Finite Group Databasebeta

← C2×#11721119 #13150671

Subgroups — C2×#11721119 #13150671

Order 1728.

1555 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
3C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
4C2×#88540C211yesmaximal, characteristic, complemented
5C2×#3750⋊C3C211yesmaximal, characteristic, complemented
6C2×#88449C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8#11721119C218yesmaximal, direct product factor
9C2×#63883—31nomaximal
10C2×D3×Dic3⋊C4—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3—24no—
19C2×C4×(C3⋊D3)⋊C3C2214yes—
20C2×((C3⋊D3)⋊C3)⋊C4—24no—
21C2×((C3⋊D3)⋊C3)⋊C4C418yescomplemented
22C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
23C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
24#88540C2214yescomplemented
25#3750⋊C3C2214yescomplemented
26#88449C2214yescomplemented
27#88448C2214yescomplemented
28C22×C4×C3⋊D3D311yescharacteristic
29C22×C3×C4×D3—31no—
30C22×#624—31no—
31C22×#624—31no—
32C22×D3×Dic3—31no—
33C22×D3×Dic3—31no—
34C2×#3750D311yescharacteristic, complemented
35C2×C3×Dic3⋊C4—31no—
36C2×C3×Dic3⋊C4—31no—
37C2×C3×Dic3⋊C4—31no—
38C2×C3×Dic3⋊C4—61no—
39C2×#3724—31no—
40C2×Dic3⋊Dic3—31no—
41C2×Dic3⋊Dic3—31no—
42C2×Dic3⋊Dic3—31no—
43#63883—38no—
44D3×Dic3⋊C4—38no—
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×(C3⋊D3)⋊C3C2×C414yes—
47C22×(C3⋊D3)⋊C3D414yes—
48C22×(C3⋊D3)⋊C3Q814yes—
49C22×(C3⋊D3)⋊C3C2312yes—
50C2×C4×H(3,𝔽3)—22no—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×#1084—22no—
53C2×#1084C2×C414yescomplemented
54C2×#1084C2312yes—
55C2×#1084C2312yes—
56C2×(C3⋊Dic3)⋊C3—22no—
57C2×(C3⋊Dic3)⋊C3—22no—
58C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C4×(C3⋊D3)⋊C3—28no—
63((C3⋊D3)⋊C3)⋊C4—28no—
64((C3⋊D3)⋊C3)⋊C4C2×C4116yescomplemented
65C22×Dic3⋊C4—91no—
66C2×D3×C4⋊C4—91no—
67C2×D3×C4⋊C4—91no—
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3×D3—31no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
76C2×C4×C3⋊D3—24no—
77C2×C4×C3⋊D3C2×D314yes—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—31no—
85C22×C3×Dic3—61no—
86C22×C3×Dic3—61no—
87C2×C3×C4×D3—34no—
88C2×C3×C4×D3—64no—
89C2×#624—38no—
90C2×#624—34no—
91C2×#624—34no—
92C2×#624—64no—
93C2×D3×Dic3—38no—
94C2×D3×Dic3—34no—
95C2×D3×Dic3—34no—
96C2×D3×Dic3—64no—
97#3750C2×D314yescomplemented
98C3×Dic3⋊C4—34no—
99C3×Dic3⋊C4—34no—
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—64no—
102#3724—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—34no—
106C22×H(3,𝔽3)C22×C412yes—
107C22×H(3,𝔽3)C2×D412yes—
108C22×H(3,𝔽3)C2×Q812yes—
109C22×H(3,𝔽3)C2411yescharacteristic
110C2×(C3⋊D3)⋊C3C4⋊C4116yescomplemented
111C2×(C3⋊D3)⋊C3C22×C414yes—
112C2×(C3⋊D3)⋊C3C2×D414yes—
113C2×(C3⋊D3)⋊C3C2×Q814yes—
114C4×H(3,𝔽3)—22no—
115#1084—22no—
116#1084C22×C414yescomplemented
117(C3⋊Dic3)⋊C3—22no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3C22×C414yescomplemented
