Finite Group Databasebeta

← #14736673

Subgroups — #14736673

Order 2816.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13078991C211yesmaximal, characteristic
2#13078655C211yesmaximal, characteristic, complemented
3C2×#10622640C211yesmaximal, characteristic
4#13068184C212yesmaximal, complemented
5#13068156C212yesmaximal, complemented
6#13068060C212yesmaximal, complemented
7#13068032C212yesmaximal, complemented
8#13067368C211yesmaximal, characteristic, complemented
9#13067307C211yesmaximal, characteristic, complemented
10#13067285C211yesmaximal, characteristic
11Q8×C11×M16C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C11C2211yescharacteristic
13C22×#4325—21no—
14C2×#66927C2211yescharacteristic
15#10623001C2211yescharacteristic
16C11×#446—22no—
17C11×#446C2212yescomplemented
18C8×Q8×C11C2212yescomplemented
19C11×#423—21no—
20C11×#423C2211yescharacteristic
21C4×C11×M16—21no—
22C4×C11×M16C2211yescharacteristic
23#10622841—21no—
24#10622744—21no—
25#10622744—21no—
26#10622644C2211yescharacteristic, complemented
27#10622644C2211yescharacteristic, complemented
28C4×#4326—21no—
29C4×#4326—21no—
30#10622640—21no—
31#10622640—21no—
32#10622640C2212yes—
33#10622640C2212yes—
34#10622634C2211yescharacteristic, complemented
35#10622634C2211yescharacteristic, complemented
36#10622583—21no—
37C2×#66873—21no—
38#10622538C2211yescharacteristic
39C2×C4⋊(C11⋊C8)C2211yescharacteristic
40#10622326—22no—
41#10622307C2212yes—
42#10622307C2212yes—
43#10622124C2212yescomplemented
44#10622123C2212yes—
45#10622123C2212yes—
46#10622119C2212yescomplemented
47#10622096C2212yescomplemented
48#10622095C2212yescomplemented
49C22×Q8×C11D411yescharacteristic
50C22×D44—21no—
51C22×D44—21no—
52C22×D44—21no—
53C2×C11×M16—21no—
54C2×C11×M16D411yescharacteristic
55C2×C11×M16D411yescharacteristic
56C4×Q8×C11—21no—
57C4×Q8×C11—21no—
58C4×Q8×C11D411yescharacteristic
59C4×Q8×C11D411yescharacteristic
60C4×Q8×C11C2311yescharacteristic
61C4×Q8×C11C2311yescharacteristic
62C2×C11×C4⋊C4—21no—
63C2×C11×C4⋊C4D411yescharacteristic
64C2×C42×C11—21no—
65C2×C42×C11C2311yescharacteristic
66C2×#4326—21no—
67C2×#4326—21no—
68C2×#4325—22no—
69C2×#4325—22no—
70C2×#4325—44no—
71#66979—21no—
72#66977—21no—
73#66977—21no—
74C22×C11⋊C8—21no—
75#66927—21no—
76#66927—21no—
77#66927—21no—
78#66927C2312yes—
79#66923—21no—
80#66922C2311yescharacteristic
81#66922C2311yescharacteristic
82C4×Dic22—21no—
83C4×Dic22—21no—
84C11×C4⋊C8—22no—
85C11×C4⋊C8—22no—
86C11×C4⋊C8—21no—
87C11×C4⋊C8C2312yes—
88C11×#97—22no—
89C11×#97D412yes—
90C4×C8×C11—22no—
91C4×C8×C11C2312yes—
92#66899—21no—
93#66899—21no—
94C22⋊(C11⋊C8)—21no—
95#66885—22no—
96#66885—22no—
97#66885—22no—
98#66885—22no—
99#66874—21no—
100#66874—21no—
101#66873—22no—
102#66873—22no—
103#66871—21no—
104#66871—21no—
105C4⋊(C11⋊C8)—21no—
106C4⋊(C11⋊C8)C2312yes—
