Finite Group Databasebeta

← #14796437

Subgroups — #14796437

Order 2816.

910 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622455C211yesmaximal, characteristic
2C2×C4×#4314C211yesmaximal, characteristic
3#13073824C211yesmaximal, characteristic, complemented
4C4⋊C4×D44C211yesmaximal, characteristic
5#13070094C212yesmaximal, complemented
6#13069996C212yesmaximal, complemented
7#13069577C211yesmaximal, characteristic, complemented
8#13068949C212yesmaximal
9#13068622C211yesmaximal, characteristic, complemented
10#13068617C212yesmaximal
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×#66881C2211yescharacteristic
25C2×#66881C2212yes—
26C2×#66880C2211yescharacteristic
27C2×#66880C2212yes—
28#10622600—21no—
29#10622598—21no—
30#10622598—21no—
31#10622548C2211yescharacteristic
32#10622548C2212yes—
33#10622542C2212yes—
34#10622541C2212yes—
35#10622455—24no—
36C4×#4314—24no—
37#10622281—22no—
38#10622269—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—22no—
60C2×C4⋊Dic11—22no—
61C2×C4⋊Dic11—21no—
62C2×C4⋊Dic11C2311yescharacteristic
63C2×C4⋊Dic11C2312yes—
64C2×C4⋊Dic11C2312yes—
65C2×C4⋊Dic11C2311yescharacteristic
66C2×Dic11⋊C4—21no—
67C2×Dic11⋊C4—22no—
68C2×C4×Dic11—21no—
69C2×#4314—22no—
70C2×#4314—22no—
71C2×#4314Q814yescomplemented
72C4⋊C4×D11—44no—
73C4×D44—22no—
74C4×D44—24no—
75C4×D44—42no—
76C4×D44D414yes—
77#66922—22no—
78#66922—22no—
79C4×Dic22—22no—
80C4×Dic22—22no—
81C11×C4⋊C8—22no—
82C11×C4⋊C8—42no—
83C4×C8×C11—22no—
84C11×#95D412yes—
85C11⋊#95—22no—
86C11⋊#95—22no—
87C11⋊#95—22no—
88#66885—22no—
89#66885—24no—
90#66881—22no—
91#66881—24no—
92#66880—22no—
93#66880—42no—
94#24815—111nomaximal, Sylow
95C23×D11—21no—
96C22×C4×C11C2×D412yes—
97C22×C4×C11C2×D411yescharacteristic
98C22×C4×C11C2×D412yes—
99C22×C4×C11C2×D411yescharacteristic
100C22×C4×C11C2411yescharacteristic
101C22×Dic11—21no—
102C22×Dic11—21no—
103C22×Dic11—22no—
104C22×Dic11—22no—
105C22×Dic11—21no—
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—42no—
115C2×C4×D11—44no—
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—81no—
125C2×Dic22—81no—
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⋊C4SD1614yescomplemented
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⋊Dic11—82no—
155C4⋊Dic11—82no—
156C4⋊Dic11C2×D412yes—
157Dic11⋊C4—42no—
158Dic11⋊C4—44no—
159C4×Dic11—42no—
160#4314—28no—
161C2×#476—111no—
162C2×C4×SD16—111no—
163#2567—111no—
164D4×C4⋊C4—111no—
165#2084—112no—
166#2066—112no—
167#1990—111no—
168#1939D1111yescharacteristic, complemented
169#1877—112no—
170#1803—111no—
171#1802—112no—
172C23×C11C22×D411yescharacteristic
173C22×D11—22no—
174C22×D11—22no—
175C22×D11—22no—
176C22×D11—41no—
177C22×D11—41no—
178C22×D11—41no—
179C22×D11—41no—
180C2×C4×C11—22no—
181C2×C4×C11—21no—
182C2×C4×C11—22no—
183C2×C4×C11—21no—
184C2×C4×C11—22no—
185C2×C4×C11—22no—
186C2×C4×C11—22no—
187C2×C4×C11#12112yes—
188C2×C4×C11#12112yes—
189C2×C4×C11C2×SD1614yes—
190C2×C4×C11C2×SD1612yes—
191C2×C4×C11#13712yes—
192C2×C4×C11C22×D411yescharacteristic
193C2×C4×C11C22×Q811yescharacteristic
194C2×C4×C11C2×#4111yescharacteristic
195C2×C4×C11C2×#4111yescharacteristic
196C2×C4×C11H(5,𝔽2)11yescharacteristic
197C2×C4×C112−1+411yescharacteristic
198C2×Dic11—22no—
199C2×Dic11—22no—
200C2×Dic11—41no—
