Finite Group Databasebeta

← C2×#11722519 #13164773

Subgroups — C2×#11722519 #13164773

Order 1728.

3279 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88748C211yesmaximal, characteristic, complemented
2C2×#88722C211yesmaximal, characteristic, complemented
3C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#11722519C2116yesmaximal, direct product factor
5C2×#88654C213yesmaximal, complemented
6C2×C4×((C3⋊D3)⋊C3)⋊C2C213yesmaximal, complemented
7C2×#88652C213yesmaximal, complemented
8C2×#88649C213yesmaximal, complemented
9C2×#64319—31nomaximal
10C2×D3×#239—31nomaximal
11C22×((C3⋊D3)⋊C3)⋊C2C2213yes—
12C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
13#88748C2218yescomplemented
14C2×C4⋊#272C2213yescomplemented
15C2×C4×#272C2213yescomplemented
16#88722C2218yescomplemented
17Q8×(C3⋊D3)⋊C3C2218yescomplemented
18C2×C4⋊((C3⋊D3)⋊C3)C2213yescomplemented
19C2×C4×(C3⋊D3)⋊C3C2213yes—
20C2×C4×(C3⋊D3)⋊C3C2213yescomplemented
21C2×#634⋊C3C2213yescomplemented
22C2×#6200C2216yescomplemented
23C2×#6199C2213yescomplemented
24C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
25C2×((C3⋊D3)⋊C3)⋊C4C2213yescomplemented
26#88654C22124yescomplemented
27C4×((C3⋊D3)⋊C3)⋊C2C22124yescomplemented
28#88652C22124yescomplemented
29#88649C22124yescomplemented
30C2×#3832D311yescharacteristic, complemented
31C2×Q8×C3⋊D3D311yescharacteristic, complemented
32C2×C3×#239—31no—
33C2×C3×#239—31no—
34C2×C3×#239—31no—
35C2×C3×#239—61no—
36C2×C3×D3×Q8—31no—
37#64319—316no—
38D3×#239—316no—
39C2×#3802—33no—
40C2×D3×D12—33no—
41C2×C4×D32—33no—
42C2×C4×D32—33no—
43C2×#3799—33no—
44C2×#3796—33no—
45C2×#3795—33no—
46C2×#3795—33no—
47C22×#272C2313yes—
48C22×(C3⋊D3)⋊C3C2313yes—
49C22×(C3⋊D3)⋊C3C2311yescharacteristic
50C2×((C3⋊D3)⋊C3)⋊C2—212no—
51C2×((C3⋊D3)⋊C3)⋊C2C23112yes—
52Q8×H(3,𝔽3)C2314yescomplemented
53C2×C4×H(3,𝔽3)C2313yes—
54C2×#1084C2311yescharacteristic
55C4⋊#272C23112yescomplemented
56C4×#272C23112yescomplemented
57C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
58C2×(C3⋊Dic3)⋊C3C2313yes—
59C4⋊((C3⋊D3)⋊C3)C23112yescomplemented
60C4×(C3⋊D3)⋊C3C23112yes—
61C4×(C3⋊D3)⋊C3C23112yescomplemented
62#634⋊C3C23112yescomplemented
63#6200C23124yescomplemented
64#6199C23112yescomplemented
65((C3⋊D3)⋊C3)⋊C4C2314yes—
66((C3⋊D3)⋊C3)⋊C4C23112yescomplemented
67C2×D3×#41—91no—
68C2×D3×#41—91no—
69C22×#239—91no—
70C22×D32—33no—
71C22×D32—33no—
72C2×C32×Q8—21no—
73C2×C32×Q8C2×D311yescharacteristic, complemented
74C2×C32×Q8C2×D311yescharacteristic, complemented
75#3832C2×D318yescomplemented
