Finite Group Databasebeta

← #13355185

Subgroups — #13355185

Order 1792.

2046 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C2×#41×Dic7C211yesmaximal, characteristic, complemented
3C2×#90004C211yesmaximal, characteristic, complemented
4C2×#89993C211yesmaximal, characteristic, complemented
5C2×#89987C211yesmaximal, characteristic, complemented
6#11729802C214yesmaximal, complemented
7#11729791C214yesmaximal, complemented
8#11729739C211yesmaximal, characteristic, complemented
9#11729733C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×D4C2211yescharacteristic
12C2×C7×AGL(1,ℤ/8)—24no—
13C2×C7×AGL(1,ℤ/8)—24no—
14C2×C7×AGL(1,ℤ/8)C418yescomplemented
15C22×C7×SD16C2212yes—
16C22×C7×D8C2212yes—
17C22×C7×M16C2211yescharacteristic
18C22×C22⋊Dic7—22no—
19#41×Dic7—24no—
20#41×Dic7—24no—
21C2×Q8×Dic7C2212yescomplemented
22C2×D4×Dic7—22no—
23C2×D4×Dic7C2212yescomplemented
24C2×#6489—21no—
25C2×#6489C2211yescharacteristic
26C22×C4⋊Dic7C2211yescharacteristic
27C22×C4×Dic7—21no—
28C22×C4×Dic7C2211yescharacteristic
29C22×#1136C2211yescharacteristic
30#90004—24no—
31C2×#6409C2212yescomplemented
32#89993—21no—
33#89993—21no—
34#89993C2214yes—
35#89987—24no—
36C2×#6406C2214yes—
37C2×#6406C2212yescomplemented
38#89346C2214yescomplemented
39#89345C2214yescomplemented
40#20928—71nomaximal, Sylow
41C25×C7—21no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41C2×C414yes—
46C2×C7×#41D414yes—
47C2×C7×#41D414yes—
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—22no—
53C22×C7×D4—21no—
54C22×C7×D4—22no—
55C22×C7×D4—22no—
56C22×C7×D4C2×C414yescomplemented
57C22×C7×D4C2312yes—
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—22no—
62C23×Dic7—21no—
63C23×Dic7D412yes—
64C7×AGL(1,ℤ/8)—232no—
65C2×C7×SD16—24no—
66C2×C7×SD16—24no—
67C2×C7×SD16C2×C418yescomplemented
68C2×C7×D8—24no—
69C2×C7×D8—24no—
70C2×C7×D8C2×C418yescomplemented
71C2×C7×M16—22no—
72C2×C7×M16—22no—
73C2×C7×M16C2×C414yescomplemented
74C22×C7×C8C2312yes—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—24no—
78C2×C22⋊Dic7—24no—
79C2×C22⋊Dic7—48no—
80Q8×Dic7—24no—
81Q8×Dic7—24no—
82D4×Dic7—216no—
83D4×Dic7—24no—
84D4×Dic7—24no—
85#6489—28no—
86#6489—22no—
87#6489D414yes—
88C2×C4⋊Dic7—22no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94C2×C4×Dic7—28no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—22no—
97C2×C4×Dic7—22no—
98C2×C4×Dic7—22no—
99C2×C4×Dic7D411yescharacteristic
100C2×C4×Dic7D411yescharacteristic
101C2×C4×Dic7C2312yes—
102C2×#1136—22no—
103C2×#1136—22no—
104C2×#1136—22no—
105C22×C7⋊C8C2312yes—
106#6409—24no—
107#6406—24no—
108#6406—28no—
109C22×AGL(1,ℤ/8)D711yescharacteristic
110C2×C4×#41—71no—
111C2×#417—71no—
112C2×#418—71no—
113C2×#390—71no—
114#1880—74no—
115#1879—74no—
116#1736—71no—
117#1728—71no—
118C24×C7—21no—
119C24×C7—24no—
120C24×C7—22no—
121C24×C7—44no—
122C24×C7—44no—
123C24×C7—42no—
124C24×C7—42no—
125C24×C7C2×D411yescharacteristic
126C7×#41—28no—
127C7×#41—28no—
128C7×#41—28no—
129C7×#41C22⋊C4116yescomplemented
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8C22×C412yes—
134C2×C7×Q8C2×D412yes—
135C2×C7×Q8C2×D412yes—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—24no—
139C2×C7×D4—28no—
140C2×C7×D4—28no—
141C2×C7×D4—24no—
