Finite Group Databasebeta

← #14788713

Subgroups — #14788713

Order 2816.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623073C211yesmaximal, characteristic
2C2×#10622644C211yesmaximal, characteristic, complemented
3#13073978C211yesmaximal, characteristic, complemented
4#13069855C212yesmaximal
5#13069708C212yesmaximal
6#13069556C211yesmaximal, characteristic, complemented
7#13069436C211yesmaximal, characteristic, complemented
8#13068954C212yesmaximal, complemented
9#13068867C212yesmaximal, complemented
10#13068626C211yesmaximal, characteristic
11C2×C11×#440C211yesmaximal, characteristic
12C22×C11×SD16C2211yescharacteristic
13C2×C4×Q8×C11C2211yescharacteristic
14C2×C4×D4×C11C2211yescharacteristic
15C22×#4326C2212yes—
16C2×#66989C2212yes—
17C2×#66979—21no—
18C2×#66977—21no—
19#10623073—24no—
20#10623073—24no—
21C2×#66940C2212yes—
22C2×#66932—22no—
23C2×#66922C2211yescharacteristic
24C2×C4×Dic22C2211yescharacteristic
25C11×#440—24no—
26C2×C11×#107C2211yescharacteristic
27C2×C11×#103C2211yescharacteristic
28C2×C11×#102C2211yescharacteristic
29C2×C11×#97C2211yescharacteristic
30#10622843C2212yes—
31C2×#66896C2212yes—
32C2×#66880C2212yes—
33C2×#66880C2212yes—
34#10622644—24no—
35#10622599—21no—
36#10622592—21no—
37#10622586C2212yes—
38C2×#66874C2212yes—
39#10622549C2211yescharacteristic
40#10622546—22no—
41C2×C4⋊(C11⋊C8)C2211yescharacteristic
42#10622188C2212yes—
43#10622188C2212yes—
44#10622167C2212yes—
45C22×Q8×C11C2311yescharacteristic
46C22×D4×C11C2311yescharacteristic
47C23×C4×C11—21no—
48C22×Dic22D411yescharacteristic
49C22×Dic22C2312yes—
50C2×C11×SD16—22no—
51C2×C11×SD16—22no—
52C2×C11×SD16—22no—
53C22×C8×C11C2311yescharacteristic
54C22×C8×C11C2311yescharacteristic
55C4×Q8×C11—22no—
56C4×Q8×C11—22no—
57C4×D4×C11—22no—
58C4×D4×C11D414yes—
59C2×C11×C4⋊C4—21no—
60C2×C11×C4⋊C4C2311yescharacteristic
61C2×C11×C4⋊C4C2311yescharacteristic
62C2×C11×C22⋊C4—21no—
63C2×C42×C11—21no—
64C2×C42×C11C2311yescharacteristic
65C2×C22⋊Dic11—22no—
66C2×C22⋊Dic11—22no—
67C2×#4326—24no—
68C2×#4326—24no—
69C2×#4326—42no—
70#66989—24no—
71#66989—24no—
72#66979—28no—
73C2×C4⋊Dic11—21no—
74C2×C4⋊Dic11—21no—
75C2×C4⋊Dic11D411yescharacteristic
76C2×C4⋊Dic11C2312yes—
77#66977—28no—
78C2×Dic11⋊C4—22no—
79C2×Dic11⋊C4—22no—
80C2×Dic11⋊C4D412yes—
81C2×C4×Dic11—22no—
82C2×C4×Dic11C2312yes—
83C22×C11⋊C8C2312yes—
84#66940—24no—
85#66940—24no—
86#66932—216no—
87#66922—22no—
88#66922D414yes—
89C4×Dic22—22no—
90C4×Dic22—22no—
91C11×#107—22no—
92C11×#103—22no—
93C11×#102—22no—
94C11×#97—22no—
95#66896—42no—
96C11⋊#95—21no—
97C11⋊#95—21no—
98C11⋊#95—22no—
99C11⋊#95—22no—
100C11⋊#95—22no—
101#66880—24no—
102#66880—42no—
103#66874—24no—
104C4⋊(C11⋊C8)—22no—
105#24201—111nomaximal, Sylow
106C24×C11—21no—
