Finite Group Databasebeta

← #14796448

Subgroups — #14796448

Order 2816.

894 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622459C211yesmaximal, characteristic
2C2×C4×#4314C211yesmaximal, characteristic
3#13073831C211yesmaximal, characteristic, complemented
4C4⋊C4×D44C211yesmaximal, characteristic
5#13070099C212yesmaximal, complemented
6#13070010C212yesmaximal, complemented
7#13069577C211yesmaximal, characteristic, complemented
8#13068949C212yesmaximal
9#13068627C212yesmaximal
10#13068622C211yesmaximal, characteristic, complemented
11C11×#1939C211yesmaximal, characteristic, complemented
12C22×#4314C2211yescharacteristic
13C2×C4⋊C4×D11—21no—
14C2×C4×D44C2212yes—
15C2×C4×D44C2211yescharacteristic
16C2×#66922C2211yescharacteristic
17C2×C4×Dic22C2211yescharacteristic
18C2×C11×C4⋊C8C2211yescharacteristic
19C2×C11×C4⋊C8C2212yes—
20C2×C4×C8×C11C2211yescharacteristic
21C4×C11×C4⋊C4C2211yescharacteristic
22C2×#66885C2211yescharacteristic
23C2×#66885C2212yes—
24C2×#66882C2212yes—
25C2×#66881C2211yescharacteristic
26C2×#66880C2211yescharacteristic
27C2×#66880C2212yes—
28#10622600—21no—
29#10622587—21no—
30#10622587—21no—
31#10622549C2212yes—
32#10622548C2211yescharacteristic
33#10622542C2212yes—
34#10622541C2212yes—
35#10622459—24no—
36C4×#4314—24no—
37#10622281—22no—
38#10622272—22no—
39C11×#336C2212yes—
40#10622189C2212yes—
41#10622189C2212yes—
42#10622189C2212yes—
43C22×D44C2311yescharacteristic
44C22×C4×D11—21no—
45C22×C4×D11—22no—
46C22×Dic22—21no—
47C22×Dic22C2311yescharacteristic
48C22×C8×C11C2312yes—
49C22×C8×C11C2311yescharacteristic
50C22×C8×C11C2311yescharacteristic
51C2×C11×C4⋊C4D411yescharacteristic
52C2×C11×C4⋊C4D411yescharacteristic
53C2×C42×C11C2312yes—
54C2×C42×C11C2311yescharacteristic
55C2×(C2×D11)⋊C4—21no—
56C2×(C2×D11)⋊C4—22no—
57C2×C4⋊Dic11—21no—
58C2×C4⋊Dic11—21no—
59C2×C4⋊Dic11—21no—
60C2×C4⋊Dic11C2311yescharacteristic
61C2×C4⋊Dic11C2312yes—
62C2×C4⋊Dic11C2312yes—
63C2×C4⋊Dic11C2311yescharacteristic
64C2×Dic11⋊C4—21no—
65C2×Dic11⋊C4—22no—
66C2×C4×Dic11—21no—
67C2×#4314—22no—
68C2×#4314—22no—
69C2×#4314Q814yescomplemented
70C4⋊C4×D11—44no—
71C4×D44—22no—
72C4×D44—24no—
73C4×D44—42no—
74C4×D44D414yes—
75#66922—22no—
76#66922—22no—
77C4×Dic22—22no—
78C4×Dic22—22no—
79C11×C4⋊C8—22no—
80C11×C4⋊C8—42no—
81C4×C8×C11—22no—
82C11×#95D412yes—
83C11⋊#95—22no—
84C11⋊#95—22no—
85C11⋊#95—22no—
86C11⋊#95—22no—
87C11⋊#95—22no—
88#66885—22no—
89#66885—24no—
90#66882—24no—
91#66881—22no—
92#66880—22no—
93#66880—42no—
94#24816—111nomaximal, Sylow
95C23×D11—21no—
96C22×C4×C11C2×D412yes—
97C22×C4×C11C2×D412yes—
98C22×C4×C11C2×D411yescharacteristic
99C22×C4×C11C2×D411yescharacteristic
100C22×C4×C11C2411yescharacteristic
101C22×Dic11—21no—
102C22×Dic11—21no—
103C22×Dic11—21no—
104C22×Dic11—22no—
105C22×Dic11—22no—