120C23×C3×C4—91no—
121C23×Dic3—91no—
122C23×Dic3—91no—
123C22×C4×D3—31no—
124C22×C4×D3—91no—
125C22×C4×D3—91no—
126C22×C4×D3—91no—
127C22×C4×D3—91no—
128C22×C4×D3—91no—
129C2×C3×C4⋊C4—91no—
130C2×C3×C4⋊C4—91no—
131C2×C4⋊Dic3—91no—
132C2×C4⋊Dic3—91no—
133C2×Dic3⋊C4—31no—
134C2×Dic3⋊C4—31no—
135C2×Dic3⋊C4—31no—
136C2×Dic3⋊C4—61no—
137C2×Dic3⋊C4—91no—
138C2×Dic3⋊C4—98no—
139C2×Dic3⋊C4—91no—
140C2×Dic3⋊C4—91no—
141C2×Dic3⋊C4—91no—
142C2×Dic3⋊C4—91no—
143D3×C4⋊C4—98no—
144D3×C4⋊C4—98no—
145C23×C32—21no—
146C23×C32C22×D311yescharacteristic
147C23×C32C22×D311yescharacteristic
148C22×C3⋊D3Dic614yes—
149C22×C3⋊D3C4×D314yes—
150C22×C3⋊D3#6714yes—
151C22×C3⋊D3C22×D312yes—
152C22×C3×D3—34no—
153C22×C3×D3—34no—
154C22×C3×D3—34no—
155C22×C3×D3—32no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—42no—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D312yes—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3C4×D314yescomplemented
165C2×C3⋊Dic3C22×D312yes—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D312yes—
168C4×C3⋊D3—28no—
169C2×C3×Dic3—34no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—34no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—64no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—122no—
187C3×C4×D3—68no—
188#624—316no—
189#624—68no—
190D3×Dic3—316no—
191D3×Dic3—68no—
192C22×C4⋊C4—271noSylow
193C2×H(3,𝔽3)C2×C4⋊C414yescomplemented
194C2×H(3,𝔽3)C23×C411yescharacteristic
195C2×H(3,𝔽3)C22×D411yescharacteristic
196C2×H(3,𝔽3)C22×Q811yescharacteristic
197(C3⋊D3)⋊C3C2×C4⋊C418yescomplemented
198C24×C3—91no—
199C23×D3—31no—
200C23×D3—91no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—61no—
204C22×C3×C4—94no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×C3×C4—184no—
211C22×C3×C4D3211yescharacteristic
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—31no—
215C22×Dic3—31no—
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C22×Dic3—61no—
220C22×Dic3—61no—
221C22×Dic3—91no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C22×Dic3—94no—
225C22×Dic3—98no—
226C22×Dic3—94no—
227C22×Dic3—184no—
228C2×C4×D3—34no—
229C2×C4×D3—64no—
230C2×C4×D3—94no—
231C2×C4×D3—94no—
232C2×C4×D3—98no—
233C2×C4×D3—94no—
234C2×C4×D3—98no—
235C2×C4×D3—94no—
236C2×C4×D3—94no—
237C2×C4×D3—184no—
238C2×C4×D3—184no—
239C2×C4×D3—184no—
240C3×C4⋊C4—94no—
241C3×C4⋊C4—94no—
242C4⋊Dic3—94no—
243C4⋊Dic3—94no—
244Dic3⋊C4—34no—
245Dic3⋊C4—34no—
246Dic3⋊C4—34no—
247Dic3⋊C4—64no—
248Dic3⋊C4—94no—
249Dic3⋊C4—94no—
250Dic3⋊C4—94no—
251Dic3⋊C4—94no—
252Dic3⋊C4—94no—
253C22×C32—21no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32C2×Dic612yes—
258C22×C32C2×C4×D312yes—
259C22×C32C2×C4×D312yes—
260C22×C32D3×D412yes—
261C22×C32D3×Q812yes—
262C22×C32C2×#6712yes—
263C22×C32C23×D311yescharacteristic
264C22×C32C23×D311yescharacteristic
265C2×C3⋊D3Dic3⋊C4116yescomplemented
266C2×C3⋊D3C2×Dic614yes—
