Finite Group Databasebeta

← C2×#11721156 #13150839

Subgroups — C2×#11721156 #13150839

Order 1728.

1727 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊#272C211yesmaximal, characteristic, complemented
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C22×#6200C212yesmaximal, complemented
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×#88451C212yesmaximal, complemented
6#11721156C218yesmaximal, direct product factor
7C2×#63916—31nomaximal
8C2×#63908—31nomaximal
9C23×#272C2212yes—
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊#272—24no—
13C2×C4⋊#272C2214yescomplemented
14C22×(C3⋊Dic3)⋊C3C2212yes—
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3—24no—
17C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
18C2×#6200—28no—
19C2×#6200C2218yescomplemented
20C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
21#88451C2218yescomplemented
22C22×C4×C3⋊D3D311yescharacteristic, complemented
23C22×C3×D12—31no—
24C22×C3×D12—31no—
25C22×C3×D12—61no—
26C22×C3×C4×D3—31no—
27C22×#626—32no—
28C22×#625—32no—
29C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
30C2×C3×C4⋊Dic3—31no—
31C2×(C2×D3)⋊Dic3—32no—
32C2×(C2×D3)⋊Dic3—32no—
33#63916—38no—
34#63908—38no—
35C23×H(3,𝔽3)C2311yescharacteristic
36C22×#272—24no—
37C22×#272—24no—
38C22×#272—24no—
39C22×#272C2314yes—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3—22no—
42C22×(C3⋊D3)⋊C3D414yescomplemented
43C22×(C3⋊D3)⋊C3C2312yes—
44C2×C4×H(3,𝔽3)D414yescomplemented
45C2×C4×H(3,𝔽3)C2312yes—
46C4⋊#272—28no—
47C2×(C3⋊Dic3)⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2314yes—
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C4×(C3⋊D3)⋊C3—28no—
52#6200—216no—
53C2×#938—91no—
54C2×#930—91no—
55C2×#895—91no—
56C23×C3⋊D3C2×D311yescharacteristic
57C23×C3×D3—31no—
58C23×C3×D3—32no—
59C23×C3×D3—32no—
60C23×C3×D3—62no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D312yes—
65C22×C3⋊Dic3C2×D311yescharacteristic
66C2×C4×C3⋊D3—24no—
67C2×C4×C3⋊D3C2×D314yescomplemented
68C22×C3×Dic3—32no—
69C22×C3×Dic3—31no—
70C2×C3×D12—34no—
71C2×C3×D12—34no—
72C2×C3×D12—64no—
73C2×C3×D12—68no—
74C2×C3×D12—64no—
75C2×C3×D12—64no—
76C2×C3×C4×D3—34no—
77C2×C3×C4×D3—64no—
78C2×#626—38no—
79C2×#626—68no—
80C2×#625—38no—
81C2×#625—68no—
82C4⋊(C3⋊Dic3)C2×D314yescomplemented
83C3×C4⋊Dic3—34no—
84(C2×D3)⋊Dic3—38no—
85(C2×D3)⋊Dic3—38no—
86C22×H(3,𝔽3)C2×D412yes—
87C22×H(3,𝔽3)C2×D412yes—
88C22×H(3,𝔽3)#4112yes—
89C22×H(3,𝔽3)C2411yescharacteristic
90C2×#272—24no—
91C2×#272—24no—
92C2×#272—24no—
93C2×#272—48no—
94C2×(C3⋊D3)⋊C3—28no—
95C2×(C3⋊D3)⋊C3—22no—
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3C2×D414yescomplemented
98C4×H(3,𝔽3)C2×D414yescomplemented
99(C3⋊Dic3)⋊C3—24no—
100(C3⋊Dic3)⋊C3—22no—
101C22×C3×D4—91no—
102C23×C3×C4—91no—
103C22×#67—92no—
104C22×#67—92no—