76Q8×C3⋊D3C2×D318yescomplemented
77C2×C4⋊(C3⋊D3)C2×D313yescomplemented
78C2×C4×C3⋊D3C2×D313yes—
79C2×C4×C3⋊D3C2×D313yescomplemented
80C2×#634C2×D313yescomplemented
81C3×#239—38no—
82C3×#239—38no—
83C3×#239—38no—
84C3×#239—68no—
85C3×D3×Q8—38no—
86C2×C3×D12—33no—
87C2×C3×D12—33no—
88C2×C3×D12—33no—
89C2×C3×D12—63no—
90C2×C3×C4×D3—33no—
91C2×C3×C4×D3—33no—
92C2×C3×C4×D3—33no—
93C2×C3×C4×D3—33no—
94C2×C3×C4×D3—63no—
95C2×C3×Dic6—33no—
96C2×#626—36no—
97C2×#625—36no—
98C2×#624—33no—
99C2×#624—31no—
100C2×D3×Dic3—33no—
101C2×D3×Dic3—33no—
102C2×D3×Dic3—33no—
103C2×D3×Dic3—31no—
104#3802—324no—
105D3×D12—324no—
106C4×D32—324no—
107C4×D32—324no—
108#3799—324no—
109#3796—324no—
110#3795—324no—
111#3795—324no—
112C22×H(3,𝔽3)C2411yescharacteristic
113C2×#272—26no—
114C2×#272C2416yes—
115C2×(C3⋊D3)⋊C3—26no—
116C2×(C3⋊D3)⋊C3#4114yescomplemented
117C2×(C3⋊D3)⋊C3C2416yes—
118C2×(C3⋊D3)⋊C3C2412yes—
119((C3⋊D3)⋊C3)⋊C2—224no—
120C4×H(3,𝔽3)C2416yes—
121#1084C2412yes—
122(C3⋊Dic3)⋊C3C2412yes—
123(C3⋊Dic3)⋊C3C2416yes—
124C2×C3×#41—91no—
125C2×C3×#41—91no—
126C22×C3×Q8—91no—
127D3×#41—916no—
128D3×#41—916no—
129C2×#239—31no—
130C2×#239—31no—
131C2×#239—31no—
132C2×#239—61no—
133C2×#239—91no—
134C2×#239—93no—
135C2×#239—93no—
136C2×#239—916no—
137C2×D3×Q8—31no—
138C2×D3×Q8—91no—
139C2×#237—93no—
140C2×#237—93no—
141C2×D3×D4—93no—
142C2×D3×D4—93no—
143C2×#235—93no—
144C2×#235—93no—
145C22×D12—93no—
146C22×C4×D3—93no—
147C22×C4×D3—93no—
148C22×C4×D3—93no—
149C22×C3⋊D3C22×D313yes—
150C22×C3⋊D3C22×D311yescharacteristic
151C22×C3×D3—31no—
152C22×C3×D3—33no—
153C22×C3×D3—33no—
154C22×C3×D3—33no—
155C22×C3×D3—63no—
156C2×D32—312no—
157C2×D32—312no—
158C2×D32—612no—
159C2×D32—612no—
160C32×Q8—24no—
161C32×Q8C22×D314yescomplemented
162C32×Q8C22×D314yescomplemented
163C2×C32×C4—23no—
164C2×C32×C4C22×D313yes—
165C2×C32×C4C22×D313yes—
166C2×C3⋊Dic3C22×D311yescharacteristic
167C2×C3⋊Dic3C22×D313yes—
168C4⋊(C3⋊D3)C22×D3112yescomplemented
169C4×C3⋊D3C22×D3112yes—
170C4×C3⋊D3C22×D3112yescomplemented
171#634C22×D3112yescomplemented
172C2×C3×Dic3—31no—
173C2×C3×Dic3—33no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—61no—
177C3×D12—312no—
178C3×D12—312no—
179C3×D12—312no—
180C3×D12—612no—
181C3×C4×D3—312no—
182C3×C4×D3—312no—
183C3×C4×D3—312no—
184C3×C4×D3—312no—
185C3×C4×D3—612no—
186C3×Dic6—312no—
187#626—324no—
188#625—324no—
189#624—34no—
190#624—312no—
191D3×Dic3—34no—
192D3×Dic3—312no—
193D3×Dic3—312no—
194D3×Dic3—312no—
195C22×#41—271noSylow
196C2×H(3,𝔽3)C2×#4112yescomplemented
