Finite Group Databasebeta

← C4⋊C4×#4324 #14772172

Subgroups — C4⋊C4×#4324 #14772172

Order 2816.

1030 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#4324C212yesmaximal
2C2×C4×#4324C211yesmaximal, characteristic
3C4⋊C4×Dic22C211yesmaximal, characteristic, complemented
4#13069341C212yesmaximal, complemented
5#13068883C212yesmaximal, complemented
6#13068877C212yesmaximal
7#13068772C211yesmaximal, characteristic, complemented
8#13068618C211yesmaximal, characteristic, complemented
9#13068616C211yesmaximal, characteristic
10C4⋊C4×C11⋊C8C211yesmaximal, characteristic, complemented
11D4×C11×C4⋊C4C211yesmaximal, characteristic
12C2×C4×D4×C11C2212yes—
13C2×C4×D4×C11C2211yescharacteristic
14C22×C11×C4⋊C4—21no—
15C22×#4324C2211yescharacteristic
16C2×C4×Dic22C2212yes—
17C2×C4×Dic22C2211yescharacteristic
18C11×#389C2212yes—
19C11×#385—22no—
20C11×#384C2211yescharacteristic
21C11×#380—21no—
22C4×C11×C4⋊C4C2211yescharacteristic
23C2×#66896C2212yes—
24C2×#66896C2211yescharacteristic
25C4×#4324—28no—
26C4×#4324C418yescomplemented
27#10622589—21no—
28C4⋊C4×Dic11—21no—
29C2×#66874C2212yes—
30C2×#66874C2211yescharacteristic
31C2×#66872C2212yes—
32C2×#66872C2212yes—
33C2×#66872C2211yescharacteristic
34#10622548C2211yescharacteristic
35#10622541C2212yes—
36C2×C4⋊(C11⋊C8)C2211yescharacteristic
37C2×C4×C11⋊C8C2212yes—
38C2×C4×C11⋊C8C2211yescharacteristic
39#10622260—22no—
40#10622167C2212yes—
41#10622167C2212yes—
42#10622162C2212yes—
43C22×D4×C11C2311yescharacteristic
44C23×C4×C11—21no—
45C23×C4×C11—22no—
46C22×Dic22C2311yescharacteristic
47C4×D4×C11—24no—
48C4×D4×C11—22no—
49C4×D4×C11—42no—
50C4×D4×C11C2×C414yes—
51C2×C11×C4⋊C4—21no—
52C2×C11×C4⋊C4—22no—
53C2×C11×C4⋊C4—21no—
54C2×C11×C4⋊C4—44no—
55C2×C11×C4⋊C4D411yescharacteristic
56C2×C11×C4⋊C4D411yescharacteristic
57C2×C11×C4⋊C4C2312yes—
58C2×C11×C4⋊C4C2311yescharacteristic
59C2×C11×C22⋊C4—21no—
60C2×C11×C22⋊C4—22no—
61C2×C42×C11C2312yes—
62C2×C42×C11C2311yescharacteristic
63C2×#4324C2×C414yes—
64C2×#4324D414yes—
65C2×#4324Q814yes—
66C2×C4⋊Dic11—21no—
67C2×C4⋊Dic11—21no—
68C2×C4⋊Dic11C2312yes—
69C2×C4⋊Dic11C2311yescharacteristic
70C2×Dic11⋊C4—21no—
71C2×Dic11⋊C4—22no—
72C2×Dic11⋊C4—22no—
73C2×C4×Dic11—21no—
74C2×C4×Dic11—22no—
75C22×C11⋊C8C2312yes—
76C22×C11⋊C8C2311yescharacteristic
77C22×C11⋊C8C2311yescharacteristic
78C4×Dic22—22no—
79C4×Dic22—24no—
80C4×Dic22—42no—
81C4×Dic22C2×C414yescomplemented
82C11×#95—22no—
83C11×#95D412yes—
84#66896—24no—
85#66896C2×C414yes—
86C11⋊#95—22no—
87#66874—24no—
88#66874C2×C414yescomplemented
89#66872—24no—
90#66872—42no—
91#66872C2×C414yescomplemented
92C4⋊(C11⋊C8)—22no—
93C4×C11⋊C8—24no—
94C4×C11⋊C8C2×C414yescomplemented
95C4⋊C4×SD16—111nomaximal, Sylow
96C24×C11—21no—
97C2×D4×C11—21no—
98C2×D4×C11—21no—
99C2×D4×C11—21no—
100C2×D4×C11C22×C412yes—
101C2×D4×C11C2×D412yes—
102C2×D4×C11C2×Q812yes—
103C22×C4×C11—21no—
104C22×C4×C11—24no—
105C22×C4×C11—22no—
106C22×C4×C11—44no—
