Finite Group Databasebeta

← #14765467

Subgroups — #14765467

Order 2816.

884 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622334C211yesmaximal, characteristic
2#13071144C211yesmaximal, characteristic
3C2×#10622586C211yesmaximal, characteristic
4#13069357C212yesmaximal, complemented
5#13069304C212yesmaximal, complemented
6#13069200C212yesmaximal, complemented
7#13068923C211yesmaximal, characteristic
8#13068867C212yesmaximal, complemented
9#13068700C211yesmaximal, characteristic
10#13068380C211yesmaximal, characteristic
11C11×#1852C211yesmaximal, characteristic
12C22×C11×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic11C2211yescharacteristic
14C2×#66977C2211yescharacteristic
15C22×Dic11⋊C4—21no—
16C22×C4×Dic11C2211yescharacteristic
17C2×#66940C2211yescharacteristic
18C2×C11×#107C2212yes—
19C2×C11×C22⋊C8C2211yescharacteristic
20C11×#380C2211yescharacteristic
21C2×C11⋊#95—21no—
22C2×C22⋊(C11⋊C8)C2211yescharacteristic
23C2×#66881C2212yes—
24C2×#66880C2212yes—
25C2×#66879C2212yes—
26C4⋊C4×Dic11C2211yescharacteristic
27#10622587C2211yescharacteristic
28#10622586—21no—
29#10622586—21no—
30#10622586C2211yescharacteristic
31#10622586C2212yes—
32#10622586C2212yes—
33C2×#66874C2212yes—
34#10622559—21no—
35C22⋊C4×Dic11—21no—
36#10622334C418yescomplemented
37#10622265—21no—
38#10622262—21no—
39C11×#340C2212yes—
40#10622188C2212yes—
41#10622167C2212yes—
42#10622167C2212yes—
43#10622167C2212yes—
44C23×C4×C11—21no—
45C23×C4×C11C2311yescharacteristic
46C23×Dic11—21no—
47C23×Dic11D411yescharacteristic
48C23×Dic11D411yescharacteristic
49C22×Dic22C2311yescharacteristic
50C22×C8×C11C2312yes—
51C2×C11×C4⋊C4—21no—
52C2×C11×C4⋊C4—42no—
53C2×C11×C4⋊C4C2311yescharacteristic
54C2×C11×C4⋊C4C2312yes—
55C2×C11×C4⋊C4C2311yescharacteristic
56C2×C11×C22⋊C4—21no—
57C2×C22⋊Dic11—21no—
58C2×C4⋊Dic11—21no—
59C2×C4⋊Dic11—22no—
60C2×C4⋊Dic11C2×C414yes—
61C2×C4⋊Dic11C2311yescharacteristic
62C2×C4⋊Dic11C2312yes—
63#66977—22no—
64#66977C2×C414yes—
65C2×Dic11⋊C4—21no—
66C2×Dic11⋊C4—21no—
67C2×Dic11⋊C4—21no—
68C2×Dic11⋊C4—22no—
69C2×Dic11⋊C4—22no—
70C2×Dic11⋊C4—44no—
71C2×C4×Dic11—21no—
72C2×C4×Dic11—21no—
73C2×C4×Dic11—42no—
74C2×C4×Dic11D411yescharacteristic
75C2×C4×Dic11D411yescharacteristic
76C2×C4×Dic11C2311yescharacteristic
77C2×C4×Dic11C2311yescharacteristic
78C22×C11⋊C8C2312yes—
79#66940—42no—
80C11×#107—42no—
81C11×C22⋊C8C2×C414yes—
82C11×#95—21no—
83C11⋊#95—21no—
84C11⋊#95—21no—
85C11⋊#95—21no—
86C11⋊#95—21no—
87C11⋊#95—22no—
88C11⋊#95—22no—
89C22⋊(C11⋊C8)—22no—
90#66881C2×C418yescomplemented
91#66880C2×C418yescomplemented
92#66879—42no—
93#66874—42no—
94#22084—111nomaximal, Sylow
95C24×C11C2×D411yescharacteristic
96C22×C4×C11—22no—
97C22×C4×C11—22no—
98C22×C4×C11—21no—