142C2×C7×D4—21no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—22no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—22no—
149C2×C7×D4—22no—
150C2×C7×D4—42no—
151C2×C7×D4—44no—
152C2×C7×D4—42no—
153C2×C7×D4—44no—
154C2×C7×D4C22×C414yes—
155C2×C7×D4C22×C412yes—
156C2×C7×D4C2×D412yes—
157C2×C7×D4C2×D412yes—
158C22×C4×C7—21no—
159C22×C4×C7—24no—
160C22×C4×C7—24no—
161C22×C4×C7—24no—
162C22×C4×C7C22×C412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D412yes—
167C22×C4×C7#4112yes—
168C22×C4×C7#4112yes—
169C22×C4×C7C2411yescharacteristic
170C22×Dic7—28no—
171C22×Dic7—24no—
172C22×Dic7—22no—
173C22×Dic7—24no—
174C22×Dic7—22no—
175C22×Dic7—42no—
176C22×Dic7—44no—
177C22×Dic7—44no—
178C22×Dic7—44no—
179C22×Dic7—42no—
180C22×Dic7C2×D412yes—
181C22×Dic7C2×D412yes—
182C7×SD16—216no—
183C7×D8—216no—
184C7×M16—28no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C2×C7×C8C22×C414yes—
188C22⋊Dic7—28no—
189C22⋊Dic7—24no—
190C22⋊Dic7—48no—
191C22⋊Dic7—48no—
192C4⋊Dic7—28no—
193C4⋊Dic7—22no—
194C4⋊Dic7—22no—
195C4⋊Dic7—22no—
196C4⋊Dic7—42no—
197C4⋊Dic7C2×D414yes—
198C4×Dic7—28no—
199C4×Dic7—22no—
200C4×Dic7—24no—
201C4×Dic7—22no—
202C4×Dic7—22no—
203C4×Dic7—22no—
204#1136—44no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C22×#41C2×D711yescharacteristic
209C23×D4C2×D711yescharacteristic
210C2×AGL(1,ℤ/8)—24no—
211C2×AGL(1,ℤ/8)—24no—
212C2×AGL(1,ℤ/8)Dic718yescomplemented
213C22×SD16C2×D712yes—
214C22×D8C2×D712yes—
215C22×M16—71no—
216C22×M16C2×D711yescharacteristic
217C4×#41—144no—
218C4×#41—144no—
219C2×C4×Q8—72no—
220C2×C4×D4—72no—
221C2×C4×D4—142no—
222C2×#117—71no—
223C2×#117—141no—
224C22×C4⋊C4—71no—
225C22×C22⋊C4—142no—
226C22×C42—71no—
227C22×C42—141no—
228#418—144no—
229#417—144no—
230C2×#103—72no—
231C2×#102—74no—
232C2×#102—72no—
233#390—74no—
234#390—141no—
235#390—141no—
236#340—74no—
237#336—74no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—24no—
244C23×C7—416no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—44no—
250C23×C7—44no—
251C23×C7—48no—
252C23×C7—88no—
253C23×C7—84no—
254C23×C7—84no—
255C23×C7C4×D412yes—
256C23×C7#12012yes—
257C23×C7#12112yes—
258C23×C7C22×D411yescharacteristic
259C7×Q8—22no—
260C7×Q8—22no—
261C7×Q8—22no—
262C7×Q8C2×C22⋊C414yescomplemented
263C7×D4—216no—
264C7×D4—24no—
265C7×D4—22no—
266C7×D4—24no—
267C7×D4—22no—
268C7×D4—22no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—48no—
272C7×D4C2×C22⋊C414yescomplemented
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—216no—
277C2×C4×C7C2×C22⋊C414yes—
278C2×C4×C7C4×D414yes—
279C2×C4×C7C4×D412yes—
280C2×C4×C7#12012yes—
281C2×C4×C7#12114yes—
282C2×C4×C7#12112yes—
283C2×C4×C7#12314yes—
284C2×C4×C7C23×C411yescharacteristic
285C2×C4×C7C22×D411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—24no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—48no—
297C2×Dic7—416no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—44no—
302C2×Dic7—84no—
303C2×Dic7AGL(1,ℤ/8)14yescomplemented
304C7×C8—24no—
305C7⋊C8—42no—
306C25—21no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41C2×Dic714yes—
311C2×#41C22⋊D714yes—
312C2×#41C22⋊D714yes—
313C22×Q8C22×D711yescharacteristic
314C22×D4—21no—
315C22×D4—22no—
316C22×D4—21no—
317C22×D4—22no—
318C22×D4—22no—
319C22×D4—22no—