107C22×C4×C11—42no—
108C22×C4×C11—42no—
109C22×C4×C11—44no—
110C22×C4×C11—82no—
111C22×C4×C11C2×D412yes—
112C22×C4×C11C2×D411yescharacteristic
113C22×C4×C11C2×D412yes—
114C22×C4×C11C2×D411yescharacteristic
115C22×C4×C11C2411yescharacteristic
116C22×Dic11—21no—
117C22×Dic11—21no—
118C22×Dic11—22no—
119C2×Dic22—21no—
120C2×Dic22—21no—
121C2×Dic22—21no—
122C2×Dic22C22×C412yes—
123C2×Dic22C2×D412yes—
124C2×Dic22C2×Q812yes—
125C11×C4⋊C4—22no—
126C11×C4⋊C4—21no—
127C11×C4⋊C4—22no—
128C11×C4⋊C4—22no—
129C11×C4⋊C4—44no—
130C11×C4⋊C4—42no—
131C11×C4⋊C4—81no—
132C11×C4⋊C4SD1614yescomplemented
133C11×C4⋊C4C22×C412yes—
134C11×C22⋊C4—24no—
135C11×C22⋊C4—44no—
136C42×C11—22no—
137C42×C11—22no—
138C42×C11C22×C412yes—
139C42×C11#4112yes—
140#4324C4⋊C4116yesdirect product factor
141C4⋊Dic11—22no—
142C4⋊Dic11—21no—
143C4⋊Dic11—22no—
144C4⋊Dic11—42no—
145C4⋊Dic11—81no—
146C4⋊Dic11C22×C412yes—
147Dic11⋊C4—24no—
148Dic11⋊C4—44no—
149Dic11⋊C4—82no—
150C4×Dic11—24no—
151C4×Dic11—44no—
152C2×C11⋊C8—22no—
153C2×C11⋊C8—22no—
154C2×C11⋊C8—21no—
155C2×C11⋊C8—21no—
156C2×C11⋊C8—22no—
157C2×C11⋊C8C22×C412yes—
158C2×C11⋊C8C22×C412yes—
159C2×C11⋊C8C2×D412yes—
160C2×C11⋊C8C2×Q812yes—
161C2×C4×SD16—111no—
162C2×C4×SD16—112no—
163Q8×C4⋊C4—111no—
164D4×C4⋊C4D1111yescharacteristic
165#1941—112no—
166#1865—112no—
167#1864—112no—
168#1844—111no—
169#1803—111no—
170#1802—111no—
171C8×C4⋊C4—111no—
172C23×C11—22no—
173C23×C11—22no—
174C23×C11—22no—
175C23×C11—41no—
176C23×C11—41no—
177C23×C11—41no—
178C23×C11—41no—
179C23×C11C22×D411yescharacteristic
180D4×C11—22no—
181D4×C11—22no—
182D4×C11—22no—
183D4×C11C2×C4⋊C414yes—
184C2×C4×C11—22no—
185C2×C4×C11—21no—
186C2×C4×C11—21no—
187C2×C4×C11—22no—
188C2×C4×C11—22no—
189C2×C4×C11—22no—
190C2×C4×C11—22no—
191C2×C4×C11—22no—
192C2×C4×C11—41no—
193C2×C4×C11—41no—
194C2×C4×C11—48no—
195C2×C4×C11—42no—
196C2×C4×C11—42no—
197C2×C4×C11—42no—
198C2×C4×C11—88no—
199C2×C4×C11C4×D412yes—
200C2×C4×C11C4×D412yes—
201C2×C4×C11C2×SD1614yes—
202C2×C4×C11C2×SD1612yes—
203C2×C4×C11#13512yes—
204C2×C4×C11C23×C411yescharacteristic
205C2×C4×C11C22×D411yescharacteristic
206C2×C4×C11C22×Q811yescharacteristic
207C2×C4×C11C2×#4111yescharacteristic
208C2×C4×C11H(5,𝔽2)11yescharacteristic
209C2×C4×C112−1+411yescharacteristic
210C2×Dic11—22no—
211C2×Dic11—22no—
212C2×Dic11—22no—
213C2×Dic11—22no—
214C2×Dic11—41no—
215C2×Dic11—41no—
216C2×Dic11—42no—
217C2×Dic11—42no—
218C2×Dic11—42no—
219Dic22—22no—
220Dic22—22no—
221Dic22—22no—
222Dic22C2×C4⋊C414yescomplemented
223C11⋊C8—22no—
224C11⋊C8—42no—
225C11⋊C8C2×C4⋊C414yescomplemented
226C22×SD16—111no—
227C2×C4×Q8—111no—
228C2×C4×Q8—112no—
229C2×C4×D4C2×D1112yes—
230C2×C4×D4C2×D1111yescharacteristic
231C22×C4⋊C4—21no—
232C4×SD16—118no—
233C4×SD16—228no—
234C2×#106—111no—
235C2×#106—112no—
236C2×#106—112no—
237C2×C4⋊C8—111no—
238C2×#103—111no—
239C2×#103—112no—
240C2×#102—111no—
241C2×#102—112no—
242C2×C4×C8—111no—
