Finite Group Databasebeta

← #14780993

Subgroups — #14780993

Order 2816.

1898 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67004C211yesmaximal, characteristic, complemented
2C22×#67004C211yesmaximal, characteristic, complemented
3C2×#10622850C211yesmaximal, characteristic, complemented
4C22×#66888C211yesmaximal, characteristic, complemented
5#13077061C211yesmaximal, characteristic, complemented
6#13069734C214yesmaximal, complemented
7#13069730C214yesmaximal, complemented
8#13068342C211yesmaximal, characteristic, complemented
9C2×C11×#575C211yesmaximal, characteristic
10C22×C11×#41C2211yescharacteristic
11C22×C11×#41C2211yescharacteristic
12C23×Dic22C2211yescharacteristic
13C11×#575—24no—
14C11×#575—24no—
15C2×C11×#137—21no—
16C2×C11×#137C2212yes—
17C2×C11×AGL(1,ℤ/8)—21no—
18C2×C11×AGL(1,ℤ/8)C2212yes—
19C2×C11×#135—22no—
20C22×C11×M16C2211yescharacteristic
21C2×#67004—22no—
22C2×#67004—24no—
23C2×#67004—24no—
24C2×#67004—24no—
25C2×#67004—24no—
26C2×#67004—22no—
27C2×#67004—22no—
28C2×#67004C2214yes—
29C22×#4326—21no—
30C22×#4326C2212yescomplemented
31C22×#4324—21no—
32C22×#4324C2212yescomplemented
33C22×#4318C2211yescharacteristic
34C22×#4318C2211yescharacteristic
35C22×#4318C2211yescharacteristic
36#10622850—24no—
37#10622847—22no—
38#10622847C2214yes—
39C2×#66900C2212yescomplemented
40C2×#66897C2212yescomplemented
41#10622820—22no—
42C2×#66888—24no—
43C2×#66888C2214yescomplemented
44#10622196C2214yescomplemented
45C2×C11×#41—21no—
46C2×C11×#41—22no—
47C2×C11×#41—21no—
48C2×C11×#41—22no—
49C2×C11×#41—22no—
50C2×C11×#41—21no—
51C2×C11×#41—21no—
52C2×C11×#41—22no—
53C2×C11×#41—21no—
54C2×C11×#41—21no—
55C2×C11×#41—22no—
56C2×C11×#41—22no—
57C2×C11×#41—22no—
58C2×C11×#41D412yes—
59C2×C11×#41D414yes—
60C2×C11×#41C2312yes—
61C22×Q8×C11D411yescharacteristic
62C22×Q8×C11C2311yescharacteristic
63C22×D4×C11—21no—
64C22×D4×C11—21no—
65C22×D4×C11D411yescharacteristic
66C22×D4×C11C2311yescharacteristic
67C23×C4×C11—21no—
68C23×C4×C11—21no—
69C23×C4×C11C2311yescharacteristic
70C23×Dic11—21no—
71C22×Dic22—21no—
72C22×Dic22—22no—
73C22×Dic22—21no—
74C22×Dic22—21no—
75C22×Dic22—21no—
76C22×Dic22—22no—
77C22×Dic22—22no—
78C22×Dic22—21no—
79C22×Dic22D414yes—
80C22×Dic22C2312yes—
81C11×#137—28no—
82C11×#137—24no—
83C11×#137—24no—
84C11×AGL(1,ℤ/8)—28no—
85C11×AGL(1,ℤ/8)—24no—
86C11×AGL(1,ℤ/8)—24no—
87C11×#135—216no—
88C2×C11×Q16—22no—
89C2×C11×SD16—22no—
90C2×C11×SD16—22no—
91C2×C11×D8—22no—
92C2×C11×M16—21no—
93C2×C11×M16—22no—
94C2×C11×M16D414yes—
95C2×C11×M16C2312yes—