267C2×C3⋊D3C2×C4×D314yes—
268C2×C3⋊D3C2×#6714yes—
269C2×C3×D3—316no—
270C2×C3×D3—34no—
271C2×C3×D3—34no—
272C2×C3×D3—34no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—42no—
276C3⋊Dic3—22no—
277C3⋊Dic3—22no—
278C3⋊Dic3C2×C4×D314yescomplemented
279C3×Dic3—34no—
280C3×Dic3—34no—
281C3×Dic3—34no—
282C3×Dic3—64no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C3×Dic3—122no—
288C23×C4—91no—
289C23×C4—271no—
290C23×C4—271no—
291C2×C4⋊C4—91no—
292C2×C4⋊C4—91no—
293C2×C4⋊C4—278no—
294C2×C4⋊C4—271no—
295C2×C4⋊C4—271no—
296H(3,𝔽3)C22×C4⋊C411yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—61no—
300C23×C3—92no—
301C23×C3—94no—
302C23×C3—94no—
303C23×C3—94no—
304C23×C3C2×D3211yescharacteristic
305C22×D3—32no—
306C22×D3—34no—
307C22×D3—34no—
308C22×D3—34no—
309C22×D3—94no—
310C22×D3—94no—
311C22×D3—94no—
312C22×D3—92no—
313C2×C3×C4—22no—
314C2×C3×C4—32no—
315C2×C3×C4—32no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—92no—
320C2×C3×C4—94no—
321C2×C3×C4—92no—
322C2×C3×C4—94no—
323C2×C3×C4—92no—
324C2×C3×C4—92no—
325C2×C3×C4—92no—
326C2×C3×C4—122no—
327C2×C3×C4—188no—
328C2×C3×C4—182no—
329C2×C3×C4—182no—
330C2×C3×C4—182no—
331C2×C3×C4C2×D3212yes—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—34no—
335C2×Dic3—34no—
336C2×Dic3—32no—
337C2×Dic3—34no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—32no—
342C2×Dic3—62no—
343C2×Dic3—64no—
344C2×Dic3—62no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—62no—
349C2×Dic3—94no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—92no—
353C2×Dic3—916no—
354C2×Dic3—122no—
355C2×Dic3—182no—
356C2×Dic3—182no—
357C2×Dic3—188no—
358C4×D3—68no—
359C4×D3—916no—
360C4×D3—916no—
361C4×D3—188no—
362C4×D3—188no—
363C4×D3—188no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—24no—
368C2×C32D3×C4⋊C414yescomplemented
369C2×C32C2×Dic3⋊C414yescomplemented
370C2×C32C22×Dic611yescharacteristic
371C2×C32C22×C4×D311yescharacteristic
372C2×C32C22×C4×D311yescharacteristic
373C2×C32C2×D3×D411yescharacteristic
374C2×C32C2×D3×Q811yescharacteristic
375C2×C32C22×#6711yescharacteristic
376C3⋊D3C2×Dic3⋊C418yescomplemented
377C3×D3—38no—
378C24—91no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—94no—
385C22×C4—184no—
386C22×C4—274no—
387C22×C4—278no—
388C22×C4—274no—
389C22×C4—544no—
390C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
391C4⋊C4—94no—
392C4⋊C4—94no—
393C4⋊C4—274no—
394C4⋊C4—274no—
395C22×C3—31no—
396C22×C3—31no—
397C22×C3—32no—
398C22×C3—32no—
399C22×C3—32no—
400C22×C3—32no—
401C22×C3—32no—
402C22×C3—32no—
403C22×C3—61no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—94no—
408C22×C3—94no—
409C22×C3—916no—
410C22×C3—94no—
411C22×C3D3×Dic612yes—
412C22×C3C4×D3212yes—
413C22×C3D3×#6712yes—
414C22×C3C22×D3211yescharacteristic
415C2×D3—34no—
416C2×D3—34no—
417C2×D3—34no—
418C2×D3—316no—
419C2×D3—94no—
420C2×D3—94no—
