Finite Group Databasebeta

← C22×#88671 #13164812

Subgroups — C22×#88671 #13164812

Order 1728.

3351 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6235C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic, complemented
3C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C22×#6200C211yesmaximal, characteristic, complemented
5C22×#6199C211yesmaximal, characteristic, complemented
6C2×#88671C2124yesmaximal, direct product factor
7C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
8C22×#6196C211yesmaximal, characteristic, complemented
9C22×#3805—31nomaximal
10C22×#3804—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C23×#272C2211yescharacteristic
13C23×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6235C22112yescomplemented
15C22×#1084C2211yescharacteristic
16C2×#638⋊C3C22112yescomplemented
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C22×(C3⋊Dic3)⋊C3C22112yescomplemented
21C2×#6200C22112yescomplemented
22C2×#6199C22112yescomplemented
23#88671C22164yesdirect product factor
24C2×((C3⋊D3)⋊C3)⋊C4C22112yescomplemented
25C2×#6196C22112yescomplemented
26C22×#638D311yescharacteristic, complemented
27C23×C3⋊Dic3D311yescharacteristic, complemented
28C22×C3×#67—31no—
29C22×C3×#67—31no—
30C22×C3×#67—31no—
31C22×C3×#67—61no—
32C23×C3×Dic3—31no—
33C22×#627—31no—
34C22×#627—31no—
35C22×#626—31no—
36C22×#625—31no—
37C22×#624—31no—
38C2×#3805—324no—
39C2×#3804—324no—
40C22×D3×Dic3—31no—
41C22×D3×Dic3—31no—
42C22×D3×Dic3—31no—
43C23×H(3,𝔽3)—24no—
44C23×H(3,𝔽3)C2316yes—
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×#272—24no—
47C22×#272C2316yes—
48C22×(C3⋊D3)⋊C3—24no—
49C22×(C3⋊D3)⋊C3C2316yes—
50#6235C23116yescomplemented
51C2×#1084C2316yes—
52#638⋊C3C23116yescomplemented
53C2×(C3⋊Dic3)⋊C3C2316yes—
54C2×(C3⋊Dic3)⋊C3C2316yes—
55C2×(C3⋊Dic3)⋊C3C2316yes—
56C2×(C3⋊Dic3)⋊C3C23116yescomplemented
57#6200C23116yescomplemented
58#6199C23116yescomplemented
59((C3⋊D3)⋊C3)⋊C4C23116yescomplemented
60#6196C23116yescomplemented
61C22×#237—91no—
62C22×#237—91no—
63C22×#235—91no—
64C24×C32—21no—
65C24×C32C2×D311yescharacteristic
66C24×C32C2×D311yescharacteristic
67C23×C3⋊D3C2×D311yescharacteristic
68C23×C3×D3—31no—
69C23×C3×D3—31no—
70C23×C3×D3—31no—
71C23×C3×D3—61no—
72C2×#638C2×D3112yescomplemented
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D3112yescomplemented
77C2×C3×#67—312no—
78C2×C3×#67—312no—
79C2×C3×#67—312no—
80C2×C3×#67—612no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3—312no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—61no—
88C2×#627—312no—
89C2×#627—312no—
90C2×#626—312no—
91C2×#625—312no—
92C2×#624—312no—
93#3805—364no—
94#3804—364no—
95C2×D3×Dic3—312no—
96C2×D3×Dic3—312no—
97C2×D3×Dic3—312no—
98C22×H(3,𝔽3)—212no—