96C22×C8×C11—21no—
97#67004—216no—
98#67004—216no—
99#67004—216no—
100#67004—28no—
101#67004—28no—
102C2×#4326—28no—
103C2×#4326—24no—
104C2×#4326—24no—
105C2×#4326—24no—
106C2×#4326—24no—
107C2×#4324—28no—
108C2×#4324—24no—
109C2×#4324—24no—
110C2×#4324—24no—
111C2×#4324—24no—
112C2×#4318—22no—
113C2×#4318—22no—
114C2×#4318—21no—
115C2×#4318—22no—
116C2×#4318—21no—
117C2×#4318—22no—
118C2×#4318—22no—
119C2×#4318D414yes—
120C2×#4318D412yes—
121C2×#4318D412yes—
122C2×#4318C2312yes—
123C2×#4318C2312yes—
124C22×C11⋊C8—21no—
125C22×C11⋊C8—21no—
126C22×C11⋊C8C2312yes—
127#66900—24no—
128#66897—24no—
129C22⋊(C11⋊C8)—28no—
130#66888—28no—
131#23440—111nomaximal, Sylow
132C24×C11—21no—
133C24×C11—21no—
134C24×C11#4111yescharacteristic
135C11×#41—24no—
136C11×#41—24no—
137C11×#41—24no—
138C11×#41—24no—
139C11×#41—24no—
140C11×#41—24no—
141C11×#41—22no—
142C11×#41—22no—
143C11×#41—22no—
144C11×#41—22no—
145C11×#41—42no—
146C11×#41—42no—
147C11×#41—42no—
148C11×#41—42no—
149C11×#41—42no—
150C11×#41—42no—
151C11×#41—42no—
152C11×#41—42no—
153C2×Q8×C11—21no—
154C2×Q8×C11—21no—
155C2×Q8×C11—22no—
156C2×Q8×C11—22no—
157C2×Q8×C11—21no—
158C2×Q8×C11—21no—
159C2×Q8×C11—21no—
160C2×Q8×C11—21no—
161C2×Q8×C11C2×D412yes—
162C2×Q8×C11C2×D412yes—
163C2×D4×C11—22no—
164C2×D4×C11—24no—
165C2×D4×C11—22no—
166C2×D4×C11—22no—
167C2×D4×C11—22no—
168C2×D4×C11—21no—
169C2×D4×C11—21no—
170C2×D4×C11—21no—
171C2×D4×C11—21no—
172C2×D4×C11—21no—
173C2×D4×C11—21no—
174C2×D4×C11—42no—
175C2×D4×C11—41no—
176C2×D4×C11—41no—
177C2×D4×C11—42no—
178C2×D4×C11—42no—
179C2×D4×C11C2×D412yes—
180C2×D4×C11C2×D412yes—
181C22×C4×C11—21no—
182C22×C4×C11—21no—
183C22×C4×C11—22no—
184C22×C4×C11—24no—
185C22×C4×C11—22no—
186C22×C4×C11—42no—
187C22×C4×C11—42no—
188C22×C4×C11—41no—
189C22×C4×C11—41no—
190C22×C4×C11—42no—
191C22×C4×C11C2×D412yes—
192C22×C4×C11C2×D411yescharacteristic
193C22×C4×C11C2×D411yescharacteristic
194C22×C4×C11C2×D412yes—
195C22×C4×C11C2×D411yescharacteristic
196C22×C4×C11C2×D411yescharacteristic
197C22×C4×C11C2×D412yes—
198C22×C4×C11C2×D412yes—
199C22×C4×C11#4111yescharacteristic
200C22×C4×C11C2411yescharacteristic
201C22×Dic11—24no—
202C22×Dic11—22no—
203C22×Dic11—41no—
204C22×Dic11—41no—
205C22×Dic11—42no—
206C2×Dic22—24no—
207C2×Dic22—24no—
208C2×Dic22—24no—
209C2×Dic22—24no—
210C2×Dic22—24no—
211C2×Dic22—22no—
212C2×Dic22—22no—
213C2×Dic22—22no—
214C2×Dic22—22no—
215C2×Dic22—22no—
216C2×Dic22—42no—
217C2×Dic22—42no—
218C2×Dic22—41no—
219C2×Dic22—42no—
220C2×Dic22—41no—
221C2×Dic22—42no—
222C2×Dic22—42no—
