Finite Group Databasebeta

← C2×#11722516 #13164764

Subgroups — C2×#11722516 #13164764

Order 1728.

2935 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×#272C211yesmaximal, characteristic, complemented
2C2×#88722C211yesmaximal, characteristic, complemented
3C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#11722516C2116yesmaximal, direct product factor
5C2×#88650C213yesmaximal, complemented
6C2×#88649C213yesmaximal, complemented
7C2×#88648C213yesmaximal, complemented
8C2×((C3⋊D3)⋊C3)⋊Q8C213yesmaximal, complemented
9C2×#64316—31nomaximal
10C2×#64315—31nomaximal
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12Q8×#272C2218yescomplemented
13C2×C4×#272C2213yes—
14C2×#6229C2213yes—
15#88722C2218yescomplemented
16Q8×(C3⋊D3)⋊C3C2218yescomplemented
17C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
18C2×C4×(C3⋊D3)⋊C3C2213yes—
19C2×C4×(C3⋊D3)⋊C3C2213yes—
20C2×#634⋊C3C2213yes—
21C2×#6200C2213yes—
22C2×#6200C2211yescharacteristic
23C2×#6198C2213yes—
24C2×((C3⋊D3)⋊C3)⋊C4C2216yes—
25C2×#6196C2213yes—
26#88650C22124yescomplemented
27#88649C22124yescomplemented
28#88648C22124yescomplemented
29((C3⋊D3)⋊C3)⋊Q8C22124yescomplemented
30C2×#3832D311yescharacteristic, complemented
31C2×Q8×C3⋊D3D311yescharacteristic, complemented
32C2×C3×#239—31no—
33C2×C3×D3×Q8—31no—
34C2×C3×D3×Q8—31no—
35C2×C3×D3×Q8—31no—
36C2×C3×D3×Q8—61no—
37#64316—316no—
38#64315—316no—
39C2×#3799—33no—
40C2×#3799—33no—
41C2×#3798—33no—
42C2×#3798—33no—
43C2×#3797—33no—
44C2×#3796—33no—
45C2×#3795—33no—
46C2×D3×Dic6—33no—
47C22×#272C2311yescharacteristic
48C22×(C3⋊D3)⋊C3C2313yes—
49C22×(C3⋊D3)⋊C3C2311yescharacteristic
50Q8×H(3,𝔽3)C2314yes—
51C2×C4×H(3,𝔽3)C2313yes—
52C2×#1084C2313yes—
53C4×#272C23112yes—
54#6229C23112yes—
55C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
56C2×(C3⋊Dic3)⋊C3C2313yes—
57C4⋊((C3⋊D3)⋊C3)C23112yes—
58C4×(C3⋊D3)⋊C3C23112yes—
59C4×(C3⋊D3)⋊C3C23112yes—
60#634⋊C3C23112yes—
61#6200C23112yes—
62#6200C2314yes—
63#6198C23112yes—
64((C3⋊D3)⋊C3)⋊C4C23124yes—
65#6196C23112yes—
66C2×#1010—91no—
67C2×#1007—91no—
68C2×#1007—91no—
69C2×C32×Q8—21no—
70C2×C32×Q8C2×D311yescharacteristic
71C2×C32×Q8C2×D311yescharacteristic
72#3832C2×D318yescomplemented
73Q8×C3⋊D3C2×D318yescomplemented
74C2×C4⋊(C3⋊D3)C2×D313yes—
75C2×C4×C3⋊D3C2×D313yes—
76C2×C4×C3⋊D3C2×D313yes—
77C2×#634C2×D313yes—
78C3×#239—38no—
79C3×D3×Q8—38no—
80C3×D3×Q8—38no—
81C3×D3×Q8—38no—
82C3×D3×Q8—68no—
83C2×C3×D12—33no—
84C2×C3×C4×D3—33no—
85C2×C3×C4×D3—33no—
86C2×C3×C4×D3—33no—
87C2×C3×C4×D3—33no—
88C2×C3×C4×D3—63no—
89C2×C3×Dic6—33no—
90C2×C3×Dic6—33no—
91C2×C3×Dic6—33no—
92C2×C3×Dic6—63no—
93C2×#627—33no—
94C2×#627—33no—
95C2×#626—33no—
96C2×#626—31no—
97C2×#626—33no—
98C2×#626—33no—
