Finite Group Databasebeta

← #34022049

Subgroups — #34022049

Order 18688.

890 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
1#22626415C211yesmaximal, characteristic, complemented
2#22626370C211yesmaximal, characteristic
3#22626010C212yesmaximal, complemented
4#22625993C212yesmaximal, complemented
5#22625924C212yesmaximal
6#22625896C212yesmaximal
7C4×#14556659C211yesmaximal, characteristic, complemented
8#22625703C211yesmaximal, characteristic, complemented
9#22625701C211yesmaximal, characteristic
10#22625649C211yesmaximal, characteristic, complemented
11C4×SD16×D73C211yesmaximal, characteristic
12C2×D4×#64408C2212yes—
13C2×#14556659C2212yes—
14C2×#14556654C2212yes—
15C2×#14556653C2212yes—
16#17989546C2211yescharacteristic
17(C2×#64408)⋊Q8C2212yes—
18#17989538C2212yes—
19#17989536—22no—
20#17989532—21no—
21#17989531—21no—
22C4⋊C4×#64408C2212yes—
23C4×C4⋊#64408C2211yescharacteristic
24C4×C4⋊#64408C2211yescharacteristic
25C22⋊C4×#64408—22no—
26C4×C22⋊#64408—21no—
27#17989523C2211yescharacteristic
28C2×(C2×#64408)⋊C4—21no—
29#17989461C2212yes—
30#17989460C2212yes—
31#17989460C2212yes—
32#17989452C2212yes—
33#17989452C2212yes—
34C2×SD16×D73C2211yescharacteristic
35C4×Q8×D73C2211yescharacteristic
36C4×D4×D73C2211yescharacteristic
37#17988678C411yescharacteristic, complemented
38C4×#11924017C411yescharacteristic, complemented
39SD16×Dic73C411yescharacteristic, complemented
40C4×#11924015C411yescharacteristic, complemented
41#17988674C411yescharacteristic, complemented
42C4×#11924016C411yescharacteristic, complemented
43#17988672C411yescharacteristic, complemented
44#106×D73C2211yescharacteristic
45#103×D73C2211yescharacteristic
46#102×D73C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48C4×SD16×C73C411yescharacteristic, complemented
49C23×#64408—22no—
50C2×#11924036C2312yes—
51D4×#64408—24no—
52D4×#64408—24no—
53C2×C4⋊#64408—22no—
54C2×C4⋊#64408C2312yes—
55C2×C4⋊#64408C2312yes—
56C2×C22⋊#64408—22no—
57C2×C4×#64408—22no—
58C2×C4×#64408—21no—
59C2×C4×#64408—21no—
60C2×C4×#64408C2312yes—
61#14556659—24no—
62#14556654—24no—
63#14556653—24no—
64#14556646—21no—
65#14556646—21no—
66(C2×#64408)⋊C4—22no—
67(C2×#64408)⋊C4—22no—
68(C2×#64408)⋊C4—22no—
69(C2×#64408)⋊C4—22no—
70(C2×#64408)⋊C4—21no—
71(C2×#64408)⋊C4—21no—
72(C2×#64408)⋊C4D411yescharacteristic
73(C2×#64408)⋊C4D411yescharacteristic
74(C2×#64408)⋊C4D412yes—
75C2×Q8×D73C2311yescharacteristic
76C2×D4×D73C2311yescharacteristic
77C22×C4×D73—21no—
78C2×#11924017C2×C411yescharacteristic
79C2×#11924016C2×C411yescharacteristic
80C2×#11924015C2×C411yescharacteristic
81SD16×D73—24no—
82C2×C8×D73C2311yescharacteristic
83C2×C8×D73C2311yescharacteristic
84Q8×Dic73C2×C411yescharacteristic, complemented
85C4×Dic146—21no—
86C4×Dic146C2×C411yescharacteristic, complemented
87#14556556—21no—
88#14556556C2×C411yescharacteristic, complemented
89C4×D292C2×C411yescharacteristic
90D4×Dic73C2×C411yescharacteristic
91#14556553C2×C411yescharacteristic
92C4×C22⋊D73—21no—
93#14556551—21no—
94C4⋊C4×D73—21no—
95C4⋊C4×D73C2311yescharacteristic
96C4⋊C4×D73C2311yescharacteristic
97C22⋊C4×D73—21no—
98C42×D73—21no—
99C42×D73C2311yescharacteristic
100#14556523C2×C411yescharacteristic, complemented