201C2×Dic11—41no—
202C2×Dic11—41no—
203C2×Dic11—41no—
204C2×Dic11—41no—
205C2×Dic11—42no—
206C2×Dic11—42no—
207C2×Dic11—42no—
208C2×Dic11—41no—
209C2×Dic11—42no—
210C2×Dic11—42no—
211C2×Dic11—41no—
212C2×Dic11—42no—
213D44—22no—
214D44—22no—
215D44—22no—
216D44C22⋊Q814yes—
217C4×D11—48no—
218C4×D11—88no—
219Dic22—22no—
220Dic22—22no—
221Dic22—41no—
222Dic22—41no—
223Dic22—84no—
224C8×C11—22no—
225C8×C11—41no—
226C8×C11—42no—
227C22×SD16—111no—
228C2×C4⋊Q8—111no—
229C2×C4×Q8—111no—
230C2×C4×D4—111no—
231C2×C4×D4—112no—
232C22×C4⋊C4—221no—
233#476—224no—
234C4×SD16—224no—
235C2×#106—112no—
236C2×#106—111no—
237C2×C4⋊C8C2×D1111yescharacteristic
238C2×C4⋊C8C2×D1112yes—
239C2×#103—111no—
240C2×#103—112no—
241C2×#102—111no—
242C2×#102—112no—
243C2×C4×C8C2×D1111yescharacteristic
244#395—222no—
245#391—112no—
246#389—112no—
247#389—221no—
248#389—221no—
249#385—222no—
250#384—112no—
251#384—111no—
252#380—221no—
253C4×C4⋊C4C2×D1111yescharacteristic
254#340—112no—
255#340—112no—
256#340—112no—
257#336C2×D1112yes—
258C22×C11C2×#12111yescharacteristic
259C22×C11#54611yescharacteristic
260C22×C11D4×Q811yescharacteristic
261C22×C11C22×SD1611yescharacteristic
262C22×C11C2×#13711yescharacteristic
263C22×C11#57611yescharacteristic
264C22×C11#57711yescharacteristic
265C2×D11—24no—
266C2×D11—42no—
267C2×D11—42no—
268C2×D11—42no—
269C2×D11—42no—
270C2×D11—42no—
271C2×D11—42no—
272C4×C11—24no—
273C4×C11—22no—
274C4×C11—42no—
275C4×C11#46114yes—
276C4×C11C2×C22⋊Q811yescharacteristic
277C4×C11#53711yescharacteristic
278C4×C11#55211yescharacteristic
279C4×C11#55411yescharacteristic
280Dic11—42no—
281Dic11—42no—
282Dic11—82no—
283Dic11—82no—
284Dic11—81no—
285C22×Q8—111no—
286C22×Q8—221no—
287C22×D4—111no—
288C23×C4—221no—
289C23×C4—222no—
290C2×SD16—114no—
291C2×SD16—222no—
292C2×SD16—222no—
293C22×C8C22×D1112yes—
294C22×C8C22×D1111yescharacteristic
295C22×C8C22×D1111yescharacteristic
296C4⋊Q8—222no—
297C4⋊Q8—222no—
298C4×Q8—222no—
299C4×Q8—222no—
300C4×D4—114no—
301C4×D4—224no—
302C4×D4—222no—
303C4×D4—442no—
304C2×C4⋊C4—111no—
305C2×C4⋊C4—112no—
306C2×C4⋊C4—111no—
307C2×C4⋊C4—112no—
308C2×C4⋊C4—221no—
309C2×C4⋊C4—222no—
310C2×C4⋊C4—221no—
311C2×C4⋊C4—221no—
312C2×C4⋊C4—221no—
313C2×C4⋊C4—222no—
314C2×C4⋊C4—222no—
315C2×C4⋊C4—444no—
316C2×C4⋊C4D4411yescharacteristic
317C2×C4⋊C4D4411yescharacteristic
318C2×C22⋊C4—221no—
319C2×C22⋊C4—222no—
320C2×C42—221no—
321C2×C42C22×D1112yes—
322C2×C42C22×D1111yescharacteristic
323#106—222no—
324#106—224no—
325C4⋊C8—22no—
326C4⋊C8—42no—
327#103—222no—
328#103—442no—
329#102—222no—
330#102—224no—
331C4×C8—22no—
332#95—222no—
333#95—222no—
334#95—222no—
335#95D4412yes—
336C2×C11#241711yescharacteristic
337C2×C11C2×#46111yescharacteristic
338C2×C11#304511yescharacteristic
339C2×C11#329511yescharacteristic
340C2×C11#333111yescharacteristic
341C2×C11Q8×SD1611yescharacteristic
342C2×C11#338511yescharacteristic
343D11—44no—
344C24—221no—
345C2×Q8—112no—
346C2×Q8—221no—
347C2×Q8—221no—
348C2×Q8—221no—
349C2×Q8—221no—
350C2×Q8—221no—
351C2×Q8—442no—
352C2×Q8—881no—
353C2×Q8—881no—
354C2×D4—112no—
355C2×D4—112no—
356C2×D4—112no—
357C2×D4—221no—
358C2×D4—221no—
359C2×D4—221no—
360C22×C4—221no—
361C22×C4—221no—
362C22×C4—224no—
363C22×C4—222no—
364C22×C4—222no—
365C22×C4—221no—
