Finite Group Databasebeta

← #14782473

Subgroups — #14782473

Order 2816.

1318 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622608C211yesmaximal, characteristic, complemented
2C2×#10622537C211yesmaximal, characteristic, complemented
3C2×#10622614C211yesmaximal, characteristic
4C2×#10622614C211yesmaximal, characteristic
5C2×#10622552C211yesmaximal, characteristic
6#13070001C214yesmaximal, complemented
7#13069987C214yesmaximal, complemented
8#13068337C211yesmaximal, characteristic, complemented
9C11×#2296C211yesmaximal, characteristic, complemented
10C23×D44C2211yescharacteristic
11C2×C11×#117C2211yescharacteristic
12C2×C11×#117C2211yescharacteristic
13C22×C42×C11C2211yescharacteristic
14C22×(C2×D11)⋊C4—22no—
15C22×C4⋊Dic11C2211yescharacteristic
16C22×#4318C2211yescharacteristic
17C22×#4318C2211yescharacteristic
18C11×#389C2212yescomplemented
19C11×#389C2212yescomplemented
20C11×#385—22no—
21C4×C11×C4⋊C4—21no—
22C4×C11×C4⋊C4—21no—
23#10622614—24no—
24#10622614—24no—
25#10622608—24no—
26C2×#66873C2212yes—
27C2×#66873C2214yes—
28C2×#66873C2212yes—
29C2×#66872C2212yescomplemented
30C2×#66871C2212yescomplemented
31#10622552—21no—
32#10622552—21no—
33#10622552C2214yes—
34#10622537—24no—
35C2×C4⋊(C11⋊C8)C2214yescomplemented
36#10622167C2214yescomplemented
37C24×D11—21no—
38C23×C4×C11D412yes—
39C23×C4×C11C2311yescharacteristic
40C23×Dic11—21no—
41C22×D44—22no—
42C22×D44—22no—
43C22×D44—22no—
44C22×D44—22no—
45C22×D44—21no—
46C22×D44—22no—
47C22×D44Q814yes—
48C22×D44C2312yes—
49C11×#117—22no—
50C11×#117—22no—
51C11×#117—22no—
52C11×#117D414yes—
53C2×C11×C4⋊C4—22no—
54C2×C11×C4⋊C4—22no—
55C2×C11×C4⋊C4C2312yes—
56C2×C11×C4⋊C4C2311yescharacteristic
57C2×C11×C4⋊C4C2311yescharacteristic
58C2×C11×C22⋊C4—21no—
59C2×C11×C22⋊C4—21no—
60C2×C42×C11—22no—
61C2×C42×C11—21no—
62C2×C42×C11—22no—
63C2×C42×C11—21no—
64C2×C42×C11D411yescharacteristic
65C2×C42×C11D411yescharacteristic
66C2×C42×C11C2312yes—
67C2×(C2×D11)⋊C4—24no—
68C2×(C2×D11)⋊C4—24no—
69C2×(C2×D11)⋊C4—48no—
70C2×C4⋊Dic11—22no—
71C2×C4⋊Dic11—22no—
72C2×C4⋊Dic11—21no—
73C2×C4⋊Dic11C2312yes—
74C2×#4318—22no—
75C2×#4318—22no—
76C2×#4318—22no—
77C2×#4318—22no—
78C2×#4318—22no—
79C2×#4318D414yes—
80C22×C11⋊C8C2312yes—
81C22×C11⋊C8C2312yes—
82C11×#95—22no—
83C11×#95—22no—
84#66873—24no—
85#66873—24no—
86#66873—28no—
87#66872—24no—
88#66871—24no—
89C4⋊(C11⋊C8)—28no—
90#23618—111nomaximal, Sylow
91C24×C11C2×D411yescharacteristic
92C23×D11—24no—
93C23×D11—22no—
94C23×D11—44no—
95C23×D11—44no—
96C23×D11—42no—
97C23×D11—42no—
98C22×C4×C11—24no—
99C22×C4×C11—21no—
100C22×C4×C11—21no—
101C22×C4×C11—24no—
102C22×C4×C11—22no—
103C22×C4×C11—42no—
104C22×C4×C11C2×D412yes—
105C22×C4×C11C2×D412yes—
106C22×C4×C11C2×D411yescharacteristic
