Finite Group Databasebeta

← #14775648

Subgroups — #14775648

Order 2816.

914 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622459C211yesmaximal, characteristic
2#13071674C211yesmaximal, characteristic, complemented
3C4⋊C4×D44C211yesmaximal, characteristic
4#13069577C211yesmaximal, characteristic, complemented
5#13069380C211yesmaximal, characteristic, complemented
6#13069354C212yesmaximal, complemented
7#13069209C212yesmaximal, complemented
8#13068880C212yesmaximal
9#13068442C212yesmaximal
10C4×#66885C211yesmaximal, characteristic
11C11×#1770C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D11—21no—
13C2×#66940—21no—
14C2×C4×D44C2212yes—
15C2×C4×D44C2211yescharacteristic
16C2×#66922C2211yescharacteristic
17C2×C11×C4⋊C8C2211yescharacteristic
18C2×C11×#97C2212yes—
19C2×C4×C8×C11C2211yescharacteristic
20C4×C11×C4⋊C4C2211yescharacteristic
21C2×#66885C2211yescharacteristic
22C2×#66885C2212yes—
23C2×#66885C2211yescharacteristic
24C2×#66882C2212yes—
25C2×#66881C2212yes—
26#10622600—21no—
27#10622587—21no—
28#10622548C2211yescharacteristic
29C4×C4⋊Dic11C2211yescharacteristic
30C4×C4⋊Dic11C2212yes—
31#10622541C2212yes—
32#10622541C2212yes—
33#10622459C418yescomplemented
34#10622281—22no—
35#10622266—22no—
36C11×#336C2212yes—
37#10622189C2212yes—
38#10622189C2212yes—
39#10622189C2211yescharacteristic
40#10622189C2212yes—
41#10622189C2211yescharacteristic
42C22×D44C2311yescharacteristic
43C22×C4×D11—21no—
44C22×C4×D11—22no—
45C22×Dic22—21no—
46C22×C8×C11C2311yescharacteristic
47C22×C8×C11C2312yes—
48C22×C8×C11C2311yescharacteristic
49C2×C11×C4⋊C4D411yescharacteristic
50C2×C11×C4⋊C4D411yescharacteristic
51C2×C42×C11C2312yes—
52C2×C42×C11C2311yescharacteristic
53C2×(C2×D11)⋊C4—21no—
54C2×(C2×D11)⋊C4—22no—
55C2×C4⋊Dic11—21no—
56C2×C4⋊Dic11—21no—
57C2×C4⋊Dic11—21no—
58C2×C4⋊Dic11C2311yescharacteristic
59C2×C4⋊Dic11C2312yes—
60C2×C4⋊Dic11C2312yes—
61C2×C4⋊Dic11C2311yescharacteristic
62C2×C4⋊Dic11C2311yescharacteristic
63C2×Dic11⋊C4—22no—
64C2×Dic11⋊C4—22no—
65C2×C4×Dic11—22no—
66C2×C4×Dic11—21no—
67C4⋊C4×D11—44no—
68#66940—44no—
69C4×D44—22no—
70C4×D44—24no—
71C4×D44—42no—
72C4×D44C2×C414yes—
73#66922—22no—
74#66922C2×C414yescomplemented
75C11×C4⋊C8C2×C414yescomplemented
76C11×#97—42no—
77C4×C8×C11—22no—
78C11×#95D412yes—
79C11⋊#95—21no—
80C11⋊#95—22no—
81C11⋊#95—22no—
82C11⋊#95—22no—
83#66885—22no—
84#66885C2×C418yes—
85#66885Q814yescomplemented
86#66882C2×C418yescomplemented
87#66881—42no—
88#22968—111nomaximal, Sylow
89C23×D11—21no—
90C22×C4×C11C2×D412yes—
91C22×C4×C11C2×D412yes—
92C22×C4×C11C2×D411yescharacteristic
93C22×C4×C11C2×D411yescharacteristic
94C22×C4×C11C2411yescharacteristic
95C22×Dic11—21no—
96C22×Dic11—22no—
97C22×Dic11—21no—
98C22×Dic11—21no—
99C22×Dic11—22no—
100C2×D44—21no—
101C2×D44—21no—