197C2×H(3,𝔽3)C2511yescharacteristic
198#272—26no—
199(C3⋊D3)⋊C3—26no—
200(C3⋊D3)⋊C3C2×#4114yescomplemented
201C23×D3—93no—
202C23×D3—93no—
203C23×D3—93no—
204C3×#41—98no—
205C3×#41—98no—
206C2×C3×Q8—31no—
207C2×C3×Q8—31no—
208C2×C3×Q8—61no—
209C2×C3×Q8—93no—
210C2×C3×Q8—98no—
211C2×C3×Q8D3211yescharacteristic
212C2×C3×D4—93no—
213C2×C3×D4—93no—
214C22×C3×C4—93no—
215C22×C3×C4—93no—
216C22×C3×C4—93no—
217C2×#67—96no—
218C2×#67—96no—
219C22×Dic3—93no—
220C22×Dic3—91no—
221C22×Dic3—93no—
222#239—38no—
223#239—38no—
224#239—38no—
225#239—68no—
226#239—98no—
227#239—924no—
228#239—924no—
229D3×Q8—38no—
230D3×Q8—98no—
231#237—924no—
232#237—924no—
233D3×D4—924no—
234D3×D4—924no—
235#235—924no—
236#235—924no—
237C2×D12—33no—
238C2×D12—33no—
239C2×D12—33no—
240C2×D12—63no—
241C2×D12—93no—
242C2×D12—924no—
243C2×D12—96no—
244C2×C4×D3—33no—
245C2×C4×D3—33no—
246C2×C4×D3—33no—
247C2×C4×D3—33no—
248C2×C4×D3—63no—
249C2×C4×D3—93no—
250C2×C4×D3—93no—
251C2×C4×D3—924no—
252C2×C4×D3—924no—
253C2×C4×D3—924no—
254C2×C4×D3—91no—
255C2×C4×D3—93no—
256C2×C4×D3—91no—
257C2×C4×D3—93no—
258C2×C4×D3—93no—
259C2×C4×D3—93no—
260C2×Dic6—33no—
261C2×Dic6—93no—
262C22×C32—21no—
263C22×C32C23×D311yescharacteristic
264C22×C32C23×D311yescharacteristic
265C2×C3⋊D3—26no—
266C2×C3⋊D3#23914yescomplemented
267C2×C3⋊D3C23×D316yes—
268C2×C3⋊D3C23×D312yes—
269C2×C3×D3—34no—
270C2×C3×D3—32no—
271C2×C3×D3—36no—
272C2×C3×D3—36no—
273C2×C3×D3—36no—
274C2×C3×D3—66no—
275C2×C3×D3—66no—
276C2×C3×D3—66no—
277C2×C3×D3—66no—
278C2×C3×D3—126no—
279D32—624no—
280D32—624no—
281C32×C4—26no—
282C32×C4C23×D316yes—
283C32×C4C23×D316yes—
284C3⋊Dic3C23×D312yes—
285C3⋊Dic3C23×D316yes—
286C3×Dic3—32no—
287C3×Dic3—36no—
288C3×Dic3—32no—
289C3×Dic3—32no—
290C3×Dic3—62no—
291C2×#41—91no—
292C2×#41—91no—
293C2×#41—273no—
294C2×#41—273no—
295C2×#41—2716no—
296C22×Q8—91no—
297C22×D4—273no—
298C23×C4—273no—
299H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
300C23×C3—91no—
301C23×C3—93no—
302C23×C3—93no—
303C22×D3—31no—
304C22×D3—33no—
305C22×D3—33no—
306C22×D3—33no—
307C22×D3—63no—
308C22×D3—91no—
309C22×D3—93no—
310C22×D3—912no—
311C22×D3—912no—
312C22×D3—912no—
313C22×D3—1812no—
314C22×D3—1812no—
315C22×D3—1812no—
316C3×Q8—34no—
317C3×Q8—34no—
318C3×Q8—64no—
319C3×Q8—912no—
320C3×Q8C2×D3214yes—
321C3×D4—912no—
322C3×D4—912no—
323C2×C3×C4—33no—
324C2×C3×C4—33no—
325C2×C3×C4—63no—
326C2×C3×C4—912no—
327C2×C3×C4—912no—
328C2×C3×C4—91no—
329C2×C3×C4—93no—
330C2×C3×C4—91no—
331C2×C3×C4—912no—
332C2×C3×C4C2×D3213yes—
333#67—924no—
334#67—924no—
