Finite Group Databasebeta

← #33097155

Subgroups — #33097155

Order 18688.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic73C211yesmaximal, characteristic, complemented
2C2×#17988490C211yesmaximal, characteristic
3#22610787C212yesmaximal, complemented
4#22610616C212yesmaximal, complemented
5#22610560C212yesmaximal, complemented
6#22610288C212yesmaximal, complemented
7#22610239C211yesmaximal, characteristic, complemented
8#22610126C211yesmaximal, characteristic, complemented
9#22609862C211yesmaximal, characteristic
10C22⋊C8×Dic73C211yesmaximal, characteristic, complemented
11C2×#450×C73C211yesmaximal, characteristic
12C2×Q8×Dic73C2211yescharacteristic, complemented
13C22×C4⋊Dic73C2211yescharacteristic
14C22×C22⋊Dic73—21no—
15C22×C22⋊Dic73—21no—
16C22×C4×Dic73C2211yescharacteristic
17C2×#14556523C2212yescomplemented
18C2×#14556515C2212yescomplemented
19C2×#14556512C2212yes—
20C2×#14556512C2212yes—
21C2×C22⋊#10596831C2211yescharacteristic
22C2×C8×Dic73C2212yescomplemented
23#17988494C2211yescharacteristic, complemented
24#17988490—21no—
25#17988490—21no—
26#17988490C2212yes—
27#17988490C2212yes—
28#17988486C2211yescharacteristic, complemented
29#17988473—21no—
30#17988462—21no—
31#17988447C2211yescharacteristic
32C4⋊C4×Dic73—21no—
33C4⋊C4×Dic73C2211yescharacteristic, complemented
34C22⋊C4×Dic73—21no—
35#17988350C2212yescomplemented
36#17988349C2212yescomplemented
37#17988341C2212yescomplemented
38C23×D4×C73C2211yescharacteristic
39C22×SD16×C73C2212yes—
40C2×C22⋊Q8×C73C2211yescharacteristic
41#450×C73C418yescomplemented
42C2×#102×C73C2212yes—
43C2×C22⋊C8×C73C2211yescharacteristic
44C23×Dic73—21no—
45C23×Dic73D411yescharacteristic
46C23×Dic73D411yescharacteristic
47C22×#10596831C2312yes—
48Q8×Dic73—42no—
49C2×C4⋊Dic73—21no—
50C2×C4⋊Dic73—21no—
51C2×C4⋊Dic73—21no—
52C2×C4⋊Dic73—42no—
53C2×C4⋊Dic73C2311yescharacteristic
54C2×C4⋊Dic73C2312yes—
55C2×C4⋊Dic73C2311yescharacteristic
56C2×C22⋊Dic73—22no—
57C2×C22⋊Dic73—22no—
58C2×C22⋊Dic73—22no—
59C2×C22⋊Dic73—22no—
60C2×C22⋊Dic73—21no—
61C2×C22⋊Dic73—44no—
62C2×C22⋊Dic73—44no—
63C2×C4×Dic73—21no—
64C2×C4×Dic73—21no—
65C2×C4×Dic73—42no—
66C2×C4×Dic73D411yescharacteristic
67C2×C4×Dic73D411yescharacteristic
68C2×C4×Dic73C2311yescharacteristic
69C2×C4×Dic73C2311yescharacteristic
70#14556523—42no—
71#14556515—42no—
72#14556512—42no—
73#14556512—42no—
74C22⋊#10596831—22no—
75C8×Dic73—42no—
76#14556498—21no—
77#14556498—21no—
78#14556498—21no—
79#14556498—21no—
80#14556498—21no—
81C25×C73—21no—
82C22×Q8×C73C2311yescharacteristic
83C22×D4×C73—21no—
84C22×D4×C73—22no—
85C22×D4×C73—42no—
86C22×D4×C73—42no—
87C22×D4×C73C2×C414yes—
88C22×D4×C73C2312yes—
89C23×C4×C73C2311yescharacteristic
90C2×SD16×C73—42no—
91C2×SD16×C73—42no—
92C2×SD16×C73C2×C418yescomplemented
93C22×C8×C73C2312yes—
94C22⋊Q8×C73—22no—
95C22⋊Q8×C73C2×C414yes—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73C2311yescharacteristic
98C2×C22⋊C4×C73—21no—
99#102×C73C2×C418yescomplemented
100C22⋊C8×C73C2×C414yes—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—21no—
104C22×Dic73—21no—
105C22×Dic73—22no—
106C22×Dic73—41no—
107C22×Dic73—41no—
108C22×Dic73—41no—
109C22×Dic73—42no—
