Finite Group Databasebeta

← C2×#13080282 #15628941

Subgroups — C2×#13080282 #15628941

Order 2816.

2794 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67004C212yesmaximal, complemented
2C2×D4×Dic22C211yesmaximal, characteristic, complemented
3#13080282C2116yesmaximal, direct product factor
4C2×#10622689C211yesmaximal, characteristic, complemented
5C2×#10622655C212yesmaximal, complemented
6C2×#10622644C211yesmaximal, characteristic, complemented
7C2×#10622688C211yesmaximal, characteristic, complemented
8C2×#10622852C212yesmaximal, complemented
9C2×#10622653C212yesmaximal, complemented
10C2×C4×#4326C211yesmaximal, characteristic, complemented
11C2×#10622625C211yesmaximal, characteristic, complemented
12C2×C11×#550C211yesmaximal, characteristic
13C22×C11×#41C2212yes—
14C23×Dic22C2212yes—
15C11×#550—24no—
16C11×#550C2218yes—
17C2×C11×#127—21no—
18C2×C11×#127C2211yescharacteristic
19C2×C11×#121—22no—
20C2×C11×#121C2212yes—
21C2×C4×Q8×C11C2211yescharacteristic
22C2×C4×D4×C11—21no—
23C2×C4×D4×C11C2211yescharacteristic
24C2×#67004—216no—
25C2×#67004C22116yescomplemented
26C22×#4326C2212yes—
27C22×#4326C2211yescharacteristic
28C2×D4×Dic11—21no—
29C22×#4324C2212yes—
30C22×#4324C2212yes—
31C22×#4324C2212yes—
32C2×#66977C2212yes—
33C22×#4318C2212yes—
34D4×Dic22—24no—
35D4×Dic22C2218yescomplemented
36C2×#66941—21no—
37C2×#66931—22no—
38C2×#66922C2211yescharacteristic
39C2×C4×Dic22C2211yescharacteristic
40#10622852C22116yescomplemented
41C2×#66899C2211yescharacteristic
42C2×#66896C2212yes—
43C2×C22⋊(C11⋊C8)C2212yes—
44#10622689C2218yescomplemented
45#10622688C2218yescomplemented
46#10622655C22116yescomplemented
47#10622653C22116yescomplemented
48#10622644C2218yescomplemented
49C4×#4326C2218yescomplemented
50#10622625C2218yescomplemented
51C2×#66874C2212yes—
52C2×#66874C2211yescharacteristic
53C2×#66871C2211yescharacteristic
54C2×C4⋊(C11⋊C8)C2211yescharacteristic
55C2×C4×C11⋊C8C2211yescharacteristic
56C2×C11×#41—24no—
57C2×C11×#41—28no—
58C2×C11×#41—44no—
59C2×C11×#41C2318yes—
60C22×Q8×C11C2311yescharacteristic
61C22×D4×C11—22no—
62C22×D4×C11—22no—
63C22×D4×C11—21no—
64C22×D4×C11—21no—
65C22×D4×C11D411yescharacteristic
66C22×D4×C11C2312yes—
67C23×C4×C11—22no—
68C23×C4×C11C2312yes—
69C23×Dic11—22no—
70C22×Dic22—28no—
71C22×Dic22—22no—
72C22×Dic22—24no—
73C22×Dic22—44no—
74C22×Dic22C2312yes—
75C22×Dic22C2318yes—
76C22×Dic22C2311yescharacteristic
77C11×#127—28no—
78C11×#127—22no—
79C11×#127C2314yes—
80C11×#121—24no—
81C11×#121—216no—
82C11×#121C2318yes—
83C4×Q8×C11—22no—
84C4×Q8×C11C2314yes—
85C4×D4×C11—28no—
86C4×D4×C11D414yes—
87C4×D4×C11C2314yes—
88C2×C11×C4⋊C4—21no—
89C2×C11×C4⋊C4D411yescharacteristic
90C2×C11×C4⋊C4C2311yescharacteristic
91C2×C11×C22⋊C4—22no—
92C2×C11×C22⋊C4D412yes—
93C2×C42×C11—21no—
