Finite Group Databasebeta

← #14803312

Subgroups — #14803312

Order 2816.

848 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10622457C211yesmaximal, characteristic
2#13074104C211yesmaximal, characteristic, complemented
3#13073974C211yesmaximal, characteristic, complemented
4#13070010C211yesmaximal, characteristic, complemented
5#13069989C211yesmaximal, characteristic, complemented
6#13069934C211yesmaximal, characteristic, complemented
7#13069927C211yesmaximal, characteristic
8#13069482C211yesmaximal, characteristic
9#13069470C211yesmaximal, characteristic, complemented
10#13069427C211yesmaximal, characteristic, complemented
11#13069386C211yesmaximal, characteristic, complemented
12#13069299C211yesmaximal, characteristic, complemented
13#13069292C211yesmaximal, characteristic, complemented
14C4×#66885C211yesmaximal, characteristic, complemented
15C2×C8×Q8×C11C211yesmaximal, characteristic, complemented
16C2×C4×Q8×C11C2211yescharacteristic
17C22×D88C2211yescharacteristic
18C22×#4314C2211yescharacteristic
19C2×#66928C2211yescharacteristic
20C2×C4×D44C2211yescharacteristic
21C8×Q8×C11—24no—
22C2×C11×C4⋊C8C2211yescharacteristic
23C2×C11×C4⋊C8C2211yescharacteristic
24C2×C11×C4⋊C8C2211yescharacteristic
25C2×C4×C8×C11C2211yescharacteristic
26C2×C4×C8×C11C2211yescharacteristic
27C2×C4×C8×C11C2211yescharacteristic
28#10622844—21no—
29#10622844—21no—
30C2×#66885C2211yescharacteristic
31C2×#66885C2211yescharacteristic
32C2×#66885C2211yescharacteristic
33C2×#66885C2211yescharacteristic
34C2×#66885C2211yescharacteristic
35C2×#66882C2211yescharacteristic
36C2×#66881C2211yescharacteristic
37C2×#66880C2211yescharacteristic
38C2×#66880C2211yescharacteristic
39C2×#66880C2211yescharacteristic
40#10622606—21no—
41#10622605—21no—
42#10622603—21no—
43#10622603—21no—
44#10622602—21no—
45#10622602—21no—
46#10622552C2211yescharacteristic
47#10622552C2211yescharacteristic
48#10622552C2211yescharacteristic
49#10622552C2211yescharacteristic
50#10622549C2211yescharacteristic
51C4×C4⋊Dic11C2211yescharacteristic
52#10622542C2211yescharacteristic
53#10622542C2211yescharacteristic
54#10622541C2211yescharacteristic
55#10622541C2211yescharacteristic
56#10622457—24no—
57#10622189C2211yescharacteristic
58#10622189C2211yescharacteristic
59#10622189C2211yescharacteristic
60#10622189C2211yescharacteristic
61C22×Q8×C11D411yescharacteristic
62C22×D44C2311yescharacteristic
63C22×D44C2311yescharacteristic
64C22×C4×D11—21no—
65C22×Dic22C2311yescharacteristic
66C22×C8×C11C2311yescharacteristic
67C22×C8×C11C2311yescharacteristic
68C22×C8×C11C2311yescharacteristic
69C22×C8×C11C2311yescharacteristic
70C4×Q8×C11—22no—
71C4×Q8×C11—22no—
72C2×C11×C4⋊C4D411yescharacteristic
73C2×C11×C4⋊C4D411yescharacteristic
74C2×C11×C4⋊C4D411yescharacteristic
75C2×C42×C11C2311yescharacteristic
76C2×C42×C11C2311yescharacteristic
77C2×C42×C11C2311yescharacteristic
78C2×(C2×D11)⋊C4—21no—
79C2×(C2×D11)⋊C4—21no—
80C2×(C2×D11)⋊C4—21no—
81C2×(C2×D11)⋊C4—21no—
82C2×(C2×D11)⋊C4—21no—
83C2×(C2×D11)⋊C4—21no—
84C2×(C2×D11)⋊C4—21no—
