Finite Group Databasebeta

← C2×#11721299 #13152055

Subgroups — C2×#11721299 #13152055

Order 1728.

1259 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C22×#6196C211yesmaximal, characteristic, complemented
3C2×#88552C211yesmaximal, characteristic
4C2×#3757⋊C3C211yesmaximal, characteristic
5C2×#88449C211yesmaximal, characteristic, complemented
6C2×#88448C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8#11721299C218yesmaximal, direct product factor
9C2×#63982—31nomaximal
10C2×#63964—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3—24no—
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2214yes—
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×#6196—24no—
21C2×#6196C2214yescomplemented
22#88552C2214yes—
23#3757⋊C3C2214yes—
24#88449C2214yescomplemented
25#88448C2214yescomplemented
26#88448C2214yescomplemented
27C23×C3⋊Dic3D311yescharacteristic
28C23×C3×Dic3—31no—
29C22×#627—31no—
30C22×#627—31no—
31C2×#3757D311yescharacteristic
32C2×C3×#217—31no—
33C2×C3×#217—31no—
34C2×C3×#217—31no—
35C2×C3×#217—61no—
36#63982—38no—
37C2×#3724—31no—
38C2×#3724—31no—
39C2×Dic3⋊Dic3—31no—
40C2×Dic3⋊Dic3—31no—
41C2×Dic3⋊Dic3—31no—
42C2×Dic3⋊Dic3—31no—
43#63964—38no—
44C23×H(3,𝔽3)—22no—
45C23×H(3,𝔽3)—22no—
46C23×H(3,𝔽3)Q814yescomplemented
47C23×H(3,𝔽3)C2312yes—
48C23×H(3,𝔽3)C2311yescharacteristic
49C2×#1084—22no—
50C2×#1084—22no—
51C2×#1084C2312yes—
52C2×#1084C2312yes—
53C2×(C3⋊Dic3)⋊C3—22no—
54C2×(C3⋊Dic3)⋊C3—22no—
55C2×(C3⋊Dic3)⋊C3—22no—
56C2×(C3⋊Dic3)⋊C3—28no—
57C2×(C3⋊Dic3)⋊C3D414yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62#6196—28no—
63C2×#924—91no—
64C2×#878—91no—
65C2×#878—91no—
66C24×C32—21no—
67C24×C32C2×D311yescharacteristic
68C24×C32C2×D311yescharacteristic
69C22×C3⋊Dic3—24no—
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D314yes—
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—34no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—64no—
85C22×C3×Dic3—61no—
86C22×C3×Dic3—61no—
87C2×#627—34no—
88C2×#627—34no—
89C2×#627—64no—
90C2×#627—64no—
91#3757C2×D314yes—
92C3×#217—34no—
93C3×#217—34no—
94C3×#217—34no—
95C3×#217—64no—
96#3724—34no—
97#3724—34no—
98Dic3⋊Dic3—34no—
99Dic3⋊Dic3—34no—
100Dic3⋊Dic3—34no—
101Dic3⋊Dic3—34no—
102C22×H(3,𝔽3)—28no—
103C22×H(3,𝔽3)—22no—
104C22×H(3,𝔽3)—22no—
105C22×H(3,𝔽3)C2×D412yes—
106C22×H(3,𝔽3)C2×Q812yes—
107C22×H(3,𝔽3)C2×Q814yescomplemented
108C22×H(3,𝔽3)#4112yes—
109C22×H(3,𝔽3)C2411yescharacteristic
110#1084—22no—
111#1084—22no—
112(C3⋊Dic3)⋊C3—22no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3C2×D414yes—
116C23×C3×C4—91no—
117C23×Dic3—31no—
118C23×Dic3—91no—
119C22×Dic6—91no—
120C22×Dic6—91no—
121C22×Dic6—91no—
122C2×C3×C22⋊C4—91no—
123C2×C3×C22⋊C4—91no—
124C2×#217—31no—
125C2×#217—31no—
126C2×#217—31no—
127C2×#217—61no—
128C2×#217—91no—
129C2×C4⋊Dic3—91no—
130C2×C4⋊Dic3—91no—
131C2×C4⋊Dic3—91no—
132#924—98no—
133C2×Dic3⋊C4—91no—
134C2×Dic3⋊C4—91no—
135C2×Dic3⋊C4—91no—
136C2×Dic3⋊C4—91no—
137C2×Dic3⋊C4—91no—
138C2×Dic3⋊C4—91no—
139#878—98no—
140#878—98no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32—22no—
145C23×C32—24no—
146C23×C32—21no—
147C23×C32—22no—
148C23×C32—42no—
149C23×C32—42no—
150C23×C32Dic614yescomplemented
151C23×C32Dic614yescomplemented
152C23×C32C22×D312yes—
153C23×C32C22×D312yes—
154C23×C32C22×D311yescharacteristic
155C23×C32C22×D311yescharacteristic
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—28no—
160C2×C3⋊Dic3#6714yes—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D312yes—