421C2×D3—916no—
422C2×D3—94no—
423C3×C4—22no—
424C3×C4—62no—
425C3×C4—62no—
426C3×C4—94no—
427C3×C4—94no—
428C3×C4—122no—
429C3×C4—182no—
430C3×C4—182no—
431C3×C4—182no—
432Dic3—34no—
433Dic3—34no—
434Dic3—34no—
435Dic3—64no—
436Dic3—62no—
437Dic3—62no—
438Dic3—62no—
439Dic3—62no—
440Dic3—94no—
441Dic3—122no—
442Dic3—182no—
443Dic3—182no—
444C32—21no—
445C32C2×D3×C4⋊C411yescharacteristic, complemented
446C32C22×Dic3⋊C411yescharacteristic, complemented
447C23—92no—
448C23—94no—
449C23—94no—
450C23—94no—
451C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
452C2×C4—22no—
453C2×C4—92no—
454C2×C4—92no—
455C2×C4—94no—
456C2×C4—94no—
457C2×C4—92no—
458C2×C4—92no—
459C2×C4—92no—
460C2×C4—188no—
461C2×C4—182no—
462C2×C4—182no—
463C2×C4—182no—
464C2×C4—2716no—
465C2×C4—548no—
466C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—34no—
470C2×C3—34no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—61no—
476C2×C3—64no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—98no—
480C2×C3D3×Dic3⋊C414yes—
481C2×C3C2×D3×Dic611yescharacteristic
482C2×C3C2×C4×D3211yescharacteristic
483C2×C3C2×D3×#6711yescharacteristic
484D3—38no—
485D3—98no—
486C22—94no—
487C22—916no—
488C22—94no—
489C22—94no—
490C22((C3⋊D3)⋊C3)⋊Q812yes—
491C22C4×((C3⋊D3)⋊C3)⋊C212yes—
492C22#8867612yes—
493C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
494C4—22no—
495C4—94no—
496C4—94no—
497C4—182no—
498C4—182no—
499C4—182no—
500C3—31no—
501C3—31no—
502C3—61no—
503C3C2×D3×Dic3⋊C411yescharacteristic
504C2—98no—
505C2#1172111914yesdirect product factor
506C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
507C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
508C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×C4⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×#88540C2semidirect
5. C2×#3750⋊C3C2semidirect
6. C2×#88449C2semidirect
7. C2×#88448C2semidirect
8. #11721119C2direct (N × H)
21. C2×((C3⋊D3)⋊C3)⋊C4C4semidirect
24. #88540C22semidirect
25. #3750⋊C3C22semidirect
26. #88449C22semidirect
27. #88448C22semidirect
34. C2×#3750D3semidirect
53. C2×#1084C2×C4semidirect
58. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
64. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
97. #3750C2×D3semidirect
110. C2×(C3⋊D3)⋊C3C4⋊C4semidirect
116. #1084C22×C4semidirect
119. (C3⋊Dic3)⋊C3C22×C4semidirect
164. C2×C3⋊Dic3C4×D3semidirect
193. C2×H(3,𝔽3)C2×C4⋊C4semidirect
197. (C3⋊D3)⋊C3C2×C4⋊C4semidirect
265. C2×C3⋊D3Dic3⋊C4semidirect
278. C3⋊Dic3C2×C4×D3semidirect
296. H(3,𝔽3)C22×C4⋊C4semidirect
368. C2×C32D3×C4⋊C4semidirect
369. C2×C32C2×Dic3⋊C4semidirect
376. C3⋊D3C2×Dic3⋊C4semidirect
445. C32C2×D3×C4⋊C4semidirect
446. C32C22×Dic3⋊C4semidirect
505. C2#11721119direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C22×((C3⋊D3)⋊C3)⋊C42
C2×#885402
C2×#3750⋊C32
C2×#884492
C2×#884482
#117211192
C2×#638833
C2×D3×Dic3⋊C43