Finite Group Databasebeta

← C2×#11721092 #13150618

Subgroups — C2×#11721092 #13150618

Order 1728.

993 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88542C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic, complemented
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#88451C211yesmaximal, characteristic
5C2×#88451C211yesmaximal, characteristic
6C2×#88449C211yesmaximal, characteristic, complemented
7C2×#88448C211yesmaximal, characteristic, complemented
8#11721092C218yesmaximal, direct product factor
9C2×#63862—31nomaximal
10C2×#63859—31nomaximal
11C23×#272C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#88542C2214yes—
19#3750⋊C3C2214yescomplemented
20C4×(C3⋊Dic3)⋊C3C2214yescomplemented
21#88451C2214yes—
22#88451C2214yes—
23#88449C2214yescomplemented
24#88448C2214yescomplemented
25C2×#3750D311yescharacteristic, complemented
26C2×C4×C3⋊Dic3D311yescharacteristic, complemented
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×(C2×D3)⋊C4—61no—
30C2×C3×Dic3⋊C4—31no—
31C2×C3×C4×Dic3—31no—
32C2×#3724—31no—
33C2×Dic3⋊Dic3—31no—
34C2×Dic3⋊Dic3—31no—
35C2×Dic3⋊Dic3—31no—
36C2×(C2×D3)⋊Dic3—31no—
37C2×(C2×D3)⋊Dic3—31no—
38C2×(C2×D3)⋊Dic3—31no—
39C2×(C2×D3)⋊Dic3—31no—
40#63862—38no—
41#63859—38no—
42C23×H(3,𝔽3)C2311yescharacteristic
43C22×#272—22no—
44C22×#272—22no—
45C22×#272—22no—
46C22×#272C2312yes—
47C2×C4×H(3,𝔽3)—22no—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×#1084—22no—
50C2×#1084C2312yes—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3—22no—
53C2×(C3⋊Dic3)⋊C3—22no—
54C2×(C3⋊Dic3)⋊C3—22no—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×#898—91no—
60C2×#879—91no—
61C2×#876—91no—
62C23×C3×D3—31no—
63C23×C3×D3—31no—
64C23×C3×D3—61no—
65C22×C32×C4—21no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C3⋊Dic3C2×D311yescharacteristic
69C22×C3⋊Dic3C2×D311yescharacteristic
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—61no—
79#3750C2×D314yescomplemented
80C4×C3⋊Dic3C2×D314yescomplemented
81C3×(C2×D3)⋊C4—34no—
82C3×(C2×D3)⋊C4—34no—
83C3×(C2×D3)⋊C4—64no—
84C3×Dic3⋊C4—34no—
85C3×C4×Dic3—34no—
86#3724—34no—
87Dic3⋊Dic3—34no—
88Dic3⋊Dic3—34no—
89Dic3⋊Dic3—34no—
90(C2×D3)⋊Dic3—34no—
91(C2×D3)⋊Dic3—34no—
92(C2×D3)⋊Dic3—34no—
93(C2×D3)⋊Dic3—34no—
94C22×H(3,𝔽3)#4112yes—
95C22×H(3,𝔽3)#4112yes—
96C22×H(3,𝔽3)#4112yes—
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×#272—22no—
99C2×#272—22no—
100C2×#272—22no—
101C2×#272—44no—
102C4×H(3,𝔽3)—22no—
103#1084—22no—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—22no—
108C2×C3×C4⋊C4—91no—
109C2×C3×C22⋊C4—91no—
110C2×C3×C42—91no—
111C2×#217—91no—
112C2×#217—91no—
113C2×(C2×D3)⋊C4—31no—
114C2×(C2×D3)⋊C4—31no—
115C2×(C2×D3)⋊C4—61no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×C4⋊Dic3—91no—
121C2×C4⋊Dic3—91no—
122C2×Dic3⋊C4—31no—
123C2×Dic3⋊C4—91no—
124C2×Dic3⋊C4—91no—
125C2×Dic3⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×C4×Dic3—31no—
129C2×C4×Dic3—91no—
130#898—98no—
131#879—98no—
132#876—98no—
133C23×C32—21no—
134C23×C32C22×D311yescharacteristic
135C23×C32C22×D311yescharacteristic
136C22×C3×D3—32no—
137C22×C3×D3—32no—
138C22×C3×D3—62no—
139C22×C3×D3—64no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—122no—