102C2×D44—21no—
103C2×D44C22×C412yes—
104C2×D44C2×Q812yes—
105C2×D44#4112yes—
106C2×C4×D11—24no—
107C2×C4×D11—44no—
108C2×C4×D11—44no—
109C2×C4×D11—42no—
110C2×C4×D11—42no—
111C2×C4×D11—82no—
112C2×Dic22—24no—
113C2×Dic22—42no—
114C2×Dic22—42no—
115C2×C8×C11—21no—
116C2×C8×C11—21no—
117C2×C8×C11—21no—
118C2×C8×C11—22no—
119C2×C8×C11—21no—
120C2×C8×C11—22no—
121C2×C8×C11—21no—
122C2×C8×C11C22×C414yes—
123C2×C8×C11C2×Q812yes—
124C11×C4⋊C4—22no—
125C11×C4⋊C4—22no—
126C42×C11—22no—
127C42×C11—22no—
128C42×C11C22×C412yes—
129C42×C11#4112yes—
130(C2×D11)⋊C4—24no—
131(C2×D11)⋊C4—44no—
132C4⋊Dic11—24no—
133C4⋊Dic11—22no—
134C4⋊Dic11—21no—
135C4⋊Dic11—21no—
136C4⋊Dic11—22no—
137C4⋊Dic11—21no—
138C4⋊Dic11—22no—
139C4⋊Dic11—21no—
140C4⋊Dic11—42no—
141C4⋊Dic11—81no—
142C4⋊Dic11C22×C412yes—
143C4⋊Dic11C22×C414yes—
144C4⋊Dic11C2×Q812yes—
145Dic11⋊C4—44no—
146Dic11⋊C4—44no—
147C4×Dic11—44no—
148C4×Dic11—42no—
149C2×#474—111no—
150#2365—111no—
151D4×C4⋊C4—111no—
152#1990—111no—
153#1948—111no—
154#1943—112no—
155#1921—112no—
156#1864—112no—
157#1770D1111yescharacteristic, complemented
158#1762—112no—
159C4×#102—111no—
160C23×C11C22×D411yescharacteristic
161C22×D11—22no—
162C22×D11—22no—
163C22×D11—22no—
164C22×D11—41no—
165C22×D11—41no—
166C22×D11—41no—
167C22×D11—41no—
168C2×C4×C11—21no—
169C2×C4×C11—21no—
170C2×C4×C11—22no—
171C2×C4×C11—22no—
172C2×C4×C11—22no—
173C2×C4×C11—22no—
174C2×C4×C11—22no—
175C2×C4×C11—22no—
176C2×C4×C11C4×D412yes—
177C2×C4×C11C4×D412yes—
178C2×C4×C11AGL(1,ℤ/8)12yes—
179C2×C4×C11AGL(1,ℤ/8)12yes—
180C2×C4×C11C23×C411yescharacteristic
181C2×C4×C11C22×Q811yescharacteristic
182C2×C4×C11C2×#4111yescharacteristic
183C2×C4×C11C2×#4111yescharacteristic
184C2×C4×C11H(5,𝔽2)11yescharacteristic
185C2×C4×C112−1+411yescharacteristic
186C2×Dic11—24no—
187C2×Dic11—22no—
188C2×Dic11—22no—
189C2×Dic11—41no—
190C2×Dic11—42no—
191C2×Dic11—41no—
192C2×Dic11—42no—
193C2×Dic11—41no—
194C2×Dic11—41no—
195C2×Dic11—42no—
196C2×Dic11—41no—
197C2×Dic11—41no—
198C2×Dic11—42no—
199C2×Dic11—42no—
200C2×Dic11—41no—
201D44—22no—
202D44—22no—
203D44—22no—
204D44C4×Q814yescomplemented
205C4×D11—48no—
206C4×D11—88no—
207Dic22—48no—
208C8×C11—24no—
209C8×C11—22no—
210C8×C11—41no—
211C2×C4⋊Q8—111no—
212C2×C4×Q8—221no—
213C2×C4×D4—111no—
214C2×C4×D4—112no—
215C22×C4⋊C4—221no—
216#474—118no—
217C2×#107—112no—
218C2×#106—112no—
219C2×C4⋊C8C2×D1111yescharacteristic
220C2×#102—112no—
221C2×#102—111no—
222C2×#102—111no—
223C2×#97C2×D1112yes—
224C2×C4×C8C2×D1111yescharacteristic
225#391—221no—
226#389—112no—
227#389—112no—
228#387—222no—
229#385—222no—
230#384—111no—
231#380—221no—
232C4×C4⋊C4—112no—
233C4×C4⋊C4—111no—
234C4×C4⋊C4C2×D1111yescharacteristic
235#340—112no—
236#340—111no—
237#340—112no—
238#340—111no—
239#340—112no—
240#336C2×D1112yes—