107C2×Q8×C11—21no—
108C2×Q8×C11—21no—
109C2×Q8×C11—21no—
110C2×Q8×C11—21no—
111C2×Q8×C11—21no—
112C2×Q8×C11#4112yes—
113C2×D4×C11—21no—
114C2×D4×C11—21no—
115C2×D4×C11—21no—
116C2×D4×C11—21no—
117C2×D4×C11—21no—
118C2×D4×C11C2×D412yes—
119C22×C4×C11—21no—
120C22×C4×C11—21no—
121C22×C4×C11—24no—
122C22×C4×C11—21no—
123C22×C4×C11—44no—
124C22×C4×C11C2×D411yescharacteristic
125C22×C4×C11C2×D411yescharacteristic
126C22×C4×C11C2411yescharacteristic
127C22×Dic11—22no—
128C22×Dic11—22no—
129C22×Dic11C2×D412yes—
130C22×Dic11C2×D412yes—
131C2×Dic22—22no—
132C2×Dic22—22no—
133C2×Dic22—22no—
134C2×Dic22—22no—
135C2×Dic22—22no—
136C2×Dic22—22no—
137C2×Dic22—42no—
138C2×Dic22C2×D414yes—
139C11×SD16—28no—
140C2×C8×C11—21no—
141C2×C8×C11—21no—
142C2×C8×C11—21no—
143C2×C8×C11—21no—
144C2×C8×C11—21no—
145C2×C8×C11#4112yes—
146C11×C4⋊C4—21no—
147C11×C4⋊C4—21no—
148C11×C4⋊C4—21no—
149C11×C4⋊C4—42no—
150C11×C4⋊C4C2×D412yes—
151C11×C22⋊C4—24no—
152C42×C11—42no—
153C42×C11C2×D412yes—
154C42×C11C2×D412yes—
155C22⋊Dic11—28no—
156C22⋊Dic11—28no—
157#4326—48no—
158C4⋊Dic11—22no—
159C4⋊Dic11—24no—
160C4⋊Dic11—24no—
161C4⋊Dic11—22no—
162C4⋊Dic11—22no—
163Dic11⋊C4—28no—
164Dic11⋊C4—24no—
165Dic11⋊C4—44no—
166C4×Dic11—28no—
167C4×Dic11—22no—
168C4×Dic11—22no—
169C2×C11⋊C8—22no—
170C2×C11⋊C8—22no—
171C2×C11⋊C8—22no—
172C2×#556—111no—
173C2×#462—111no—
174C2×#440D1111yescharacteristic
175#2581—111no—
176#2042—112no—
177#2015—112no—
178#1985—111no—
179#1960—111no—
180#1878—112no—
181#1863—112no—
182#1804—111no—
183C23×C11—22no—
184C23×C11—41no—
185C23×C11—41no—
186C23×C11—42no—
187C23×C11—42no—
188C23×C11C22×D411yescharacteristic
189Q8×C11—22no—
190Q8×C11—22no—
191Q8×C11—41no—
192Q8×C11—41no—
193D4×C11—22no—
194D4×C11—22no—
195D4×C11—41no—
196D4×C11—41no—
197C2×C4×C11—22no—
198C2×C4×C11—22no—
199C2×C4×C11—22no—
200C2×C4×C11—22no—
201C2×C4×C11—48no—
202C2×C4×C11—41no—
203C2×C4×C11—41no—
204C2×C4×C11—42no—
205C2×C4×C11—41no—
206C2×C4×C11—42no—
207C2×C4×C11#12012yes—
208C2×C4×C11#12012yes—
209C2×C4×C11C22×D411yescharacteristic
210C2×C4×C11C22×D411yescharacteristic
211C2×C4×C11C2×#4111yescharacteristic
212C2×C4×C11C2×#4111yescharacteristic
213C2×C4×C11H(5,𝔽2)11yescharacteristic
214C2×C4×C112−1+411yescharacteristic
215C2×Dic11—22no—
216C2×Dic11—22no—
217C2×Dic11—24no—
218C2×Dic11—22no—
219C2×Dic11—22no—
220C2×Dic11—24no—
221C2×Dic11—42no—
222C2×Dic11—42no—
223C2×Dic11—42no—
224C2×Dic11—42no—
225C2×Dic11—42no—
226Dic22—24no—
227Dic22—24no—
228Dic22—44no—
229Dic22—42no—
230Dic22—42no—
231C8×C11—22no—
232C8×C11—41no—
233C11⋊C8—42no—
234C22×Q16—112no—
235C22×SD16C2×D1111yescharacteristic
236#556—224no—
237#556—224no—
238C2×C4⋊Q8—111no—
239C2×#126—222no—
240C2×#124—112no—
241C2×C22⋊Q8—221no—
242C2×C4×Q8—111no—