99C2×#625—33no—
100C2×#625—31no—
101C2×#624—36no—
102C2×D3×Dic3—36no—
103#3799—324no—
104#3799—324no—
105#3798—324no—
106#3798—324no—
107#3797—324no—
108#3796—324no—
109#3795—324no—
110D3×Dic6—324no—
111C22×H(3,𝔽3)C2411yescharacteristic
112C2×#272—22no—
113C2×#272C2412yes—
114C2×(C3⋊D3)⋊C3—26no—
115C2×(C3⋊D3)⋊C3—22no—
116C2×(C3⋊D3)⋊C3C2416yes—
117C2×(C3⋊D3)⋊C3C2412yes—
118C4×H(3,𝔽3)C2416yes—
119#1084C2416yes—
120(C3⋊Dic3)⋊C3C2412yes—
121(C3⋊Dic3)⋊C3C2416yes—
122C2×C3×#41—91no—
123C22×C3×Q8—91no—
124C22×C3×Q8—91no—
125#1010—916no—
126#1007—916no—
127#1007—916no—
128C2×#239—31no—
129C2×#239—91no—
130C2×#239—93no—
131C2×#239—93no—
132C2×D3×Q8—31no—
133C2×D3×Q8—31no—
134C2×D3×Q8—31no—
135C2×D3×Q8—61no—
136C2×D3×Q8—91no—
137C2×D3×Q8—93no—
138C2×D3×Q8—93no—
139C2×#237—93no—
140C2×#237—93no—
141C2×#235—93no—
142C2×#235—93no—
143C2×#235—93no—
144C2×#235—93no—
145C2×#235—93no—
146C22×Dic6—93no—
147C22×C3⋊D3C22×D313yes—
148C22×C3⋊D3C22×D311yescharacteristic
149C22×C3×D3—33no—
150C22×C3×D3—31no—
151C22×C3×D3—31no—
152C22×C3×D3—31no—
153C22×C3×D3—61no—
154C32×Q8—24no—
155C32×Q8C22×D314yes—
156C32×Q8C22×D314yes—
157C2×C32×C4—23no—
158C2×C32×C4C22×D313yes—
159C2×C32×C4C22×D313yes—
160C2×C3⋊Dic3C22×D313yes—
161C2×C3⋊Dic3C22×D311yescharacteristic
162C4⋊(C3⋊D3)C22×D3112yes—
163C4×C3⋊D3C22×D3112yes—
164C4×C3⋊D3C22×D3112yes—
165#634C22×D3112yes—
166C2×C3×Dic3—33no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—33no—
169C2×C3×Dic3—33no—
170C2×C3×Dic3—63no—
171C3×D12—312no—
172C3×C4×D3—312no—
173C3×C4×D3—312no—
174C3×C4×D3—312no—
175C3×C4×D3—312no—
176C3×C4×D3—612no—
177C3×Dic6—312no—
178C3×Dic6—312no—
179C3×Dic6—312no—
180C3×Dic6—612no—
181#627—312no—
182#627—312no—
183#626—312no—
184#626—312no—
185#626—34no—
186#626—312no—
187#625—312no—
188#625—34no—
189#624—324no—
190D3×Dic3—324no—
191C2×2−1+4—271noSylow
192C2×H(3,𝔽3)2−1+412yescomplemented
193C2×H(3,𝔽3)C2511yescharacteristic
194#272—22no—
195(C3⋊D3)⋊C3—26no—
196(C3⋊D3)⋊C3—22no—
197C3×#41—98no—
198C2×C3×Q8—31no—
199C2×C3×Q8—31no—
200C2×C3×Q8—61no—
201C2×C3×Q8—93no—
202C2×C3×Q8—98no—
203C2×C3×Q8—98no—
204C2×C3×Q8—93no—
205C2×C3×Q8D3211yescharacteristic
206C2×C3×D4—93no—
207C22×C3×C4—93no—
208C22×C3×C4—93no—
209C22×C3×C4—93no—
210C2×#67—93no—
211C2×#67—91no—
212C2×#67—93no—
213C2×#67—93no—
214C2×#67—91no—
215C2×#67—93no—
216C22×Dic3—96no—
217#239—38no—
218#239—98no—
219#239—924no—
220#239—924no—
221D3×Q8—38no—
222D3×Q8—38no—
223D3×Q8—38no—
224D3×Q8—68no—
225D3×Q8—98no—
226D3×Q8—924no—
227D3×Q8—924no—
228#237—924no—
229#237—924no—
230#235—924no—
231#235—924no—
232#235—924no—
233#235—924no—
234#235—924no—
235C2×D12—33no—
236C2×D12—93no—
237C2×D12—93no—
238C2×D12—91no—
239C2×D12—93no—