147C22×C3×D3—122no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—42no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D312yes—
154C2×C3⋊Dic3—22no—
155C2×C3⋊Dic3—22no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3⋊Dic3C22×D312yes—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—122no—
176C2×#126—271noSylow
177C2×H(3,𝔽3)#12614yescomplemented
178C2×H(3,𝔽3)C2×#4111yescharacteristic
179C2×H(3,𝔽3)C2×#4111yescharacteristic
180C2×H(3,𝔽3)C2×#4111yescharacteristic
181#272—42no—
182C24×C3—91no—
183C23×D3—31no—
184C23×D3—31no—
185C23×D3—61no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—61no—
189C22×C3×C4—91no—
190C22×C3×C4—91no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4—91no—
194C22×C3×C4D3211yescharacteristic
195C22×Dic3—31no—
196C22×Dic3—31no—
197C22×Dic3—31no—
198C22×Dic3—31no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—61no—
202C22×Dic3—91no—
203C22×Dic3—91no—
204C22×Dic3—91no—
205C22×Dic3—91no—
206C3×C4⋊C4—94no—
207C3×C22⋊C4—94no—
208C3×C42—94no—
209#217—94no—
210#217—94no—
211(C2×D3)⋊C4—34no—
212(C2×D3)⋊C4—34no—
213(C2×D3)⋊C4—64no—
214(C2×D3)⋊C4—94no—
215(C2×D3)⋊C4—94no—
216(C2×D3)⋊C4—94no—
217(C2×D3)⋊C4—94no—
218C4⋊Dic3—94no—
219C4⋊Dic3—94no—
220Dic3⋊C4—34no—
221Dic3⋊C4—94no—
222Dic3⋊C4—94no—
223Dic3⋊C4—94no—
224Dic3⋊C4—94no—
225Dic3⋊C4—94no—
226C4×Dic3—34no—
227C4×Dic3—94no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32#23512yes—
233C22×C32#23512yes—
234C22×C32#23512yes—
235C22×C32#23512yes—
236C22×C32#23712yes—
237C22×C32#23912yes—
238C22×C32C23×D311yescharacteristic
239C22×C32C23×D311yescharacteristic
240C2×C3×D3—64no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—122no—
248C2×C3×D3—122no—
249C2×C3×D3—128no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—42no—
255C3⋊Dic3—22no—
256C3⋊Dic3—22no—
257C3⋊Dic3—22no—
258C3⋊Dic3—22no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C3×Dic3—62no—
262C3×Dic3—62no—
263C3×Dic3—62no—
264C3×Dic3—62no—
265C3×Dic3—122no—
266#126—278no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—271no—
269C2×C4⋊C4—271no—
270C2×C22⋊C4—91no—
271C2×C22⋊C4—271no—
272C2×C22⋊C4—271no—
273C2×C42—91no—
274H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
275C23×C3—31no—
276C23×C3—31no—
277C23×C3—61no—
278C23×C3—92no—
279C23×C3—182no—
280C23×C3—182no—
281C23×C3—182no—
282C23×C3C2×D3211yescharacteristic
283C22×D3—32no—
284C22×D3—32no—
285C22×D3—62no—
286C22×D3—62no—
287C22×D3—62no—
288C22×D3—62no—
289C22×D3—62no—
290C22×D3—62no—
291C22×D3—62no—
292C22×D3—64no—
293C22×D3—122no—
294C22×D3—122no—
295C2×C3×C4—22no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—62no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—92no—
306C2×C3×C4—122no—
307C2×C3×C4—182no—
308C2×C3×C4—182no—
309C2×C3×C4—182no—
310C2×C3×C4—182no—
311C2×C3×C4—182no—
312C2×C3×C4C2×D3212yes—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—92no—
330C2×Dic3—122no—
331C2×Dic3—182no—
332C2×Dic3—182no—
333C2×Dic3—182no—
334C2×Dic3—182no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—24no—
339C2×C32#87614yescomplemented
340C2×C32#89814yescomplemented
341C2×C32C2×#23511yescharacteristic
342C2×C32C2×#23511yescharacteristic
343C2×C32C2×#23511yescharacteristic
344C2×C32C2×#23511yescharacteristic
345C2×C32C2×#23711yescharacteristic