85C2×(C2×D11)⋊C4—21no—
86C2×(C2×D11)⋊C4—21no—
87C2×(C2×D11)⋊C4—21no—
88C2×(C2×D11)⋊C4—21no—
89C2×C4⋊Dic11C2311yescharacteristic
90C2×C4⋊Dic11C2311yescharacteristic
91C2×C4⋊Dic11C2311yescharacteristic
92C2×C4⋊Dic11C2311yescharacteristic
93C2×C4⋊Dic11C2311yescharacteristic
94C2×Dic11⋊C4—21no—
95C2×Dic11⋊C4—21no—
96C2×C4×Dic11—21no—
97C2×D88—22no—
98C2×D88—22no—
99C2×D88—41no—
100C2×#4314—22no—
101C2×#4314—22no—
102C2×#4314—41no—
103#66928—22no—
104#66928—22no—
105C4×D44—22no—
106C4×D44—22no—
107C11×C4⋊C8—22no—
108C11×C4⋊C8—22no—
109C11×C4⋊C8D414yes—
110C4×C8×C11—22no—
111C4×C8×C11—22no—
112C4×C8×C11—22no—
113C11⋊#95—21no—
114C11⋊#95—21no—
115C11⋊#95—21no—
116C11⋊#95—21no—
117C11⋊#95—21no—
118C11⋊#95—21no—
119C11⋊#95—21no—
120C11⋊#95—21no—
121C11⋊#95—21no—
122#66885—22no—
123#66885—22no—
124#66885—22no—
125#66885—22no—
126#66885—41no—
127#66882—41no—
128#66881—41no—
129#66880—22no—
130#66880—22no—
131#66880—41no—
132#25296—111nomaximal, Sylow
133C23×D11—21no—
134C23×D11—21no—
135C2×Q8×C11—22no—
136C2×Q8×C11—22no—
137C2×Q8×C11—22no—
138C22×C4×C11C2×D411yescharacteristic
139C22×C4×C11C2×D411yescharacteristic
140C22×C4×C11C2×D411yescharacteristic
141C22×C4×C11C2×D411yescharacteristic
142C22×C4×C11C2×D411yescharacteristic
143C22×C4×C11C2×D411yescharacteristic
144C22×C4×C11C2411yescharacteristic
145C22×Dic11—21no—
146C22×Dic11—21no—
147C22×Dic11—21no—
148C22×Dic11—21no—
149C22×Dic11—21no—
150C22×Dic11—21no—
151C2×D44—21no—
152C2×D44—21no—
153C2×D44—21no—
154C2×D44—21no—
155C2×D44—21no—
156C2×D44—21no—
157C2×D44—21no—
158C2×D44—21no—
159C2×D44—21no—
160C2×D44—21no—
161C2×D44—21no—
162C2×D44#4112yes—
163C2×C4×D11—42no—
164C2×C4×D11—81no—
165C2×C4×D11—81no—
166C2×Dic22—21no—
167C2×Dic22—21no—
168C2×Dic22—21no—
169C2×Dic22—21no—
170C2×Dic22—21no—
171C2×Dic22—21no—
172C2×C8×C11—21no—
173C2×C8×C11—21no—
174C2×C8×C11—21no—
175C2×C8×C11—21no—
176C2×C8×C11—21no—
177C2×C8×C11—21no—
178C2×C8×C11—21no—
179C2×C8×C11—21no—
180C2×C8×C11—21no—
181C2×C8×C11C2×D412yes—
182C2×C8×C11C2×D412yes—
183C2×C8×C11#4112yes—
184C11×C4⋊C4—22no—
185C11×C4⋊C4—22no—
186C11×C4⋊C4—22no—
187C42×C11—21no—
188C42×C11—21no—
189C42×C11—21no—
190C42×C11—21no—
191C42×C11C2×D412yes—
192C42×C11#4112yes—
193(C2×D11)⋊C4—42no—
194(C2×D11)⋊C4—42no—
195(C2×D11)⋊C4—42no—
196(C2×D11)⋊C4—42no—
197(C2×D11)⋊C4—42no—
198(C2×D11)⋊C4—42no—
199(C2×D11)⋊C4—42no—
200(C2×D11)⋊C4—81no—
201(C2×D11)⋊C4—81no—
202(C2×D11)⋊C4—81no—
203(C2×D11)⋊C4—81no—
204C4⋊Dic11—21no—
205C4⋊Dic11—21no—
206C4⋊Dic11—21no—
207C4⋊Dic11—21no—
208C4⋊Dic11—21no—
209C4⋊Dic11—21no—
210C4⋊Dic11—21no—
211C4⋊Dic11—21no—
212C4⋊Dic11—21no—
213C4⋊Dic11—21no—
214Dic11⋊C4—81no—
215Dic11⋊C4—81no—
216C4×Dic11—81no—
217D88—44no—
218#4314—44no—
219C2×#463—111no—
220C2×C8×Q8D1111yescharacteristic, complemented
221#2592—111no—
222#2580—111no—
223#2068—111no—
224#2065—111no—
225#2055—111no—
226#2054—111no—
227#1970—111no—
228#1968—111no—
229#1958—111no—
230#1949—111no—
231#1934—111no—
232#1933—111no—
233C4×#102—111no—
234C23×C11C22×D411yescharacteristic
235C22×D11—22no—
236C22×D11—41no—