110C22×Dic73—42no—
111C22×Dic73—42no—
112C22×Dic73SD1614yescomplemented
113C22×Dic73C2×D411yescharacteristic
114C22×Dic73C2×D411yescharacteristic
115C22×Dic73C2×D411yescharacteristic
116C22×Dic73C2×D411yescharacteristic
117C2×#10596831—22no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C4⋊Dic73—22no—
121C4⋊Dic73—22no—
122C4⋊Dic73—22no—
123C4⋊Dic73—22no—
124C4⋊Dic73—42no—
125C4⋊Dic73—42no—
126C4⋊Dic73—42no—
127C22⋊Dic73—44no—
128C22⋊Dic73—42no—
129C22⋊Dic73—44no—
130C22⋊Dic73—44no—
131C22⋊Dic73—44no—
132C4×Dic73—22no—
133C4×Dic73—22no—
134C4×Dic73—41no—
135C4×Dic73—41no—
136C4×Dic73—42no—
137C4×Dic73—42no—
138C24×C73—24no—
139C24×C73—22no—
140C24×C73—42no—
141C24×C73—42no—
142C24×C73—42no—
143C24×C73—42no—
144C24×C73—42no—
145C24×C73—42no—
146C24×C73C2×D411yescharacteristic
147C2×Q8×C73—21no—
148C2×Q8×C73—22no—
149C2×Q8×C73—22no—
150C2×Q8×C73C22×C412yes—
151C2×D4×C73—24no—
152C2×D4×C73—22no—
153C2×D4×C73—22no—
154C2×D4×C73—22no—
155C2×D4×C73—22no—
156C2×D4×C73—22no—
157C2×D4×C73—48no—
158C2×D4×C73—42no—
159C2×D4×C73—42no—
160C2×D4×C73—48no—
161C2×D4×C73C22×C414yes—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73C22×C412yes—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73#4112yes—
169C22×C4×C73C2411yescharacteristic
170SD16×C73—48no—
171C2×C8×C73—22no—
172C2×C8×C73—22no—
173C2×C8×C73C22×C414yes—
174C4⋊C4×C73—24no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73C22×C412yes—
177C22⋊C4×C73—24no—
178C23×C73—21no—
179C23×C73—21no—
180C23×C73—21no—
181C23×C73—21no—
182C23×C73—24no—
183C23×C73—48no—
184C23×C73—48no—
185C23×C73—42no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73—42no—
189C23×C73—42no—
190C23×C73—42no—
191C23×C73—48no—
192C23×C73—84no—
193C23×C73—84no—
194C23×C73—84no—
195C23×C73—84no—
196C23×C73C4×D412yes—
197C23×C73C2×SD1612yes—
198C23×C73#13512yes—
199C23×C73C22×D411yescharacteristic
200Q8×C73—24no—
201Q8×C73—42no—
202D4×C73—24no—
203D4×C73—24no—
204D4×C73—48no—
205D4×C73—42no—
206D4×C73—42no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73C4×D412yes—
218C2×C4×C73C23×C411yescharacteristic
219C2×C4×C73C2×#4111yescharacteristic
220C2×C4×C73H(5,𝔽2)12yes—
221C2×C4×C73H(5,𝔽2)12yes—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—44no—
232C2×Dic73—44no—
233C2×Dic73—41no—
234C2×Dic73—42no—
235C2×Dic73—41no—
236C2×Dic73—42no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—42no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—84no—
243C2×Dic73C2×SD1612yes—
244C2×Dic73C2×SD1612yes—
245C2×Dic73AGL(1,ℤ/8)12yes—
246C8×C73—24no—
247#10596831—42no—
248C22×C73—22no—
249C22×C73—22no—
250C22×C73—22no—
251C22×C73—22no—
252C22×C73—22no—
253C22×C73—22no—
254C22×C73—44no—
255C22×C73—44no—
256C22×C73—44no—
257C22×C73—816no—
258C22×C73—82no—
259C22×C73—82no—
260C22×C73—82no—
261C22×C73—82no—
262C22×C73C4×SD1614yes—
263C22×C73C2×C4×D411yescharacteristic
264C22×C73#53411yescharacteristic
265C22×C73#53511yescharacteristic
266C22×C73C22×SD1611yescharacteristic
267C22×C73C2×#13511yescharacteristic
268C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C73#57511yescharacteristic
270C4×C73—22no—
271C4×C73—42no—
272C4×C73—42no—
273C4×C73#51812yes—