107C4⋊(C11⋊C8)C2311yescharacteristic
108C4⋊(C11⋊C8)C2311yescharacteristic
109C4×C11⋊C8—21no—
110#19174—111nomaximal, Sylow
111C23×D11—41no—
112C2×Q8×C11—21no—
113C2×Q8×C11—22no—
114C2×Q8×C11—41no—
115C2×Q8×C11C2×D411yescharacteristic
116C2×Q8×C11C2×D411yescharacteristic
117C22×C4×C11—21no—
118C22×C4×C11—21no—
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11#4111yescharacteristic
122C2×D44—22no—
123C2×D44—22no—
124C2×D44—22no—
125C2×D44—42no—
126C2×D44—42no—
127C2×D44—41no—
128C2×D44—41no—
129C2×D44—41no—
130C2×D44—42no—
131C2×D44—42no—
132C2×D44—82no—
133C2×Dic22—21no—
134C2×Dic22—21no—
135C11×M16—22no—
136C11×M16—22no—
137C11×M16—42no—
138C2×C8×C11—22no—
139C2×C8×C11C2×D412yes—
140C2×C8×C11C2×D412yes—
141C11×C4⋊C4—21no—
142C11×C4⋊C4—21no—
143C11×C4⋊C4—21no—
144C11×C4⋊C4—21no—
145C11×C4⋊C4—21no—
146C11×C4⋊C4C2×D411yescharacteristic
147C11×C4⋊C4C2×D411yescharacteristic
148C42×C11—21no—
149C42×C11—21no—
150C42×C11—21no—
151C42×C11—21no—
152C42×C11C2×D411yescharacteristic
153C42×C11C2×D411yescharacteristic
154C42×C11#4111yescharacteristic
155C42×C11C2411yescharacteristic
156C22⋊Dic11—41no—
157#4326—42no—
158#4326—42no—
159#4325—44no—
160#4325—44no—
161C4⋊Dic11—21no—
162C4⋊Dic11—21no—
163C4⋊Dic11—21no—
164C4⋊Dic11—21no—
165Dic11⋊C4—41no—
166Dic11⋊C4—41no—
167C4×Dic11—41no—
168C2×C11⋊C8—21no—
169C2×C11⋊C8—21no—
170C2×C11⋊C8—22no—
171C2×C11⋊C8—42no—
172#3138—111no—
173#3104—111no—
174C2×#460—111no—
175Q8×M16D1111yescharacteristic, complemented
176#1702—112no—
177#1698—112no—
178#1682—112no—
179#1678—112no—
180#1565—111no—
181#1550—111no—
182#1545—111no—
183C23×C11#12111yescharacteristic
184C22×D11—42no—
185C22×D11—82no—
186C22×D11—82no—
187C22×D11—82no—
188Q8×C11—21no—
189Q8×C11—21no—
190Q8×C11—21no—
191Q8×C11—42no—
192Q8×C11AGL(1,ℤ/8)12yes—
193C2×C4×C11—21no—
194C2×C4×C11—21no—
195C2×C4×C11—21no—
196C2×C4×C11—21no—
197C2×C4×C11—21no—
198C2×C4×C11—21no—
199C2×C4×C11—41no—
200C2×C4×C11—42no—
201C2×C4×C11—81no—
202C2×C4×C11#12111yescharacteristic
203C2×C4×C11#12111yescharacteristic
204C2×C4×C11#12111yescharacteristic
205C2×C4×C11#13512yes—
206C2×C4×C11#13712yes—
207C2×C4×C11C22×D411yescharacteristic
208C2×C4×C11C22×D411yescharacteristic
209C2×C4×C11C2×#4111yescharacteristic
210C2×Dic11—41no—
211C2×Dic11—41no—
212D44—42no—
213D44—42no—
214D44—42no—
215D44—42no—
216D44—82no—
217D44—82no—
218D44—82no—
219D44—82no—
220Dic22—41no—
221Dic22—41no—
222Dic22—81no—
223C8×C11—22no—
224C8×C11—22no—
225C8×C11—42no—
226C11⋊C8—42no—
227C11⋊C8—41no—
228C11⋊C8—41no—
229C22×SD16—221no—
230#533—111no—
231C2×#127—111no—
232C2×C4×Q8C2×D1111yescharacteristic
233#490—222no—
234#486—112no—
235#486—112no—
236#484—221no—
237#477—111no—
238#477—111no—
239#462—111no—
240#462—111no—
241#460—112no—
242#460—112no—
243#460—221no—
244#460—221no—
245#451—221no—