101#14556522C2×C411yescharacteristic, complemented
102#14556522C2×C411yescharacteristic, complemented
103#14556516C2×C411yescharacteristic, complemented
104#14556515C2×C411yescharacteristic, complemented
105#14556514C2×C411yescharacteristic, complemented
106#14556513C2×C411yescharacteristic
107#14556512C2×C411yescharacteristic
108#14556511C2×C411yescharacteristic
109C4×#10596831C2×C411yescharacteristic, complemented
110C8×Dic73C2×C411yescharacteristic, complemented
111C8×Dic73C2×C411yescharacteristic, complemented
112C2×SD16×C73C2×C411yescharacteristic
113C4×Q8×C73C2×C411yescharacteristic, complemented
114C4×D4×C73C2×C411yescharacteristic
115#106×C73C2×C411yescharacteristic, complemented
116#103×C73C2×C411yescharacteristic, complemented
117#102×C73C2×C411yescharacteristic
118C4×C8×C73C2×C411yescharacteristic, complemented
119C22×#64408—22no—
120C22×#64408—28no—
121C22×#64408—22no—
122C22×#64408—82no—
123C22×#64408—82no—
124C22×#64408C2×D412yes—
125C22×#64408C2×D412yes—
126#11924036—22no—
127#11924036—22no—
128C2×#10596833—22no—
129C4⋊#64408—22no—
130C4⋊#64408—22no—
131C4⋊#64408—22no—
132C4⋊#64408—82no—
133C4⋊#64408#4114yes—
134C22⋊#64408—28no—
135C4×#64408—22no—
136C4×#64408—22no—
137C4×#64408—81no—
138C4×#64408—81no—
139C4×#64408—82no—
140C23×D73—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C411yescharacteristic
143Q8×D73—22no—
144Q8×D73—22no—
145C2×D292C22×C411yescharacteristic
146C2×C22⋊D73—21no—
147D4×D73—22no—
148D4×D73#4114yes—
149C22×Dic73—21no—
150C2×C4×D73—21no—
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—82no—
156C2×C4×D73C2×D411yescharacteristic
157C2×C4×D73C2×D411yescharacteristic
158C2×C4×D73C2411yescharacteristic
159#11924017—22no—
160#11924016—22no—
161#11924015—22no—
162C2×#10596831C22×C411yescharacteristic
163C2×#10596831C22×C411yescharacteristic
164C8×D73—22no—
165C8×D73—22no—
166C4⋊Dic73—21no—
167C4⋊Dic73C22×C411yescharacteristic
168C4⋊Dic73C22×C411yescharacteristic
169#11924007—21no—
170#11924007—21no—
171#11924007—21no—
172#11924007—21no—
173C22⋊Dic73—21no—
174#11924005—21no—
175#11924005—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73C22×C411yescharacteristic
180C4×Dic73#4111yescharacteristic
181C4×Dic73#4111yescharacteristic
182C2×Q8×C73C22×C411yescharacteristic
183C2×D4×C73C22×C411yescharacteristic
184C22×C4×C73—21no—
185SD16×C73—22no—
186C2×C8×C73C22×C411yescharacteristic
187C2×C8×C73C22×C411yescharacteristic
188C4⋊C4×C73—21no—
189C4⋊C4×C73C22×C411yescharacteristic
190C4⋊C4×C73C22×C411yescharacteristic
191C22⋊C4×C73—21no—
192C42×C73—21no—
193C42×C73C22×C411yescharacteristic
194C23×C73—21no—
195C22×D73—21no—
196C22×D73—24no—
197C22×D73—42no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73C22×D411yescharacteristic
201C2×#64408—22no—
202C2×#64408—22no—
203C2×#64408—24no—
204C2×#64408—22no—
205C2×#64408—22no—
206C2×#64408—22no—
207C2×#64408—22no—
208C2×#64408—42no—
209C2×#64408—42no—
210C2×#64408—42no—
211C2×#64408—42no—
212C2×#64408—42no—
213C2×#64408—88no—
214Q8×C73—21no—
215Q8×C73—21no—
216D4×C73—21no—
217D4×C73#11712yes—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—81no—
222C2×C4×C73C4×D411yescharacteristic
