Finite Group Databasebeta

← #14789916

Subgroups — #14789916

Order 2816.

1066 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623001C211yesmaximal, characteristic
2C2×#10622635C212yesmaximal, complemented
3#13069949C211yesmaximal, characteristic, complemented
4#13069944C211yesmaximal, characteristic, complemented
5#13069757C211yesmaximal, characteristic, complemented
6#13069757C211yesmaximal, characteristic, complemented
7#13069573C212yesmaximal, complemented
8#13068929C212yesmaximal
9#13068813C212yesmaximal
10#13068596C211yesmaximal, characteristic
11C4×C11×C22⋊Q8C211yesmaximal, characteristic
12C2×C11×C22⋊Q8C2211yescharacteristic
13C2×C4×Q8×C11C2211yescharacteristic
14C22×C42×C11C2211yescharacteristic
15C2×#66979C2212yes—
16C2×#66977—21no—
17C2×#66977—21no—
18#10623001—28no—
19C2×#66923C2212yes—
20C2×#66922C2211yescharacteristic
21C2×#66922C2211yescharacteristic
22C2×C4×Dic22—21no—
23C2×C4×Dic22—21no—
24C11×#391C2211yescharacteristic
25C11×#389C2211yescharacteristic
26C11×#387—21no—
27C11×#385—21no—
28C11×#380C2211yescharacteristic
29C4×C11×C4⋊C4—21no—
30C4×C11×C4⋊C4C2211yescharacteristic
31C4×C11×C22⋊C4—21no—
32C2×#66899C2212yes—
33C2×#66899C2212yes—
34C2×C22⋊(C11⋊C8)C2212yes—
35#10622635—28no—
36C2×#66872C2212yes—
37C2×#66871C2212yes—
38C2×C4⋊(C11⋊C8)C2212yes—
39#10622167C2212yes—
40#10622167C2212yes—
41#10622167C2212yes—
42#10622167C2212yes—
43#10622162C2211yescharacteristic
44#10622162C2211yescharacteristic
45C22×Q8×C11C2311yescharacteristic
46C23×C4×C11D411yescharacteristic
47C23×C4×C11D411yescharacteristic
48C23×C4×C11C2311yescharacteristic
49C22×Dic22—21no—
50C22×Dic22—21no—
51C11×C22⋊Q8—22no—
52C11×C22⋊Q8Q814yescomplemented
53C4×Q8×C11—24no—
54C2×C11×C4⋊C4—21no—
55C2×C11×C4⋊C4—21no—
56C2×C11×C4⋊C4—21no—
57C2×C11×C4⋊C4C2311yescharacteristic
58C2×C11×C4⋊C4C2311yescharacteristic
59C2×C11×C4⋊C4C2311yescharacteristic
60C2×C11×C22⋊C4—21no—
61C2×C11×C22⋊C4—21no—
62C2×C42×C11—24no—
63C2×C42×C11—21no—
64C2×C42×C11—21no—
65C2×C42×C11D411yescharacteristic
66C2×C42×C11D411yescharacteristic
67C2×C42×C11C2311yescharacteristic
68C2×C42×C11C2311yescharacteristic
69C2×C22⋊Dic11—22no—
70#66979—24no—
71#66979—24no—
72C2×C4⋊Dic11—21no—
73C2×C4⋊Dic11—21no—
74C2×C4⋊Dic11C2312yes—
75C2×C4⋊Dic11C2312yes—
76#66977—28no—
77#66977—28no—
78C2×Dic11⋊C4—22no—
79C2×Dic11⋊C4—22no—
80C2×C4×Dic11—22no—
81C22×C11⋊C8C2312yes—
82C22×C11⋊C8C2312yes—
83#66923—24no—
84#66923D418yes—
85#66922—24no—
86#66922—24no—
87C4×Dic22—28no—
88C4×Dic22—28no—
89C11×#95—21no—
90C11×#95—21no—
91C11×#95—21no—
92C11×#95—21no—
93C11×#95—21no—
94#66899—24no—
95#66899—24no—
96C22⋊(C11⋊C8)—24no—
97#66872—24no—
98#66871—24no—
99C4⋊(C11⋊C8)—24no—
100#24306—111nomaximal, Sylow
101C24×C11C2×D411yescharacteristic
