Finite Group Databasebeta

← #33091650

Subgroups — #33091650

Order 18688.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic73C211yesmaximal, characteristic, complemented
2C2×#17988688C212yesmaximal, complemented
3#121×Dic73C211yesmaximal, characteristic
4#22610288C212yesmaximal, complemented
5#22610249C211yesmaximal, characteristic
6#22610187C212yesmaximal, complemented
7#22610121C211yesmaximal, characteristic, complemented
8#22609866C211yesmaximal, characteristic, complemented
9#22609763C212yesmaximal, complemented
10#22609695C211yesmaximal, characteristic, complemented
11C2×#448×C73C211yesmaximal, characteristic
12C2×Q8×Dic73—28no—
13C2×Q8×Dic73—21no—
14C2×Q8×Dic73—21no—
15C2×Q8×Dic73C2212yes—
16C2×Q8×Dic73C2212yes—
17C2×D4×Dic73—21no—
18C2×D4×Dic73C2211yescharacteristic
19C22×C4⋊Dic73—21no—
20C22×C4⋊Dic73C2211yescharacteristic
21C22×C4×Dic73—21no—
22C22×C4×Dic73C2211yescharacteristic
23#17988688—28no—
24C2×#14556525C2212yes—
25C2×#14556515C2212yes—
26C2×#14556515C2212yes—
27C2×#14556512C2212yes—
28C2×C22⋊#10596831C2211yescharacteristic
29C2×#14556501C2212yes—
30#17988490C2211yescharacteristic
31#17988486C2211yescharacteristic
32#17988480—21no—
33#17988467—21no—
34#17988447C2211yescharacteristic
35C4⋊C4×Dic73C2211yescharacteristic
36C22⋊C4×Dic73—21no—
37#17988350C2212yes—
38#17988349C2212yes—
39#17988341C2212yes—
40C23×Q8×C73C2211yescharacteristic
41C22×SD16×C73C2212yes—
42C2×#121×C73C2211yescharacteristic
43#448×C73C418yescomplemented
44C2×#103×C73C2212yes—
45C2×C22⋊C8×C73C2211yescharacteristic
46C23×Dic73—21no—
47C23×Dic73—21no—
48C23×Dic73D411yescharacteristic
49C23×Dic73D411yescharacteristic
50C22×#10596831C2312yes—
51Q8×Dic73—28no—
52Q8×Dic73—28no—
53Q8×Dic73—24no—
54Q8×Dic73—24no—
55Q8×Dic73—24no—
56Q8×Dic73D418yescomplemented
57D4×Dic73—24no—
58D4×Dic73—44no—
59C2×C4⋊Dic73—28no—
60C2×C4⋊Dic73—22no—
61C2×C4⋊Dic73—22no—
62C2×C4⋊Dic73—24no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2312yes—
68C2×C22⋊Dic73—21no—
69C2×C22⋊Dic73—21no—
70C2×C22⋊Dic73—21no—
71C2×C22⋊Dic73—21no—
72C2×C4×Dic73—28no—
73C2×C4×Dic73—22no—
74C2×C4×Dic73—22no—
75C2×C4×Dic73—24no—
76C2×C4×Dic73—21no—
77C2×C4×Dic73D411yescharacteristic
78C2×C4×Dic73D411yescharacteristic
79C2×C4×Dic73C2311yescharacteristic
80C2×C4×Dic73C2311yescharacteristic
81#14556525—24no—
82#14556515—24no—
83#14556515—24no—
84#14556512—24no—
85C22⋊#10596831—22no—
86#14556501—24no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90C22×Q8×C73—24no—
91C22×Q8×C73—24no—
92C22×Q8×C73—21no—
93C22×Q8×C73—22no—
94C22×Q8×C73C2×C414yescomplemented
95C22×Q8×C73C2312yes—
96C22×D4×C73—21no—
97C22×D4×C73C2311yescharacteristic
98C23×C4×C73—21no—
99C23×C4×C73C2311yescharacteristic
100C2×SD16×C73—24no—
101C2×SD16×C73—24no—
102C2×SD16×C73C2×C418yescomplemented
103C22×C8×C73C2312yes—
104#121×C73—22no—
105#121×C73C2×C414yes—
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C22⋊C4×C73—21no—
108#103×C73C2×C418yescomplemented
109C22⋊C8×C73C2×C414yescomplemented
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—24no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—21no—
118C22×Dic73—21no—