106C2×D44—21no—
107C2×D44—21no—
108C2×D44—21no—
109C2×D44C2×D412yes—
110C2×D44C2×Q812yes—
111C2×D44#4112yes—
112C2×C4×D11—24no—
113C2×C4×D11—44no—
114C2×C4×D11—44no—
115C2×C4×D11—42no—
116C2×C4×D11—42no—
117C2×C4×D11—82no—
118C2×Dic22—21no—
119C2×Dic22—21no—
120C2×Dic22—21no—
121C2×Dic22—21no—
122C2×Dic22—21no—
123C2×Dic22—42no—
124C2×Dic22—42no—
125C2×Dic22—42no—
126C2×Dic22C2×Q812yes—
127C2×C8×C11—21no—
128C2×C8×C11—21no—
129C2×C8×C11—22no—
130C2×C8×C11—21no—
131C2×C8×C11—22no—
132C2×C8×C11—21no—
133C2×C8×C11—22no—
134C2×C8×C11—21no—
135C2×C8×C11C2×Q812yes—
136C11×C4⋊C4—22no—
137C11×C4⋊C4—22no—
138C42×C11—22no—
139C42×C11—22no—
140C42×C11C2×D412yes—
141C42×C11#4112yes—
142(C2×D11)⋊C4—24no—
143(C2×D11)⋊C4—44no—
144C4⋊Dic11—21no—
145C4⋊Dic11—22no—
146C4⋊Dic11—21no—
147C4⋊Dic11—21no—
148C4⋊Dic11—22no—
149C4⋊Dic11—22no—
150C4⋊Dic11—22no—
151C4⋊Dic11—42no—
152C4⋊Dic11—42no—
153C4⋊Dic11—81no—
154C4⋊Dic11C2×D412yes—
155Dic11⋊C4—44no—
156Dic11⋊C4—42no—
157C4×Dic11—42no—
158#4314—28no—
159C2×#474—111no—
160C2×C4×SD16—111no—
161#2568—111no—
162D4×C4⋊C4—111no—
163#2085—112no—
164#2068—112no—
165#1990—111no—
166#1939D1111yescharacteristic, complemented
167#1877—112no—
168#1804—112no—
169#1803—111no—
170C23×C11C22×D411yescharacteristic
171C22×D11—22no—
172C22×D11—22no—
173C22×D11—22no—
174C22×D11—41no—
175C22×D11—41no—
176C22×D11—41no—
177C22×D11—41no—
178C2×C4×C11—21no—
179C2×C4×C11—21no—
180C2×C4×C11—22no—
181C2×C4×C11—22no—
182C2×C4×C11—22no—
183C2×C4×C11—22no—
184C2×C4×C11—22no—
185C2×C4×C11—22no—
186C2×C4×C11#12112yes—
187C2×C4×C11#12112yes—
188C2×C4×C11#13512yes—
189C2×C4×C11AGL(1,ℤ/8)12yes—
190C2×C4×C11C22×D411yescharacteristic
191C2×C4×C11C22×Q811yescharacteristic
192C2×C4×C11C2×#4111yescharacteristic
193C2×C4×C11C2×#4111yescharacteristic
194C2×C4×C11H(5,𝔽2)11yescharacteristic
195C2×C4×C112−1+411yescharacteristic
196C2×Dic11—22no—
197C2×Dic11—22no—
198C2×Dic11—41no—
199C2×Dic11—41no—
200C2×Dic11—41no—
201C2×Dic11—41no—
202C2×Dic11—41no—
203C2×Dic11—42no—
204C2×Dic11—42no—
205C2×Dic11—42no—
206C2×Dic11—42no—
207C2×Dic11—42no—
208C2×Dic11—42no—
209C2×Dic11—41no—
210C2×Dic11—41no—
211D44—22no—
212D44—22no—
213D44—22no—
214D44C22⋊Q814yes—
215C4×D11—48no—
216C4×D11—88no—
217Dic22—22no—
218Dic22—22no—
219Dic22—41no—
220Dic22—41no—
221Dic22—84no—
222C8×C11—22no—
223C8×C11—41no—
224C8×C11—42no—
225C22×SD16—111no—
226C2×C4⋊Q8—111no—
227C2×C4×Q8—111no—
228C2×C4×D4—111no—
229C2×C4×D4—112no—
230C22×C4⋊C4—221no—
231#474—224no—
232C4×SD16—224no—
233C2×#107—112no—
234C2×#106—111no—
235C2×C4⋊C8C2×D1111yescharacteristic
236C2×C4⋊C8C2×D1112yes—
237C2×#103—111no—
238C2×#103—112no—
239C2×#102—111no—
240C2×#102—112no—
241C2×C4×C8C2×D1111yescharacteristic
242#400—222no—
243#391—112no—
244#391—221no—
245#391—221no—
246#389—112no—