223C2×Dic22C2×D414yes—
224C11×Q16—28no—
225C11×SD16—28no—
226C11×SD16—28no—
227C11×D8—28no—
228C11×M16—24no—
229C11×M16—22no—
230C11×M16C2×D414yes—
231C2×C8×C11—24no—
232C2×C8×C11—22no—
233#4326—216no—
234#4326—216no—
235#4324—216no—
236#4324—216no—
237#4318—24no—
238#4318—24no—
239#4318—24no—
240#4318—22no—
241#4318—22no—
242#4318—22no—
243#4318—42no—
244#4318C2×D414yes—
245C2×C11⋊C8—24no—
246C2×C11⋊C8—22no—
247C2×C11⋊C8—22no—
248C2×C11⋊C8—22no—
249C2×C11⋊C8—22no—
250C2×C11⋊C8—42no—
251C2×C11⋊C8C2×D414yes—
252C2×#575D1111yescharacteristic
253C22×#137—111no—
254C22×#137—111no—
255C2×#413—111no—
256C22×#101—111no—
257#2879—111no—
258#2020—114no—
259#2019—114no—
260#1738—111no—
261C23×C11—22no—
262C23×C11—22no—
263C23×C11—42no—
264C23×C11—41no—
265C23×C11—41no—
266C23×C11—42no—
267C23×C11—82no—
268C23×C11—82no—
269C23×C11#12112yes—
270C23×C11#12111yescharacteristic
271C23×C11#12111yescharacteristic
272C23×C11#12112yes—
273C23×C11C2×#4111yescharacteristic
274Q8×C11—22no—
275Q8×C11—22no—
276Q8×C11—22no—
277Q8×C11—22no—
278Q8×C11—41no—
279Q8×C11—41no—
280Q8×C11—42no—
281D4×C11—22no—
282D4×C11—22no—
283D4×C11—22no—
284D4×C11—22no—
285D4×C11—42no—
286D4×C11—42no—
287D4×C11—44no—
288D4×C11—44no—
289D4×C11—44no—
290D4×C11—41no—
291D4×C11—41no—
292D4×C11—42no—
293C2×C4×C11—22no—
294C2×C4×C11—22no—
295C2×C4×C11—41no—
296C2×C4×C11—41no—
297C2×C4×C11—42no—
298C2×C4×C11—42no—
299C2×C4×C11—42no—
300C2×C4×C11—44no—
301C2×C4×C11—44no—
302C2×C4×C11—44no—
303C2×C4×C11#12012yes—
304C2×C4×C11#12012yes—
305C2×C4×C11#12012yes—
306C2×C4×C11#12012yes—
307C2×C4×C11#12012yes—
308C2×C4×C11#12012yes—
309C2×C4×C11#12112yes—
310C2×C4×C11#12111yescharacteristic
311C2×C4×C11#12112yes—
312C2×C4×C11#12111yescharacteristic
313C2×C4×C11#12712yes—
314C2×C4×C11#12712yes—
315C2×C4×C11C22×D411yescharacteristic
316C2×C4×C11C22×D411yescharacteristic
317C2×C4×C11C22×D411yescharacteristic
318C2×C4×C11C22×D411yescharacteristic
319C2×C4×C11C22×D411yescharacteristic
320C2×C4×C11C22×D411yescharacteristic
321C2×Dic11—24no—
322C2×Dic11—42no—
323C2×Dic11—42no—
324C2×Dic11—44no—
325C2×Dic11—44no—
326Dic22—24no—
327Dic22—24no—
328Dic22—42no—
329Dic22—42no—
330Dic22—44no—
331Dic22—44no—
332C8×C11—24no—
333C11⋊C8—24no—
334C11⋊C8—24no—
335C11⋊C8—42no—
336C22×#41C2×D1111yescharacteristic
337C22×#41C2×D1111yescharacteristic
338C23×Q8—111no—
339#575—24no—
340#575—24no—
341C2×#137—21no—
342C2×#137—114no—
343C2×#137—222no—
344C2×#137—222no—
345C2×#137—224no—
346C2×#137—224no—
347C2×#137—224no—
348C2×#137—224no—
349C2×#137—222no—