320C22×D4—22no—
321C22×D4C2×Dic714yescomplemented
322C22×D4C22×D712yes—
323C22×D4C22×D711yescharacteristic
324C23×C4—21no—
325C23×C4—72no—
326C23×C4—142no—
327C23×C4—141no—
328C23×C4C22×D711yescharacteristic
329AGL(1,ℤ/8)—232no—
330C2×SD16—24no—
331C2×SD16—24no—
332C2×SD16C2×Dic718yescomplemented
333C2×D8—24no—
334C2×D8—24no—
335C2×D8C2×Dic718yescomplemented
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—142no—
339C2×M16—142no—
340C2×M16—142no—
341C2×M16C2×Dic714yescomplemented
342C22×C8—72no—
343C22×C8C22×D712yes—
344C4×Q8—144no—
345C4×Q8—144no—
346C4×D4—144no—
347C4×D4—1416no—
348C4×D4—144no—
349#117—74no—
350#117—148no—
351#117—142no—
352C2×C4⋊C4—72no—
353C2×C4⋊C4—72no—
354C2×C4⋊C4—142no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4—141no—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—144no—
362C2×C22⋊C4—288no—
363C2×C42—71no—
364C2×C42—71no—
365C2×C42—72no—
366C2×C42—142no—
367C2×C42—148no—
368C2×C42—141no—
369C2×C42—142no—
370C2×C42—142no—
371#103—144no—
372#102—148no—
373#102—144no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—24no—
379C22×C7—44no—
380C22×C7—44no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—48no—
384C22×C7—44no—
385C22×C7—816no—
386C22×C7—84no—
387C22×C7—82no—
388C22×C7—82no—
389C22×C7#38614yes—
390C22×C7C2×C4×D411yescharacteristic
391C22×C7C2×#12011yescharacteristic
392C22×C7C2×#12111yescharacteristic
393C22×C7C2×AGL(1,ℤ/8)12yes—
394C22×C7#57512yes—
395C4×C7—24no—
396C4×C7#38614yes—
397C4×C7C22×C22⋊C411yescharacteristic
398C4×C7C2×C4×D411yescharacteristic
399C4×C7C2×#12111yescharacteristic
400C4×C7C2×#12311yescharacteristic
401Dic7—24no—
402Dic7—42no—
403Dic7—44no—
404Dic7—82no—
405C24—21no—
406C24—24no—
407C24—22no—
408C24—44no—
409C24—44no—
410C24—42no—
411C24—42no—
412C24D4×D711yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41C22⋊Dic7116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C22×Dic712yes—
421C2×Q8C2×C22⋊D712yes—
422C2×Q8C2×C22⋊D712yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—28no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4C22×Dic714yes—
442C2×D4C22×Dic712yes—
443C2×D4C2×C22⋊D712yes—
444C2×D4C2×C22⋊D712yes—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—72no—
450C22×C4—72no—
451C22×C4—144no—
452C22×C4—148no—
453C22×C4—144no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—284no—
457C22×C4—282no—
458C22×C4—284no—
459C22×C4—284no—
460C22×C4—282no—
461C22×C4D4×D711yescharacteristic
462C22×C4D4×D712yes—
463C22×C4#115912yes—
464C22×C4#115912yes—
465C22×C4C22×Dic712yes—
466C22×C4C2×C22⋊D712yes—
467C22×C4C2×C22⋊D712yes—
468C22×C4C23×D711yescharacteristic
469SD16—216no—
470D8—216no—
471M16—28no—
472M16—284no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—142no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8C22×Dic714yes—
479C4⋊C4—74no—
480C4⋊C4—148no—
481C4⋊C4—142no—
482C4⋊C4—142no—
483C4⋊C4—142no—
484C4⋊C4—282no—
485C22⋊C4—148no—
486C22⋊C4—144no—
487C22⋊C4—288no—
488C22⋊C4—288no—
489C42—148no—
490C42—142no—
491C42—142no—
492C42—144no—
493C42—142no—
494C42—142no—
495C2×C7—24no—
496C2×C7—44no—
497C2×C7—84no—
498C2×C7C2×#38611yescharacteristic
499C2×C7C4×AGL(1,ℤ/8)12yes—
500C2×C7#300612yes—
501C2×C7#303811yescharacteristic
502C2×C7#304011yescharacteristic