99C22×C4×C11—22no—
100C22×C4×C11—21no—
101C22×C4×C11—42no—
102C22×C4×C11—42no—
103C22×C4×C11—42no—
104C22×C4×C11C22×C412yes—
105C22×C4×C11C2×D411yescharacteristic
106C22×C4×C11#4112yes—
107C22×C4×C11C2411yescharacteristic
108C22×Dic11—22no—
109C22×Dic11—24no—
110C22×Dic11—22no—
111C22×Dic11—22no—
112C22×Dic11—21no—
113C22×Dic11—21no—
114C22×Dic11—22no—
115C22×Dic11—41no—
116C22×Dic11—41no—
117C22×Dic11—41no—
118C22×Dic11—42no—
119C22×Dic11—42no—
120C22×Dic11C2×D411yescharacteristic
121C22×Dic11C2×D411yescharacteristic
122C22×Dic11C2×D411yescharacteristic
123C22×Dic11C2×D411yescharacteristic
124C2×Dic22—21no—
125C2×Dic22—22no—
126C2×Dic22—22no—
127C2×Dic22C22×C412yes—
128C2×C8×C11—22no—
129C2×C8×C11—22no—
130C2×C8×C11C22×C414yes—
131C11×C4⋊C4—22no—
132C11×C4⋊C4—22no—
133C11×C4⋊C4—22no—
134C11×C4⋊C4—22no—
135C11×C4⋊C4—42no—
136C11×C22⋊C4—42no—
137C22⋊Dic11—24no—
138C4⋊Dic11—24no—
139C4⋊Dic11—21no—
140C4⋊Dic11—22no—
141C4⋊Dic11C22×C414yes—
142C4⋊Dic11C22×C412yes—
143Dic11⋊C4—24no—
144Dic11⋊C4—42no—
145Dic11⋊C4—42no—
146Dic11⋊C4—44no—
147Dic11⋊C4—44no—
148C4×Dic11—22no—
149C4×Dic11—22no—
150C4×Dic11—41no—
151C4×Dic11—41no—
152C4×Dic11—42no—
153C4×Dic11—42no—
154C2×C11⋊C8—22no—
155C2×C11⋊C8—22no—
156C2×C11⋊C8—22no—
157C2×#483—111no—
158C4×C22⋊Q8—111no—
159C2×#389—111no—
160#1944—112no—
161#1935—112no—
162#1920—112no—
163#1872—111no—
164#1863—112no—
165#1852D1111yescharacteristic
166#1823—111no—
167#1749—111no—
168C23×C11—21no—
169C23×C11—21no—
170C23×C11—21no—
171C23×C11—21no—
172C23×C11C4×D412yes—
173C23×C11AGL(1,ℤ/8)12yes—
174C23×C11#13712yes—
175C23×C11C22×D411yescharacteristic
176C2×C4×C11—21no—
177C2×C4×C11—21no—
178C2×C4×C11—24no—
179C2×C4×C11—24no—
180C2×C4×C11—41no—
181C2×C4×C11—42no—
182C2×C4×C11—41no—
183C2×C4×C11—42no—
184C2×C4×C11—41no—
185C2×C4×C11—41no—
186C2×C4×C11—41no—
187C2×C4×C11—41no—
188C2×C4×C11—42no—
189C2×C4×C11—84no—
190C2×C4×C11C4×D412yes—
191C2×C4×C11C23×C411yescharacteristic
192C2×C4×C11C2×#4111yescharacteristic
193C2×C4×C11H(5,𝔽2)12yes—
194C2×C4×C112−1+412yes—
195C2×Dic11—21no—
196C2×Dic11—21no—
197C2×Dic11—22no—
198C2×Dic11—22no—
199C2×Dic11—24no—
200C2×Dic11—21no—
201C2×Dic11—21no—
202C2×Dic11—21no—
203C2×Dic11—21no—
204C2×Dic11—21no—
205C2×Dic11—21no—
206C2×Dic11—21no—
207C2×Dic11—21no—
208C2×Dic11—21no—
209C2×Dic11—44no—
210C2×Dic11—44no—
211C2×Dic11—41no—
212C2×Dic11—41no—
213C2×Dic11—42no—
214C2×Dic11—41no—
215C2×Dic11—42no—
216C2×Dic11—41no—
217C2×Dic11—48no—
218C2×Dic11#13712yes—
219Dic22—24no—
220Dic22—42no—
221C8×C11—24no—
222C11⋊C8—42no—
223C2×C22⋊Q8—111no—
224C2×C4×Q8—111no—
225C22×C4⋊C4—111no—
226C22×C4⋊C4—221no—
227C22×C4⋊C4C2×D1111yescharacteristic
228C22×C42—111no—
229#483—118no—
230C2×#107C2×D1112yes—
231C2×#106—112no—