247#385—222no—
248#384—111no—
249#382—112no—
250#380—221no—
251C4×C4⋊C4C2×D1111yescharacteristic
252#340—112no—
253#340—112no—
254#340—112no—
255#336C2×D1112yes—
256C22×C11C2×#12111yescharacteristic
257C22×C11#54611yescharacteristic
258C22×C11D4×Q811yescharacteristic
259C22×C11C2×#13511yescharacteristic
260C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C11#57711yescharacteristic
262C22×C11#57811yescharacteristic
263C2×D11—24no—
264C2×D11—42no—
265C2×D11—42no—
266C2×D11—42no—
267C2×D11—42no—
268C2×D11—42no—
269C2×D11—42no—
270C4×C11—22no—
271C4×C11—42no—
272C4×C11—42no—
273C4×C11#46314yes—
274C4×C11C2×C22⋊Q811yescharacteristic
275C4×C11#53711yescharacteristic
276C4×C11#55211yescharacteristic
277C4×C11#55411yescharacteristic
278Dic11—42no—
279Dic11—42no—
280Dic11—82no—
281Dic11—81no—
282Dic11—82no—
283C22×Q8—111no—
284C22×Q8—221no—
285C22×D4—111no—
286C23×C4—221no—
287C23×C4—222no—
288C2×SD16—114no—
289C2×SD16—222no—
290C2×SD16—222no—
291C22×C8C22×D1112yes—
292C22×C8C22×D1111yescharacteristic
293C22×C8C22×D1111yescharacteristic
294C4⋊Q8—222no—
295C4⋊Q8—222no—
296C4×Q8—222no—
297C4×Q8—222no—
298C4×D4—114no—
299C4×D4—224no—
300C4×D4—222no—
301C4×D4—442no—
302C2×C4⋊C4—111no—
303C2×C4⋊C4—112no—
304C2×C4⋊C4—111no—
305C2×C4⋊C4—112no—
306C2×C4⋊C4—221no—
307C2×C4⋊C4—222no—
308C2×C4⋊C4—221no—
309C2×C4⋊C4—221no—
310C2×C4⋊C4—221no—
311C2×C4⋊C4—444no—
312C2×C4⋊C4D4411yescharacteristic
313C2×C4⋊C4D4411yescharacteristic
314C2×C22⋊C4—221no—
315C2×C22⋊C4—222no—
316C2×C42—221no—
317C2×C42C22×D1112yes—
318C2×C42C22×D1111yescharacteristic
319#107—224no—
320#106—222no—
321C4⋊C8—22no—
322C4⋊C8—42no—
323#103—222no—
324#103—442no—
325#102—222no—
326#102—224no—
327C4×C8—22no—
328#95—222no—
329#95—222no—
330#95—222no—
331#95—222no—
332#95—222no—
333#95D4412yes—
334C2×C11#241711yescharacteristic
335C2×C11C2×#46311yescharacteristic
336C2×C11#304611yescharacteristic
337C2×C11#330211yescharacteristic
338C2×C11#332411yescharacteristic
339C2×C11#338111yescharacteristic
340C2×C11#338611yescharacteristic
341D11—44no—
342C24—221no—
343C2×Q8—112no—
344C2×Q8—221no—
345C2×Q8—221no—
346C2×Q8—221no—
347C2×Q8—221no—
348C2×Q8—221no—
349C2×Q8—442no—
350C2×Q8—442no—
351C2×Q8—442no—
352C2×D4—112no—
353C2×D4—112no—
354C2×D4—112no—
355C2×D4—221no—
356C2×D4—221no—
357C2×D4—221no—
358C22×C4—221no—
359C22×C4—221no—
360C22×C4—224no—
361C22×C4—221no—
362C22×C4—222no—
363C22×C4—222no—
364C22×C4—444no—
365C22×C4—444no—
366C22×C4—442no—
367C22×C4—442no—
368C22×C4—882no—
369C22×C4C2×D4412yes—
370C22×C4C2×D4412yes—
371C22×C4C2×D4411yescharacteristic
372C22×C4C2×D4411yescharacteristic
373C22×C4C23×D1111yescharacteristic
374SD16—228no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8Q8×D1112yes—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—112no—