102C2×Q8×C11—21no—
103C2×Q8×C11—22no—
104C2×Q8×C11—22no—
105C2×Q8×C11C2×Q812yes—
106C22×C4×C11—22no—
107C22×C4×C11—22no—
108C22×C4×C11—21no—
109C22×C4×C11—21no—
110C22×C4×C11—22no—
111C22×C4×C11—22no—
112C22×C4×C11—22no—
113C22×C4×C11—22no—
114C22×C4×C11—21no—
115C22×C4×C11—21no—
116C22×C4×C11—41no—
117C22×C4×C11—41no—
118C22×C4×C11C2×D411yescharacteristic
119C22×C4×C11C2×D411yescharacteristic
120C22×C4×C11C2×D411yescharacteristic
121C22×C4×C11C2×D411yescharacteristic
122C22×C4×C11C2×D411yescharacteristic
123C22×C4×C11C2×Q812yes—
124C22×C4×C11#4112yes—
125C22×C4×C11C2411yescharacteristic
126C22×Dic11—22no—
127C22×Dic11—22no—
128C2×Dic22—24no—
129C2×Dic22—24no—
130C2×Dic22—42no—
131C2×Dic22—42no—
132C2×Dic22—42no—
133C2×Dic22—42no—
134C11×C4⋊C4—24no—
135C11×C4⋊C4—21no—
136C11×C4⋊C4—22no—
137C11×C4⋊C4—22no—
138C11×C4⋊C4—42no—
139C11×C4⋊C4—42no—
140C11×C4⋊C4C2×Q812yes—
141C11×C22⋊C4—24no—
142C11×C22⋊C4—42no—
143C42×C11—22no—
144C42×C11—22no—
145C42×C11—22no—
146C42×C11—42no—
147C42×C11—42no—
148C42×C11C2×D414yes—
149C22⋊Dic11—28no—
150C4⋊Dic11—24no—
151C4⋊Dic11—24no—
152C4⋊Dic11—22no—
153C4⋊Dic11—22no—
154C4⋊Dic11C2×D414yes—
155C4⋊Dic11C2×D414yes—
156Dic11⋊C4—28no—
157Dic11⋊C4—28no—
158C4×Dic11—28no—
159C2×C11⋊C8—22no—
160C2×C11⋊C8—22no—
161C2×C11⋊C8—22no—
162C2×C11⋊C8—22no—
163C2×C11⋊C8—22no—
164C2×C11⋊C8—22no—
165C2×#533—111no—
166C2×#479—112no—
167C4×C22⋊Q8D1111yescharacteristic
168#2058—111no—
169#2057—111no—
170#2024—111no—
171#2024—111no—
172#1989—112no—
173#1873—112no—
174#1854—112no—
175#1797—111no—
176C23×C11—21no—
177C23×C11—21no—
178C23×C11—21no—
179C23×C11—21no—
180C23×C11—21no—
181C23×C11—21no—
182C23×C11C22⋊Q812yes—
183C23×C11C22×D411yescharacteristic
184Q8×C11—24no—
185Q8×C11—42no—
186C2×C4×C11—21no—
187C2×C4×C11—21no—
188C2×C4×C11—21no—
189C2×C4×C11—21no—
190C2×C4×C11—21no—
191C2×C4×C11—21no—
192C2×C4×C11—21no—
193C2×C4×C11—21no—
194C2×C4×C11—21no—
195C2×C4×C11—21no—
196C2×C4×C11—24no—
197C2×C4×C11—24no—
198C2×C4×C11—22no—
199C2×C4×C11—22no—
200C2×C4×C11—44no—
201C2×C4×C11—44no—
202C2×C4×C11—41no—
203C2×C4×C11—41no—
204C2×C4×C11—41no—
205C2×C4×C11—41no—
206C2×C4×C11—41no—
207C2×C4×C11—41no—
208C2×C4×C11—41no—
209C2×C4×C11—41no—
210C2×C4×C11—41no—
211C2×C4×C11—41no—
212C2×C4×C11#12012yes—
213C2×C4×C11#12012yes—
214C2×C4×C11C22⋊Q812yes—
215C2×C4×C11#13512yes—
216C2×C4×C11#13712yes—
217C2×C4×C11C22×D412yes—
218C2×C4×C11C22×Q811yescharacteristic
219C2×C4×C11C2×#4111yescharacteristic
220C2×C4×C112−1+412yes—
221C2×Dic11—24no—
222C2×Dic11—24no—
223C2×Dic11—42no—
224C2×Dic11—42no—
225C2×Dic11—42no—
226C2×Dic11—42no—
227Dic22—48no—
228Dic22—48no—
229C11⋊C8—42no—
230C11⋊C8—42no—
231#533—228no—
232C2×C4⋊Q8—111no—
233C2×C4⋊Q8—111no—
234C2×#125—112no—
235C2×C22⋊Q8—221no—