99C22×H(3,𝔽3)#4114yescomplemented
100C22×H(3,𝔽3)C2414yes—
101C22×H(3,𝔽3)C2413yes—
102C2×#272—212no—
103C2×#272C2414yes—
104C2×(C3⋊D3)⋊C3—212no—
105C2×(C3⋊D3)⋊C3C2414yes—
106#1084C2414yes—
107(C3⋊Dic3)⋊C3C2414yes—
108(C3⋊Dic3)⋊C3C2414yes—
109(C3⋊Dic3)⋊C3C2414yes—
110C22×C3×D4—91no—
111C22×C3×D4—91no—
112C23×C3×C4—91no—
113C22×#67—31no—
114C22×#67—31no—
115C22×#67—31no—
116C22×#67—61no—
117C22×#67—91no—
118C22×#67—91no—
119C22×#67—91no—
120C23×Dic3—31no—
121C23×Dic3—91no—
122C23×Dic3—91no—
123C23×Dic3—91no—
124C2×#237—924no—
125C2×#237—924no—
126C2×#235—924no—
127C22×D12—91no—
128C22×C4×D3—91no—
129C22×C4×D3—91no—
130C22×C4×D3—91no—
131C22×C4×D3—91no—
132C22×Dic6—91no—
133C22×Dic6—91no—
134C22×Dic6—91no—
135C23×C32—24no—
136C23×C32—24no—
137C23×C32—21no—
138C23×C32—26no—
139C23×C32—44no—
140C23×C32C22×D316yes—
141C23×C32C22×D316yes—
142C23×C32C22×D311yescharacteristic
143C23×C32C22×D311yescharacteristic
144C22×C3⋊D3—24no—
145C22×C3⋊D3C22×D316yes—
146C22×C3×D3—36no—
147C22×C3×D3—36no—
148C22×C3×D3—36no—
149C22×C3×D3—66no—
150C22×C3×D3—64no—
151C22×C3×D3—64no—
152C22×C3×D3—64no—
153C22×C3×D3—124no—
154#638C22×D3116yescomplemented
155C2×C3⋊Dic3C22×D316yes—
156C2×C3⋊Dic3C22×D316yes—
157C2×C3⋊Dic3C22×D316yes—
158C2×C3⋊Dic3C22×D3116yescomplemented
159C3×#67—316no—
160C3×#67—316no—
161C3×#67—316no—
162C3×#67—616no—
163C2×C3×Dic3—36no—
164C2×C3×Dic3—36no—
165C2×C3×Dic3—36no—
166C2×C3×Dic3—316no—
167C2×C3×Dic3—36no—
168C2×C3×Dic3—36no—
169C2×C3×Dic3—66no—
170#627—316no—
171#627—316no—
172#626—316no—
173#625—316no—
174#624—316no—
175D3×Dic3—316no—
176D3×Dic3—316no—
177D3×Dic3—316no—
178C22×#41—271noSylow
179C2×H(3,𝔽3)—24no—
180C2×H(3,𝔽3)C2×#4116yescomplemented
181C2×H(3,𝔽3)C2511yescharacteristic
182#272—24no—
183(C3⋊D3)⋊C3—24no—
184C24×C3—31no—
185C24×C3—31no—
186C24×C3—61no—
187C24×C3—91no—
188C24×C3—91no—
189C24×C3D3211yescharacteristic
190C23×D3—31no—
191C23×D3—31no—
192C23×D3—31no—
193C23×D3—61no—
194C23×D3—91no—
195C2×C3×D4—912no—
196C2×C3×D4—912no—
197C22×C3×C4—91no—
198C22×C3×C4—91no—
199C22×C3×C4—91no—
200C22×C3×C4—912no—
201C22×C3×C4—91no—
202C2×#67—312no—
203C2×#67—312no—
204C2×#67—312no—
205C2×#67—612no—
206C2×#67—912no—
207C2×#67—912no—
208C2×#67—912no—
209C22×Dic3—31no—
210C22×Dic3—31no—
211C22×Dic3—31no—
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—312no—
215C22×Dic3—61no—
216C22×Dic3—91no—
217C22×Dic3—91no—
218C22×Dic3—91no—
219C22×Dic3—912no—
220C22×Dic3—912no—
221C22×Dic3—912no—
222#237—964no—
223#237—964no—
224#235—964no—
225C2×D12—912no—
226C2×C4×D3—912no—
227C2×C4×D3—912no—
228C2×C4×D3—912no—
229C2×C4×D3—912no—
230C2×Dic6—912no—
231C2×Dic6—912no—
232C2×Dic6—912no—
233C22×C32—212no—
234C22×C32—212no—
235C22×C32—23no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—412no—