241C22×C11C2×C4×D411yescharacteristic
242C22×C11#54611yescharacteristic
243C22×C11D4×Q811yescharacteristic
244C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
245C22×C11C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C11#57711yescharacteristic
247C22×C11#57811yescharacteristic
248C2×D11—24no—
249C2×D11—42no—
250C2×D11—42no—
251C2×D11—42no—
252C2×D11—42no—
253C2×D11—42no—
254C2×D11—42no—
255C4×C11—22no—
256C4×C11—42no—
257C4×C11—42no—
258C4×C11#44214yes—
259C4×C11C2×C4×Q811yescharacteristic
260C4×C11#52011yescharacteristic
261C4×C11#55211yescharacteristic
262C4×C11#55411yescharacteristic
263Dic11—42no—
264Dic11—44no—
265Dic11—42no—
266Dic11—81no—
267Dic11—82no—
268C22×Q8—221no—
269C22×D4—111no—
270C23×C4—221no—
271C23×C4—222no—
272C22×C8C22×D1111yescharacteristic
273C22×C8C22×D1112yes—
274C22×C8C22×D1111yescharacteristic
275C4⋊Q8—114no—
276C4⋊Q8—222no—
277C4×Q8—444no—
278C4×D4—114no—
279C4×D4—224no—
280C4×D4—222no—
281C4×D4—442no—
282C2×C4⋊C4—111no—
283C2×C4⋊C4—112no—
284C2×C4⋊C4—111no—
285C2×C4⋊C4—112no—
286C2×C4⋊C4—111no—
287C2×C4⋊C4—221no—
288C2×C4⋊C4—221no—
289C2×C4⋊C4—222no—
290C2×C4⋊C4—222no—
291C2×C4⋊C4—221no—
292C2×C4⋊C4—444no—
293C2×C4⋊C4D4411yescharacteristic
294C2×C4⋊C4D4411yescharacteristic
295C2×C22⋊C4—221no—
296C2×C22⋊C4—222no—
297C2×C42—222no—
298C2×C42—221no—
299C2×C42C22×D1112yes—
300C2×C42C22×D1111yescharacteristic
301#107—118no—
302#106—442no—
303C4⋊C8C4×D1114yescomplemented
304#102—118no—
305#102—114no—
306#102—222no—
307#97—42no—
308C4×C8—22no—
309#95—221no—
310#95—222no—
311#95—222no—
312#95—222no—
313#95D4412yes—
314C2×C11#236011yescharacteristic
315C2×C11C2×#44211yescharacteristic
316C2×C11#294711yescharacteristic
317C2×C11#332411yescharacteristic
318C2×C11#333011yescharacteristic
319C2×C11#338411yescharacteristic
320C2×C11#338611yescharacteristic
321D11—44no—
322C24—221no—
323C2×Q8—224no—
324C2×Q8—442no—
325C2×Q8—442no—
326C2×D4—112no—
327C2×D4—112no—
328C2×D4—112no—
329C2×D4—221no—
330C2×D4—221no—
331C2×D4—221no—
332C22×C4—221no—
333C22×C4—222no—
334C22×C4—221no—
335C22×C4—224no—
336C22×C4—221no—
337C22×C4—222no—
338C22×C4—444no—
339C22×C4—444no—
340C22×C4—442no—
341C22×C4—442no—
342C22×C4—882no—
343C22×C4C2×D4412yes—
344C22×C4C2×D4412yes—
345C22×C4C2×D4411yescharacteristic
346C22×C4C2×D4411yescharacteristic
347C22×C4C23×D1111yescharacteristic
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8C2×C4×D1114yes—
356C2×C8Q8×D1112yes—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—114no—
360C4⋊C4—112no—
361C4⋊C4—112no—
362C4⋊C4—224no—
363C4⋊C4—221no—
364C4⋊C4—222no—
365C4⋊C4—221no—
366C4⋊C4—221no—
367C4⋊C4—222no—
368C4⋊C4—221no—
369C4⋊C4—222no—
370C4⋊C4—444no—
371C4⋊C4—444no—
372C4⋊C4—442no—
373C4⋊C4—881no—
374C22⋊C4—224no—
375C22⋊C4—444no—
376C42—22no—
377C42—22no—