223C2×C4×C73C4×D411yescharacteristic
224C2×C4×C73C23×C411yescharacteristic
225C22⋊D73—24no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—81no—
230C2×Dic73C4×D411yescharacteristic
231C2×Dic73C4×D411yescharacteristic
232C2×Dic73C2×#4111yescharacteristic
233D292—21no—
234D292#11712yes—
235C4×D73—22no—
236C4×D73—21no—
237C4×D73—21no—
238C4×D73—42no—
239C4×D73—41no—
240C4×D73—42no—
241C4×D73—41no—
242C4×D73—81no—
243C4×D73—81no—
244C4×D73C2×#4112yes—
245C4×D73C2×#4112yes—
246Dic146—21no—
247Dic146—21no—
248Dic146—42no—
249#10596833—42no—
250C8×C73—21no—
251C8×C73—21no—
252#10596831—21no—
253#10596831—21no—
254C22×C73—22no—
255C22×C73—41no—
256C22×C73—41no—
257C22×C73C2×C4×D411yescharacteristic
258C2×D73—22no—
259C2×D73—41no—
260C2×D73—41no—
261C2×D73—44no—
262C2×D73—44no—
263C2×D73#52512yes—
264C2×D73#57711yescharacteristic
265C2×D73#57712yes—
266C2×D73#57711yescharacteristic
267#64408—42no—
268#64408—44no—
269#64408—42no—
270#64408—82no—
271C4×C73—21no—
272C4×C73—21no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73—41no—
276C4×C73C2×#11711yescharacteristic
277C4×C73C4×#4111yescharacteristic
278Dic73—21no—
279Dic73—21no—
280Dic73—41no—
281Dic73—41no—
282Dic73—41no—
283Dic73#52912yes—
284#21133—731nomaximal, Sylow
285C2×C73—42no—
286C2×C73#229011yescharacteristic
287C2×C73#294911yescharacteristic
288C2×C73#295211yescharacteristic
289D73—42no—
290D73#312312yes—
291D73#312512yes—
292C2×C4×SD16—731no—
293#2320—731no—
294#2275—731no—
295#1927—732no—
296#1919—732no—
297#1877—732no—
298#1864—732no—
299C4×#106—731no—
300#1763—731no—
301#1762—731no—
302#1746—731no—
303C73#2113311yescharacteristic, complemented, Sylow
304C22×SD16—731no—
305C2×C4×Q8—731no—
306C2×C4×D4—731no—
307C2×C4×D4—732no—
308C4×SD16—731no—
309C4×SD16—731no—
310C4×SD16—731no—
311C4×SD16—731no—
312C4×SD16—731no—
313C4×SD16—731no—
314C4×SD16—731no—
315C4×SD16#6440811yescharacteristic, complemented
316C2×#106—731no—
317C2×#106—732no—
318C2×#103—731no—
319C2×#103—732no—
320C2×#102—731no—
321C2×#102—732no—
322C2×#97—731no—
323C2×C4×C8—731no—
324#391—732no—
325#389—732no—
326#387—1462no—
327#383—1461no—
328#382—1461no—
329C4×C4⋊C4—731no—
330C4×C4⋊C4—732no—
331C4×C4⋊C4—731no—
332C4×C22⋊C4—1461no—
333C4×C22⋊C4—1462no—
334#376—731no—
335C2×#95—1461no—
336#340—732no—
337#340—732no—
338#340—732no—
339#336—732no—
340#336—732no—
341C22×Q8—731no—
342C22×D4—731no—
343C23×C4—1461no—
344C23×C4—1462no—
345C2×SD16—731no—
346C2×SD16—731no—
347C2×SD16—731no—
348C2×SD16—1464no—
349C2×SD16C2×#6440811yescharacteristic
350C22×C8—731no—
351C22×C8—731no—
352C22×C8—732no—
353C4×Q8—731no—
354C4×Q8—731no—
355C4×Q8—731no—
356C4×Q8—1461no—
357C4×Q8—1461no—
358C4×Q8C2×#6440811yescharacteristic, complemented
359C4×D4—731no—
360C4×D4—731no—
361C4×D4—731no—
362C4×D4—1461no—
363C4×D4—1461no—
364C4×D4—1464no—
365C4×D4—1464no—
366C4×D4C2×#6440811yescharacteristic
367C2×C4⋊C4—732no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—732no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1462no—
373C2×C22⋊C4—1461no—
374C2×C22⋊C4—1462no—
375C2×C42—731no—
376C2×C42—732no—
377C2×C42—1461no—
378C2×C42—1461no—
379C2×C42—1462no—
380C2×C42—1461no—
381#106—731no—
382#106—731no—
383#106—731no—