94C2×C42×C11C2311yescharacteristic
95C2×C22⋊Dic11—22no—
96#67004—264no—
97C2×#4326—28no—
98C2×#4326D418yescomplemented
99C2×#4326C2318yes—
100C2×#4326C2314yes—
101D4×Dic11—28no—
102C2×#4324—28no—
103C2×#4324—28no—
104C2×#4324—44no—
105C2×#4324C2318yes—
106C2×#4324C2318yes—
107C2×#4324C2318yes—
108C2×C4⋊Dic11—21no—
109C2×C4⋊Dic11—21no—
110C2×C4⋊Dic11C2311yescharacteristic
111#66977—24no—
112#66977C2318yes—
113C2×Dic11⋊C4—21no—
114C2×Dic11⋊C4—22no—
115C2×C4×Dic11—21no—
116C2×#4318—28no—
117C2×#4318C2318yes—
118C22×C11⋊C8C2311yescharacteristic
119C22×C11⋊C8C2312yes—
120C22×C11⋊C8C2311yescharacteristic
121#66941—28no—
122#66931—216no—
123#66922—22no—
124#66922C2314yes—
125C4×Dic22—22no—
126C4×Dic22C2314yes—
127#66899C2314yes—
128#66896C2318yes—
129C22⋊(C11⋊C8)C2318yes—
130#66874C2314yes—
131#66874C2318yes—
132#66871C2314yes—
133C4⋊(C11⋊C8)C2314yes—
134C4×C11⋊C8C2314yes—
135C2×#3278—111nomaximal, Sylow
136C24×C11—22no—
137C24×C11C2×D412yes—
138C11×#41—216no—
139C11×#41—48no—
140C11×#41—416no—
141C2×Q8×C11—21no—
142C2×Q8×C11—22no—
143C2×Q8×C11C2×D414yes—
144C2×Q8×C11C2412yes—
145C2×D4×C11—28no—
146C2×D4×C11—28no—
147C2×D4×C11—24no—
148C2×D4×C11—24no—
149C2×D4×C11—24no—
150C2×D4×C11—22no—
151C2×D4×C11—24no—
152C2×D4×C11—24no—
153C2×D4×C11—48no—
154C2×D4×C11—44no—
155C2×D4×C11—82no—
156C2×D4×C11—84no—
157C2×D4×C11C2×D414yes—
158C2×D4×C11C2414yes—
159C22×C4×C11—21no—
160C22×C4×C11—24no—
161C22×C4×C11—24no—
162C22×C4×C11—21no—
163C22×C4×C11—28no—
164C22×C4×C11—48no—
165C22×C4×C11C2×D411yescharacteristic
166C22×C4×C11C2×D412yes—
167C22×C4×C11C2×D414yes—
168C22×C4×C11C2×D411yescharacteristic
169C22×C4×C11C2414yes—
170C22×C4×C11C2411yescharacteristic
171C22×Dic11—21no—
172C22×Dic11—28no—
173C22×Dic11—21no—
174C22×Dic11—22no—
175C22×Dic11—48no—
176C2×Dic22—216no—
177C2×Dic22—21no—
178C2×Dic22—24no—
179C2×Dic22—22no—
180C2×Dic22—28no—
181C2×Dic22—22no—
182C2×Dic22—24no—
183C2×Dic22—48no—
184C2×Dic22—416no—
185C2×Dic22—48no—
186C2×Dic22C2×D414yes—
187C2×Dic22C2414yes—
188C2×Dic22C2412yes—
189C11×C4⋊C4—24no—
190C11×C4⋊C4—21no—
191C11×C4⋊C4C2×D414yes—
192C11×C4⋊C4C2412yes—
193C11×C22⋊C4—28no—
194C11×C22⋊C4C2×D418yes—
195C42×C11—24no—
196C42×C11C2×D412yes—
197C42×C11C2412yes—
198C22⋊Dic11—28no—
199#4326—216no—
200#4326C2×D4116yescomplemented
201#4324—216no—
202#4324—216no—
203#4324—48no—
204C4⋊Dic11—21no—
205C4⋊Dic11—24no—
206C4⋊Dic11—24no—
207C4⋊Dic11C2412yes—
208Dic11⋊C4—24no—
209Dic11⋊C4—28no—
210C4×Dic11—24no—
211#4318—216no—
212C2×C11⋊C8—24no—
213C2×C11⋊C8—22no—
214C2×C11⋊C8C2×D414yes—
215C2×C11⋊C8C2412yes—
216C2×C11⋊C8C2414yes—
217C2×C11⋊C8C2412yes—
218C22×#137—112no—
219C2×#550D1111yescharacteristic
220C2×D4×Q8—111no—
221#3278—1116no—
222C2×#492—111no—
223C2×#468—112no—
224C2×#462—111no—