350C2×#137C2×D1112yes—
351C2×AGL(1,ℤ/8)—21no—
352C2×AGL(1,ℤ/8)C2×D1112yes—
353C2×#135—22no—
354C22×Q16—112no—
355C22×Q16—221no—
356C22×SD16—112no—
357C22×SD16—221no—
358C22×M16—111no—
359C22×M16—111no—
360C22×M16—111no—
361C22×M16C2×D1111yescharacteristic
362#413—224no—
363C2×#101—112no—
364C2×#101—114no—
365C2×#101—224no—
366C2×#100—112no—
367#408—114no—
368#408—222no—
369#407—222no—
370#341—114no—
371C22×C11—22no—
372C22×C11—22no—
373C22×C11—22no—
374C22×C11—42no—
375C22×C11—42no—
376C22×C11—42no—
377C22×C11—42no—
378C22×C11—44no—
379C22×C11—82no—
380C22×C11—82no—
381C22×C11—84no—
382C22×C11—82no—
383C22×C11#39212yes—
384C22×C11#39212yes—
385C22×C11C2×#12111yescharacteristic
386C22×C11C2×#12111yescharacteristic
387C22×C11C2×#12111yescharacteristic
388C4×C11—42no—
389C4×C11—42no—
390C4×C11#39212yes—
391C4×C11#39212yes—
392C4×C11C2×#12011yescharacteristic
393C4×C11C2×#12011yescharacteristic
394C4×C11C2×#12011yescharacteristic
395C4×C11C2×#12711yescharacteristic
396Dic11—44no—
397C2×#41—21no—
398C2×#41—22no—
399C2×#41—21no—
400C2×#41—22no—
401C2×#41—22no—
402C2×#41—21no—
403C2×#41—21no—
404C2×#41—22no—
405C2×#41—21no—
406C2×#41—21no—
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—22no—
410C2×#41C22⋊D1112yes—
411C2×#41C22⋊D1114yes—
412C2×#41C22×D1112yes—
413C22×Q8—114no—
414C22×Q8—112no—
415C22×Q8—221no—
416C22×Q8—222no—
417C22×Q8—221no—
418C22×Q8—221no—
419C22×Q8—222no—
420C22×Q8—221no—
421C22×Q8—221no—
422C22×Q8—222no—
423C22×Q8C22⋊D1111yescharacteristic
424C22×Q8C22×D1111yescharacteristic
425C22×D4—21no—
426C22×D4—21no—
427C22×D4C22⋊D1111yescharacteristic
428C22×D4C22×D1111yescharacteristic
429C23×C4—21no—
430C23×C4—21no—
431C23×C4—221no—
432C23×C4C22×D1111yescharacteristic
433#137—28no—
434#137—24no—
435#137—24no—
436#137—228no—
437#137—228no—
438#137—2216no—
439#137—2216no—
440#137—2216no—
441AGL(1,ℤ/8)—28no—
442AGL(1,ℤ/8)—24no—
443AGL(1,ℤ/8)—24no—
444#135—216no—
445C2×Q16—22no—
446C2×Q16—224no—
447C2×Q16—224no—
448C2×Q16—228no—
449C2×Q16—224no—
450C2×Q16—224no—
451C2×SD16—22no—
452C2×SD16—22no—
453C2×SD16—224no—
454C2×SD16—224no—
455C2×SD16—228no—
456C2×SD16—224no—
457C2×SD16—224no—
458C2×D8—22no—
459C2×M16—21no—
460C2×M16—22no—
461C2×M16—112no—
462C2×M16—112no—
463C2×M16—114no—
464C2×M16—112no—
465C2×M16—112no—
466C2×M16—222no—
467C2×M16—222no—
468C2×M16—222no—
469C2×M16—221no—
470C2×M16—221no—
471C2×M16—222no—
472C2×M16—222no—
473C2×M16C22⋊D1114yes—
474C2×M16C22×D1112yes—
475C22×C8—21no—
476C22×C8—112no—
477C22×C8—221no—
478C22×C8—221no—
479#101—224no—