243C2×C4×Q8—112no—
244C2×C4×Q8C2×D1111yescharacteristic
245C2×C4×D4C2×D1111yescharacteristic
246C2×#117—221no—
247#462—224no—
248#440—24no—
249C2×#107C2×D1111yescharacteristic
250C2×C4⋊C8—111no—
251C2×#103—112no—
252C2×#103—112no—
253C2×#103—112no—
254C2×#103C2×D1111yescharacteristic
255C2×#102—112no—
256C2×#102C2×D1111yescharacteristic
257C2×#97C2×D1111yescharacteristic
258#400—221no—
259#395—221no—
260#391—112no—
261#389—112no—
262#387—222no—
263#382—111no—
264#340—112no—
265#336—112no—
266#336—112no—
267C22×C11—42no—
268C22×C11—42no—
269C22×C11—42no—
270C22×C11—82no—
271C22×C11—82no—
272C22×C11C2×#12011yescharacteristic
273C22×C11#54811yescharacteristic
274C22×C11#55011yescharacteristic
275C22×C11#57512yes—
276C22×C11#57812yes—
277C4×C11—22no—
278C4×C11—22no—
279C4×C11—42no—
280C4×C11—42no—
281C4×C11—81no—
282C4×C11#52511yescharacteristic
283C4×C11#53711yescharacteristic
284C4×C11#54611yescharacteristic
285C4×C11#54711yescharacteristic
286Dic11—44no—
287Dic11—42no—
288Dic11—42no—
289Dic11—82no—
290C22×Q8—111no—
291C22×Q8—112no—
292C22×Q8C22×D1111yescharacteristic
293C22×D4C22×D1111yescharacteristic
294C23×C4—21no—
295C2×Q16—224no—
296C2×Q16—224no—
297C2×Q16—442no—
298C2×SD16—22no—
299C2×SD16—22no—
300C2×SD16—22no—
301C22×C8—112no—
302C22×C8C22×D1111yescharacteristic
303C22×C8C22×D1111yescharacteristic
304C4⋊Q8—114no—
305C4⋊Q8—222no—
306#126—2216no—
307#124—224no—
308#124—224no—
309C22⋊Q8—228no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—224no—
313C4×Q8—224no—
314C4×Q8—222no—
315C4×Q8—222no—
316C4×D4—22no—
317C4×D4C22⋊D1114yes—
318#117—228no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—111no—
321C2×C4⋊C4—112no—
322C2×C4⋊C4—112no—
323C2×C4⋊C4—222no—
324C2×C4⋊C4—221no—
325C2×C4⋊C4—221no—
326C2×C4⋊C4—222no—
327C2×C4⋊C4C22×D1111yescharacteristic
328C2×C4⋊C4C22×D1111yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—222no—
331C2×C22⋊C4—222no—
332C2×C42—21no—
333C2×C42—112no—
334C2×C42—222no—
335C2×C42C22×D1111yescharacteristic
336#107—22no—
337C4⋊C8—222no—
338#103—22no—
339#103—224no—
340#103—224no—
341#103—442no—
342#102—22no—
343#102—442no—
344#97—22no—
345#95—222no—
346#95—222no—
347#95—222no—
348#95—221no—
349#95—221no—
350C2×C11—82no—
351C2×C11#240711yescharacteristic
352C2×C11#300511yescharacteristic
353C2×C11#301111yescharacteristic
354C2×C11#337712yes—
355C2×C11#340212yes—
356C24—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—114no—
363C2×Q8—222no—
364C2×Q8—222no—
365C2×Q8—222no—
366C2×Q8—222no—
367C2×Q8—222no—
368C2×Q8—222no—
369C2×Q8—442no—
370C2×Q8#433912yes—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4C2×C22⋊D1112yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—44no—
382C22×C4—112no—
383C22×C4—112no—
384C22×C4—222no—