236C2×C22⋊Q8—221no—
237C2×C22⋊Q8C2×D1111yescharacteristic
238C2×C4×Q8—221no—
239C2×C4×Q8—221no—
240C2×C4×Q8C2×D1111yescharacteristic
241C2×#117—112no—
242C22×C42C2×D1111yescharacteristic
243#479—228no—
244C2×#107—112no—
245C2×#106—112no—
246C2×C4⋊C8—112no—
247C2×#103—112no—
248C2×#103—112no—
249C2×C22⋊C8—112no—
250#391C2×D1111yescharacteristic
251#389C2×D1111yescharacteristic
252#387—21no—
253#385—21no—
254#380C2×D1111yescharacteristic
255C4×C4⋊C4—21no—
256C4×C4⋊C4C2×D1111yescharacteristic
257C4×C22⋊C4—21no—
258#340—112no—
259#340—112no—
260#340—112no—
261#340—112no—
262#336—111no—
263#336—111no—
264C22×C11—22no—
265C22×C11—22no—
266C22×C11—22no—
267C22×C11—41no—
268C22×C11—41no—
269C22×C11—41no—
270C22×C11—41no—
271C22×C11C2×#12011yescharacteristic
272C22×C11C2×C22⋊Q811yescharacteristic
273C22×C11#53611yescharacteristic
274C22×C11C2×#13511yescharacteristic
275C22×C11C2×#13711yescharacteristic
276C22×C11#57511yescharacteristic
277C22×C11#57511yescharacteristic
278C4×C11—22no—
279C4×C11—22no—
280C4×C11—22no—
281C4×C11—22no—
282C4×C11—22no—
283C4×C11—42no—
284C4×C11—42no—
285C4×C11—81no—
286C4×C11—81no—
287C4×C11#54812yes—
288C4×C11D4×Q812yes—
289Dic11—44no—
290Dic11—44no—
291C22×Q8—221no—
292C22×Q8—221no—
293C22×Q8C22×D1111yescharacteristic
294C23×C4C22⋊D1111yescharacteristic
295C23×C4C22⋊D1111yescharacteristic
296C23×C4C22×D1111yescharacteristic
297C22×C8—112no—
298C22×C8—112no—
299C4⋊Q8—224no—
300C4⋊Q8—224no—
301#125—118no—
302#125—224no—
303C22⋊Q8—22no—
304C22⋊Q8—228no—
305C22⋊Q8—228no—
306C22⋊Q8Dic2214yescomplemented
307C4×Q8—24no—
308C4×Q8—228no—
309C4×Q8—228no—
310#117—224no—
311#117—224no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—112no—
316C2×C4⋊C4—112no—
317C2×C4⋊C4—222no—
318C2×C4⋊C4—222no—
319C2×C4⋊C4—221no—
320C2×C4⋊C4—221no—
321C2×C4⋊C4C22×D1111yescharacteristic
322C2×C4⋊C4C22×D1111yescharacteristic
323C2×C4⋊C4C22×D1111yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—222no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—222no—
331C2×C42C22⋊D1111yescharacteristic
332C2×C42C22⋊D1111yescharacteristic
333C2×C42C22×D1111yescharacteristic
334C2×C42C22×D1111yescharacteristic
335#107—224no—
336#106—224no—
337C4⋊C8—224no—
338#103—224no—
339#103—224no—
340C22⋊C8—224no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346C2×C11—42no—
347C2×C11#240911yescharacteristic
348C2×C11#300211yescharacteristic
349C2×C11#300711yescharacteristic
350C2×C11#307611yescharacteristic
351C2×C11#307811yescharacteristic
352C2×C11#323111yescharacteristic
353C2×C11#324011yescharacteristic
354C24C2×C22⋊D1111yescharacteristic
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—224no—
359C2×Q8—224no—
360C2×Q8—442no—
361C2×Q8—442no—
362C2×Q8—442no—
363C2×Q8—442no—