274C4×C73#53812yes—
275Dic73—22no—
276Dic73—22no—
277Dic73—22no—
278Dic73—41no—
279Dic73—81no—
280Dic73—82no—
281Dic73—81no—
282#22043—731nomaximal, Sylow
283C2×C73—24no—
284C2×C73—84no—
285C2×C73#232111yescharacteristic
286C2×C73C2×C4×SD1611yescharacteristic
287C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
288C2×C73#318711yescharacteristic
289C2×C73#318811yescharacteristic
290C2×C73#320811yescharacteristic
291C2×C73#321111yescharacteristic
292C2×#450D7311yescharacteristic
293C4×C22⋊Q8—731no—
294C2×#390—731no—
295#1968—732no—
296#1934—732no—
297#1926—732no—
298#1880—732no—
299#1872—731no—
300#1853—731no—
301#1786—731no—
302C4×C22⋊C8—731no—
303C73#2204311yescharacteristic, complemented, Sylow
304C23×D4C2×D7311yescharacteristic
305C22×SD16C2×D7312yes—
306C2×C22⋊Q8C2×D7311yescharacteristic
307C2×C4×Q8—731no—
308C22×C4⋊C4—731no—
309C22×C22⋊C4—1461no—
310C22×C22⋊C4—1461no—
311C22×C42—731no—
312#450Dic7318yescomplemented
313C2×#106—732no—
314C2×#103—732no—
315C2×#102—732no—
316C2×#102—732no—
317C2×#102C2×D7312yes—
318C2×C22⋊C8—731no—
319C2×C22⋊C8C2×D7311yescharacteristic
320C2×C4×C8—732no—
321#391—731no—
322#390—732no—
323#390—732no—
324#390—1461no—
325#390—1461no—
326#389—731no—
327#387—1461no—
328#385—1461no—
329#380—731no—
330C4×C4⋊C4—731no—
331C4×C4⋊C4—1461no—
332C4×C22⋊C4—1461no—
333#340—732no—
334#340—732no—
335#336—732no—
336C25—21no—
337C22×Q8C22×D7311yescharacteristic
338C22×D4—21no—
339C22×D4—22no—
340C22×D4—42no—
341C22×D4—42no—
342C22×D4C2×Dic7314yes—
343C22×D4C22×D7312yes—
344C23×C4—731no—
345C23×C4—731no—
346C23×C4—1461no—
347C23×C4C22×D7311yescharacteristic
348C2×SD16—42no—
349C2×SD16—42no—
350C2×SD16C2×Dic7318yescomplemented
351C22×C8—732no—
352C22×C8C22×D7312yes—
353C22⋊Q8—22no—
354C22⋊Q8C2×Dic7314yes—
355C4×Q8—2922no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—2922no—
358C2×C4⋊C4—732no—
359C2×C4⋊C4—731no—
360C2×C4⋊C4—731no—
361C2×C4⋊C4—1461no—
362C2×C4⋊C4—1461no—
363C2×C4⋊C4—1461no—
364C2×C4⋊C4C22×D7311yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—2924no—
367C2×C22⋊C4—2924no—
368C2×C22⋊C4—1462no—
369C2×C22⋊C4—1462no—
370C2×C22⋊C4—1462no—
371C2×C22⋊C4—1462no—
372C2×C22⋊C4—1461no—
373C2×C42—2922no—
374C2×C42—731no—
375C2×C42—731no—
376C2×C42—731no—
377C2×C42—731no—
378C2×C42—1461no—
379C2×C42—1461no—
380#106—2922no—
381#103—2922no—
382#102—2922no—
383#102—2922no—
384#102C2×Dic7318yescomplemented
385C22⋊C8—1462no—
386C22⋊C8C2×Dic7314yes—
387C4×C8—2922no—
388#95—1461no—
389#95—1461no—
390#95—1461no—
391#95—1461no—
392#95—1461no—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24D4×D7311yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C22×Dic7312yes—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4C22×Dic7314yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—2921no—
422C22×C4—2921no—
423C22×C4—2921no—
424C22×C4—2922no—
425C22×C4—2922no—
426C22×C4—2922no—
427C22×C4—734no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1462no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4C22×Dic7312yes—
438C22×C4D4×D7311yescharacteristic
439C22×C4#1192402712yes—
440C22×C4C23×D7311yescharacteristic
441SD16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—1462no—