105C22×D12—31no—
106C22×D12—31no—
107C22×D12—61no—
108C22×D12—92no—
109C22×C4×D3—31no—
110C22×C4×D3—91no—
111C2×C3×C4⋊C4—91no—
112C2×#217—92no—
113#938—98no—
114#930—98no—
115C2×(C2×D3)⋊C4—92no—
116C2×(C2×D3)⋊C4—92no—
117C2×C4⋊Dic3—31no—
118C2×C4⋊Dic3—91no—
119#895—98no—
120C23×C32—21no—
121C23×C32C22×D311yescharacteristic
122C23×C32C22×D311yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3D1214yescomplemented
126C22×C3⋊D3C22×D312yes—
127C22×C3×D3—34no—
128C22×C3×D3—32no—
129C22×C3×D3—34no—
130C22×C3×D3—34no—
131C22×C3×D3—64no—
132C22×C3×D3—62no—
133C22×C3×D3—62no—
134C22×C3×D3—68no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C22×C3×D3—64no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3—124no—
142C22×C3×D3—124no—
143C2×C32×C4—24no—
144C2×C32×C4—22no—
145C2×C32×C4D1214yescomplemented
146C2×C32×C4#6714yescomplemented
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D312yes—
149C2×C3⋊Dic3—24no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3C22×D312yes—
152C2×C3⋊Dic3C22×D314yes—
153C4×C3⋊D3—28no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3—62no—
158C3×D12—616no—
159C3×D12—68no—
160C3×D12—68no—
161C3×C4×D3—68no—
162#626—616no—
163#625—616no—
164C2×#121—271noSylow
165C2×H(3,𝔽3)#12114yescomplemented
166C2×H(3,𝔽3)C22×D411yescharacteristic
167C2×H(3,𝔽3)C22×D411yescharacteristic
168C2×H(3,𝔽3)C2×#4111yescharacteristic
169#272—44no—
170(C3⋊D3)⋊C3—24no—
171C24×C3—91no—
172C24×C3—92no—
173C23×D3—31no—
174C23×D3—32no—
175C23×D3—32no—
176C23×D3—62no—
177C23×D3—91no—
178C2×C3×D4—94no—
179C2×C3×D4—184no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×C3×C4—61no—
183C22×C3×C4—94no—
184C22×C3×C4—91no—
185C22×C3×C4—92no—
186C22×C3×C4—184no—
187C22×C3×C4D3211yescharacteristic
188C2×#67—98no—
189C2×#67—98no—
190C2×#67—188no—
191C2×#67—188no—
192C22×Dic3—32no—
193C22×Dic3—31no—
194C22×Dic3—92no—
195C22×Dic3—91no—
196C2×D12—34no—
197C2×D12—34no—
198C2×D12—64no—
199C2×D12—64no—
200C2×D12—64no—
201C2×D12—68no—
202C2×D12—98no—
203C2×D12—188no—
204C2×C4×D3—34no—
205C2×C4×D3—64no—
206C2×C4×D3—94no—
207C2×C4×D3—184no—
208C3×C4⋊C4—94no—
209#217—98no—
210(C2×D3)⋊C4—98no—
211(C2×D3)⋊C4—98no—
212C4⋊Dic3—34no—
213C4⋊Dic3—94no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32C2×D1212yes—
219C22×C32C2×D1212yes—
220C22×C32#23512yes—
221C22×C32D3×D412yes—
222C22×C32#23912yes—
223C22×C32C2×#6712yes—
224C22×C32C23×D311yescharacteristic
225C22×C32C23×D311yescharacteristic
226C2×C3⋊D3—28no—
227C2×C3⋊D3—22no—
228C2×C3⋊D3—22no—
229C2×C3⋊D3C2×D1214yescomplemented
230C2×C3×D3—34no—
231C2×C3×D3—68no—
232C2×C3×D3—62no—
233C2×C3×D3—62no—
234C2×C3×D3—68no—
235C2×C3×D3—64no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—124no—
242C2×C3×D3—124no—
243C2×C3×D3—1216no—
244C2×C3×D3—128no—
245C2×C3×D3—128no—
246C32×C4—24no—
