Finite Group Databasebeta

← C2×#11721125 #13150677

Subgroups — C2×#11721125 #13150677

Order 1728.

1289 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
3C2×#88540C211yesmaximal, characteristic, complemented
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5C2×#88450C211yesmaximal, characteristic, complemented
6C2×#88450C211yesmaximal, characteristic, complemented
7C2×#88450C211yesmaximal, characteristic, complemented
8#11721125C218yesmaximal, direct product factor
9C2×#63897—31nomaximal
10C2×#63884—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C4⋊((C3⋊D3)⋊C3)—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
20C2×((C3⋊D3)⋊C3)⋊C4—24no—
21C2×((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
22#88540C2214yescomplemented
23#3750⋊C3C2214yescomplemented
24#88450C2214yescomplemented
25#88450C2214yescomplemented
26#88450C2214yescomplemented
27C22×C4⋊(C3⋊D3)D311yescharacteristic
28C22×C3×D12—31no—
29C22×#624—31no—
30C22×D3×Dic3—31no—
31C2×#3750D311yescharacteristic, complemented
32C2×C3×Dic3⋊C4—31no—
33C2×C3×Dic3⋊C4—31no—
34C2×C3×Dic3⋊C4—31no—
35C2×C3×Dic3⋊C4—61no—
36C2×#3722—31no—
37C2×#3722—31no—
38C2×#3722—31no—
39C2×(C2×D3)⋊Dic3—31no—
40C2×(C2×D3)⋊Dic3—31no—
41C2×(C2×D3)⋊Dic3—31no—
42#63897—38no—
43#63884—38no—
44C23×H(3,𝔽3)C2311yescharacteristic
45C22×(C3⋊D3)⋊C3—22no—
46C22×(C3⋊D3)⋊C3—22no—
47C22×(C3⋊D3)⋊C3—22no—
48C22×(C3⋊D3)⋊C3—22no—
49C22×(C3⋊D3)⋊C3—22no—
50C22×(C3⋊D3)⋊C3D414yes—
51C22×(C3⋊D3)⋊C3C2312yes—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)C2312yes—
55C2×#1084—22no—
56C2×#1084—22no—
57C2×#1084C2312yes—
58C2×#1084C2312yes—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3—22no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C4⋊((C3⋊D3)⋊C3)—28no—
64((C3⋊D3)⋊C3)⋊C4—28no—
65C2×#935—91no—
66C2×#894—91no—
67C2×#894—91no—
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3⋊D3C2×D311yescharacteristic
70C23×C3×D3—31no—
71C23×C3×D3—31no—
72C22×C32×C4—21no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C2×C4⋊(C3⋊D3)—24no—
78C2×C4⋊(C3⋊D3)C2×D314yes—
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×D12—34no—
88C2×C3×D12—64no—
89C2×#624—34no—
90C2×#624—64no—
91C2×D3×Dic3—34no—
92C2×D3×Dic3—64no—
93#3750C2×D314yescomplemented
94C3×Dic3⋊C4—34no—
95C3×Dic3⋊C4—34no—
96C3×Dic3⋊C4—34no—
97C3×Dic3⋊C4—64no—
98#3722—34no—
99#3722—34no—
100#3722—34no—
101(C2×D3)⋊Dic3—34no—
102(C2×D3)⋊Dic3—34no—
103(C2×D3)⋊Dic3—34no—
104C22×H(3,𝔽3)C2×D412yes—
105C22×H(3,𝔽3)#4112yes—
106C22×H(3,𝔽3)#4112yes—
107C22×H(3,𝔽3)C2411yescharacteristic
108C2×(C3⋊D3)⋊C3—28no—
109C2×(C3⋊D3)⋊C3—22no—
110C2×(C3⋊D3)⋊C3—22no—
111C2×(C3⋊D3)⋊C3—22no—
112C2×(C3⋊D3)⋊C3—22no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—44no—
115C2×(C3⋊D3)⋊C3C2×D414yes—
116C4×H(3,𝔽3)—22no—
117#1084—22no—
118#1084—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—22no—
121C22×C3×D4—91no—
122C23×Dic3—91no—
123C22×D12—31no—
124C22×D12—91no—
125C22×C4×D3—91no—
126C22×C4×D3—91no—
127C2×C3×C4⋊C4—91no—
128C2×C3×C4⋊C4—91no—
129C2×#217—91no—
130C2×#217—91no—
131C2×#217—91no—
132#935—98no—
133C2×(C2×D3)⋊C4—91no—
134C2×(C2×D3)⋊C4—91no—
135C2×(C2×D3)⋊C4—91no—
136C2×(C2×D3)⋊C4—91no—
137C2×(C2×D3)⋊C4—91no—
138C2×(C2×D3)⋊C4—91no—
139C2×Dic3⋊C4—31no—
140C2×Dic3⋊C4—31no—
141C2×Dic3⋊C4—31no—
142C2×Dic3⋊C4—61no—
143C2×Dic3⋊C4—91no—
144#894—98no—
145#894—98no—
146C23×C32—21no—
147C23×C32C22×D311yescharacteristic
148C23×C32C22×D311yescharacteristic
149C22×C3⋊D3—22no—
150C22×C3⋊D3—22no—
151C22×C3⋊D3—22no—
152C22×C3⋊D3—22no—
153C22×C3⋊D3—22no—
154C22×C3⋊D3#6714yes—
155C22×C3⋊D3C22×D312yes—
156C22×C3⋊D3C22×D312yes—
157C22×C3×D3—34no—
158C22×C3×D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C22×C3×D3—62no—
