Finite Group Databasebeta

← C4×#88672 #13150675

Subgroups — C4×#88672 #13150675

Order 1728.

1193 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88672C211yesmaximal, characteristic
2C2×C4×#1084C211yesmaximal, characteristic, complemented
3C4×#638⋊C3C212yesmaximal, complemented
4#11721315C212yesmaximal, complemented
5#11721298C211yesmaximal, characteristic, complemented
6#11721138C211yesmaximal, characteristic, complemented
7C4×#6198C211yesmaximal, characteristic, complemented
8#11721121C212yesmaximal, complemented
9C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
10C4×#6196C211yesmaximal, characteristic, complemented
11#11721078C211yesmaximal, characteristic, complemented
12C4×#3804—32nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C2×#638⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#6198C2211yescharacteristic
19#88672C418yesdirect product factor
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22C2×#6196C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25C4×#1084C2211yescharacteristic
26C4×#1084C2211yescharacteristic
27#3757⋊C3C2212yes—
28#3752⋊C3C2212yes—
29#3750⋊C3C2212yes—
30C4×(C3⋊Dic3)⋊C3C2212yes—
31#88450C2212yes—
32#88449C2211yescharacteristic
33#88449C2211yescharacteristic
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37C2×#3804—32no—
38C4×#638D312yescomplemented
39C3×C4×#67—32no—
40C2×C3×C4×Dic3—32no—
41C2×C3×C4×Dic3—61no—
42Dic3×#67—32no—
43#63955—32no—
44#63952—32no—
45C4×#627—32no—
46C4×#626—32no—
47C4×#624—32no—
48#63880—32no—
49#63877—32no—
50C4×D3×Dic3—32no—
51#63837—32no—
52#63836—32no—
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×(C3⋊D3)⋊C3C2312yes—
55C2×C4×H(3,𝔽3)—22no—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×#1084—22no—
59C2×#1084C2×C414yescomplemented
60C2×#1084C2311yescharacteristic
61C2×#1084C2311yescharacteristic
62C2×#1084C2311yescharacteristic
63C2×#1084C2311yescharacteristic
64#638⋊C3C2×C418yescomplemented
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C4×(C3⋊D3)⋊C3—24no—
69#6198C2×C414yescomplemented
70((C3⋊D3)⋊C3)⋊C4—24no—
71((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
72#6196C2×C414yescomplemented
73C4×#237—92no—
74C4×#235—91no—
75C22×C32×C4—21no—
76C22×C32×C4C2×D312yes—
77C2×#638C2×D312yes—
78C2×C4×C3⋊D3C2×D312yes—
79C2×C3×#67—32no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—32no—
82C22×C3×Dic3—61no—
83C22×C3×Dic3—61no—
84C2×C3×C4×D3—32no—
85C2×#627—32no—
86C2×#626—32no—
87C2×#624—32no—
88C2×#624—32no—
89#3804—316no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92#3757C2×D312yes—
93#3752C2×D312yes—
94#3750C2×D312yes—
95C4×C3⋊Dic3C2×D312yes—
96C3×#217—32no—
97C3×(C2×D3)⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—32no—
104C3×C4×Dic3—61no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107C3×C4×Dic3—61no—
108#3724—32no—
109Dic3⋊Dic3—32no—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114Dic32—32no—
115Dic32—32no—
116Dic32—32no—
117Dic32—32no—
118C22×H(3,𝔽3)—21no—
119C22×H(3,𝔽3)—21no—
120C22×H(3,𝔽3)C22×C412yes—
121C22×H(3,𝔽3)C2411yescharacteristic
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3C22×C414yes—
125C4×H(3,𝔽3)—21no—
126C4×H(3,𝔽3)#4112yescomplemented
127#1084—21no—
128#1084C22×C412yes—
129#1084C22×C412yes—
130#1084#4112yescomplemented
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3—22no—
133(C3⋊Dic3)⋊C3C22×C414yes—
134C2×#237—92no—
135C2×#235—91no—
136C3×C4×D4—92no—
137C2×C3×C42—91no—
138D4×Dic3—92no—
139C4×#67—32no—
140C4×#67—92no—
141C4×#67—92no—
142C4×#67—92no—
143#926—91no—
144C2×C4×Dic3—32no—
145C2×C4×Dic3—61no—
146C2×C4×Dic3—92no—
147#892—92no—
