3⋊D3—24no—
387C3×D3—216no—
388C3×D3—616no—
389#41—38no—
390#41—98no—
391#41—2716no—
392#41—2716no—
393#41—2732no—
394C2×Q8—31no—
395C2×Q8—91no—
396C2×Q8—274no—
397C2×Q8—2716no—
398C2×Q8—2716no—
399C2×Q8—272no—
400C2×Q8—272no—
401C2×Q8—278no—
402C2×D4—32no—
403C2×D4—31no—
404C2×D4—92no—
405C2×D4—91no—
406C2×D4—274no—
407C22×C4—32no—
408C22×C4—92no—
409C22×C4—274no—
410C22×C4—274no—
411C22×C4—272no—
412C22×C4D3×C3⋊D311yescharacteristic
413C22×C3—216no—
414C22×C3—28no—
415C22×C3—22no—
416C22×C3—28no—
417C22×C3—24no—
418C22×C3—316no—
419C22×C3—616no—
420C22×C3—94no—
421C22×C3—184no—
422C22×C3C22×D3218yes—
423C22×C3C22×D3214yes—
424C22×C3C23×C3⋊D312yes—
425C22×C3C23×C3⋊D311yescharacteristic
426C2×D3—24no—
427C2×D3—316no—
428C2×D3—616no—
429C2×D3C23×C3⋊D314yes—
430C3×C4—28no—
431C3×C4—28no—
432C3×C4—316no—
433C3×C4—94no—
434C3×C4C22×D3218yes—
435C3×C4C22×D3218yes—
436C3×C4C23×C3⋊D312yes—
437C3×C4C23×C3⋊D312yes—
438Dic3—316no—
439Dic3—932no—
440Dic3—98no—
441Dic3—1832no—
442Dic3C23×C3⋊D314yes—
443C32—24no—
444C32C2×#101014yescomplemented
445C32C2×#101011yescharacteristic, complemented
446C23—32no—
447C23—92no—
448C23C2×D3×C3⋊D311yescharacteristic
449Q8—34no—
450Q8—94no—
451Q8—2716no—
452Q8—278no—
453Q8—278no—
454D4—38no—
455D4—34no—
456D4—98no—
457D4—94no—
458D4—2716no—
459C2×C4—32no—
460C2×C4—38no—
461C2×C4—92no—
462C2×C4—98no—
463C2×C4—2716no—
464C2×C4—2716no—
465C2×C4—278no—
466C2×C4—274no—
467C2×C4C2×D3×C3⋊D314yes—
468C2×C4C2×D3×C3⋊D311yescharacteristic
469C2×C4C2×D3×C3⋊D311yescharacteristic
470C2×C3—216no—
471C2×C3—28no—
472C2×C3—22no—
473C2×C3—28no—
474C2×C3—24no—
475C2×C3—616no—
476C2×C3—184no—
477C2×C3#6429818yescomplemented
478C2×C3#6436712yescomplemented
479C2×C3C23×D3214yes—
480C2×C3C24×C3⋊D311yescharacteristic
481D3—24no—
482D3—616no—
483C22—22no—
484C22—34no—
485C22—64no—
486C22—94no—
487C22—184no—
488C22C22×D3×C3⋊D312yes—
489C22C22×D3×C3⋊D311yescharacteristic
490C4—34no—
491C4—94no—
492C4—278no—
493C4C22×D3×C3⋊D312yes—
494C4C22×D3×C3⋊D312yes—
495C3—24no—
496C3C2×#6429814yescomplemented
497C3C2×#6436711yescharacteristic, complemented
498C2—22no—
499C2—64no—
500C2—184no—
501C2#1172456312yesdirect product factor
502C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
centerC22C22×D3×C3⋊D3
commutator, cosocleC2×C33C25
FrattiniC2C23×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C24
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×2−1+4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6308C2semidirect
2. C2×C3×#3828C2semidirect
3. C2×C32×#235C2semidirect
4. C2×#89029C2semidirect
5. C2×#89021C2semidirect
6. C2×C3⋊D3×Dic6C2semidirect
7. #11724563C2direct (N × H)
8. C2×#89014C2semidirect
9. C2×D3×#634C2semidirect
16. C2×#6308C22semidirect
18. C3×#3828C22semidirect
23. C32×#235C22semidirect
30. #89029C22semidirect
32. #89021C22semidirect
33. C3⋊D3×Dic6C22semidirect
34. #89014C22semidirect
35. D3×#634C22semidirect
42. C2×#3828D3semidirect
48. C2×C3×#235D3semidirect
99. #3828C2×D3semidirect
118. C3×#235C2×D3semidirect
159. C2×#235C3⋊D3semidirect
223. C2×C332−1+4semidirect
258. #235C2×C3⋊D3semidirect
319. C33C2×2−1+4semidirect
382. C2×C32#1010semidirect
383. C2×C32#1010semidirect
444. C32C2×#1010semidirect
445. C32C2×#1010semidirect
477. C2×C3#64298semidirect
478. C2×C3#64367semidirect
496. C3C2×#64298semidirect
497. C3C2×#64367semidirect
501. C2#11724563direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#63082
C2×C3×#38282
C2×C32×#2352
C2×#890292
C2×#890212
C2×C3⋊D3×Dic62
#117245632
C2×#890142
C2×D3×#6342
C2×#643673
C2×#642983