164C22×C3×D3—62no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—42no—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C22×D312yes—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3C22×D312yes—
174C2×C3⋊Dic3C22×D312yes—
175C4⋊(C3⋊D3)—28no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—122no—
192C3×D12—68no—
193#624—68no—
194D3×Dic3—68no—
195C2×#123—271noSylow
196C2×H(3,𝔽3)#12314yescomplemented
197C2×H(3,𝔽3)C22×D411yescharacteristic
198C2×H(3,𝔽3)C2×#4111yescharacteristic
199C2×H(3,𝔽3)C2×#4111yescharacteristic
200(C3⋊D3)⋊C3—24no—
201(C3⋊D3)⋊C3—42no—
202C24×C3—91no—
203C24×C3—91no—
204C23×D3—31no—
205C23×D3—31no—
206C23×D3—91no—
207C23×D3—91no—
208C2×C3×D4—94no—
209C2×C3×D4—184no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—61no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4D3211yescharacteristic
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—31no—
221C22×Dic3—31no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C22×Dic3—61no—
225C22×Dic3—61no—
226C22×Dic3—91no—
227C22×Dic3—91no—
228C22×Dic3—94no—
229C22×Dic3—184no—
230C2×D12—34no—
231C2×D12—64no—
232C2×D12—94no—
233C2×D12—184no—
234C2×C4×D3—94no—
235C2×C4×D3—94no—
236C2×C4×D3—184no—
237C2×C4×D3—184no—
238C3×C4⋊C4—94no—
239C3×C4⋊C4—94no—
240#217—94no—
241#217—94no—
242#217—94no—
243(C2×D3)⋊C4—94no—
244(C2×D3)⋊C4—94no—
245(C2×D3)⋊C4—94no—
246(C2×D3)⋊C4—94no—
247(C2×D3)⋊C4—94no—
248(C2×D3)⋊C4—94no—
249Dic3⋊C4—34no—
250Dic3⋊C4—34no—
251Dic3⋊C4—34no—
252Dic3⋊C4—64no—
253Dic3⋊C4—94no—
254C22×C32—21no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32#23512yes—
259C22×C32D3×D412yes—
260C22×C32#23712yes—
261C22×C32#23712yes—
262C22×C32#23912yes—
263C22×C32C2×#6712yes—
264C22×C32C23×D311yescharacteristic
265C22×C32C23×D311yescharacteristic
266C2×C3⋊D3—28no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—22no—
270C2×C3⋊D3—22no—
271C2×C3⋊D3—22no—
272C2×C3⋊D3—44no—
273C2×C3⋊D3C2×#6714yes—
274C2×C3×D3—34no—
275C2×C3×D3—68no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—124no—
282C32×C4—22no—
283C32×C4—22no—
284C32×C4—42no—
285C3⋊Dic3—22no—
286C3⋊Dic3—22no—
287C3×Dic3—64no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C3×Dic3—122no—
295C22×D4—91no—
296C23×C4—271no—
297#123—278no—
298C2×C4⋊C4—91no—
299C2×C4⋊C4—91no—
300C2×C22⋊C4—271no—
301C2×C22⋊C4—271no—
302C2×C22⋊C4—271no—
303H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
304C23×C3—31no—
305C23×C3—31no—
306C23×C3—61no—
307C23×C3—92no—
308C23×C3—94no—
309C23×C3—92no—
310C23×C3—182no—
311C23×C3—182no—
312C23×C3—182no—
313C23×C3—182no—
314C23×C3—182no—
315C23×C3C2×D3211yescharacteristic
316C22×D3—32no—
317C22×D3—34no—
318C22×D3—32no—
319C22×D3—62no—
320C22×D3—62no—
321C22×D3—62no—
322C22×D3—62no—
323C22×D3—62no—
324C22×D3—94no—
325C22×D3—92no—
326C22×D3—92no—
327C22×D3—182no—
328C22×D3—182no—
329C22×D3—182no—
330C22×D3—182no—
331C22×D3—182no—
332C3×D4—188no—
333C2×C3×C4—22no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—92no—
340C2×C3×C4—92no—
341C2×C3×C4—92no—
342C2×C3×C4—92no—
343C2×C3×C4—122no—
344C2×C3×C4—182no—
345C2×C3×C4—182no—
346C2×C3×C4—182no—
347C2×C3×C4—182no—
348C2×C3×C4C2×D3212yes—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—62no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—64no—
362C2×Dic3—62no—
363C2×Dic3—62no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—122no—
367C2×Dic3—182no—
368C2×Dic3—182no—
369C2×Dic3—188no—
370D12—68no—
371D12—188no—
372C4×D3—188no—
373C4×D3—188no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—24no—
378C2×C32#89414yescomplemented
379C2×C32#93514yescomplemented
380C2×C32C2×#23511yescharacteristic
381C2×C32C2×D3×D411yescharacteristic