163C2×C3⋊Dic3C22×D312yes—
164C2×C3⋊Dic3C22×D312yes—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—68no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—122no—
185C2×C3×Dic3—122no—
186#627—68no—
187#627—68no—
188C2×C22⋊Q8—271noSylow
189C2×H(3,𝔽3)—24no—
190C2×H(3,𝔽3)C22⋊Q814yescomplemented
191C2×H(3,𝔽3)C22×D411yescharacteristic
192C2×H(3,𝔽3)C22×Q811yescharacteristic
193C2×H(3,𝔽3)C2×#4111yescharacteristic
194C24×C3—31no—
195C24×C3—31no—
196C24×C3—61no—
197C24×C3D3211yescharacteristic
198C22×C3×C4—91no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C22×C3×C4—94no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—184no—
206C22×Dic3—31no—
207C22×Dic3—31no—
208C22×Dic3—31no—
209C22×Dic3—31no—
210C22×Dic3—31no—
211C22×Dic3—34no—
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—31no—
215C22×Dic3—61no—
216C22×Dic3—61no—
217C22×Dic3—64no—
218C22×Dic3—91no—
219C22×Dic3—94no—
220C22×Dic3—91no—
221C22×Dic3—91no—
222C22×Dic3—91no—
223C22×Dic3—184no—
224C2×Dic6—94no—
225C2×Dic6—94no—
226C2×Dic6—94no—
227C2×Dic6—184no—
228C2×Dic6—184no—
229C2×Dic6—184no—
230C3×C22⋊C4—94no—
231C3×C22⋊C4—94no—
232#217—34no—
233#217—34no—
234#217—34no—
235#217—64no—
236#217—94no—
237C4⋊Dic3—94no—
238C4⋊Dic3—94no—
239C4⋊Dic3—94no—
240Dic3⋊C4—94no—
241Dic3⋊C4—94no—
242Dic3⋊C4—94no—
243Dic3⋊C4—94no—
244Dic3⋊C4—94no—
245Dic3⋊C4—94no—
246C22×C32—28no—
247C22×C32—28no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—24no—
256C22×C32—22no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—48no—
260C22×C32C2×Dic612yes—
261C22×C32C2×Dic612yes—
262C22×C32C2×Dic614yescomplemented
263C22×C32C2×Dic614yescomplemented
264C22×C32#23512yes—
265C22×C32D3×D412yes—
266C22×C32#23712yes—
267C22×C32C2×#6712yes—
268C22×C32C23×D311yescharacteristic
269C22×C32C23×D311yescharacteristic
270C3⋊Dic3—22no—
271C3⋊Dic3—22no—
272C3⋊Dic3—22no—
273C3⋊Dic3C2×#6714yes—
274C3×Dic3—34no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—122no—
283C3×Dic3—122no—
284C22×Q8—271no—
285C23×C4—91no—
286C22⋊Q8—278no—
287C2×C4⋊C4—271no—
288C2×C4⋊C4—271no—
289C2×C4⋊C4—271no—
290C2×C22⋊C4—91no—
291C2×C22⋊C4—91no—
292H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—34no—
296C23×C3—34no—
297C23×C3—32no—
298C23×C3—32no—
299C23×C3—31no—
300C23×C3—31no—
301C23×C3—62no—
302C23×C3—62no—
303C23×C3—62no—
304C23×C3—62no—
305C23×C3—64no—
306C23×C3—62no—
307C23×C3—61no—
308C23×C3—122no—
309C23×C3—122no—
310C23×C3#62714yes—
311C23×C3C2×D3212yes—
312C23×C3C2×D3211yescharacteristic
313C2×C3×C4—92no—
314C2×C3×C4—94no—
315C2×C3×C4—92no—
316C2×C3×C4—92no—
317C2×C3×C4—92no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4—188no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323C2×C3×C4—182no—
324C2×C3×C4—182no—
325C2×C3×C4—182no—
326C2×Dic3—32no—
327C2×Dic3—34no—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—32no—
334C2×Dic3—32no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—68no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—94no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—92no—
350C2×Dic3—122no—
351C2×Dic3—122no—
352C2×Dic3—188no—
353C2×Dic3—182no—
354C2×Dic3—182no—
355C2×Dic3—182no—
356Dic6—188no—
357Dic6—188no—
358Dic6—188no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—24no—
365C2×C32—44no—
366C2×C32#87814yescomplemented
367C2×C32#92414yescomplemented
368C2×C32C22×Dic611yescharacteristic
369C2×C32C22×Dic611yescharacteristic
370C2×C32C2×#23511yescharacteristic