225C2×#460—111no—
226C2×#452—112no—
227C2×#448—112no—
228C2×C4×Q16—111no—
229C2×#436—111no—
230C23×C11—24no—
231C23×C11—24no—
232C23×C11—24no—
233C23×C11—24no—
234C23×C11—44no—
235C23×C11—44no—
236C23×C11—44no—
237C23×C11C22×D414yes—
238C23×C11C22×D411yescharacteristic
239Q8×C11—24no—
240Q8×C11—22no—
241Q8×C11C22×D414yes—
242D4×C11—28no—
243D4×C11—24no—
244D4×C11—24no—
245D4×C11—416no—
246D4×C11—44no—
247D4×C11—48no—
248D4×C11—84no—
249D4×C11—88no—
250C2×C4×C11—24no—
251C2×C4×C11—24no—
252C2×C4×C11—22no—
253C2×C4×C11—22no—
254C2×C4×C11—28no—
255C2×C4×C11—24no—
256C2×C4×C11—24no—
257C2×C4×C11—22no—
258C2×C4×C11—416no—
259C2×C4×C11—42no—
260C2×C4×C11#13714yescomplemented
261C2×C4×C11C22×D412yes—
262C2×C4×C11C22×D412yes—
263C2×C4×C11C22×D414yes—
264C2×C4×C11C22×D411yescharacteristic
265C2×C4×C11C22×D412yes—
266C2×C4×C11H(5,𝔽2)12yes—
267C2×C4×C11C2511yescharacteristic
268C2×Dic11—24no—
269C2×Dic11—22no—
270C2×Dic11—22no—
271C2×Dic11—24no—
272C2×Dic11—416no—
273C2×Dic11—44no—
274C2×Dic11—42no—
275Dic22—24no—
276Dic22—22no—
277Dic22—24no—
278Dic22—24no—
279Dic22—416no—
280Dic22—44no—
281Dic22C22×D414yes—
282C11⋊C8—24no—
283C11⋊C8—22no—
284C11⋊C8C22×D414yes—
285C22×#41C2×D1112yes—
286C23×Q8—112no—
287C2×#137—1116no—
288C2×#137—2216no—
289C22×Q16—111no—
290C22×Q16—112no—
291C22×SD16—112no—
292C22×SD16—112no—
293C22×SD16—112no—
294C22×M16—112no—
295#550—24no—
296#550C2×D1118yes—
297D4×Q8—118no—
298D4×Q8—224no—
299C2×C4⋊Q8—111no—
300C2×C4⋊Q8—221no—
301C2×#127—21no—
302C2×#127C2×D1111yescharacteristic
303C2×C22⋊Q8—112no—
304C2×C22⋊Q8—222no—
305C2×#121—22no—
306C2×#121C2×D1112yes—
307C2×C4×Q8—111no—
308C2×C4×Q8C2×D1111yescharacteristic
309C2×C4×D4—21no—
310C2×C4×D4—221no—
311C2×C4×D4C2×D1111yescharacteristic
312#492—118no—
313#468—1116no—
314#462—118no—
315#460—118no—
316#452—1116no—
317#448—1116no—
318C4×Q16—118no—
319#436—118no—
320C2×#107—111no—
321C2×C4⋊C8—111no—
322C2×#103—111no—
323C2×#103—111no—
324C2×#103—112no—
325C2×#102—112no—
326C2×C22⋊C8—112no—
327C2×C4×C8—111no—
328C22×C11—24no—
329C22×C11—24no—
330C22×C11—24no—
331C22×C11—48no—
332C22×C11—44no—
333C22×C11—44no—
334C22×C11—44no—
335C22×C11—88no—
336C22×C11D4212yes—
337C22×C11C2×#13712yes—
338C22×C11#57712yes—
339C22×C11C23×D411yescharacteristic
340C4×C11—24no—
341C4×C11—24no—
342C4×C11—22no—
343C4×C11—42no—
344C4×C11D4212yes—
345C4×C11C2×#13714yescomplemented
346C4×C11C23×D411yescharacteristic
347C4×C11C2×H(5,𝔽2)11yescharacteristic
348Dic11—24no—
349Dic11—44no—
350Dic11—42no—
351C2×#41—24no—
352C2×#41—28no—
353C2×#41—44no—
354C2×#41C22×D1118yes—
355C22×Q8—111no—
356C22×Q8—112no—
357C22×Q8—118no—
358C22×Q8—228no—
359C22×Q8—222no—
360C22×Q8—224no—
361C22×Q8—444no—
362C22×Q8C22×D1111yescharacteristic
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—21no—