237C22×D11—41no—
238C22×D11—41no—
239C22×D11—41no—
240C22×D11—41no—
241C22×D11—41no—
242C22×D11—41no—
243C22×D11—41no—
244C22×D11—41no—
245C22×D11—41no—
246C22×D11—41no—
247C22×D11—41no—
248C22×D11—41no—
249Q8×C11—28no—
250C2×C4×C11—21no—
251C2×C4×C11—21no—
252C2×C4×C11—21no—
253C2×C4×C11—21no—
254C2×C4×C11—21no—
255C2×C4×C11—21no—
256C2×C4×C11—21no—
257C2×C4×C11—21no—
258C2×C4×C11—21no—
259C2×C4×C11—21no—
260C2×C4×C11—21no—
261C2×C4×C11—21no—
262C2×C4×C11—21no—
263C2×C4×C11—21no—
264C2×C4×C11#12112yes—
265C2×C4×C11#12112yes—
266C2×C4×C11#13512yes—
267C2×C4×C11#13512yes—
268C2×C4×C11C22×D411yescharacteristic
269C2×C4×C11C2×#4111yescharacteristic
270C2×C4×C11C2×#4111yescharacteristic
271C2×C4×C11C2×#4111yescharacteristic
272C2×C4×C11H(5,𝔽2)11yescharacteristic
273C2×C4×C112−1+411yescharacteristic
274C2×Dic11—41no—
275C2×Dic11—41no—
276C2×Dic11—41no—
277C2×Dic11—41no—
278C2×Dic11—41no—
279C2×Dic11—41no—
280C2×Dic11—41no—
281C2×Dic11—41no—
282C2×Dic11—41no—
283C2×Dic11—41no—
284C2×Dic11—41no—
285C2×Dic11—41no—
286C2×Dic11—41no—
287C2×Dic11—41no—
288C2×Dic11—41no—
289C2×Dic11—41no—
290C2×Dic11—41no—
291C2×Dic11—41no—
292D44—22no—
293D44—22no—
294D44—41no—
295D44—41no—
296D44—41no—
297D44—41no—
298D44—41no—
299D44—41no—
300C4×D11—84no—
301Dic22—41no—
302Dic22—41no—
303Dic22—41no—
304Dic22—41no—
305C8×C11—22no—
306C8×C11—22no—
307C8×C11—22no—
308C8×C11—41no—
309C22×SD16—111no—
310C22×D8—111no—
311C2×#124—111no—
312C2×C4×Q8C2×D1111yescharacteristic
313C2×C4×D4—111no—
314#463—224no—
315C8×Q8—24no—
316C2×#107—111no—
317C2×#106—111no—
318C2×C4⋊C8C2×D1111yescharacteristic
319C2×C4⋊C8C2×D1111yescharacteristic
320C2×C4⋊C8C2×D1111yescharacteristic
321C2×#103—111no—
322C2×#103—111no—
323C2×#103—111no—
324C2×#102—111no—
325C2×#102—111no—
326C2×#102—111no—
327C2×#102—111no—
328C2×#102—111no—
329C2×C4×C8C2×D1111yescharacteristic
330C2×C4×C8C2×D1111yescharacteristic
331C2×C4×C8C2×D1111yescharacteristic
332#399—221no—
333#397—221no—
334#393—221no—
335#393—221no—
336#391—111no—
337#391—111no—
338#390—111no—
339#390—111no—
340#390—111no—
341#390—111no—
342#389—111no—
343#389—111no—
344#388—221no—
345#388—221no—
346#385—221no—
347#385—221no—
348#382—111no—
349C4×C4⋊C4—111no—
350#340—111no—
351#340—111no—
352#340—111no—
353#340—111no—
354C22×C11C2×#12111yescharacteristic
355C22×C11#54811yescharacteristic
356C22×C11#55011yescharacteristic
357C22×C11C2×#13511yescharacteristic
358C22×C11C2×#13511yescharacteristic
359C22×C11#57611yescharacteristic
360C22×C11#57711yescharacteristic
361C2×D11—42no—
362C2×D11—42no—
363C2×D11—42no—
364C2×D11—81no—
365C2×D11—81no—
366C2×D11—81no—
367C2×D11—81no—
368C2×D11—81no—
369C2×D11—81no—
370C2×D11—81no—
371C2×D11—81no—
372C2×D11—81no—
373C2×D11—81no—
374C2×D11—81no—
375C4×C11—22no—
376C4×C11—22no—
377C4×C11—22no—
378C4×C11—22no—
379C4×C11—41no—
380C4×C11—41no—
381C4×C11#52911yescharacteristic
382C4×C11#54211yescharacteristic
383C4×C11#54611yescharacteristic
384C4×C11#54811yescharacteristic
385Dic11—81no—
386Dic11—81no—
387Dic11—81no—
388Dic11—81no—
389Dic11—81no—
390Dic11—81no—