346C2×C32C2×#23911yescharacteristic
347C3×D3—124no—
348C3×D3—122no—
349C3×D3—122no—
350C24—91no—
351C22×C4—91no—
352C22×C4—91no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C4⋊C4—94no—
358C4⋊C4—274no—
359C4⋊C4—274no—
360C22⋊C4—94no—
361C22⋊C4—274no—
362C22⋊C4—274no—
363C42—94no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—32no—
367C22×C3—32no—
368C22×C3—32no—
369C22×C3—32no—
370C22×C3—32no—
371C22×C3—32no—
372C22×C3—61no—
373C22×C3—62no—
374C22×C3—62no—
375C22×C3—62no—
376C22×C3—182no—
377C22×C3—182no—
378C22×C3—182no—
379C22×C3—364no—
380C22×C3#379812yes—
381C22×C3#379912yes—
382C22×C3#380412yes—
383C22×C3C22×D3211yescharacteristic
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—62no—
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—64no—
391C2×D3—122no—
392C2×D3—122no—
393C2×D3—124no—
394C2×D3—124no—
395C2×D3—128no—
396C3×C4—22no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—122no—
400C3×C4—182no—
401C3×C4—182no—
402C3×C4—182no—
403C3×C4—182no—
404C3×C4—182no—
405Dic3—62no—
406Dic3—62no—
407Dic3—62no—
408Dic3—62no—
409Dic3—62no—
410Dic3—62no—
411Dic3—122no—
412Dic3—182no—
413Dic3—182no—
414Dic3—182no—
415Dic3—182no—
416C32—21no—
417C32C2×#87611yescharacteristic, complemented
418C32C2×#89811yescharacteristic, complemented
419C23—92no—
420C23—182no—
421C23—182no—
422C23—182no—
423C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
424C2×C4—22no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—182no—
433C2×C4—182no—
434C2×C4—182no—
435C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—34no—
439C2×C3—34no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—61no—
445C2×C3—64no—
446C2×C3—61no—
447C2×C3—61no—
448C2×C3—362no—
449C2×C3#6386814yes—
450C2×C3C2×#379811yescharacteristic
451C2×C3C2×#379911yescharacteristic
452C2×C3C2×#380411yescharacteristic
453D3—122no—
454D3—122no—
455D3—124no—
456C22—182no—
457C22—182no—
458C22—182no—
459C22—364no—
460C22#8865012yes—
461C22#8865212yes—
462C22#8867112yes—
463C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
464C4—22no—
465C4—182no—
466C4—182no—
467C4—182no—
468C4—182no—
469C4—182no—
470C3—31no—
471C3—31no—
472C3—61no—
473C3C2×#6386811yescharacteristic
474C2—362no—
475C2#1172109214yesdirect product factor
476C2C2×#8865011yescharacteristic
477C2C2×#8865211yescharacteristic
478C2C2×#8867111yescharacteristic
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 radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#126
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. C2×#3750⋊C3C2semidirect
3. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
6. C2×#88449C2semidirect
7. C2×#88448C2semidirect
8. #11721092C2direct (N × H)
19. #3750⋊C3C22semidirect
20. C4×(C3⋊Dic3)⋊C3C22semidirect
23. #88449C22semidirect
24. #88448C22semidirect
25. C2×#3750D3semidirect
26. C2×C4×C3⋊Dic3D3semidirect
79. #3750C2×D3semidirect
80. C4×C3⋊Dic3C2×D3semidirect
177. C2×H(3,𝔽3)#126semidirect
274. H(3,𝔽3)C2×#126semidirect
339. C2×C32#876semidirect
340. C2×C32#898semidirect
417. C32C2×#876semidirect
418. C32C2×#898semidirect
475. C2#11721092direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#885422
C2×#3750⋊C32
C2×C4×(C3⋊Dic3)⋊C32
C2×#884512
C2×#884492
C2×#884482
#117210922
C2×#638623
C2×#638593