366C22×C4—444no—
367C22×C4—442no—
368C22×C4—442no—
369C22×C4—444no—
370C22×C4—882no—
371C22×C4C2×D4412yes—
372C22×C4C2×D4411yescharacteristic
373C22×C4C2×D4412yes—
374C22×C4C2×D4411yescharacteristic
375C22×C4C23×D1111yescharacteristic
376SD16—228no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8Q8×D1112yes—
386C4⋊C4—22no—
387C4⋊C4—112no—
388C4⋊C4—221no—
389C4⋊C4—221no—
390C4⋊C4—222no—
391C4⋊C4—222no—
392C4⋊C4—221no—
393C4⋊C4—222no—
394C4⋊C4—222no—
395C4⋊C4—442no—
396C4⋊C4—442no—
397C4⋊C4—444no—
398C4⋊C4—442no—
399C4⋊C4—882no—
400C4⋊C4—881no—
401C4⋊C4—882no—
402C4⋊C4#431414yescomplemented
403C22⋊C4—224no—
404C22⋊C4—444no—
405C42—22no—
406C42—22no—
407C42—442no—
408C42D4×D1112yes—
409C42#434312yes—
410C11#2481511yescharacteristic, complemented, Sylow
411C23—222no—
412C23—222no—
413C23—222no—
414C23—441no—
415C23—441no—
416C23—441no—
417C23—441no—
418C23C22×D4411yescharacteristic
419Q8—222no—
420Q8—222no—
421Q8—441no—
422Q8—441no—
423Q8—884no—
424D4—114no—
425D4—222no—
426D4—222no—
427D4—222no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—222no—
436C2×C4—222no—
437C2×C4—441no—
438C2×C4—441no—
439C2×C4—441no—
440C2×C4—441no—
441C2×C4—448no—
442C2×C4—442no—
443C2×C4—441no—
444C2×C4—442no—
445C2×C4—441no—
446C2×C4—442no—
447C2×C4—442no—
448C2×C4—441no—
449C2×C4—442no—
450C2×C4—442no—
451C2×C4—888no—
452C2×C4#6694812yes—
453C2×C4#6694812yes—
454C2×C4C2×#431414yes—
455C2×C4C2×#431412yes—
456C2×C4#6696212yes—
457C2×C4C2×D4×D1111yescharacteristic
458C2×C4#6703711yescharacteristic
459C2×C4C2×Q8×D1111yescharacteristic
460C2×C4C2×#434311yescharacteristic
461C2×C4#41×D1111yescharacteristic
462C2×C4#6704311yescharacteristic
463C8—22no—
464C8—41no—
465C8—42no—
466C22—224no—
467C22—442no—
468C22—442no—
469C22—442no—
470C22—442no—
471C22—442no—
472C22—442no—
473C22C2×#6694811yescharacteristic
474C22#1062308511yescharacteristic
475C22Q8×D4411yescharacteristic
476C22C22×#431411yescharacteristic
477C22C2×#6696211yescharacteristic
478C22#1062328211yescharacteristic
479C22#1062328311yescharacteristic
480C4—24no—
481C4—22no—
482C4—42no—
483C4—442no—
484C4—442no—
485C4—882no—
486C4—882no—
487C4—881no—
488C4#1062245414yes—
489C4#1062313911yescharacteristic
490C4D11×C22⋊Q811yescharacteristic
491C4#1062322511yescharacteristic
492C4#1062326111yescharacteristic
493C2—444no—
494C2#1307206511yescharacteristic
495C2C2×#1062245411yescharacteristic
496C2#1307815711yescharacteristic
497C2#1308044111yescharacteristic
498C2#1308093311yescharacteristic
499C2Q8×#431411yescharacteristic
500C2#1308152811yescharacteristic
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC11×#1939C2
socleC23×C11C22×D4
prime radicalC22×D44C23
hypercenter#1939D11
hypocenterC11#24815
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. #13073824C2semidirect
5. #13070094C2semidirect
6. #13069996C2semidirect
7. #13069577C2semidirect
9. #13068622C2semidirect
11. C11×#1939C2semidirect
71. C2×#4314Q8semidirect
137. C11×C4⋊C4SD16semidirect
168. #1939D11semidirect
402. C4⋊C4#4314semidirect
410. C11#24815semidirect

Maximal subgroups

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

subgroupindex
C2×#106224552
C2×C4×#43142
#130738242
C4⋊C4×D442
#130700942
#130699962
#130695772
#130689492
#130686222
#130686172
C11×#19392
#2481511