148#887—92no—
149#67⋊C4—92no—
150#67⋊C4—92no—
151#877—92no—
152#877—92no—
153C4×D12—91no—
154C4×D12—91no—
155#872—92no—
156C42×D3—92no—
157C42×D3—92no—
158C4×Dic6—91no—
159C4×Dic6—92no—
160C4×Dic6—91no—
161C23×C32—21no—
162C23×C32C22×D312yes—
163C22×C3⋊D3C22×D312yes—
164C22×C3×D3—32no—
165C2×C32×C4—24no—
166C2×C32×C4—24no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C22×D312yes—
171#638C4×D318yescomplemented
172C2×C3⋊Dic3C22×D312yes—
173C2×C3⋊Dic3C22×D312yes—
174C2×C3⋊Dic3C22×D312yes—
175C4×C3⋊D3—24no—
176C3×#67—38no—
177C2×C3×Dic3—38no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3—64no—
192C3×C4×D3—64no—
193#627—38no—
194#626—38no—
195#624—38no—
196#624—64no—
197D3×Dic3—38no—
198D3×Dic3—64no—
199C4×#41—271noSylow
200C2×H(3,𝔽3)—22no—
201C2×H(3,𝔽3)C23×C411yescharacteristic
202C2×H(3,𝔽3)C2×#4111yescharacteristic
203C2×H(3,𝔽3)C2×#4111yescharacteristic
204(C3⋊D3)⋊C3—24no—
205C2×C3×D4—92no—
206C22×C3×C4—32no—
207C22×C3×C4—61no—
208C22×C3×C4—92no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4D3211yescharacteristic
212C2×#67—32no—
213C2×#67—92no—
214C2×#67—92no—
215C22×Dic3—32no—
216C22×Dic3—32no—
217C22×Dic3—61no—
218C22×Dic3—61no—
219C22×Dic3—92no—
220C22×Dic3—92no—
221#237—916no—
222#235—98no—
223C2×D12—91no—
224C2×C4×D3—32no—
225C2×C4×D3—92no—
226C2×C4×D3—92no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×C4×D3—92no—
230C2×Dic6—91no—
231C2×Dic6—92no—
232C3×C4⋊C4—92no—
233C3×C22⋊C4—92no—
234C3×C22⋊C4—92no—
235C3×C42—92no—
236C3×C42—91no—
237C3×C42—91no—
238C3×C42—91no—
239C3×C42—91no—
240#217—32no—
241#217—92no—
242#217—92no—
243(C2×D3)⋊C4—32no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247C4⋊Dic3—91no—
248C4⋊Dic3—91no—
249C4⋊Dic3—92no—
250Dic3⋊C4—32no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256C4×Dic3—32no—
257C4×Dic3—32no—
258C4×Dic3—32no—
259C4×Dic3—32no—
260C4×Dic3—32no—
261C4×Dic3—61no—
262C4×Dic3—61no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C4×Dic3—92no—
268C4×Dic3—92no—
269C4×Dic3—92no—
270C4×Dic3—92no—
271C4×Dic3—92no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32C2×C4×D314yes—
279C22×C32C23×D312yes—
280C2×C3⋊D3—22no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3C2×C4×D314yes—
283C2×C3×D3—34no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4#23714yescomplemented
290C3⋊Dic3—22no—
291C3⋊Dic3—22no—
292C3⋊Dic3C2×C4×D314yes—
293C3×Dic3—34no—
294C3×Dic3—34no—
295C3×Dic3—34no—
296C3×Dic3—34no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C2×#41—271no—
305C4×Q8—271no—
306C4×Q8—271no—
307C4×D4—92no—
308C4×D4—271no—
309C4×D4—271no—
310C4×D4—272no—
311#117—272no—
312#117—271no—
313C2×C42—91no—
314C2×C42—272no—
315H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
316C23×C3—32no—
317C23×C3—61no—
318C23×C3—92no—
319C23×C3C2×D3211yescharacteristic
320C22×D3—32no—
321C22×D3—92no—
322C3×D4—98no—
323C2×C3×C4—22no—
324C2×C3×C4—32no—
325C2×C3×C4—32no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—92no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—94no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—91no—
338C2×C3×C4—184no—
339C2×C3×C4—182no—
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C2×D3211yescharacteristic
342#67—38no—
343#67—98no—
344#67—98no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—32no—
348C2×Dic3—38no—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—64no—
358C2×Dic3—64no—
359C2×Dic3—64no—
360C2×Dic3—92no—
361C2×Dic3—92no—
362C2×Dic3—92no—
363C2×Dic3—98no—
364C2×Dic3—184no—
365D12—94no—
366C4×D3—64no—
367C4×D3—98no—
368C4×D3—98no—
369C4×D3—184no—
370C4×D3—184no—
371C4×D3—184no—