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—416no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—48no—
517C23—84no—
518C23—84no—
519C23—88no—
520C23D4×Dic712yes—
521C23#650012yes—
522C23#650212yes—
523C23C2×D4×D711yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8C2×C22⋊Dic714yescomplemented
528D4—216no—
529D4—22no—
530D4—24no—
531D4—22no—
532D4—22no—
533D4—24no—
534D4—42no—
535D4—42no—
536D4—48no—
537D4C2×C22⋊Dic714yescomplemented
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—216no—
542C2×C4—74no—
543C2×C4—142no—
544C2×C4—142no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—144no—
548C2×C4—142no—
549C2×C4—288no—
550C2×C4—2816no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—284no—
554C2×C4—282no—
555C2×C4—564no—
556C2×C4D4×Dic714yes—
557C2×C4D4×Dic712yes—
558C2×C4#649814yes—
559C2×C4#650012yes—
560C2×C4#650214yes—
561C2×C4#650212yes—
562C2×C4C2×C22⋊Dic714yes—
563C2×C4C2×D4×D711yescharacteristic
564C2×C4C2×#115911yescharacteristic
565C2×C4C2×#115911yescharacteristic
566C2×C4C23×Dic711yescharacteristic
567C2×C4C22×C22⋊D711yescharacteristic
568C2×C4C22×C22⋊D711yescharacteristic
569C8—24no—
570C8—282no—
571C7#2092811yescharacteristic, complemented, Sylow
572C22—24no—
573C22—22no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—48no—
583C22—816no—
584C22—82no—
585C22—82no—
586C22—84no—
587C22#8999214yes—
588C22D7×AGL(1,ℤ/8)12yes—
589C22#9046412yes—
590C22C2×D4×Dic711yescharacteristic
591C22C2×#650011yescharacteristic
592C22C2×#650211yescharacteristic
593C4—24no—
594C4—144no—
595C4—282no—
596C4—284no—
597C4—562no—
598C4#8999214yes—
599C4C2×D4×Dic711yescharacteristic
600C4C2×#649811yescharacteristic
601C4C2×#650211yescharacteristic
602C4C22×C22⋊Dic711yescharacteristic
603C2—24no—
604C2—44no—
605C2—84no—
606C2Dic7×AGL(1,ℤ/8)12yes—
607C2#1174137711yescharacteristic
608C2#1174137811yescharacteristic
609C2#1174139112yes—
610C2C2×#8999211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC22×C7×AGL(1,ℤ/8)C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC22×AGL(1,ℤ/8)D7
hypocenterC7#20928
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. C2×#41×Dic7C2semidirect
3. C2×#90004C2semidirect
4. C2×#89993C2semidirect
5. C2×#89987C2semidirect
6. #11729802C2semidirect
7. #11729791C2semidirect
8. #11729739C2semidirect
9. #11729733C2semidirect
14. C2×C7×AGL(1,ℤ/8)C4semidirect
21. C2×Q8×Dic7C22semidirect
23. C2×D4×Dic7C22semidirect
31. C2×#6409C22semidirect
37. C2×#6406C22semidirect
38. #89346C22semidirect
39. #89345C22semidirect
56. C22×C7×D4C2×C4semidirect
67. C2×C7×SD16C2×C4semidirect
70. C2×C7×D8C2×C4semidirect
73. C2×C7×M16C2×C4semidirect
129. C7×#41C22⋊C4semidirect
212. C2×AGL(1,ℤ/8)Dic7semidirect
262. C7×Q8C2×C22⋊C4semidirect
272. C7×D4C2×C22⋊C4semidirect
303. C2×Dic7AGL(1,ℤ/8)semidirect
321. C22×D4C2×Dic7semidirect
332. C2×SD16C2×Dic7semidirect
335. C2×D8C2×Dic7semidirect
341. C2×M16C2×Dic7semidirect
416. #41C22⋊Dic7semidirect
527. Q8C2×C22⋊Dic7semidirect
537. D4C2×C22⋊Dic7semidirect
571. C7#20928semidirect

Maximal subgroups

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

subgroupindex
C22×C7×AGL(1,ℤ/8)2
C2×#41×Dic72
C2×#900042
C2×#899932
C2×#899872
#117298022
#117297912
#117297392
#117297332
#209287