445C2×C8—1462no—
446C2×C8—1462no—
447C2×C8C22×Dic7314yes—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—2922no—
451C4⋊C4—2922no—
452C4⋊C4—2922no—
453C4⋊C4—1462no—
454C4⋊C4—1462no—
455C4⋊C4—1462no—
456C4⋊C4—1462no—
457C4⋊C4C22×Dic7312yes—
458C22⋊C4—24no—
459C22⋊C4—2922no—
460C22⋊C4—2924no—
461C22⋊C4—2924no—
462C22⋊C4—2924no—
463C22⋊C4—2924no—
464C42—2921no—
465C42—2921no—
466C42—2922no—
467C42—2922no—
468C42—1462no—
469C42—1462no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—24no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—48no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23D4×Dic7312yes—
489C23SD16×D7312yes—
490C23#1455661212yes—
491C23C2×D4×D7311yescharacteristic
492Q8—24no—
493Q8—42no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—2924no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2922no—
513C2×C4—2921no—
514C2×C4—2922no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2922no—
518C2×C4—2921no—
519C2×C4—2924no—
520C2×C4—5844no—
521C2×C4—732no—
522C2×C4—732no—
523C2×C4—732no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4D4×Dic7312yes—
534C2×C4C23×Dic7311yescharacteristic
535C2×C4C2×#1192402711yescharacteristic
536C2×C4#1455663612yes—
537C2×C4#1455663612yes—
538C8—24no—
539C8—2922no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22SD16×Dic7314yes—
555C22C2×D4×Dic7311yescharacteristic
556C22#1798914711yescharacteristic
557C22#1798915211yescharacteristic
558C22C2×SD16×D7311yescharacteristic
559C22C2×#1455661211yescharacteristic
560C22AGL(1,ℤ/8)×D7311yescharacteristic
561C22#1798938311yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—2921no—
566C4—5841no—
567C4—5842no—
568C4—5841no—
569C4—1462no—
570C4—1462no—
571C4—1462no—
572C4#1798906212yes—
573C4#1798917212yes—
574C2—24no—
575C2—84no—
576C2#2261259211yescharacteristic
577C2C2×SD16×Dic7311yescharacteristic
578C2AGL(1,ℤ/8)×Dic7311yescharacteristic
579C2#2262068511yescharacteristic
580C2#2262069911yescharacteristic
581C2#2262087011yescharacteristic
582C2#2262089911yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#450×C73C2
socleC23×C73C22×D4
prime radicalC23×D4×C73C22
cosocleC22×C4×C73C24
hypercenterC2×#450D73
hypocenterC73#22043
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×Dic73C2semidirect
3. #22610787C2semidirect
4. #22610616C2semidirect
5. #22610560C2semidirect
6. #22610288C2semidirect
7. #22610239C2semidirect
8. #22610126C2semidirect
10. C22⋊C8×Dic73C2semidirect
12. C2×Q8×Dic73C22semidirect
17. C2×#14556523C22semidirect
18. C2×#14556515C22semidirect
22. C2×C8×Dic73C22semidirect
23. #17988494C22semidirect
28. #17988486C22semidirect
33. C4⋊C4×Dic73C22semidirect
35. #17988350C22semidirect
36. #17988349C22semidirect
37. #17988341C22semidirect
41. #450×C73C4semidirect
92. C2×SD16×C73C2×C4semidirect
99. #102×C73C2×C4semidirect
112. C22×Dic73SD16semidirect
303. C73#22043semidirect
312. #450Dic73semidirect
350. C2×SD16C2×Dic73semidirect
384. #102C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic732
C2×#179884902
#226107872
#226106162
#226105602
#226102882
#226102392
#226101262
#226098622
C22⋊C8×Dic732
C2×#450×C732
#2204373