335C2×Dic3—31no—
336C2×Dic3—31no—
337C2×Dic3—33no—
338C2×Dic3—31no—
339C2×Dic3—61no—
340C2×Dic3—93no—
341C2×Dic3—91no—
342C2×Dic3—912no—
343C2×Dic3—94no—
344C2×Dic3—912no—
345D12—312no—
346D12—312no—
347D12—312no—
348D12—612no—
349D12—912no—
350D12—924no—
351C4×D3—312no—
352C4×D3—312no—
353C4×D3—312no—
354C4×D3—312no—
355C4×D3—612no—
356C4×D3—912no—
357C4×D3—912no—
358C4×D3—94no—
359C4×D3—912no—
360C4×D3—94no—
361C4×D3—912no—
362C4×D3—912no—
363C4×D3—912no—
364Dic6—312no—
365Dic6—912no—
366C2×C32—22no—
367C2×C32—21no—
368C2×C32C2×#23912yescomplemented
369C2×C32D3×#4112yescomplemented
370C2×C32C24×D311yescharacteristic
371C2×C32C24×D311yescharacteristic
372C3⋊D3—26no—
373C3⋊D3C2×#23914yescomplemented
374C3×D3—34no—
375C3×D3—66no—
376C3×D3—66no—
377C3×D3—66no—
378C3×D3—126no—
379C24—273no—
380#41—98no—
381#41—98no—
382#41—2724no—
383#41—2724no—
384C2×Q8—98no—
385C2×Q8—93no—
386C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
387C2×D4—93no—
388C2×D4—93no—
389C2×D4—2724no—
390C2×D4—276no—
391C22×C4—93no—
392C22×C4—93no—
393C22×C4—93no—
394C22×C4—2724no—
395C22×C4—271no—
396C22×C4—273no—
397C22×C4—273no—
398C22×C3—31no—
399C22×C3—31no—
400C22×C3—61no—
401C22×C3—92no—
402C22×C3—94no—
403C22×C3—96no—
404C22×C3—96no—
405C22×C3—186no—
406C22×C3—186no—
407C22×C3C22×D3211yescharacteristic
408C2×D3—34no—
409C2×D3—32no—
410C2×D3—36no—
411C2×D3—36no—
412C2×D3—36no—
413C2×D3—66no—
414C2×D3—66no—
415C2×D3—66no—
416C2×D3—66no—
417C2×D3—92no—
418C2×D3—94no—
419C2×D3—96no—
420C2×D3—126no—
421C2×D3—186no—
422C2×D3—1824no—
423C2×D3—1824no—
424C2×D3—1824no—
425C3×C4—36no—
426C3×C4—36no—
427C3×C4—66no—
428C3×C4—92no—
429C3×C4—96no—
430C3×C4—92no—
431C3×C4C22×D3216yes—
432Dic3—32no—
433Dic3—32no—
434Dic3—36no—
435Dic3—32no—
436Dic3—62no—
437Dic3—96no—
438Dic3—92no—
439C32—21no—
440C32C22×#23911yescharacteristic, complemented
441C32C2×D3×#4111yescharacteristic, complemented
442C23—91no—
443C23—93no—
444C23—93no—
445C23—2712no—
446C23—5412no—
447Q8—912no—
448Q8C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
449D4—912no—
450D4—912no—
451D4—2724no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—93no—
455C2×C4—912no—
456C2×C4—912no—
457C2×C4—912no—
458C2×C4—274no—
459C2×C4—2712no—
460C2×C4—2712no—
461C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
462C2×C3—32no—
463C2×C3—32no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—62no—
467C2×C3—61no—
468C2×C3—94no—
469C2×C3—186no—
470C2×C3—186no—
471C2×C3D3×#23912yes—
472C2×C3C23×D3211yescharacteristic
473D3—34no—