107C22×C4×C11C2×D412yes—
108C22×C4×C11C2×D412yes—
109C22×C4×C11C2×Q812yes—
110C22×C4×C11#4112yes—
111C22×C4×C11#4112yes—
112C22×C4×C11#4112yes—
113C22×C4×C11C2411yescharacteristic
114C22×Dic11—22no—
115C22×Dic11—44no—
116C22×Dic11—42no—
117C2×D44—24no—
118C2×D44—28no—
119C2×D44—28no—
120C2×D44—22no—
121C2×D44—22no—
122C2×D44—22no—
123C2×D44—22no—
124C2×D44—22no—
125C2×D44—44no—
126C2×D44—42no—
127C2×D44—42no—
128C2×D44—44no—
129C2×D44C2×Q814yes—
130C11×C4⋊C4—21no—
131C11×C4⋊C4—21no—
132C11×C4⋊C4—22no—
133C11×C4⋊C4—22no—
134C11×C4⋊C4—44no—
135C11×C4⋊C4—44no—
136C11×C4⋊C4C2×D412yes—
137C11×C4⋊C4C2×D412yes—
138C11×C22⋊C4—24no—
139C11×C22⋊C4—42no—
140C42×C11—22no—
141C42×C11—22no—
142C42×C11—24no—
143C42×C11—22no—
144C42×C11—22no—
145C42×C11—42no—
146(C2×D11)⋊C4—48no—
147(C2×D11)⋊C4—48no—
148C4⋊Dic11—22no—
149C4⋊Dic11—22no—
150C4⋊Dic11—42no—
151#4318—28no—
152#4318—44no—
153C2×C11⋊C8—22no—
154C2×C11⋊C8—22no—
155C2×C11⋊C8—22no—
156C2×C11⋊C8—22no—
157C2×C11⋊C8—22no—
158C2×C11⋊C8C2×D414yes—
159C2×#428—111no—
160C2×#423—111no—
161C2×#418—111no—
162C2×#418—111no—
163#2296D1111yescharacteristic, complemented
164C2×#390—111no—
165#2067—114no—
166#2065—114no—
167#1736—111no—
168C23×C11—22no—
169C23×C11—22no—
170C23×C11#12112yes—
171C23×C11#12112yes—
172C23×C11C22⋊Q812yes—
173C23×C11C22×D411yescharacteristic
174C22×D11—24no—
175C22×D11—416no—
176C22×D11—42no—
177C22×D11—42no—
178C22×D11—44no—
179C22×D11—44no—
180C22×D11—48no—
181C22×D11—84no—
182C22×D11—84no—
183C22×D11—88no—
184C2×C4×C11—22no—
185C2×C4×C11—22no—
186C2×C4×C11—22no—
187C2×C4×C11—22no—
188C2×C4×C11—22no—
189C2×C4×C11—22no—
190C2×C4×C11—22no—
191C2×C4×C11—48no—
192C2×C4×C11—42no—
193C2×C4×C11—42no—
194C2×C4×C11—42no—
195C2×C4×C11—42no—
196C2×C4×C11—42no—
197C2×C4×C11#12114yes—
198C2×C4×C11#12112yes—
199C2×C4×C11#12112yes—
200C2×C4×C11C22⋊Q814yes—
201C2×C4×C11C22⋊Q814yes—
202C2×C4×C11C22⋊Q812yes—
203C2×C4×C11#12614yes—
204C2×C4×C11AGL(1,ℤ/8)14yes—
205C2×C4×C11C22×D411yescharacteristic
206C2×C4×C11C22×D411yescharacteristic
207C2×C4×C11C22×Q811yescharacteristic
208C2×C4×C11C2×#4111yescharacteristic
209C2×C4×C11C2×#4111yescharacteristic
210C2×C4×C11C2×#4111yescharacteristic
211C2×Dic11—42no—
212C2×Dic11—44no—
213C2×Dic11—84no—
214D44—24no—
215D44—24no—
216D44—42no—
217D44—42no—
218D44—48no—
219C11⋊C8—24no—
220C11⋊C8—42no—
221C23×D4—111no—
222C22×M16—111no—
223C22×M16—111no—
224C2×#117C2×D1111yescharacteristic
225C2×#117C2×D1111yescharacteristic
226C22×C4⋊C4—111no—
227C22×C22⋊C4—222no—
228C22×C42C2×D1111yescharacteristic
229#428—224no—
230C2×#107—112no—
231C2×#106—112no—
232#423—224no—
233C2×C4⋊C8—114no—
234#418—224no—
235#418—224no—
236C2×#102—112no—
237C2×#102—114no—
238C2×#102—112no—
239#390—114no—
240#390—221no—
241#390—221no—