387C4⋊C4—221no—
388C4⋊C4—221no—
389C4⋊C4—222no—
390C4⋊C4—222no—
391C4⋊C4—221no—
392C4⋊C4—222no—
393C4⋊C4—222no—
394C4⋊C4—444no—
395C4⋊C4—442no—
396C4⋊C4—442no—
397C4⋊C4—442no—
398C4⋊C4—881no—
399C22⋊C4—224no—
400C22⋊C4—444no—
401C42—22no—
402C42—22no—
403C42—442no—
404C42D4×D1112yes—
405C42#434312yes—
406C11#2481611yescharacteristic, complemented, Sylow
407C23—222no—
408C23—222no—
409C23—222no—
410C23—441no—
411C23—441no—
412C23—441no—
413C23—441no—
414C23C22×D4411yescharacteristic
415Q8—222no—
416Q8—222no—
417Q8—441no—
418Q8—441no—
419Q8—884no—
420D4—114no—
421D4—222no—
422D4—222no—
423D4—222no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—222no—
433C2×C4—222no—
434C2×C4—441no—
435C2×C4—441no—
436C2×C4—441no—
437C2×C4—441no—
438C2×C4—448no—
439C2×C4—442no—
440C2×C4—442no—
441C2×C4—442no—
442C2×C4—442no—
443C2×C4—441no—
444C2×C4—441no—
445C2×C4—441no—
446C2×C4—442no—
447C2×C4—442no—
448C2×C4—888no—
449C2×C4#6694812yes—
450C2×C4#6694812yes—
451C2×C4#6695712yes—
452C2×C4#6696112yes—
453C2×C4C2×D4×D1111yescharacteristic
454C2×C4#6703711yescharacteristic
455C2×C4C2×Q8×D1111yescharacteristic
456C2×C4C2×#434311yescharacteristic
457C2×C4#41×D1111yescharacteristic
458C2×C4#6704311yescharacteristic
459C8—22no—
460C8—41no—
461C8—42no—
462C22—224no—
463C22—442no—
464C22—442no—
465C22—442no—
466C22—442no—
467C22—442no—
468C22—442no—
469C22C2×#6694811yescharacteristic
470C22#1062308511yescharacteristic
471C22Q8×D4411yescharacteristic
472C22C2×#6695711yescharacteristic
473C22C2×#6696111yescharacteristic
474C22#1062328211yescharacteristic
475C22#1062328411yescharacteristic
476C4—22no—
477C4—42no—
478C4—42no—
479C4—442no—
480C4—442no—
481C4—882no—
482C4—882no—
483C4—881no—
484C4#1062245714yes—
485C4#1062313911yescharacteristic
486C4D11×C22⋊Q811yescharacteristic
487C4#1062322511yescharacteristic
488C4#1062326111yescharacteristic
489C2—444no—
490C2#1307206511yescharacteristic
491C2C2×#1062245711yescharacteristic
492C2#1307816811yescharacteristic
493C2#1308053411yescharacteristic
494C2#1308084811yescharacteristic
495C2#1308149311yescharacteristic
496C2#1308153911yescharacteristic
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1939C2
socleC23×C11C22×D4
prime radicalC22×D44C23
hypercenter#1939D11
hypocenterC11#24816
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
3. #13073831C2semidirect
5. #13070099C2semidirect
6. #13070010C2semidirect
7. #13069577C2semidirect
10. #13068622C2semidirect
11. C11×#1939C2semidirect
69. C2×#4314Q8semidirect
166. #1939D11semidirect
406. C11#24816semidirect

Maximal subgroups

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

subgroupindex
C2×#106224592
C2×C4×#43142
#130738312
C4⋊C4×D442
#130700992
#130700102
#130695772
#130689492
#130686272
#130686222
C11×#19392
#2481611