232C2×#103—112no—
233C2×#103—112no—
234C2×C22⋊C8—111no—
235C2×C22⋊C8C2×D1111yescharacteristic
236C2×#97—112no—
237#391—111no—
238#389—112no—
239#389—112no—
240#389—111no—
241#389—221no—
242#389—221no—
243#387—221no—
244#385—221no—
245#380C2×D1111yescharacteristic
246C4×C4⋊C4—111no—
247C4×C4⋊C4—221no—
248C4×C22⋊C4—221no—
249C2×#95—221no—
250#340—112no—
251#340—112no—
252#340—112no—
253#340C2×D1112yes—
254#336—112no—
255C22×C11—22no—
256C22×C11—22no—
257C22×C11—22no—
258C22×C11—22no—
259C22×C11—22no—
260C22×C11—22no—
261C22×C11#44014yes—
262C22×C11C2×C4×D411yescharacteristic
263C22×C11#53411yescharacteristic
264C22×C11#53611yescharacteristic
265C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C11C2×#13711yescharacteristic
267C22×C11C2×#13711yescharacteristic
268C22×C11#57511yescharacteristic
269C4×C11—22no—
270C4×C11—81no—
271C4×C11—82no—
272C4×C11—81no—
273C4×C11#52012yes—
274C4×C11#53912yes—
275Dic11—22no—
276Dic11—42no—
277Dic11—41no—
278Dic11—41no—
279Dic11—41no—
280Dic11—44no—
281Dic11—42no—
282C22×Q8—111no—
283C23×C4—21no—
284C23×C4—111no—
285C23×C4—111no—
286C23×C4—221no—
287C23×C4C22×D1111yescharacteristic
288C22×C8—112no—
289C22×C8C22×D1112yes—
290C22⋊Q8—114no—
291C22⋊Q8—222no—
292C4×Q8—442no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—114no—
296C2×C4⋊C4—112no—
297C2×C4⋊C4—111no—
298C2×C4⋊C4—221no—
299C2×C4⋊C4—221no—
300C2×C4⋊C4—222no—
301C2×C4⋊C4—222no—
302C2×C4⋊C4—222no—
303C2×C4⋊C4—221no—
304C2×C4⋊C4—221no—
305C2×C4⋊C4—444no—
306C2×C4⋊C4C22×D1112yes—
307C2×C4⋊C4C22×D1111yescharacteristic
308C2×C4⋊C4C22×D1111yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—221no—
311C2×C42—111no—
312C2×C42—111no—
313C2×C42—111no—
314C2×C42—111no—
315C2×C42—221no—
316C2×C42—221no—
317C2×C42—442no—
318#107—42no—
319#106—118no—
320#103—118no—
321#103—442no—
322C22⋊C8—222no—
323C22⋊C8C4×D1114yes—
324#97—442no—
325#95—21no—
326#95—222no—
327#95—222no—
328#95—221no—
329#95—221no—
330#95—221no—
331#95—221no—
332C2×C11—24no—
333C2×C11#232211yescharacteristic
334C2×C11C2×#44011yescharacteristic
335C2×C11C4×#13711yescharacteristic
336C2×C11#319411yescharacteristic
337C2×C11#319511yescharacteristic
338C2×C11#323611yescharacteristic
339C2×C11#323811yescharacteristic
340C24D4×D1111yescharacteristic
341C2×Q8—112no—
342C2×Q8—221no—
343C2×Q8—222no—
344C2×Q8—222no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—111no—
354C22×C4—111no—
355C22×C4—111no—
356C22×C4—111no—
357C22×C4—222no—
358C22×C4—224no—
359C22×C4—222no—
360C22×C4—222no—
361C22×C4—221no—
362C22×C4—221no—
363C22×C4—222no—
364C22×C4—441no—
365C22×C4—441no—
366C22×C4—441no—
367C22×C4—442no—
368C22×C4—442no—
369C22×C4C2×C4×D1112yes—
370C22×C4D4×D1111yescharacteristic
371C22×C4#434312yes—