480#101—228no—
481#100—224no—
482C22⋊C8—228no—
483C2×C11—21no—
484C2×C11—21no—
485C2×C11—22no—
486C2×C11—22no—
487C2×C11—82no—
488C2×C11—82no—
489C2×C11C2×#39211yescharacteristic
490C2×C11#306912yes—
491C24—21no—
492C24—21no—
493C24#434111yescharacteristic
494#41—24no—
495#41—24no—
496#41—24no—
497#41—24no—
498#41—24no—
499#41—24no—
500#41—22no—
501#41—22no—
502#41—22no—
503#41—22no—
504#41—42no—
505#41—42no—
506#41—42no—
507#41—42no—
508#41—42no—
509#41—42no—
510#41—42no—
511#41—42no—
512C2×Q8—21no—
513C2×Q8—21no—
514C2×Q8—22no—
515C2×Q8—22no—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—21no—
519C2×Q8—21no—
520C2×Q8—114no—
521C2×Q8—224no—
522C2×Q8—222no—
523C2×Q8—224no—
524C2×Q8—224no—
525C2×Q8—224no—
526C2×Q8—224no—
527C2×Q8—222no—
528C2×Q8—222no—
529C2×Q8—222no—
530C2×Q8—222no—
531C2×Q8—442no—
532C2×Q8—441no—
533C2×Q8—441no—
534C2×Q8—442no—
535C2×Q8—442no—
536C2×Q8—442no—
537C2×Q8—442no—
538C2×Q8C2×C22⋊D1112yes—
539C2×Q8C2×C22⋊D1112yes—
540C2×D4—24no—
541C2×D4—22no—
542C2×D4—22no—
543C2×D4—22no—
544C2×D4—21no—
545C2×D4—22no—
546C2×D4—21no—
547C2×D4—21no—
548C2×D4—21no—
549C2×D4—21no—
550C2×D4—21no—
551C2×D4—42no—
552C2×D4—42no—
553C2×D4—41no—
554C2×D4—41no—
555C2×D4—42no—
556C2×D4C2×C22⋊D1112yes—
557C2×D4C2×C22⋊D1112yes—
558C22×C4—21no—
559C22×C4—21no—
560C22×C4—24no—
561C22×C4—22no—
562C22×C4—22no—
563C22×C4—42no—
564C22×C4—42no—
565C22×C4—41no—
566C22×C4—41no—
567C22×C4—42no—
568C22×C4—224no—
569C22×C4—222no—
570C22×C4—442no—
571C22×C4—441no—
572C22×C4—441no—
573C22×C4D4×D1112yes—
574C22×C4D4×D1111yescharacteristic
575C22×C4D4×D1111yescharacteristic
576C22×C4D4×D1112yes—
577C22×C4#434111yescharacteristic
578C22×C4C2×C22⋊D1111yescharacteristic
579C22×C4C2×C22⋊D1111yescharacteristic
580C22×C4C2×C22⋊D1112yes—
581C22×C4C2×C22⋊D1112yes—
582C22×C4C23×D1111yescharacteristic
583Q16—28no—
584Q16—2216no—
585Q16—2216no—
586SD16—28no—
587SD16—28no—
588SD16—2216no—
589SD16—2216no—
590D8—28no—
591M16—24no—
592M16—22no—
593M16—114no—
594M16—224no—
595M16—224no—
596M16—224no—
597M16—222no—
598M16—222no—
599M16—222no—
600M16—442no—
601M16C2×C22⋊D1114yes—
602C2×C8—24no—
603C2×C8—22no—
604C2×C8—114no—
605C2×C8—224no—
606C2×C8—222no—
607C2×C8—222no—
608C2×C8—222no—
609C2×C8—222no—
610C2×C8—442no—
611C11#2344011yescharacteristic, complemented, Sylow
612C23—22no—
613C23—22no—
614C23—42no—
615C23—41no—
616C23—41no—
617C23—42no—
618C23—82no—
619C23—82no—
620C23#6699112yes—
621C23#6699211yescharacteristic
622C23#6699211yescharacteristic
623C23#6699212yes—
624C23C2×#434111yescharacteristic