243C2×C4×C8—112no—
244#389—112no—
245#389—222no—
246#389C2×D1112yes—
247#385—22no—
248#384—111no—
249#384—221no—
250#384C2×D1111yescharacteristic
251#380—21no—
252C4×C4⋊C4—221no—
253C4×C4⋊C4C2×D1111yescharacteristic
254#340—112no—
255#340—112no—
256#336—112no—
257C22×C11—24no—
258C22×C11—42no—
259C22×C11—42no—
260C22×C11—42no—
261C22×C11—42no—
262C22×C11—42no—
263C22×C11—42no—
264C22×C11C2×C4×D411yescharacteristic
265C22×C11D4211yescharacteristic
266C22×C11D4×Q811yescharacteristic
267C22×C11C22×SD1611yescharacteristic
268C22×C11C2×#13511yescharacteristic
269C22×C11#57711yescharacteristic
270C22×C11#57711yescharacteristic
271C4×C11—22no—
272C4×C11—24no—
273C4×C11—42no—
274C4×C11—42no—
275C4×C11—82no—
276C4×C11C4×SD1614yescomplemented
277C4×C11C22×C4⋊C411yescharacteristic
278C4×C11#52011yescharacteristic
279C4×C11#54711yescharacteristic
280C4×C11#55411yescharacteristic
281Dic11—24no—
282Dic11—42no—
283Dic11—82no—
284C22×Q8—111no—
285C22×D4C22×D1111yescharacteristic
286C23×C4—21no—
287C23×C4—22no—
288C2×SD16—114no—
289C2×SD16—114no—
290C2×SD16—114no—
291C22×C8—111no—
292C22×C8—111no—
293C22×C8—112no—
294C4×Q8—114no—
295C4×Q8—224no—
296C4×Q8—222no—
297C4×Q8—442no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—42no—
301C4×D4C4×D1114yes—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—111no—
307C2×C4⋊C4—112no—
308C2×C4⋊C4—221no—
309C2×C4⋊C4—222no—
310C2×C4⋊C4—221no—
311C2×C4⋊C4—221no—
312C2×C4⋊C4—222no—
313C2×C4⋊C4C22⋊D1111yescharacteristic
314C2×C4⋊C4C22⋊D1111yescharacteristic
315C2×C4⋊C4C22×D1112yes—
316C2×C4⋊C4C22×D1111yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C42—221no—
320C2×C42—222no—
321C2×C42C22×D1112yes—
322C2×C42C22×D1111yescharacteristic
323#106—114no—
324#106—224no—
325#106—442no—
326C4⋊C8—222no—
327#103—114no—
328#103—224no—
329#102—114no—
330#102—224no—
331C4×C8—114no—
332C4×C8—224no—
333#95—22no—
334#95—222no—
335#95C22⋊D1112yes—
336C2×C11—44no—
337C2×C11D4×C4⋊C411yescharacteristic
338C2×C11C2×C4×SD1611yescharacteristic
339C2×C11#295211yescharacteristic
340C2×C11D4×SD1611yescharacteristic
341C2×C11#328411yescharacteristic
342C2×C11Q8×SD1611yescharacteristic
343C2×C11#338111yescharacteristic
344C24—21no—
345C2×Q8—112no—
346C2×Q8—112no—
347C2×Q8—112no—
348C2×Q8—221no—
349C2×Q8—221no—
350C2×Q8—221no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4C2×C4×D1112yes—
355C2×D4D4×D1112yes—
356C2×D4Q8×D1112yes—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—82no—
365C22×C4—221no—
366C22×C4—221no—
367C22×C4—222no—
368C22×C4C2×C22⋊D1112yes—
369C22×C4C2×C22⋊D1111yescharacteristic
370C22×C4C2×C22⋊D1112yes—
371C22×C4C2×C22⋊D1111yescharacteristic
372C22×C4C23×D1111yescharacteristic
373SD16—1116no—
374C2×C8—112no—
375C2×C8—112no—
376C2×C8—112no—
377C2×C8—112no—
378C2×C8—221no—
379C2×C8—221no—
380C2×C8—222no—
381C2×C8—222no—
382C2×C8—222no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—81no—
390C4⋊C4—112no—