391C22×Q8—111no—
392C22×Q8D4411yescharacteristic
393C22×D4—111no—
394C22×D4—111no—
395C23×C4—221no—
396C2×SD16—222no—
397C2×SD16—222no—
398C2×SD16—441no—
399C2×D8—222no—
400C2×D8—222no—
401C2×D8—441no—
402C22×C8C22×D1111yescharacteristic
403C22×C8C22×D1111yescharacteristic
404C22×C8C22×D1111yescharacteristic
405C22×C8C22×D1111yescharacteristic
406#124—222no—
407#124—222no—
408C4×Q8—22no—
409C4×Q8—22no—
410C4×D4—222no—
411C4×D4—222no—
412C2×C4⋊C4—111no—
413C2×C4⋊C4—111no—
414C2×C4⋊C4—111no—
415C2×C4⋊C4—111no—
416C2×C4⋊C4—111no—
417C2×C4⋊C4—221no—
418C2×C4⋊C4—221no—
419C2×C4⋊C4D4411yescharacteristic
420C2×C4⋊C4D4411yescharacteristic
421C2×C4⋊C4D4411yescharacteristic
422C2×C22⋊C4—221no—
423C2×C22⋊C4—221no—
424C2×C22⋊C4—221no—
425C2×C22⋊C4—221no—
426C2×C22⋊C4—221no—
427C2×C22⋊C4—221no—
428C2×C22⋊C4—221no—
429C2×C22⋊C4—221no—
430C2×C22⋊C4—221no—
431C2×C22⋊C4—221no—
432C2×C22⋊C4—221no—
433C2×C42—221no—
434C2×C42C22×D1111yescharacteristic
435C2×C42C22×D1111yescharacteristic
436C2×C42C22×D1111yescharacteristic
437#107—441no—
438#106—441no—
439C4⋊C8—22no—
440C4⋊C8—22no—
441C4⋊C8C22⋊D1114yes—
442#103—222no—
443#103—222no—
444#103—441no—
445#102—222no—
446#102—222no—
447#102—222no—
448#102—222no—
449#102—441no—
450C4×C8—22no—
451C4×C8—22no—
452C4×C8—22no—
453#95—221no—
454#95—221no—
455#95—221no—
456#95—221no—
457#95—221no—
458#95—221no—
459#95—221no—
460#95—221no—
461#95—221no—
462C2×C11#242611yescharacteristic
463C2×C11#305011yescharacteristic
464C2×C11#306211yescharacteristic
465C2×C11#336011yescharacteristic
466C2×C11#336111yescharacteristic
467C2×C11#339411yescharacteristic
468C2×C11#339611yescharacteristic
469D11—82no—
470D11—161no—
471C24—221no—
472C24—221no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—221no—
477C2×Q8—221no—
478C2×Q8—221no—
479C2×Q8—221no—
480C2×Q8—221no—
481C2×Q8—221no—
482C2×D4—112no—
483C2×D4—221no—
484C2×D4—221no—
485C2×D4—221no—
486C2×D4—221no—
487C2×D4—221no—
488C2×D4—221no—
489C2×D4—221no—
490C2×D4—221no—
491C2×D4—221no—
492C2×D4—221no—
493C2×D4—221no—
494C22×C4—221no—
495C22×C4—221no—
496C22×C4—221no—
497C22×C4—221no—
498C22×C4—221no—
499C22×C4—221no—
500C22×C4—442no—
501C22×C4—881no—
502C22×C4—881no—
503C22×C4C2×D4411yescharacteristic
504C22×C4C2×D4411yescharacteristic
505C22×C4C2×D4411yescharacteristic
506C22×C4C2×D4411yescharacteristic
507C22×C4C2×D4411yescharacteristic
508C22×C4C2×D4411yescharacteristic
509C22×C4C23×D1111yescharacteristic
510SD16—444no—
511D8—444no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8#434312yes—
522C2×C8C2×C22⋊D1112yes—
523C2×C8C2×C22⋊D1112yes—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—221no—
528C4⋊C4—221no—
529C4⋊C4—221no—
530C4⋊C4—221no—
531C4⋊C4—221no—
532C4⋊C4—221no—
533C4⋊C4—221no—
534C4⋊C4—221no—
535C4⋊C4—221no—
536C4⋊C4—221no—
537C4⋊C4—881no—
538C4⋊C4—881no—
539C22⋊C4—442no—
540C22⋊C4—442no—
541C22⋊C4—442no—
542C22⋊C4—442no—
543C22⋊C4—442no—
544C22⋊C4—442no—
545C22⋊C4—442no—
546C22⋊C4—881no—
547C22⋊C4—881no—
548C22⋊C4—881no—
549C22⋊C4—881no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—881no—