625Q8—22no—
626Q8—22no—
627Q8—22no—
628Q8—22no—
629Q8—41no—
630Q8—41no—
631Q8—42no—
632Q8—224no—
633Q8—224no—
634Q8—442no—
635Q8—444no—
636Q8—444no—
637Q8—442no—
638D4—22no—
639D4—22no—
640D4—22no—
641D4—22no—
642D4—42no—
643D4—42no—
644D4—44no—
645D4—44no—
646D4—44no—
647D4—41no—
648D4—41no—
649D4—42no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—224no—
661C2×C4—442no—
662C2×C4—442no—
663C2×C4—444no—
664C2×C4—444no—
665C2×C4#6699012yes—
666C2×C4#6699012yes—
667C2×C4#6699012yes—
668C2×C4#6699012yes—
669C2×C4#6699112yes—
670C2×C4#6699212yes—
671C2×C4#6699211yescharacteristic
672C2×C4#6699211yescharacteristic
673C2×C4#6699312yes—
674C2×C4#6699312yes—
675C2×C4#6700612yes—
676C2×C4#6700612yes—
677C2×C4C2×D4×D1111yescharacteristic
678C2×C4C2×D4×D1111yescharacteristic
679C2×C4C2×D4×D1111yescharacteristic
680C2×C4C22×C22⋊D1111yescharacteristic
681C2×C4C22×C22⋊D1111yescharacteristic
682C2×C4C22×C22⋊D1111yescharacteristic
683C8—24no—
684C8—224no—
685C8—224no—
686C8—442no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—42no—
694C22—42no—
695C22—82no—
696C22—82no—
697C22—84no—
698C22—82no—
699C22#1062283612yes—
700C22#1062283612yes—
701C22C2×#6699111yescharacteristic
702C22C2×#6699211yescharacteristic
703C22C2×#6699211yescharacteristic
704C4—42no—
705C4—42no—
706C4—444no—
707C4#1062283612yes—
708C4#1062283612yes—
709C4C2×#6699011yescharacteristic
710C4C2×#6699011yescharacteristic
711C4C2×#6699311yescharacteristic
712C4C2×#6700611yescharacteristic
713C2—21no—
714C2—21no—
715C2—22no—
716C2—22no—
717C2—82no—
718C2—82no—
719C2C2×#1062283611yescharacteristic
720C2#1307835512yes—
subgroupG/H
centerC22C2×#66992
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
Fitting, prime radicalC2×C11×#575C2
socleC22×C11C2×#121
hypercenterC2×#575D11
hypocenterC11#23440
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#575

Complements

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

NH ≅ G/Ntype
1. C22×#67004C2semidirect
2. C22×#67004C2semidirect
3. C2×#10622850C2semidirect
4. C22×#66888C2semidirect
5. #13077061C2semidirect
6. #13069734C2semidirect
7. #13069730C2semidirect
8. #13068342C2semidirect
30. C22×#4326C22semidirect
32. C22×#4324C22semidirect
39. C2×#66900C22semidirect
40. C2×#66897C22semidirect
43. C2×#66888C22semidirect
44. #10622196C22semidirect
611. C11#23440semidirect

Maximal subgroups

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

subgroupindex
C22×#670042
C2×#106228502
C22×#668882
#130770612
#130697342
#130697302
#130683422
C2×C11×#5752
#2344011