239C22×C32#23514yescomplemented
240C22×C32#23714yescomplemented
241C22×C32C23×D314yes—
242C22×C32C23×D314yes—
243C22×C32C23×D313yes—
244C22×C32C23×D313yes—
245C2×C3⋊D3—212no—
246C2×C3⋊D3C23×D314yes—
247C2×C3×D3—34no—
248C2×C3×D3—34no—
249C2×C3×D3—34no—
250C2×C3×D3—64no—
251C2×C3×D3—612no—
252C2×C3×D3—612no—
253C2×C3×D3—612no—
254C2×C3×D3—1212no—
255C3⋊Dic3C23×D314yes—
256C3⋊Dic3C23×D314yes—
257C3⋊Dic3C23×D314yes—
258C3×Dic3—34no—
259C3×Dic3—34no—
260C3×Dic3—34no—
261C3×Dic3—34no—
262C3×Dic3—34no—
263C3×Dic3—64no—
264C2×#41—2724no—
265C22×Q8—271no—
266C22×D4—91no—
267C22×D4—91no—
268C22×D4—271no—
269C23×C4—91no—
270C23×C4—271no—
271C23×C4—271no—
272H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
273C23×C3—24no—
274C23×C3—36no—
275C23×C3—36no—
276C23×C3—31no—
277C23×C3—31no—
278C23×C3—64no—
279C23×C3—64no—
280C23×C3—66no—
281C23×C3—61no—
282C23×C3—96no—
283C23×C3—96no—
284C23×C3—124no—
285C23×C3—184no—
286C23×C3—184no—
287C23×C3C2×D3216yes—
288C23×C3C2×D3211yescharacteristic
289C22×D3—36no—
290C22×D3—36no—
291C22×D3—36no—
292C22×D3—66no—
293C22×D3—64no—
294C22×D3—64no—
295C22×D3—64no—
296C22×D3—96no—
297C22×D3—124no—
298C22×D3—184no—
299C3×D4—916no—
300C3×D4—916no—
301C2×C3×C4—96no—
302C2×C3×C4—96no—
303C2×C3×C4—96no—
304C2×C3×C4—916no—
305C2×C3×C4—96no—
306#67—316no—
307#67—316no—
308#67—316no—
309#67—616no—
310#67—916no—
311#67—916no—
312#67—916no—
313C2×Dic3—36no—
314C2×Dic3—36no—
315C2×Dic3—36no—
316C2×Dic3—36no—
317C2×Dic3—36no—
318C2×Dic3—316no—
319C2×Dic3—66no—
320C2×Dic3—96no—
321C2×Dic3—96no—
322C2×Dic3—96no—
323C2×Dic3—916no—
324C2×Dic3—916no—
325C2×Dic3—916no—
326D12—916no—
327C4×D3—916no—
328C4×D3—916no—
329C4×D3—916no—
330C4×D3—916no—
331Dic6—916no—
332Dic6—916no—
333Dic6—916no—
334C2×C32—24no—
335C2×C32—24no—
336C2×C32—26no—
337C2×C32—21no—
338C2×C32—44no—
339C2×C32C2×#23516yescomplemented
340C2×C32C2×#23716yescomplemented
341C2×C32C24×D311yescharacteristic
342C2×C32C24×D311yescharacteristic
343C3⋊D3—24no—
344C3×D3—64no—
345C3×D3—64no—
346C3×D3—64no—
347C3×D3—124no—
348C24—91no—
349C24—91no—
350C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
351#41—2764no—
352C2×Q8—2712no—
353C2×D4—912no—
354C2×D4—912no—
355C2×D4—2712no—
356C22×C4—912no—
357C22×C4—91no—
358C22×C4—91no—
359C22×C4—91no—
360C22×C4—91no—
361C22×C4—2712no—
362C22×C4—2712no—
363C22×C3—212no—
364C22×C3—33no—
365C22×C3—33no—
366C22×C3—34no—
367C22×C3—34no—
368C22×C3—34no—
369C22×C3—34no—
370C22×C3—612no—
371C22×C3—612no—
372C22×C3—63no—
373C22×C3—64no—
374C22×C3—64no—
375C22×C3—94no—
376C22×C3—94no—
377C22×C3—1212no—
378C22×C3—1812no—
379C22×C3—1812no—
380C22×C3#380414yes—
381C22×C3C22×D3214yes—
382C22×C3C22×D3213yes—
383C2×D3—34no—
384C2×D3—34no—
385C2×D3—34no—
386C2×D3—64no—
387C2×D3—612no—