366C22×D4—21no—
367C22×D4C22⋊D1111yescharacteristic
368C22×D4C22×D1112yes—
369C23×C4—22no—
370C23×C4—222no—
371C23×C4C22×D1112yes—
372#137—2264no—
373C2×Q16—118no—
374C2×Q16—114no—
375C2×Q16—118no—
376C2×Q16—228no—
377C2×SD16—118no—
378C2×SD16—118no—
379C2×SD16—118no—
380C2×SD16—228no—
381C2×SD16—228no—
382C2×SD16—444no—
383C2×M16—118no—
384C2×M16—228no—
385C22×C8—111no—
386C22×C8—112no—
387C22×C8—111no—
388C4⋊Q8—114no—
389C4⋊Q8—228no—
390C4⋊Q8—222no—
391#127—28no—
392#127—22no—
393#127C22×D1114yes—
394C22⋊Q8—118no—
395C22⋊Q8—224no—
396C22⋊Q8—2216no—
397#121—24no—
398#121—216no—
399#121C22×D1118yes—
400C4×Q8—22no—
401C4×Q8—114no—
402C4×Q8—222no—
403C4×Q8C22×D1114yes—
404C4×D4—28no—
405C4×D4—228no—
406C4×D4C22⋊D1114yes—
407C4×D4C22×D1114yes—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—111no—
410C2×C4⋊C4—221no—
411C2×C4⋊C4—221no—
412C2×C4⋊C4—222no—
413C2×C4⋊C4—221no—
414C2×C4⋊C4C22⋊D1111yescharacteristic
415C2×C4⋊C4C22×D1111yescharacteristic
416C2×C22⋊C4—22no—
417C2×C22⋊C4—222no—
418C2×C22⋊C4C22⋊D1112yes—
419C2×C42—21no—
420C2×C42—221no—
421C2×C42C22×D1111yescharacteristic
422#107—114no—
423C4⋊C8—114no—
424#103—114no—
425#103—114no—
426#103—118no—
427#102—118no—
428C22⋊C8—118no—
429C4×C8—114no—
430C2×C11—44no—
431C2×C11—84no—
432C2×C11#327814yescomplemented
433C2×C11C2×D4211yescharacteristic
434C2×C11C22×#13711yescharacteristic
435C2×C11C2×#57711yescharacteristic
436C24—22no—
437C24C2×C22⋊D1112yes—
438#41—216no—
439#41—48no—
440#41—416no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—114no—
444C2×Q8—112no—
445C2×Q8—114no—
446C2×Q8—2216no—
447C2×Q8—222no—
448C2×Q8—221no—
449C2×Q8—228no—
450C2×Q8—222no—
451C2×Q8—224no—
452C2×Q8—224no—
453C2×Q8—448no—
454C2×Q8—448no—
455C2×Q8—4416no—
456C2×Q8D4×D1114yes—
457C2×Q8C23×D1112yes—
458C2×D4—28no—
459C2×D4—28no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—48no—
467C2×D4—44no—
468C2×D4—82no—
469C2×D4—84no—
470C2×D4C2×C22⋊D1114yes—
471C2×D4C23×D1114yes—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—28no—
477C22×C4—48no—
478C22×C4—221no—
479C22×C4—228no—
480C22×C4—221no—
481C22×C4—222no—
482C22×C4—448no—
483C22×C4C2×C22⋊D1114yes—
484C22×C4C2×C22⋊D1111yescharacteristic
485C22×C4C2×C22⋊D1112yes—
486C22×C4C2×C22⋊D1111yescharacteristic
487C22×C4C23×D1114yes—
488C22×C4C23×D1111yescharacteristic
489Q16—1116no—
490Q16—2216no—
491SD16—2216no—
492SD16—2216no—
493SD16—448no—
494M16—2216no—
495C2×C8—114no—
496C2×C8—112no—
497C2×C8—112no—
498C2×C8—114no—
499C2×C8—224no—
500C2×C8—222no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—112no—
504C4⋊C4—221no—
505C4⋊C4—224no—
506C4⋊C4—224no—
507C4⋊C4—228no—
508C4⋊C4—224no—
509C4⋊C4C2×C22⋊D1114yes—
510C4⋊C4C23×D1112yes—
511C22⋊C4—28no—
512C22⋊C4—228no—
513C22⋊C4C2×C22⋊D1118yes—