247C32×C4C2×D1214yescomplemented
248C32×C4C2×#6714yescomplemented
249C3⋊Dic3—24no—
250C3⋊Dic3—22no—
251C3×Dic3—64no—
252C3×Dic3—62no—
253C22×D4—91no—
254C22×D4—272no—
255C23×C4—91no—
256#121—278no—
257C2×C4⋊C4—91no—
258C2×C22⋊C4—272no—
259H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
260C23×C3—31no—
261C23×C3—31no—
262C23×C3—61no—
263C23×C3—92no—
264C23×C3—94no—
265C23×C3—94no—
266C23×C3—182no—
267C23×C3—182no—
268C23×C3—184no—
269C23×C3—184no—
270C23×C3—184no—
271C23×C3C2×D3211yescharacteristic
272C22×D3—32no—
273C22×D3—34no—
274C22×D3—34no—
275C22×D3—34no—
276C22×D3—64no—
277C22×D3—62no—
278C22×D3—62no—
279C22×D3—64no—
280C22×D3—64no—
281C22×D3—64no—
282C22×D3—64no—
283C22×D3—64no—
284C22×D3—64no—
285C22×D3—68no—
286C22×D3—94no—
287C22×D3—92no—
288C22×D3—124no—
289C22×D3—124no—
290C22×D3—182no—
291C22×D3—182no—
292C3×D4—188no—
293C2×C3×C4—34no—
294C2×C3×C4—34no—
295C2×C3×C4—32no—
296C2×C3×C4—32no—
297C2×C3×C4—64no—
298C2×C3×C4—62no—
299C2×C3×C4—92no—
300C2×C3×C4—94no—
301C2×C3×C4—188no—
302C2×C3×C4—184no—
303C2×C3×C4—182no—
304C2×C3×C4#62614yes—
305C2×C3×C4C2×D3212yes—
306#67—1816no—
307#67—1816no—
308C2×Dic3—34no—
309C2×Dic3—32no—
310C2×Dic3—64no—
311C2×Dic3—62no—
312C2×Dic3—94no—
313C2×Dic3—92no—
314C2×Dic3—182no—
315C2×Dic3—184no—
316D12—68no—
317D12—68no—
318D12—616no—
319D12—1816no—
320C4×D3—68no—
321C4×D3—188no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—24no—
326C2×C32#89514yescomplemented
327C2×C32#93014yescomplemented
328C2×C32C22×D1211yescharacteristic
329C2×C32C22×D1211yescharacteristic
330C2×C32C2×#23511yescharacteristic
331C2×C32C2×D3×D411yescharacteristic
332C2×C32C2×#23911yescharacteristic
333C2×C32C22×#6711yescharacteristic
334C3⋊D3—24no—
335C3×D3—64no—
336C3×D3—128no—
337C3×D3—124no—
338C3×D3—124no—
339C24—91no—
340C24—92no—
341C2×D4—94no—
342C2×D4—184no—
343C2×D4—278no—
344C2×D4—548no—
345C22×C4—91no—
346C22×C4—92no—
347C22×C4—94no—
348C22×C4—184no—
349C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C4⋊C4—94no—
351C22⋊C4—278no—
352C22×C3—31no—
353C22×C3—31no—
354C22×C3—32no—
355C22×C3—32no—
356C22×C3—32no—
357C22×C3—32no—
358C22×C3—32no—
359C22×C3—32no—
360C22×C3—61no—
361C22×C3—62no—
362C22×C3—62no—
363C22×C3—62no—
364C22×C3—94no—
365C22×C3—182no—
366C22×C3—188no—
367C22×C3—182no—
368C22×C3—184no—
369C22×C3—184no—
370C22×C3—184no—
371C22×C3—368no—
372C22×C3#379912yes—
373C22×C3D3×D1212yes—
374C22×C3C2×#62612yes—
375C22×C3C22×D3211yescharacteristic
376C2×D3—34no—
377C2×D3—62no—
378C2×D3—62no—
379C2×D3—68no—
380C2×D3—64no—
381C2×D3—64no—
382C2×D3—64no—
383C2×D3—64no—
384C2×D3—64no—
385C2×D3—64no—
386C2×D3—68no—
387C2×D3—94no—
388C2×D3—124no—
389C2×D3—124no—
390C2×D3—128no—
391C2×D3—128no—
392C2×D3—1216no—
393C2×D3—182no—
394C2×D3—188no—
395C2×D3—182no—
396C3×C4—34no—
397C3×C4—34no—
398C3×C4—64no—