364C2×Q8C2×Dic2212yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—222no—
378C22×C4—222no—
379C22×C4C2×Dic2212yes—
380C22×C4#433912yes—
381C22×C4C2×C22⋊D1111yescharacteristic
382C22×C4C2×C22⋊D1111yescharacteristic
383C22×C4C2×C22⋊D1111yescharacteristic
384C22×C4C2×C22⋊D1111yescharacteristic
385C22×C4C2×C22⋊D1111yescharacteristic
386C22×C4C23×D1111yescharacteristic
387C2×C8—222no—
388C2×C8—222no—
389C2×C8—222no—
390C2×C8—222no—
391C2×C8—222no—
392C2×C8—222no—
393C4⋊C4—24no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—114no—
400C4⋊C4—114no—
401C4⋊C4—224no—
402C4⋊C4—224no—
403C4⋊C4—228no—
404C4⋊C4—228no—
405C4⋊C4—222no—
406C4⋊C4—222no—
407C4⋊C4C2×Dic2212yes—
408C22⋊C4—24no—
409C22⋊C4—42no—
410C22⋊C4—228no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—42no—
415C42—42no—
416C42—228no—
417C42D4×D1114yes—
418C11#2430611yescharacteristic, complemented, Sylow
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23#6697712yes—
426C23C22×C22⋊D1111yescharacteristic
427Q8—24no—
428Q8—42no—
429Q8—448no—
430Q8—448no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—224no—
458C2×C4—224no—
459C2×C4—442no—
460C2×C4—442no—
461C2×C4—442no—
462C2×C4—442no—
463C2×C4#6697712yes—
464C2×C4#6699012yes—
465C2×C4#6699012yes—
466C2×C4#6699512yes—
467C2×C4#6700312yes—
468C2×C4C22×Dic2211yescharacteristic
469C2×C4C2×#433911yescharacteristic
470C2×C4C2×D4×D1112yes—
471C2×C4#6704312yes—
472C8—442no—
473C8—442no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22C2×#6697711yescharacteristic
482C22C2×#6699011yescharacteristic
483C22C2×#6699511yescharacteristic
484C22C2×#6700311yescharacteristic
485C22#1062335311yescharacteristic
486C22#1062335311yescharacteristic
487C22#1062335611yescharacteristic
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—444no—
498C4—444no—
499C4D4×Dic2212yes—
500C4#1062308312yes—
501C2—42no—
502C2#1307198111yescharacteristic
503C2#1307783211yescharacteristic
504C2#1307787411yescharacteristic
505C2#1307840411yescharacteristic
506C2#1307842111yescharacteristic
507C2#1307977111yescharacteristic
508C2#1307985211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC4×C11×C22⋊Q8C2
socleC23×C11C22×D4
prime radicalC24×C11C2×D4
hypercenterC4×C22⋊Q8D11
hypocenterC11#24306
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#10622635C2semidirect
3. #13069949C2semidirect
4. #13069944C2semidirect
5. #13069757C2semidirect
6. #13069757C2semidirect
7. #13069573C2semidirect
52. C11×C22⋊Q8Q8semidirect
306. C22⋊Q8Dic22semidirect
418. C11#24306semidirect

Maximal subgroups

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

subgroupindex
C2×#106230012
C2×#106226352
#130699492
#130699442
#130697572
#130695732
#130689292
#130688132
#130685962
C4×C11×C22⋊Q82
#2430611