240C2×C4×D3—33no—
241C2×C4×D3—33no—
242C2×C4×D3—33no—
243C2×C4×D3—33no—
244C2×C4×D3—63no—
245C2×C4×D3—93no—
246C2×C4×D3—93no—
247C2×C4×D3—96no—
248C2×C4×D3—96no—
249C2×Dic6—33no—
250C2×Dic6—33no—
251C2×Dic6—33no—
252C2×Dic6—63no—
253C2×Dic6—93no—
254C2×Dic6—924no—
255C2×Dic6—93no—
256C2×Dic6—93no—
257C2×Dic6—93no—
258C22×C32—21no—
259C22×C32C23×D311yescharacteristic
260C22×C32C23×D311yescharacteristic
261C2×C3⋊D3—22no—
262C2×C3⋊D3—26no—
263C2×C3⋊D3C23×D316yes—
264C2×C3⋊D3C23×D312yes—
265C2×C3×D3—36no—
266C2×C3×D3—32no—
267C2×C3×D3—32no—
268C2×C3×D3—32no—
269C2×C3×D3—62no—
270C2×C3×D3—64no—
271C2×C3×D3—66no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C32×C4—26no—
276C32×C4C23×D316yes—
277C32×C4C23×D316yes—
278C3⋊Dic3C23×D316yes—
279C3⋊Dic3C23×D312yes—
280C3×Dic3—36no—
281C3×Dic3—32no—
282C3×Dic3—36no—
283C3×Dic3—36no—
284C3×Dic3—66no—
2852−1+4—2716no—
286C2×#41—91no—
287C2×#41—273no—
288C2×#41—273no—
289C2×#41—273no—
290C22×Q8—91no—
291C22×Q8—91no—
292C22×Q8—273no—
293H(3,𝔽3)C2×2−1+411yescharacteristic, complemented, Sylow
294C23×C3—93no—
295C23×C3—91no—
296C23×C3—91no—
297C22×D3—31no—
298C22×D3—31no—
299C22×D3—31no—
300C22×D3—33no—
301C22×D3—61no—
302C22×D3—93no—
303C22×D3—91no—
304C3×Q8—34no—
305C3×Q8—34no—
306C3×Q8—64no—
307C3×Q8—912no—
308C3×Q8—912no—
309C3×Q8C2×D3214yes—
310C3×D4—912no—
311C2×C3×C4—33no—
312C2×C3×C4—33no—
313C2×C3×C4—63no—
314C2×C3×C4—912no—
315C2×C3×C4—912no—
316C2×C3×C4—91no—
317C2×C3×C4—93no—
318C2×C3×C4—93no—
319C2×C3×C4—912no—
320C2×C3×C4C2×D3213yes—
321#67—912no—
322#67—94no—
323#67—912no—
324#67—912no—
325#67—94no—
326#67—912no—
327C2×Dic3—33no—
328C2×Dic3—33no—
329C2×Dic3—33no—
330C2×Dic3—31no—
331C2×Dic3—63no—
332C2×Dic3—91no—
333C2×Dic3—93no—
334C2×Dic3—924no—
335D12—312no—
336D12—912no—
337D12—912no—
338D12—94no—
339D12—912no—
340C4×D3—312no—
341C4×D3—312no—
342C4×D3—312no—
343C4×D3—312no—
344C4×D3—612no—
345C4×D3—912no—
346C4×D3—912no—
347C4×D3—924no—
348C4×D3—924no—
349Dic6—312no—
350Dic6—312no—
351Dic6—312no—
352Dic6—612no—
353Dic6—912no—
354Dic6—912no—
355Dic6—912no—
356Dic6—912no—
357C2×C32—22no—
358C2×C32—21no—
359C2×C32#100712yescomplemented
360C2×C32#101012yescomplemented
361C2×C32C24×D311yescharacteristic
362C2×C32C24×D311yescharacteristic
363C3⋊D3—22no—
364C3⋊D3—26no—
365C3×D3—64no—
366C3×D3—66no—
367C3×D3—62no—
368C3×D3—62no—
369C3×D3—62no—
370#41—98no—
371#41—2724no—
372#41—2724no—
373#41—2724no—
374C2×Q8—98no—
375C2×Q8—98no—
376C2×Q8—93no—
377C2×Q8—93no—
378C2×Q8—273no—
379C2×Q8—2724no—
380C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C2×D4—93no—
382C2×D4—271no—
383C2×D4—273no—
384C2×D4—273no—
385C22×C4—93no—
386C22×C4—93no—
387C22×C4—93no—