378C42—444no—
379C42—442no—
380C42C2×C4×D1112yes—
381C42#433912yes—
382C11#2296811yescharacteristic, complemented, Sylow
383C23—222no—
384C23—222no—
385C23—222no—
386C23—441no—
387C23—441no—
388C23—441no—
389C23—441no—
390C23C22×D4411yescharacteristic
391Q8—448no—
392D4—114no—
393D4—222no—
394D4—222no—
395D4—222no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—222no—
405C2×C4—224no—
406C2×C4—222no—
407C2×C4—441no—
408C2×C4—442no—
409C2×C4—441no—
410C2×C4—441no—
411C2×C4—442no—
412C2×C4—441no—
413C2×C4—448no—
414C2×C4—441no—
415C2×C4—442no—
416C2×C4—442no—
417C2×C4—441no—
418C2×C4—442no—
419C2×C4—441no—
420C2×C4—888no—
421C2×C4C4×D4412yes—
422C2×C4C4×D4412yes—
423C2×C4#6696112yes—
424C2×C4#6696112yes—
425C2×C4C22×C4×D1111yescharacteristic
426C2×C4C2×#433911yescharacteristic
427C2×C4#6703711yescharacteristic
428C2×C4C2×Q8×D1111yescharacteristic
429C2×C4#41×D1111yescharacteristic
430C2×C4#6704311yescharacteristic
431C8—22no—
432C8—24no—
433C8—41no—
434C22—224no—
435C22—442no—
436C22—442no—
437C22—442no—
438C22—442no—
439C22—442no—
440C22—442no—
441C22C2×C4×D4411yescharacteristic
442C22#1062308511yescharacteristic
443C22Q8×D4411yescharacteristic
444C22C2×#6696111yescharacteristic
445C22C2×#6696111yescharacteristic
446C22#1062328211yescharacteristic
447C22#1062328411yescharacteristic
448C4—22no—
449C4—42no—
450C4—42no—
451C4—442no—
452C4—444no—
453C4—442no—
454C4—882no—
455C4—881no—
456C4#1062232414yes—
457C4#1062304211yescharacteristic
458C4#1062306011yescharacteristic
459C4#1062307711yescharacteristic
460C4C4×Q8×D1111yescharacteristic
461C2—444no—
462C2#1307162311yescharacteristic
463C2C2×#1062232411yescharacteristic
464C2#1307746111yescharacteristic
465C2#1308084811yescharacteristic
466C2#1308092211yescharacteristic
467C2#1308151711yescharacteristic
468C2#1308153911yescharacteristic
subgroupG/H
centerC23C22×D44
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingC11×#1770C2
socleC23×C11C22×D4
prime radicalC22×D44C23
cosocleC22×C4×C11C24
hypercenter#1770D11
hypocenterC11#22968
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1770

Complements

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

NH ≅ G/Ntype
2. #13071674C2semidirect
4. #13069577C2semidirect
5. #13069380C2semidirect
6. #13069354C2semidirect
7. #13069209C2semidirect
11. C11×#1770C2semidirect
33. #10622459C4semidirect
74. #66922C2×C4semidirect
75. C11×C4⋊C8C2×C4semidirect
85. #66885Q8semidirect
86. #66882C2×C4semidirect
157. #1770D11semidirect
204. D44C4×Q8semidirect
303. C4⋊C8C4×D11semidirect
382. C11#22968semidirect

Maximal subgroups

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

subgroupindex
C2×#106224592
#130716742
C4⋊C4×D442
#130695772
#130693802
#130693542
#130692092
#130688802
#130684422
C4×#668852
C11×#17702
#2296811