119C22×Dic73—41no—
120C22×Dic73—42no—
121C22×Dic73—42no—
122C22×Dic73—42no—
123C22×Dic73—42no—
124C22×Dic73—42no—
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C2×#10596831—22no—
130C2×#10596831—22no—
131C2×#10596831—22no—
132C4⋊Dic73—28no—
133C4⋊Dic73—24no—
134C4⋊Dic73—22no—
135C4⋊Dic73—28no—
136C4⋊Dic73—22no—
137C4⋊Dic73—22no—
138C4⋊Dic73—42no—
139C4⋊Dic73C2×D414yes—
140C22⋊Dic73—42no—
141C22⋊Dic73—42no—
142C22⋊Dic73—42no—
143C22⋊Dic73—42no—
144C4×Dic73—28no—
145C4×Dic73—22no—
146C4×Dic73—28no—
147C4×Dic73—22no—
148C4×Dic73—22no—
149C4×Dic73—42no—
150C4×Dic73C2×D414yes—
151C24×C73—21no—
152C24×C73C2×D411yescharacteristic
153C2×Q8×C73—216no—
154C2×Q8×C73—24no—
155C2×Q8×C73—24no—
156C2×Q8×C73—216no—
157C2×Q8×C73—24no—
158C2×Q8×C73—22no—
159C2×Q8×C73—22no—
160C2×Q8×C73—22no—
161C2×Q8×C73C22×C414yes—
162C2×Q8×C73C2×D414yes—
163C2×Q8×C73#4114yes—
164C2×D4×C73—24no—
165C2×D4×C73—22no—
166C2×D4×C73—22no—
167C2×D4×C73—21no—
168C2×D4×C73—42no—
169C2×D4×C73—42no—
170C2×D4×C73C22×C412yes—
171C22×C4×C73—24no—
172C22×C4×C73—24no—
173C22×C4×C73—24no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—21no—
178C22×C4×C73C22×C412yes—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73#4112yes—
181C22×C4×C73C2411yescharacteristic
182SD16×C73—216no—
183C2×C8×C73—22no—
184C2×C8×C73—22no—
185C2×C8×C73C22×C414yes—
186C4⋊C4×C73—21no—
187C4⋊C4×C73C22×C412yes—
188C22⋊C4×C73—24no—
189C23×C73—21no—
190C23×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73—22no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73C4×D412yes—
201C23×C73AGL(1,ℤ/8)12yes—
202C23×C73#13712yes—
203C23×C73C22×D411yescharacteristic
204Q8×C73—216no—
205Q8×C73—24no—
206Q8×C73—24no—
207Q8×C73—24no—
208Q8×C73C4×D418yescomplemented
209D4×C73—24no—
210D4×C73—48no—
211D4×C73—42no—
212C2×C4×C73—216no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—21no—
216C2×C4×C73—24no—
217C2×C4×C73—21no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—22no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73C4×D412yes—
224C2×C4×C73C23×C411yescharacteristic
225C2×C4×C73C22×D412yes—
226C2×C4×C73C2×#4112yes—
227C2×C4×C73C2×#4111yescharacteristic
228C2×Dic73—28no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—24no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—44no—
240C2×Dic73—48no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—84no—
250C2×Dic73#12012yes—
251C2×Dic73#12012yes—
252C2×Dic73#13712yes—
253C8×C73—24no—
254#10596831—42no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73—42no—
262C22×C73—42no—
263C22×C73—42no—
264C22×C73—81no—
265C22×C73—81no—
266C22×C73—81no—
267C22×C73—81no—
268C22×C73#44014yescomplemented
269C22×C73C2×C4×D411yescharacteristic
270C22×C73C2×#12011yescharacteristic
271C22×C73#52511yescharacteristic
272C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C73C2×#13711yescharacteristic
274C22×C73C2×#13711yescharacteristic
275C22×C73#57511yescharacteristic
276C4×C73—28no—
277C4×C73—22no—
278C4×C73—42no—
279C4×C73C2×C4×D412yes—
280C4×C73#52512yes—
281Dic73—22no—
282Dic73—22no—
283Dic73—22no—
284Dic73—41no—
285Dic73—44no—
286Dic73—81no—
287Dic73—81no—