246#446—22no—
247#446C2×D1112yescomplemented
248C8×Q8C2×D1112yescomplemented
249C4×Q16—221no—
250C4×Q16—221no—
251#431—111no—
252#423—21no—
253#423C2×D1111yescharacteristic
254C2×C4⋊C8—111no—
255#418—221no—
256#418—221no—
257C2×#102—221no—
258C4×M16—21no—
259C4×M16C2×D1111yescharacteristic
260#335—112no—
261#334—112no—
262#333—112no—
263#332—112no—
264#331—112no—
265#331—112no—
266C22×C11—21no—
267C22×C11—21no—
268C22×C11#46412yes—
269C22×C11C2×#12111yescharacteristic
270C2×D11—82no—
271C2×D11—82no—
272C2×D11—82no—
273C2×D11—164no—
274C4×C11—21no—
275C4×C11—22no—
276C4×C11—21no—
277C4×C11—21no—
278C4×C11—41no—
279C4×C11—41no—
280C4×C11—41no—
281C4×C11#46412yes—
282C4×C11C2×#13511yescharacteristic
283C4×C11C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C11C2×#13711yescharacteristic
285C4×C11#57511yescharacteristic
286Dic11—81no—
287Dic11—81no—
288C22×Q8D4411yescharacteristic
289C22×D4—221no—
290C22×D4—221no—
291C22×D4—221no—
292C2×Q16—221no—
293C2×Q16—221no—
294C2×SD16—222no—
295C2×SD16—222no—
296C2×SD16—444no—
297C2×M16—21no—
298C2×M16C22⋊D1111yescharacteristic
299C2×M16C22⋊D1111yescharacteristic
300C22×C8—221no—
301C4⋊Q8—111no—
302C4⋊Q8—111no—
303#127—112no—
304#127—221no—
305#127—221no—
306#127—221no—
307#125—221no—
308C22⋊Q8—221no—
309C22⋊Q8—221no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—221no—
313C4×Q8—221no—
314C4×Q8D4411yescharacteristic
315C4×Q8D4411yescharacteristic
316C4×Q8C22×D1111yescharacteristic
317C4×Q8C22×D1111yescharacteristic
318#117—221no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4D4411yescharacteristic
321C2×C42—21no—
322C2×C42C22×D1111yescharacteristic
323#107—221no—
324#107—221no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—21no—
328C4⋊C8—111no—
329C4⋊C8—111no—
330C4⋊C8—112no—
331C4⋊C8—221no—
332C4⋊C8C22×D1112yes—
333#103—221no—
334#103—221no—
335#103—221no—
336#103—221no—
337#102—222no—
338#102—222no—
339#102—222no—
340#102—222no—
341#102—222no—
342#102—222no—
343C22⋊C8—221no—
344#97—22no—
345#97C22⋊D1112yes—
346C4×C8—22no—
347C4×C8—221no—
348C4×C8C22×D1112yes—
349C2×C11—22no—
350C2×C11C2×#46411yescharacteristic
351C2×C11#305511yescharacteristic
352C2×C11#306811yescharacteristic
353D11—162no—
354C24—441no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—41no—
358C2×Q8—221no—
359C2×Q8—221no—
360C2×Q8C2×D4411yescharacteristic
361C2×Q8C2×D4411yescharacteristic
362C2×D4—222no—
363C2×D4—222no—
364C2×D4—222no—
365C2×D4—442no—
366C2×D4—442no—
367C2×D4—441no—
368C2×D4—441no—
369C2×D4—441no—
370C2×D4—442no—
371C2×D4—442no—
372C2×D4—882no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C2×D4411yescharacteristic
376C22×C4#433911yescharacteristic
377C22×C4C2×C22⋊D1111yescharacteristic
378Q16—442no—
379Q16—442no—
380SD16—444no—
381SD16—444no—
382M16—22no—
383M16—22no—
384M16—42no—
385C2×C8—22no—