371C2×C32C2×D3×D411yescharacteristic
372C2×C32C2×#23711yescharacteristic
373C2×C32C22×#6711yescharacteristic
374C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
375C2×Q8—274no—
376C2×Q8—544no—
377C22×C4—91no—
378C22×C4—94no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4—91no—
383C22×C4—91no—
384C22×C4—184no—
385C4⋊C4—274no—
386C4⋊C4—274no—
387C4⋊C4—274no—
388C22⋊C4—94no—
389C22⋊C4—94no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—28no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—32no—
396C22×C3—32no—
397C22×C3—34no—
398C22×C3—34no—
399C22×C3—32no—
400C22×C3—32no—
401C22×C3—32no—
402C22×C3—32no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—68no—
408C22×C3—68no—
409C22×C3—61no—
410C22×C3—62no—
411C22×C3—64no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—122no—
415C22×C3—122no—
416C22×C3—128no—
417C22×C3#380412yes—
418C22×C3C2×#62712yes—
419C22×C3C2×#62714yes—
420C22×C3D3×#6712yes—
421C22×C3C22×D3211yescharacteristic
422C3×C4—94no—
423C3×C4—182no—
424C3×C4—182no—
425C3×C4—182no—
426C3×C4—182no—
427C3×C4—182no—
428Dic3—34no—
429Dic3—62no—
430Dic3—62no—
431Dic3—62no—
432Dic3—62no—
433Dic3—62no—
434Dic3—62no—
435Dic3—62no—
436Dic3—94no—
437Dic3—122no—
438Dic3—122no—
439Dic3—182no—
440Dic3—182no—
441Dic3—182no—
442C32—21no—
443C32C2×#87811yescharacteristic, complemented
444C32C2×#92411yescharacteristic, complemented
445C23—22no—
446C23—22no—
447C23#619614yescomplemented
448C23C2×((C3⋊D3)⋊C3)⋊C212yes—
449C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
450Q8—548no—
451C2×C4—92no—
452C2×C4—94no—
453C2×C4—92no—
454C2×C4—92no—
455C2×C4—92no—
456C2×C4—92no—
457C2×C4—92no—
458C2×C4—188no—
459C2×C4—182no—
460C2×C4—182no—
461C2×C4—182no—
462C2×C4—182no—
463C2×C4—182no—
464C2×C3—24no—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—31no—
471C2×C3—34no—
472C2×C3—34no—
473C2×C3—64no—
474C2×C3—64no—
475C2×C3—61no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—64no—
479C2×C3—124no—
480C2×C3#6396414yes—
481C2×C3C2×#380411yescharacteristic
482C2×C3C22×#62711yescharacteristic
483C2×C3C2×D3×#6711yescharacteristic
484C22—22no—
485C22—22no—
486C22—28no—
487C22C2×#619612yes—
488C22C2×#619614yescomplemented
489C22#8867112yes—
490C22#8867612yes—
491C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
492C4—94no—
493C4—182no—
494C4—182no—
495C4—182no—
496C4—182no—
497C4—182no—
498C3—31no—
499C3—31no—
500C3—61no—
501C3C2×#6396411yescharacteristic
502C2—24no—
503C2#1172129914yesdirect product factor
504C2C22×#619611yescharacteristic
505C2C2×#8867111yescharacteristic
506C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6196C2semidirect
5. C2×#88449C2semidirect
6. C2×#88448C2semidirect
7. C2×#88448C2semidirect
8. #11721299C2direct (N × H)
21. C2×#6196C22semidirect
24. #88449C22semidirect
25. #88448C22semidirect
26. #88448C22semidirect
46. C23×H(3,𝔽3)Q8semidirect
107. C22×H(3,𝔽3)C2×Q8semidirect
150. C23×C32Dic6semidirect
151. C23×C32Dic6semidirect
190. C2×H(3,𝔽3)C22⋊Q8semidirect
262. C22×C32C2×Dic6semidirect
263. C22×C32C2×Dic6semidirect
292. H(3,𝔽3)C2×C22⋊Q8semidirect
366. C2×C32#878semidirect
367. C2×C32#924semidirect
443. C32C2×#878semidirect
444. C32C2×#924semidirect
447. C23#6196semidirect
488. C22C2×#6196semidirect
503. C2#11721299direct (N × H)

Maximal subgroups

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

subgroupindex
C23×(C3⋊Dic3)⋊C32
C22×#61962
C2×#885522
C2×#3757⋊C32
C2×#884492
C2×#884482
#117212992
C2×#639823
C2×#639643