372C22×C4C23×D1111yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—222no—
376C2×C8—222no—
377C2×C8—222no—
378C2×C8C2×C4×D1114yes—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—114no—
385C4⋊C4—112no—
386C4⋊C4—224no—
387C4⋊C4—224no—
388C4⋊C4—222no—
389C4⋊C4—221no—
390C4⋊C4—444no—
391C4⋊C4—444no—
392C4⋊C4—442no—
393C4⋊C4—442no—
394C22⋊C4—42no—
395C22⋊C4—224no—
396C42—222no—
397C42—222no—
398C42—441no—
399C42—441no—
400C42—442no—
401C42—442no—
402C11#2208411yescharacteristic, complemented, Sylow
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23#6694612yes—
408C23#6696412yes—
409C23#6697112yes—
410C23C2×D4×D1111yescharacteristic
411Q8—224no—
412Q8—442no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—84no—
427C2×C4—112no—
428C2×C4—221no—
429C2×C4—221no—
430C2×C4—221no—
431C2×C4—221no—
432C2×C4—222no—
433C2×C4—222no—
434C2×C4—224no—
435C2×C4—221no—
436C2×C4—221no—
437C2×C4—221no—
438C2×C4—221no—
439C2×C4—221no—
440C2×C4—221no—
441C2×C4—221no—
442C2×C4—444no—
443C2×C4—444no—
444C2×C4—448no—
445C2×C4—442no—
446C2×C4—442no—
447C2×C4—441no—
448C2×C4—441no—
449C2×C4—441no—
450C2×C4—441no—
451C2×C4#6694612yes—
452C2×C4C22×C4×D1111yescharacteristic
453C2×C4#6703712yes—
454C2×C4C2×#434311yescharacteristic
455C2×C4#6704312yes—
456C8—24no—
457C8—442no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22#1062250214yes—
465C22C2×#6694611yescharacteristic
466C22#1062312011yescharacteristic
467C22#1062317911yescharacteristic
468C22C2×#6696411yescharacteristic
469C22C2×#6697111yescharacteristic
470C22#1062330411yescharacteristic
471C22D11×#13711yescharacteristic
472C4—22no—
473C4—81no—
474C4—82no—
475C4—81no—
476C4—222no—
477C4—441no—
478C4—441no—
479C4—441no—
480C4—444no—
481C4—442no—
482C4—442no—
483C4#1062307712yes—
484C4#1062317612yes—
485C2—24no—
486C2#1307128411yescharacteristic
487C2C2×#1062250211yescharacteristic
488C2#1307743411yescharacteristic
489C2#1307942711yescharacteristic
490C2#1307943811yescharacteristic
491C2#1307982111yescharacteristic
492C2#1307983511yescharacteristic
subgroupG/H
centerC23C2×D4×D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1852C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
cosocleC22×C4×C11C24
hypercenter#1852D11
hypocenterC11#22084
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #13069357C2semidirect
5. #13069304C2semidirect
6. #13069200C2semidirect
8. #13068867C2semidirect
36. #10622334C4semidirect
90. #66881C2×C4semidirect
91. #66880C2×C4semidirect
402. C11#22084semidirect

Maximal subgroups

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

subgroupindex
C2×#106223342
#130711442
C2×#106225862
#130693572
#130693042
#130692002
#130689232
#130688672
#130687002
#130683802
C11×#18522
#2208411