385C22×C4—222no—
386C22×C4D4×D1111yescharacteristic
387C22×C4D4×D1111yescharacteristic
388C22×C4C23×D1111yescharacteristic
389Q16—448no—
390SD16—28no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—222no—
397C2×C8—222no—
398C2×C8—222no—
399C2×C8#433912yes—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—42no—
404C4⋊C4—228no—
405C4⋊C4—222no—
406C4⋊C4—224no—
407C4⋊C4—224no—
408C4⋊C4—224no—
409C4⋊C4—222no—
410C4⋊C4—222no—
411C4⋊C4—444no—
412C4⋊C4C2×C22⋊D1112yes—
413C22⋊C4—24no—
414C22⋊C4—228no—
415C22⋊C4—228no—
416C42—42no—
417C42—228no—
418C42—222no—
419C42—222no—
420C42D4×D1112yes—
421C42C2×C22⋊D1112yes—
422C11#2420111yescharacteristic, complemented, Sylow
423C23—22no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23C2×D4×D1111yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—41no—
432Q8—41no—
433Q8—224no—
434Q8—224no—
435Q8—444no—
436Q8—442no—
437Q8—442no—
438D4—22no—
439D4—22no—
440D4—41no—
441D4—41no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—222no—
453C2×C4—222no—
454C2×C4—224no—
455C2×C4—222no—
456C2×C4—222no—
457C2×C4—224no—
458C2×C4—442no—
459C2×C4—442no—
460C2×C4—442no—
461C2×C4—442no—
462C2×C4—442no—
463C2×C4#6699012yes—
464C2×C4#6699012yes—
465C2×C4C2×#433911yescharacteristic
466C2×C4C2×#433911yescharacteristic
467C2×C4C2×D4×D1111yescharacteristic
468C2×C4#6704211yescharacteristic
469C2×C4#6704311yescharacteristic
470C2×C4C22×C22⋊D1111yescharacteristic
471C8—22no—
472C8—41no—
473C8—442no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—82no—
478C22—82no—
479C22#1062308311yescharacteristic
480C22#1062308411yescharacteristic
481C22#1062330412yes—
482C22#1062331012yes—
483C22C2×#6699011yescharacteristic
484C4—22no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—444no—
490C4—442no—
491C4—442no—
492C4—882no—
493C4#1062308111yescharacteristic
494C4#1062308211yescharacteristic
495C4#1062332211yescharacteristic
496C4#1062336411yescharacteristic
497C2—82no—
498C2#1307196111yescharacteristic
499C2#1307785511yescharacteristic
500C2#1307791011yescharacteristic
501C2#1308145912yes—
502C2#1308169612yes—
subgroupG/H
centerC23C2×D4×D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C11×#440C2
socleC23×C11C22×D4
prime radicalC22×D4×C11C23
hypercenterC2×#440D11
hypocenterC11#24201
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C2×#10622644C2semidirect
3. #13073978C2semidirect
6. #13069556C2semidirect
7. #13069436C2semidirect
8. #13068954C2semidirect
9. #13068867C2semidirect
422. C11#24201semidirect

Maximal subgroups

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

subgroupindex
C2×#106230732
C2×#106226442
#130739782
#130698552
#130697082
#130695562
#130694362
#130689542
#130688672
#130686262
C2×C11×#4402
#2420111