391C4⋊C4—224no—
392C4⋊C4—221no—
393C4⋊C4—222no—
394C4⋊C4—222no—
395C4⋊C4—444no—
396C4⋊C4—442no—
397C4⋊C4—882no—
398C4⋊C4—881no—
399C4⋊C4#432414yesdirect product factor
400C4⋊C4C2×C4×D1112yes—
401C22⋊C4—24no—
402C22⋊C4—44no—
403C42—22no—
404C42—22no—
405C42—224no—
406C42—444no—
407C42C2×C4×D1112yes—
408C42#433912yes—
409C11C4⋊C4×SD1611yescharacteristic, complemented, Sylow
410C23—22no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23C22×C22⋊D1111yescharacteristic
418Q8—114no—
419Q8—222no—
420Q8—222no—
421Q8—222no—
422D4—22no—
423D4—22no—
424D4—22no—
425D4C4⋊C4×D1114yes—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—88no—
441C2×C4—222no—
442C2×C4—222no—
443C2×C4—222no—
444C2×C4—222no—
445C2×C4—441no—
446C2×C4—441no—
447C2×C4—442no—
448C2×C4—442no—
449C2×C4—442no—
450C2×C4C4×C22⋊D1112yes—
451C2×C4C4×C22⋊D1112yes—
452C2×C4C2×#432414yes—
453C2×C4C2×#432412yes—
454C2×C4#6700312yes—
455C2×C4C22×C4×D1111yescharacteristic
456C2×C4C2×#433911yescharacteristic
457C2×C4C2×D4×D1111yescharacteristic
458C2×C4#6703711yescharacteristic
459C2×C4C2×Q8×D1111yescharacteristic
460C2×C4#6704011yescharacteristic
461C8—114no—
462C8—222no—
463C8—442no—
464C22—24no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22C2×C4×C22⋊D1111yescharacteristic
472C22C22×#432411yescharacteristic
473C22D4×C22⋊D1111yescharacteristic
474C22Q8×C22⋊D1111yescharacteristic
475C22C2×#6700311yescharacteristic
476C22#1062336711yescharacteristic
477C22#1062336811yescharacteristic
478C4—24no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—224no—
484C4—442no—
485C4—882no—
486C4C4×#432414yescomplemented
487C4C2×C4⋊C4×D1111yescharacteristic
488C4#1062303511yescharacteristic
489C4#1062303811yescharacteristic
490C4#1062304211yescharacteristic
491C2—44no—
492C2C4⋊C4×C22⋊D1111yescharacteristic
493C2C2×C4×#432411yescharacteristic
494C2#1307749711yescharacteristic
495C2D4×#432411yescharacteristic
496C2#1308034211yescharacteristic
497C2Q8×#432411yescharacteristic
498C2#1308149111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingD4×C11×C4⋊C4C2
socleC23×C11C22×D4
prime radicalC22×D4×C11C23
cosocleC22×C4×C11C24
hypercenterD4×C4⋊C4D11
hypocenterC11C4⋊C4×SD16
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×Dic22C2semidirect
4. #13069341C2semidirect
5. #13068883C2semidirect
7. #13068772C2semidirect
8. #13068618C2semidirect
10. C4⋊C4×C11⋊C8C2semidirect
26. C4×#4324C4semidirect
81. C4×Dic22C2×C4semidirect
88. #66874C2×C4semidirect
91. #66872C2×C4semidirect
94. C4×C11⋊C8C2×C4semidirect
132. C11×C4⋊C4SD16semidirect
140. #4324C4⋊C4direct (N × H)
222. Dic22C2×C4⋊C4semidirect
225. C11⋊C8C2×C4⋊C4semidirect
276. C4×C11C4×SD16semidirect
399. C4⋊C4#4324direct (N × H)
409. C11C4⋊C4×SD16semidirect
486. C4C4×#4324semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#43242
C4⋊C4×Dic222
#130693412
#130688832
#130688772
#130687722
#130686182
#130686162
C4⋊C4×C11⋊C82
D4×C11×C4⋊C42
C4⋊C4×SD1611