382C2×C32C2×#23711yescharacteristic
383C2×C32C2×#23711yescharacteristic
384C2×C32C2×#23911yescharacteristic
385C2×C32C22×#6711yescharacteristic
386C3⋊D3—24no—
387C3⋊D3—42no—
388C3×D3—64no—
389C3×D3—122no—
390C24—91no—
391C24—91no—
392C2×D4—94no—
393C2×D4—184no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—91no—
398C22×C4—274no—
399C22×C4—544no—
400C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C4⋊C4—94no—
402C4⋊C4—94no—
403C22⋊C4—274no—
404C22⋊C4—274no—
405C22⋊C4—274no—
406C22×C3—31no—
407C22×C3—31no—
408C22×C3—32no—
409C22×C3—32no—
410C22×C3—32no—
411C22×C3—32no—
412C22×C3—32no—
413C22×C3—32no—
414C22×C3—61no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—94no—
419C22×C3—188no—
420C22×C3—182no—
421C22×C3—182no—
422C22×C3—182no—
423C22×C3—182no—
424C22×C3—182no—
425C22×C3—364no—
426C22×C3#379512yes—
427C22×C3#379612yes—
428C22×C3D3×#6712yes—
429C22×C3C22×D3211yescharacteristic
430C2×D3—34no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—68no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—94no—
438C2×D3—124no—
439C2×D3—188no—
440C2×D3—182no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—182no—
444C2×D3—182no—
445C2×D3—364no—
446C3×C4—22no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—122no—
450C3×C4—182no—
451C3×C4—182no—
452C3×C4—182no—
453C3×C4—182no—
454Dic3—62no—
455Dic3—62no—
456Dic3—62no—
457Dic3—62no—
458Dic3—64no—
459Dic3—62no—
460Dic3—62no—
461Dic3—122no—
462Dic3—182no—
463Dic3—182no—
464C32—21no—
465C32C2×#89411yescharacteristic, complemented
466C32C2×#93511yescharacteristic, complemented
467C23—92no—
468C23—94no—
469C23—92no—
470C23—182no—
471C23—182no—
472C23—182no—
473C23—182no—
474C23—182no—
475C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
476D4—188no—
477C2×C4—22no—
478C2×C4—92no—
479C2×C4—92no—
480C2×C4—92no—
481C2×C4—92no—
482C2×C4—182no—
483C2×C4—182no—
484C2×C4—182no—
485C2×C4—182no—
486C2×C4—548no—
487C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—34no—
491C2×C3—34no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—31no—
495C2×C3—31no—
496C2×C3—61no—
497C2×C3—64no—
498C2×C3—61no—
499C2×C3—61no—
500C2×C3—184no—
501C2×C3—362no—
502C2×C3#6388414yes—
503C2×C3C2×#379511yescharacteristic
504C2×C3C2×#379611yescharacteristic
505C2×C3C2×D3×#6711yescharacteristic
506D3—64no—
507D3—122no—
508D3—184no—
509D3—362no—
510C22—94no—
511C22—188no—
512C22—182no—
513C22—182no—
514C22—182no—
515C22—182no—
516C22—182no—
517C22—364no—
518C22#8864812yes—
519C22#8864912yes—
520C22#8867612yes—
521C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C4—22no—
523C4—182no—
524C4—182no—
525C4—182no—
526C4—182no—
527C3—31no—
528C3—31no—
529C3—61no—
530C3C2×#6388411yescharacteristic
531C2—184no—
532C2—362no—
533C2#1172112514yesdirect product factor
534C2C2×#8864811yescharacteristic
535C2C2×#8864911yescharacteristic
536C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#123
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
3. C2×#88540C2semidirect
4. C2×#3750⋊C3C2semidirect
5. C2×#88450C2semidirect
6. C2×#88450C2semidirect
7. C2×#88450C2semidirect
8. #11721125C2direct (N × H)
21. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
22. #88540C22semidirect
23. #3750⋊C3C22semidirect
24. #88450C22semidirect
25. #88450C22semidirect
26. #88450C22semidirect
31. C2×#3750D3semidirect
93. #3750C2×D3semidirect
196. C2×H(3,𝔽3)#123semidirect
303. H(3,𝔽3)C2×#123semidirect
378. C2×C32#894semidirect
379. C2×C32#935semidirect
465. C32C2×#894semidirect
466. C32C2×#935semidirect
533. C2#11721125direct (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)⋊C3)2
C22×((C3⋊D3)⋊C3)⋊C42
C2×#885402
C2×#3750⋊C32
C2×#884502
#117211252
C2×#638973
C2×#638843