386C2×C8—221no—
387C2×C8—221no—
388C2×C8—222no—
389C2×C8—442no—
390C2×C8C2×C22⋊D1112yes—
391C2×C8C2×C22⋊D1112yes—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—221no—
398C4⋊C4—221no—
399C4⋊C4—221no—
400C4⋊C4—221no—
401C4⋊C4—441no—
402C4⋊C4—441no—
403C4⋊C4C2×D4411yescharacteristic
404C4⋊C4C2×D4411yescharacteristic
405C22⋊C4—441no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—441no—
411C42C2×D4411yescharacteristic
412C42#433911yescharacteristic
413C42C2×C22⋊D1111yescharacteristic
414C42C23×D1111yescharacteristic
415C11#1917411yescharacteristic, complemented, Sylow
416C23—442no—
417C23—882no—
418C23—882no—
419C23—882no—
420C23#6698311yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—441no—
426Q8—441no—
427Q8—881no—
428Q8#6696112yes—
429D4—442no—
430D4—442no—
431D4—442no—
432D4—442no—
433D4—882no—
434D4—882no—
435D4—882no—
436D4—882no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—441no—
447C2×C4—441no—
448C2×C4#6698311yescharacteristic
449C2×C4#6698311yescharacteristic
450C2×C4#6698311yescharacteristic
451C2×C4#6699512yes—
452C2×C4#6700312yes—
453C2×C4C22×D4411yescharacteristic
454C2×C4C2×#433911yescharacteristic
455C2×C4C22×C22⋊D1111yescharacteristic
456C8—22no—
457C8—22no—
458C8—42no—
459C8—442no—
460C8—441no—
461C8—441no—
462C22—21no—
463C22—21no—
464C22—882no—
465C22—882no—
466C22—882no—
467C22—1764no—
468C22#1062264112yes—
469C22C2×#6698311yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—881no—
478C4—881no—
479C4#1062264112yes—
480C4C2×#6696111yescharacteristic
481C4#1062327911yescharacteristic
482C4C2×#6699511yescharacteristic
483C4C2×#6700311yescharacteristic
484C2—22no—
485C2—1762no—
486C2C2×#1062264111yescharacteristic
487C2#1307823511yescharacteristic
488C2#1307834411yescharacteristic
subgroupG/H
centerC22C2×#66983
commutator, cosocleC42×C11C24
FrattiniC42C23×D11
FittingQ8×C11×M16C2
socleC22×C11C2×#121
prime radicalC2×#66927C22
hypercenterQ8×M16D11
hypocenterC11#19174
LCSG ▷ C42×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #13078655C2semidirect
4. #13068184C2semidirect
5. #13068156C2semidirect
6. #13068060C2semidirect
7. #13068032C2semidirect
8. #13067368C2semidirect
9. #13067307C2semidirect
11. Q8×C11×M16C2semidirect
17. C11×#446C22semidirect
18. C8×Q8×C11C22semidirect
26. #10622644C22semidirect
27. #10622644C22semidirect
34. #10622634C22semidirect
35. #10622634C22semidirect
43. #10622124C22semidirect
46. #10622119C22semidirect
47. #10622096C22semidirect
48. #10622095C22semidirect
175. Q8×M16D11semidirect
247. #446C2×D11semidirect
248. C8×Q8C2×D11semidirect
415. C11#19174semidirect

Maximal subgroups

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

subgroupindex
#130789912
#130786552
C2×#106226402
#130681842
#130681562
#130680602
#130680322
#130673682
#130673072
#130672852
Q8×C11×M162
#1917411