372Dic6—94no—
373Dic6—98no—
374C2×C32—24no—
375C2×C32—24no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32C22×C4×D312yes—
380C2×C32C2×#23712yes—
381C2×C32D3×#4112yes—
382C3⋊D3—24no—
383C3×D3—64no—
384#41—278no—
385C2×Q8—271no—
386C2×D4—92no—
387C2×D4—271no—
388C22×C4—91no—
389C22×C4—91no—
390C22×C4—92no—
391C22×C4—272no—
392C22×C4—272no—
393C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
394C4⋊C4—92no—
395C4⋊C4—271no—
396C4⋊C4—271no—
397C4⋊C4—272no—
398C22⋊C4—92no—
399C22⋊C4—92no—
400C22⋊C4—272no—
401C42—91no—
402C42—91no—
403C42—91no—
404C42—91no—
405C42—92no—
406C42—272no—
407C42—272no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—32no—
411C22×C3—34no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—61no—
417C22×C3—62no—
418C22×C3—94no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3C4×D3212yes—
422C22×C3C22×D3211yescharacteristic
423C2×D3—34no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—94no—
427C2×D3—182no—
428C2×D3—182no—
429C3×C4—21no—
430C3×C4—34no—
431C3×C4—62no—
432C3×C4—62no—
433C3×C4—62no—
434C3×C4—94no—
435C3×C4—92no—
436C3×C4—92no—
437C3×C4—92no—
438C3×C4—182no—
439C3×C4—182no—
440C3×C4—181no—
441C3×C4#380512yes—
442Dic3—34no—
443Dic3—34no—
444Dic3—34no—
445Dic3—34no—
446Dic3—62no—
447Dic3—62no—
448Dic3—62no—
449Dic3—62no—
450Dic3—62no—
451Dic3—62no—
452Dic3—62no—
453Dic3—94no—
454Dic3—182no—
455Dic3—182no—
456C32—21no—
457C32C4×#23712yescomplemented
458C23—92no—
459C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460Q8—274no—
461D4—98no—
462D4—274no—
463C2×C4—22no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—92no—
468C2×C4—94no—
469C2×C4—91no—
470C2×C4—92no—
471C2×C4—92no—
472C2×C4—182no—
473C2×C4—184no—
474C2×C4—278no—
475C2×C4—544no—
476C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
477C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
478C2×C3—22no—
479C2×C3—32no—
480C2×C3—32no—
481C2×C3—32no—
482C2×C3—64no—
483C2×C3—64no—
484C2×C3—61no—
485C2×C3—61no—
486C2×C3—61no—
487C2×C3—184no—
488C2×C3C2×C4×D3211yescharacteristic
489C2×C3#6430611yescharacteristic
490C2×C3C2×#380511yescharacteristic
491D3—64no—
492D3—184no—
493C22—21no—
494C22—21no—
495C22—94no—
496C22—182no—
497C22—182no—
498C22C4×((C3⋊D3)⋊C3)⋊C212yes—
499C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
500C4—21no—
501C4—92no—
502C4—92no—
503C4—94no—
504C4—92no—
505C4—182no—
506C4—181no—
507C4—182no—
508C4#8867212yesdirect product factor
509C3—32no—
510C3—61no—
511C3C4×#380511yescharacteristic
512C2—22no—
513C2—184no—
514C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
515C2#1172249111yescharacteristic
516C2C2×#8867211yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#88672C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1084C2semidirect
3. C4×#638⋊C3C2semidirect
4. #11721315C2semidirect
5. #11721298C2semidirect
6. #11721138C2semidirect
7. C4×#6198C2semidirect
8. #11721121C2semidirect
9. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
10. C4×#6196C2semidirect
11. #11721078C2semidirect
19. #88672C4direct (N × H)
38. C4×#638D3semidirect
59. C2×#1084C2×C4semidirect
64. #638⋊C3C2×C4semidirect
69. #6198C2×C4semidirect
71. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
72. #6196C2×C4semidirect
126. C4×H(3,𝔽3)#41semidirect
130. #1084#41semidirect
171. #638C4×D3semidirect
289. C32×C4#237semidirect
315. H(3,𝔽3)C4×#41semidirect
457. C32C4×#237semidirect
508. C4#88672direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886722
C2×C4×#10842
C4×#638⋊C32
#117213152
#117212982
#117211382
C4×#61982
#117211212
C4×((C3⋊D3)⋊C3)⋊C42
C4×#61962
#117210782
C4×#38043