555C42#433912yes—
556C42C2×C22⋊D1112yes—
557C11#2529611yescharacteristic, complemented, Sylow
558C23—222no—
559C23—441no—
560C23—441no—
561C23—441no—
562C23—441no—
563C23—441no—
564C23—441no—
565C23—441no—
566C23—441no—
567C23—441no—
568C23—441no—
569C23—441no—
570C23—441no—
571C23—441no—
572C23C22×D4411yescharacteristic
573Q8—28no—
574Q8—441no—
575Q8—441no—
576Q8—441no—
577Q8—441no—
578D4—222no—
579D4—222no—
580D4—441no—
581D4—441no—
582D4—441no—
583D4—441no—
584D4—441no—
585D4—441no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—441no—
601C2×C4—441no—
602C2×C4—441no—
603C2×C4—441no—
604C2×C4—441no—
605C2×C4—441no—
606C2×C4—441no—
607C2×C4—441no—
608C2×C4—441no—
609C2×C4—441no—
610C2×C4—441no—
611C2×C4—441no—
612C2×C4—441no—
613C2×C4—441no—
614C2×C4—441no—
615C2×C4—441no—
616C2×C4—441no—
617C2×C4—441no—
618C2×C4—884no—
619C2×C4#6695712yes—
620C2×C4#6695712yes—
621C2×C4#6698312yes—
622C2×C4#6698312yes—
623C2×C4C2×#433911yescharacteristic
624C2×C4C2×#434311yescharacteristic
625C2×C4#6704011yescharacteristic
626C2×C4#41×D1111yescharacteristic
627C2×C4#6704211yescharacteristic
628C2×C4C22×C22⋊D1111yescharacteristic
629C8—22no—
630C8—22no—
631C8—22no—
632C8—41no—
633C22—442no—
634C22—442no—
635C22—442no—
636C22—881no—
637C22—881no—
638C22—881no—
639C22—881no—
640C22—881no—
641C22—881no—
642C22—881no—
643C22—881no—
644C22—881no—
645C22—881no—
646C22—881no—
647C22#1062310711yescharacteristic
648C22#1062310811yescharacteristic
649C22C2×#6695711yescharacteristic
650C22C2×#6695711yescharacteristic
651C22#1062328211yescharacteristic
652C22#1062328311yescharacteristic
653C22C2×#6698311yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—881no—
661C4—881no—
662C4—881no—
663C4—881no—
664C4—881no—
665C4—881no—
666C4#1062311111yescharacteristic
667C4#1062311311yescharacteristic
668C4#1062334811yescharacteristic
669C4#1062336011yescharacteristic
670C2—882no—
671C2—1761no—
672C2#1307217611yescharacteristic
673C2#1307819711yescharacteristic
674C2#1307829611yescharacteristic
675C2#1308126711yescharacteristic
676C2#1308127811yescharacteristic
677C2#1308161411yescharacteristic
678C2#1308162911yescharacteristic
subgroupG/H
centerC23C22×D44
commutator, cosocleC22×C4×C11C24
FrattiniC22×C4C23×D11
FittingC2×C8×Q8×C11C2
socleC23×C11C22×D4
prime radicalC22×D88C22
hypercenterC2×C8×Q8D11
hypocenterC11#25296
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #13074104C2semidirect
3. #13073974C2semidirect
4. #13070010C2semidirect
5. #13069989C2semidirect
6. #13069934C2semidirect
9. #13069470C2semidirect
10. #13069427C2semidirect
11. #13069386C2semidirect
12. #13069299C2semidirect
13. #13069292C2semidirect
14. C4×#66885C2semidirect
15. C2×C8×Q8×C11C2semidirect
220. C2×C8×Q8D11semidirect
557. C11#25296semidirect

Maximal subgroups

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

subgroupindex
C2×#106224572
#130741042
#130739742
#130700102
#130699892
#130699342
#130699272
#130694822
#130694702
#130694272
#130693862
#130692992
#130692922
C4×#668852
C2×C8×Q8×C112
#2529611