388C2×D3—612no—
389C2×D3—612no—
390C2×D3—94no—
391C2×D3—1212no—
392C2×D3—1812no—
393C3×C4—94no—
394C3×C4—94no—
395C3×C4—94no—
396C3×C4—94no—
397Dic3—34no—
398Dic3—34no—
399Dic3—34no—
400Dic3—34no—
401Dic3—34no—
402Dic3—64no—
403Dic3—94no—
404Dic3—94no—
405Dic3—94no—
406C32—21no—
407C32C22×#23511yescharacteristic, complemented
408C32C22×#23711yescharacteristic, complemented
409C23—24no—
410C23—96no—
411C23—96no—
412C23—184no—
413C23—184no—
414C23C2×((C3⋊D3)⋊C3)⋊C216yes—
415C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416Q8—2716no—
417D4—916no—
418D4—916no—
419D4—2716no—
420C2×C4—96no—
421C2×C4—96no—
422C2×C4—916no—
423C2×C4—96no—
424C2×C4—96no—
425C2×C4—2716no—
426C2×C4—2716no—
427C2×C3—24no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—36no—
431C2×C3—36no—
432C2×C3—64no—
433C2×C3—64no—
434C2×C3—61no—
435C2×C3—66no—
436C2×C3—124no—
437C2×C3—184no—
438C2×C3—184no—
439C2×C3C2×#380416yes—
440C2×C3C23×D3211yescharacteristic
441D3—64no—
442D3—64no—
443D3—64no—
444D3—124no—
445D3—184no—
446C22—212no—
447C22—94no—
448C22—94no—
449C22—1812no—
450C22—1812no—
451C22#8867114yesdirect product factor
452C22C22×((C3⋊D3)⋊C3)⋊C214yes—
453C22C22×((C3⋊D3)⋊C3)⋊C213yes—
454C4—94no—
455C4—94no—
456C4—94no—
457C4—94no—
458C3—31no—
459C3—31no—
460C3—61no—
461C3C22×#380411yescharacteristic
462C2—24no—
463C2—184no—
464C2—184no—
465C2C2×#8867116yesdirect product factor
466C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6235C2semidirect
2. C22×#638⋊C3C2semidirect
3. C23×(C3⋊Dic3)⋊C3C2semidirect
4. C22×#6200C2semidirect
5. C22×#6199C2semidirect
6. C2×#88671C2direct (N × H)
7. C22×((C3⋊D3)⋊C3)⋊C4C2semidirect
8. C22×#6196C2semidirect
14. C2×#6235C22semidirect
16. C2×#638⋊C3C22semidirect
20. C22×(C3⋊Dic3)⋊C3C22semidirect
21. C2×#6200C22semidirect
22. C2×#6199C22semidirect
23. #88671C22direct (N × H)
24. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
25. C2×#6196C22semidirect
26. C22×#638D3semidirect
27. C23×C3⋊Dic3D3semidirect
50. #6235C23semidirect
52. #638⋊C3C23semidirect
56. C2×(C3⋊Dic3)⋊C3C23semidirect
57. #6200C23semidirect
58. #6199C23semidirect
59. ((C3⋊D3)⋊C3)⋊C4C23semidirect
60. #6196C23semidirect
72. C2×#638C2×D3semidirect
76. C22×C3⋊Dic3C2×D3semidirect
99. C22×H(3,𝔽3)#41semidirect
154. #638C22×D3semidirect
158. C2×C3⋊Dic3C22×D3semidirect
180. C2×H(3,𝔽3)C2×#41semidirect
239. C22×C32#235semidirect
240. C22×C32#237semidirect
272. H(3,𝔽3)C22×#41semidirect
339. C2×C32C2×#235semidirect
340. C2×C32C2×#237semidirect
407. C32C22×#235semidirect
408. C32C22×#237semidirect
451. C22#88671direct (N × H)
465. C2C2×#88671direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C22×#638⋊C32
C23×(C3⋊Dic3)⋊C32
C22×#62002
C22×#61992
C2×#886712
C22×((C3⋊D3)⋊C3)⋊C42
C22×#61962
C22×#38053
C22×#38043