288#21454—731nomaximal, Sylow
289C2×C73—24no—
290C2×C73—82no—
291C2×C73C4×#12011yescharacteristic
292C2×C73C2×#44011yescharacteristic
293C2×C73C4×#13711yescharacteristic
294C2×C73#300411yescharacteristic
295C2×C73#300611yescharacteristic
296C2×C73#311211yescharacteristic
297C2×C73#311711yescharacteristic
298C22×C4×Q8—731no—
299C2×#448D7311yescharacteristic
300C2×#440—732no—
301C4×#121—731no—
302#1880—732no—
303#1874—731no—
304#1863—732no—
305#1852—731no—
306#1787—731no—
307#1763—732no—
308#1749—731no—
309C73#2145411yescharacteristic, complemented, Sylow
310C23×Q8C2×D7311yescharacteristic
311C22×SD16C2×D7312yes—
312C2×#121C2×D7311yescharacteristic
313C2×C4×Q8—732no—
314C2×C4×Q8—732no—
315C2×C4×Q8—1461no—
316C2×C4×Q8—1461no—
317C2×C4×Q8—1468no—
318C2×C4×D4—731no—
319C2×C4×D4—1461no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—1461no—
322C22×C42—731no—
323C22×C42—1461no—
324#448Dic7318yescomplemented
325#440—1468no—
326C2×#107—732no—
327C2×#103—732no—
328C2×#103—732no—
329C2×#103C2×D7312yes—
330C2×#102—732no—
331C2×C22⋊C8—731no—
332C2×C22⋊C8C2×D7311yescharacteristic
333C2×#97—732no—
334#390—731no—
335#389—731no—
336#388—1461no—
337#386—1461no—
338#380—731no—
339C4×C4⋊C4—731no—
340C4×C22⋊C4—1461no—
341#340—732no—
342#340—732no—
343#336—732no—
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C2×Dic7314yescomplemented
349C22×Q8C22×D7312yes—
350C22×D4—21no—
351C22×D4C22×D7311yescharacteristic
352C23×C4—21no—
353C23×C4—731no—
354C23×C4—731no—
355C23×C4—1461no—
356C23×C4—1461no—
357C23×C4C22×D7311yescharacteristic
358C2×SD16—24no—
359C2×SD16—24no—
360C2×SD16C2×Dic7318yescomplemented
361C22×C8—732no—
362C22×C8C22×D7312yes—
363#121—22no—
364#121C2×Dic7314yes—
365C4×Q8—738no—
366C4×Q8—1464no—
367C4×Q8—1464no—
368C4×Q8—1468no—
369C4×Q8—1468no—
370C4×Q8—1464no—
371C4×D4—2924no—
372C4×D4—1464no—
373C2×C4⋊C4—732no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1464no—
378C2×C4⋊C4—1462no—
379C2×C4⋊C4—1462no—
380C2×C4⋊C4—1468no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4C22×D7311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C42—731no—
389C2×C42—731no—
390C2×C42—731no—
391C2×C42—731no—
392C2×C42—1461no—
393C2×C42—1464no—
394C2×C42—1462no—
395C2×C42—1462no—
396C2×C42—1468no—
397#107—1464no—
398#103—1464no—
399#103—1464no—
400#103C2×Dic7318yescomplemented
401#102—1464no—
402C22⋊C8—1462no—
403C22⋊C8C2×Dic7314yescomplemented
404#97—1464no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408C24—21no—
409C24D4×D7311yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8C22×Dic7314yes—
419C2×Q8D4×D7314yes—
420C2×Q8#1192402714yes—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×Dic7312yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—2922no—
436C22×C4—2922no—
437C22×C4—2921no—
438C22×C4—2922no—
439C22×C4—2922no—
440C22×C4—2922no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1462no—
448C22×C4—1464no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4C22×Dic7312yes—
455C22×C4D4×D7311yescharacteristic
456C22×C4#1192402712yes—
457C22×C4C23×D7311yescharacteristic
458SD16—216no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—1462no—