399C3×C4—184no—
400C3×C4—182no—
401C3×C4C2×#62614yes—
402Dic3—64no—
403Dic3—62no—
404Dic3—182no—
405Dic3—184no—
406C32—21no—
407C32C2×#89511yescharacteristic, complemented
408C32C2×#93011yescharacteristic, complemented
409C23—92no—
410C23—94no—
411C23—94no—
412C23—182no—
413C23—182no—
414C23—184no—
415C23—184no—
416C23—184no—
417C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
418D4—188no—
419D4—5416no—
420C2×C4—92no—
421C2×C4—94no—
422C2×C4—184no—
423C2×C4—182no—
424C2×C4—188no—
425C2×C4#620014yescomplemented
426C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—34no—
430C2×C3—34no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—61no—
436C2×C3—64no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—184no—
440C2×C3—364no—
441C2×C3#6390914yes—
442C2×C3C2×#379911yescharacteristic
443C2×C3C2×D3×D1211yescharacteristic
444C2×C3C22×#62611yescharacteristic
445D3—64no—
446D3—124no—
447D3—124no—
448D3—128no—
449D3—184no—
450C22—94no—
451C22—182no—
452C22—188no—
453C22—182no—
454C22—184no—
455C22—184no—
456C22—184no—
457C22—368no—
458C22#8865212yes—
459C22#8865412yes—
460C22C2×#620012yes—
461C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
462C4—182no—
463C4—184no—
464C4C2×#620014yescomplemented
465C3—31no—
466C3—31no—
467C3—61no—
468C3C2×#6390911yescharacteristic
469C2—184no—
470C2—364no—
471C2#1172115614yesdirect product factor
472C2C2×#8865211yescharacteristic
473C2C2×#8865411yescharacteristic
474C2C22×#620011yescharacteristic
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
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#121
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
1. C22×C4⋊#272C2semidirect
2. C22×C4×(C3⋊D3)⋊C3C2semidirect
3. C22×#6200C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88451C2semidirect
6. #11721156C2direct (N × H)
13. C2×C4⋊#272C22semidirect
17. C2×C4×(C3⋊D3)⋊C3C22semidirect
19. C2×#6200C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. #88451C22semidirect
22. C22×C4×C3⋊D3D3semidirect
29. C2×C4⋊(C3⋊Dic3)D3semidirect
42. C22×(C3⋊D3)⋊C3D4semidirect
44. C2×C4×H(3,𝔽3)D4semidirect
67. C2×C4×C3⋊D3C2×D3semidirect
82. C4⋊(C3⋊Dic3)C2×D3semidirect
97. C2×(C3⋊D3)⋊C3C2×D4semidirect
98. C4×H(3,𝔽3)C2×D4semidirect
125. C22×C3⋊D3D12semidirect
145. C2×C32×C4D12semidirect
146. C2×C32×C4#67semidirect
165. C2×H(3,𝔽3)#121semidirect
229. C2×C3⋊D3C2×D12semidirect
247. C32×C4C2×D12semidirect
248. C32×C4C2×#67semidirect
259. H(3,𝔽3)C2×#121semidirect
326. C2×C32#895semidirect
327. C2×C32#930semidirect
407. C32C2×#895semidirect
408. C32C2×#930semidirect
425. C2×C4#6200semidirect
464. C4C2×#6200semidirect
471. C2#11721156direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C22×C4×(C3⋊D3)⋊C32
C22×#62002
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884512
#117211562
C2×#639163
C2×#639083