388C22×C4—276no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—61no—
392C22×C3—96no—
393C22×C3—92no—
394C22×C3—92no—
395C22×C3—182no—
396C22×C3—186no—
397C22×C3—182no—
398C22×C3C22×D3211yescharacteristic
399C2×D3—32no—
400C2×D3—32no—
401C2×D3—32no—
402C2×D3—36no—
403C2×D3—62no—
404C2×D3—62no—
405C2×D3—62no—
406C2×D3—64no—
407C2×D3—62no—
408C2×D3—66no—
409C2×D3—96no—
410C2×D3—92no—
411C2×D3—186no—
412C2×D3—182no—
413C3×C4—36no—
414C3×C4—36no—
415C3×C4—66no—
416C3×C4—92no—
417C3×C4—96no—
418C3×C4—96no—
419C3×C4C22×D3216yes—
420Dic3—36no—
421Dic3—36no—
422Dic3—36no—
423Dic3—32no—
424Dic3—66no—
425Dic3—92no—
426Dic3—96no—
427C32—21no—
428C32C2×#100711yescharacteristic, complemented
429C32C2×#101011yescharacteristic, complemented
430C23—91no—
431C23—91no—
432C23—93no—
433Q8—912no—
434Q8—912no—
435Q8—2712no—
436Q8C2×((C3⋊D3)⋊C3)⋊C214yes—
437D4—912no—
438D4—274no—
439D4—2712no—
440D4—2712no—
441C2×C4—91no—
442C2×C4—93no—
443C2×C4—93no—
444C2×C4—912no—
445C2×C4—912no—
446C2×C4—912no—
447C2×C4—2724no—
448C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
449C2×C3—32no—
450C2×C3—32no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—62no—
454C2×C3—61no—
455C2×C3—182no—
456C2×C3—186no—
457C2×C3—182no—
458C2×C3#6431512yes—
459C2×C3C23×D3211yescharacteristic
460D3—62no—
461D3—62no—
462D3—64no—
463D3—62no—
464D3—66no—
465D3—186no—
466D3—182no—
467C22—92no—
468C22—92no—
469C22—96no—
470C22—182no—
471C22—182no—
472C22—186no—
473C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
474C4—92no—
475C4—96no—
476C4—96no—
477C4C22×((C3⋊D3)⋊C3)⋊C216yes—
478C3—31no—
479C3—31no—
480C3—61no—
481C3C2×#6431511yescharacteristic
482C2—182no—
483C2—182no—
484C2—186no—
485C2#1172251612yesdirect product factor
486C2C23×((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)C2×2−1+4
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×Q8×#272C2semidirect
2. C2×#88722C2semidirect
3. C2×Q8×(C3⋊D3)⋊C3C2semidirect
4. #11722516C2direct (N × H)
5. C2×#88650C2semidirect
6. C2×#88649C2semidirect
7. C2×#88648C2semidirect
8. C2×((C3⋊D3)⋊C3)⋊Q8C2semidirect
12. Q8×#272C22semidirect
15. #88722C22semidirect
16. Q8×(C3⋊D3)⋊C3C22semidirect
26. #88650C22semidirect
27. #88649C22semidirect
28. #88648C22semidirect
29. ((C3⋊D3)⋊C3)⋊Q8C22semidirect
30. C2×#3832D3semidirect
31. C2×Q8×C3⋊D3D3semidirect
72. #3832C2×D3semidirect
73. Q8×C3⋊D3C2×D3semidirect
192. C2×H(3,𝔽3)2−1+4semidirect
293. H(3,𝔽3)C2×2−1+4semidirect
359. C2×C32#1007semidirect
360. C2×C32#1010semidirect
428. C32C2×#1007semidirect
429. C32C2×#1010semidirect
485. C2#11722516direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×#887222
C2×Q8×(C3⋊D3)⋊C32
#117225162
C2×#886502
C2×#886492
C2×#886482
C2×((C3⋊D3)⋊C3)⋊Q82
C2×#643163
C2×#643153