474D3—66no—
475D3—66no—
476D3—66no—
477D3—94no—
478D3—126no—
479D3—186no—
480C22—94no—
481C22—92no—
482C22—96no—
483C22—96no—
484C22—186no—
485C22—186no—
486C22—5424no—
487C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
488C4—92no—
489C4—92no—
490C4—96no—
491C4C22×((C3⋊D3)⋊C3)⋊C216yes—
492C3—31no—
493C3—31no—
494C3—61no—
495C3C2×D3×#23911yescharacteristic
496C2—94no—
497C2—186no—
498C2—186no—
499C2#1172251912yesdirect product factor
500C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3C22×D32
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88748C2semidirect
2. C2×#88722C2semidirect
3. C2×Q8×(C3⋊D3)⋊C3C2semidirect
4. #11722519C2direct (N × H)
5. C2×#88654C2semidirect
6. C2×C4×((C3⋊D3)⋊C3)⋊C2C2semidirect
7. C2×#88652C2semidirect
8. C2×#88649C2semidirect
12. C2×Q8×H(3,𝔽3)C22semidirect
13. #88748C22semidirect
14. C2×C4⋊#272C22semidirect
15. C2×C4×#272C22semidirect
16. #88722C22semidirect
17. Q8×(C3⋊D3)⋊C3C22semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. C2×C4×(C3⋊D3)⋊C3C22semidirect
21. C2×#634⋊C3C22semidirect
22. C2×#6200C22semidirect
23. C2×#6199C22semidirect
25. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
26. #88654C22semidirect
27. C4×((C3⋊D3)⋊C3)⋊C2C22semidirect
28. #88652C22semidirect
29. #88649C22semidirect
30. C2×#3832D3semidirect
31. C2×Q8×C3⋊D3D3semidirect
52. Q8×H(3,𝔽3)C23semidirect
55. C4⋊#272C23semidirect
56. C4×#272C23semidirect
59. C4⋊((C3⋊D3)⋊C3)C23semidirect
61. C4×(C3⋊D3)⋊C3C23semidirect
62. #634⋊C3C23semidirect
63. #6200C23semidirect
64. #6199C23semidirect
66. ((C3⋊D3)⋊C3)⋊C4C23semidirect
73. C2×C32×Q8C2×D3semidirect
74. C2×C32×Q8C2×D3semidirect
75. #3832C2×D3semidirect
76. Q8×C3⋊D3C2×D3semidirect
77. C2×C4⋊(C3⋊D3)C2×D3semidirect
79. C2×C4×C3⋊D3C2×D3semidirect
80. C2×#634C2×D3semidirect
116. C2×(C3⋊D3)⋊C3#41semidirect
161. C32×Q8C22×D3semidirect
162. C32×Q8C22×D3semidirect
168. C4⋊(C3⋊D3)C22×D3semidirect
170. C4×C3⋊D3C22×D3semidirect
171. #634C22×D3semidirect
196. C2×H(3,𝔽3)C2×#41semidirect
200. (C3⋊D3)⋊C3C2×#41semidirect
266. C2×C3⋊D3#239semidirect
299. H(3,𝔽3)C22×#41semidirect
368. C2×C32C2×#239semidirect
369. C2×C32D3×#41semidirect
373. C3⋊D3C2×#239semidirect
386. C2×Q8((C3⋊D3)⋊C3)⋊C2semidirect
440. C32C22×#239semidirect
441. C32C2×D3×#41semidirect
448. Q8C2×((C3⋊D3)⋊C3)⋊C2semidirect
499. C2#11722519direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887482
C2×#887222
C2×Q8×(C3⋊D3)⋊C32
#117225192
C2×#886542
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#886522
C2×#886492
C2×#643193
C2×D3×#2393