514C42—24no—
515C42—224no—
516C42C2×C22⋊D1112yes—
517C42C23×D1112yes—
518C11C2×#327811yescharacteristic, complemented, Sylow
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C22×C22⋊D1114yes—
527C23C22×C22⋊D1111yescharacteristic
528Q8—24no—
529Q8—22no—
530Q8—114no—
531Q8—224no—
532Q8—222no—
533Q8—224no—
534Q8—224no—
535Q8—4416no—
536Q8—444no—
537Q8C2×D4×D1114yes—
538D4—28no—
539D4—24no—
540D4—24no—
541D4—416no—
542D4—44no—
543D4—48no—
544D4—84no—
545D4—88no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—416no—
555C2×C4—42no—
556C2×C4—224no—
557C2×C4—222no—
558C2×C4—222no—
559C2×C4—224no—
560C2×C4—4416no—
561C2×C4—444no—
562C2×C4—442no—
563C2×C4#6700414yescomplemented
564C2×C4C2×D4×D1112yes—
565C2×C4#6703712yes—
566C2×C4C22×C22⋊D1112yes—
567C2×C4C22×C22⋊D1114yes—
568C2×C4C22×C22⋊D1111yescharacteristic
569C2×C4C22×C22⋊D1112yes—
570C2×C4C24×D1111yescharacteristic
571C8—114no—
572C8—224no—
573C8—222no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—48no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—88no—
582C22D4×C22⋊D1112yes—
583C22C2×#6700412yes—
584C22#1062336712yes—
585C22C23×C22⋊D1111yescharacteristic
586C4—24no—
587C4—24no—
588C4—22no—
589C4—42no—
590C4—224no—
591C4—444no—
592C4—442no—
593C4D4×C22⋊D1112yes—
594C4C2×#6700414yescomplemented
595C4C22×D4×D1111yescharacteristic
596C4C2×#6703711yescharacteristic
597C2—44no—
598C2—84no—
599C2#1308028214yesdirect product factor
600C2C2×D4×C22⋊D1111yescharacteristic
601C2C22×#6700411yescharacteristic
602C2C2×#1062336711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
Fitting, prime radicalC2×C11×#550C2
socleC23×C11C22×D4
hypercenterC2×#550D11
hypocenterC11C2×#3278
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C22×#67004C2semidirect
2. C2×D4×Dic22C2semidirect
3. #13080282C2direct (N × H)
4. C2×#10622689C2semidirect
5. C2×#10622655C2semidirect
6. C2×#10622644C2semidirect
7. C2×#10622688C2semidirect
8. C2×#10622852C2semidirect
9. C2×#10622653C2semidirect
10. C2×C4×#4326C2semidirect
11. C2×#10622625C2semidirect
25. C2×#67004C22semidirect
35. D4×Dic22C22semidirect
40. #10622852C22semidirect
44. #10622689C22semidirect
45. #10622688C22semidirect
46. #10622655C22semidirect
47. #10622653C22semidirect
48. #10622644C22semidirect
49. C4×#4326C22semidirect
50. #10622625C22semidirect
98. C2×#4326D4semidirect
200. #4326C2×D4semidirect
260. C2×C4×C11#137semidirect
345. C4×C11C2×#137semidirect
432. C2×C11#3278semidirect
518. C11C2×#3278semidirect
563. C2×C4#67004semidirect
594. C4C2×#67004semidirect
599. C2#13080282direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#670042
C2×D4×Dic222
#130802822
C2×#106226892
C2×#106226552
C2×#106226442
C2×#106226882
C2×#106228522
C2×#106226532
C2×C4×#43262
C2×#106226252
C2×C11×#5502
C2×#327811