462C2×C8—1462no—
463C2×C8—1462no—
464C2×C8C22×Dic7314yes—
465C4⋊C4—21no—
466C4⋊C4—2922no—
467C4⋊C4—734no—
468C4⋊C4—1468no—
469C4⋊C4—1468no—
470C4⋊C4—1464no—
471C4⋊C4—1462no—
472C4⋊C4—1462no—
473C4⋊C4—1462no—
474C4⋊C4C22×Dic7312yes—
475C22⋊C4—24no—
476C22⋊C4—2922no—
477C22⋊C4—2922no—
478C22⋊C4—2922no—
479C22⋊C4—2922no—
480C42—2922no—
481C42—734no—
482C42—1462no—
483C42—1462no—
484C42—1468no—
485C42—1468no—
486C42—1462no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23D4×Dic7312yes—
499C23#1455661812yes—
500C23#1455662012yes—
501C23C2×D4×D7311yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8D4×Dic7318yescomplemented
507D4—24no—
508D4—48no—
509D4—42no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—2928no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2924no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—5844no—
532C2×C4—732no—
533C2×C4—732no—
534C2×C4—732no—
535C2×C4—1461no—
536C2×C4—1464no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4—1468no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1461no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4D4×Dic7312yes—
547C2×C4C23×Dic7311yescharacteristic
548C2×C4C2×D4×D7312yes—
549C2×C4C2×#1192402712yes—
550C2×C4C2×#1192402711yescharacteristic
551C8—24no—
552C8—2922no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22#1798868814yescomplemented
567C22C2×D4×Dic7311yescharacteristic
568C22#120×D7311yescharacteristic
569C22#1798909311yescharacteristic
570C22C2×#1455661811yescharacteristic
571C22#137×D7311yescharacteristic
572C22C2×#1455662011yescharacteristic
573C22#1798938611yescharacteristic
574C4—28no—
575C4—22no—
576C4—42no—
577C4—2924no—
578C4—2921no—
579C4—5841no—
580C4—5841no—
581C4—1462no—
582C4—1462no—
583C4—1462no—
584C4C2×D4×Dic7312yes—
585C4#1798909312yes—
586C2—24no—
587C2—82no—
588C2#120×Dic7311yescharacteristic
589C2C2×#1798868811yescharacteristic
590C2#137×Dic7311yescharacteristic
591C2#2261916211yescharacteristic
592C2#2261918211yescharacteristic
593C2#2262004811yescharacteristic
594C2#2262009711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#448×C73C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
cosocleC22×C4×C73C24
hypercenterC2×#448D73
hypocenterC73#21454
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic73C2semidirect
2. C2×#17988688C2semidirect
4. #22610288C2semidirect
6. #22610187C2semidirect
7. #22610121C2semidirect
8. #22609866C2semidirect
9. #22609763C2semidirect
10. #22609695C2semidirect
43. #448×C73C4semidirect
56. Q8×Dic73D4semidirect
94. C22×Q8×C73C2×C4semidirect
102. C2×SD16×C73C2×C4semidirect
108. #103×C73C2×C4semidirect
109. C22⋊C8×C73C2×C4semidirect
208. Q8×C73C4×D4semidirect
268. C22×C73#440semidirect
309. C73#21454semidirect
324. #448Dic73semidirect
348. C22×Q8C2×Dic73semidirect
360. C2×SD16C2×Dic73semidirect
400. #103C2×Dic73semidirect
403. C22⋊C8C2×Dic73semidirect
506. Q8D4×Dic73semidirect
566. C22#17988688semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic732
C2×#179886882
#121×Dic732
#226102882
#226102492
#226101872
#226101212
#226098662
#226097632
#226096952
C2×#448×C732
#2145473