242#389C2×D1112yescomplemented
243#389C2×D1112yescomplemented
244#385—22no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247#340—114no—
248C22×C11—24no—
249C22×C11—22no—
250C22×C11—22no—
251C22×C11—24no—
252C22×C11#39614yes—
253C22×C11C2×#12111yescharacteristic
254C22×C11C2×#12111yescharacteristic
255C22×C11C2×C22⋊Q811yescharacteristic
256C22×C11C2×AGL(1,ℤ/8)12yes—
257C22×C11#57512yes—
258C2×D11—48no—
259C2×D11—44no—
260C2×D11—816no—
261C2×D11—82no—
262C2×D11—82no—
263C2×D11—84no—
264C4×C11—24no—
265C4×C11—42no—
266C4×C11—42no—
267C4×C11—42no—
268C4×C11—82no—
269C4×C11#39614yes—
270C4×C11C2×#12111yescharacteristic
271C4×C11C2×C22⋊Q811yescharacteristic
272C4×C11C2×C22⋊Q811yescharacteristic
273C4×C11C2×#12611yescharacteristic
274Dic11—82no—
275C25—221no—
276C22×D4—114no—
277C22×D4—112no—
278C22×D4—222no—
279C22×D4—222no—
280C22×D4—222no—
281C22×D4—222no—
282C22×D4—222no—
283C22×D4—221no—
284C23×C4—221no—
285C23×C4C22⋊D1112yes—
286C23×C4C22×D1111yescharacteristic
287C2×M16—114no—
288C2×M16—222no—
289C2×M16—222no—
290C2×M16—222no—
291C2×M16—222no—
292C2×M16—222no—
293C22×C8—112no—
294C22×C8—112no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117D4414yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—112no—
302C2×C4⋊C4—222no—
303C2×C4⋊C4—222no—
304C2×C4⋊C4—221no—
305C2×C4⋊C4C22×D1112yes—
306C2×C4⋊C4C22×D1111yescharacteristic
307C2×C4⋊C4C22×D1111yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—224no—
311C2×C22⋊C4—224no—
312C2×C22⋊C4—448no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42C22⋊D1111yescharacteristic
318C2×C42C22⋊D1111yescharacteristic
319C2×C42C22×D1112yes—
320#107—224no—
321#106—224no—
322C4⋊C8—228no—
323#102—228no—
324#102—224no—
325#102—224no—
326#95—22no—
327#95—22no—
328C2×C11—24no—
329C2×C11C2×#39611yescharacteristic
330C2×C11#303811yescharacteristic
331C2×C11#304011yescharacteristic
332C2×C11#305512yes—
333C2×C11#307712yes—
334D11—84no—
335C24—224no—
336C24—222no—
337C24—444no—
338C24—444no—
339C24—442no—
340C24—442no—
341C24C2×C22⋊D1111yescharacteristic
342C2×D4—114no—
343C2×D4—228no—
344C2×D4—228no—
345C2×D4—224no—
346C2×D4—222no—
347C2×D4—222no—
348C2×D4—222no—
349C2×D4—222no—
350C2×D4—222no—
351C2×D4—444no—
352C2×D4—444no—
353C2×D4—442no—
354C2×D4—442no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—222no—
362C22×C4—444no—
363C22×C4—442no—
364C22×C4C2×D4412yes—
365C22×C4#433912yes—
366C22×C4D4×D1112yes—
367C22×C4#434112yes—
368C22×C4Q8×D1112yes—
369C22×C4#434312yes—
370C22×C4C2×C22⋊D1112yes—
371C22×C4C2×C22⋊D1112yes—
372C22×C4C2×C22⋊D1111yescharacteristic
373C22×C4C23×D1111yescharacteristic
374M16—228no—
375M16—444no—
376C2×C8—114no—
377C2×C8—222no—
378C2×C8—222no—
379C2×C8—222no—
380C2×C8—222no—
381C2×C8—222no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—222no—
389C4⋊C4—222no—
390C4⋊C4—442no—
391C4⋊C4C2×D4412yes—
392C4⋊C4C2×D4412yes—
393C22⋊C4—24no—
394C22⋊C4—42no—
395C22⋊C4—448no—
396C22⋊C4—448no—
397C42—24no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—42no—
403C11#2361811yescharacteristic, complemented, Sylow
404C23—22no—
405C23—22no—
406C23—224no—
407C23—4416no—
408C23—442no—
409C23—442no—
410C23—444no—
411C23—444no—
412C23—448no—
413C23—884no—
414C23—884no—
415C23—888no—
416C23#6698312yes—
417C23#6699212yes—
418C23#6699912yes—
419C23C22×C22⋊D1111yescharacteristic
420D4—224no—
421D4—224no—
422D4—442no—
423D4—442no—
424D4—448no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—48no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—442no—
439C2×C4—444no—
440C2×C4—884no—
441C2×C4#6694814yes—
442C2×C4#6694914yes—
443C2×C4#6695014yes—
444C2×C4#6695114yes—
445C2×C4#6698312yes—
446C2×C4#6699212yes—
447C2×C4#6699912yes—
448C2×C4#6700214yes—
449C2×C4C22×D4411yescharacteristic
450C2×C4C2×#433911yescharacteristic
451C2×C4C2×D4×D1111yescharacteristic
452C2×C4C2×#434111yescharacteristic
453C2×C4C2×Q8×D1111yescharacteristic
454C2×C4C2×#434311yescharacteristic
455C8—224no—
456C8—442no—
457C22—22no—
458C22—22no—
459C22—24no—
460C22—24no—
461C22—448no—
462C22—444no—
463C22—8816no—
464C22—882no—
465C22—882no—
466C22—884no—
467C22#1062260714yes—
468C22C2×#6698311yescharacteristic
469C22C2×#6699211yescharacteristic
470C22C2×#6699911yescharacteristic
471C22C2×#6700212yes—
472C22#1062335312yes—
473C4—24no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—82no—
478C4—882no—
479C4#1062260714yes—
480C4C2×#6694811yescharacteristic
481C4C2×#6694911yescharacteristic
482C4C2×#6695011yescharacteristic
483C4C2×#6695111yescharacteristic
484C2—24no—
485C2—884no—
486C2C2×#1062260711yescharacteristic
487C2#1307809611yescharacteristic
488C2#1307811011yescharacteristic
489C2#1307823712yes—
490C2#1307840912yes—
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#2296C2
socleC23×C11C22×D4
prime radicalC23×D44C22
hypercenter#2296D11
hypocenterC11#23618
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
1. C2×#10622608C2semidirect
2. C2×#10622537C2semidirect
6. #13070001C2semidirect
7. #13069987C2semidirect
8. #13068337C2semidirect
9. C11×#2296C2semidirect
18. C11×#389C22semidirect
19. C11×#389C22semidirect
29. C2×#66872C22semidirect
30. C2×#66871C22semidirect
35. C2×C4⋊(C11⋊C8)C22semidirect
36. #10622167C22semidirect
163. #2296D11semidirect
242. #389C2×D11semidirect
243. #389C2×D11semidirect
403. C11#23618semidirect

Maximal subgroups

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

subgroupindex
C2×#106226082
C2×#106225372
C2×#106226142
C2×#106225522
#130700012
#130699872
#130683372
C11×#22962
#2361811