384#106—1464no—
385#106C2×#6440811yescharacteristic, complemented
386#103—731no—
387#103—731no—
388#103—731no—
389#103—1464no—
390#103C2×#6440811yescharacteristic, complemented
391#102—731no—
392#102—731no—
393#102—731no—
394#102—1464no—
395#102C2×#6440811yescharacteristic
396#97—1461no—
397#97—1461no—
398C4×C8—731no—
399C4×C8—731no—
400C4×C8—731no—
401C4×C8C2×#6440811yescharacteristic, complemented
402#95—731no—
403#95—731no—
404#95—732no—
405#95—1461no—
406#95—1462no—
407#95—1461no—
408#95—1462no—
409#95—1462no—
410#95—1462no—
411C24—1461no—
412C2×Q8—731no—
413C2×Q8—1461no—
414C2×Q8—1462no—
415C2×Q8—1462no—
416C2×Q8C22×#6440811yescharacteristic
417C2×D4—734no—
418C2×D4—731no—
419C2×D4—1462no—
420C2×D4—1461no—
421C2×D4C22×#6440811yescharacteristic
422C22×C4—21no—
423C22×C4—5842no—
424C22×C4—5842no—
425C22×C4—5842no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—732no—
430C22×C4—732no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1461no—
438C22×C4—1468no—
439C22×C4—1461no—
440SD16—22no—
441SD16—1462no—
442SD16—1462no—
443SD16—1462no—
444C2×C8—731no—
445C2×C8—731no—
446C2×C8—1462no—
447C2×C8—1462no—
448C2×C8—1462no—
449C2×C8—1462no—
450C2×C8—1462no—
451C2×C8C22×#6440811yescharacteristic
452C2×C8C22×#6440811yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—5842no—
455C4⋊C4—734no—
456C4⋊C4—731no—
457C4⋊C4—731no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4—1461no—
462C4⋊C4—1462no—
463C4⋊C4—1462no—
464C4⋊C4—1462no—
465C4⋊C4—1461no—
466C4⋊C4C22×#6440811yescharacteristic
467C4⋊C4C22×#6440811yescharacteristic
468C22⋊C4—21no—
469C22⋊C4—1461no—
470C22⋊C4—1461no—
471C22⋊C4—1468no—
472C22⋊C4—1461no—
473C42—21no—
474C42—5842no—
475C42—5841no—
476C42—5841no—
477C42—731no—
478C42—731no—
479C42—731no—
480C42—1461no—
481C42—1461no—
482C42—1462no—
483C42—1462no—
484C42—1461no—
485C42C22×#6440811yescharacteristic
486C23—21no—
487C23—2922no—
488C23—2921no—
489C23—2921no—
490C23—731no—
491C23—1464no—
492C23—1461no—
493Q8—21no—
494Q8—21no—
495Q8—2922no—
496Q8—1461no—
497Q8—1461no—
498D4—21no—
499D4—732no—
500D4—1464no—
501D4—1461no—
502D4#1455667012yes—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—81no—
507C2×C4—2922no—
508C2×C4—2921no—
509C2×C4—2922no—
510C2×C4—2921no—
511C2×C4—2922no—
512C2×C4—2922no—
513C2×C4—2922no—
514C2×C4—2922no—
515C2×C4—2922no—
516C2×C4—5841no—
517C2×C4—5841no—
518C2×C4—5848no—
519C2×C4—5841no—
520C2×C4—731no—
521C2×C4—732no—
522C2×C4—732no—
523C2×C4—731no—
524C2×C4—731no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1462no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1462no—
531C2×C4—1464no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4D4×#6440811yescharacteristic
539C2×C4D4×#6440811yescharacteristic
540C2×C4C23×#6440811yescharacteristic
541C8—21no—
542C8—21no—
543C8—2922no—
544C8—1461no—
545C8—1461no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—2924no—
550C22—2924no—
551C22—2921no—
552C22—2921no—
553C22—732no—
554C22—732no—
555C22—731no—
556C22—731no—
557C22—1462no—
558C22C2×D4×#6440811yescharacteristic
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—2924no—
565C4—2922no—
566C4—2922no—
567C4—2921no—
568C4—2921no—
569C4—2921no—
570C4—5842no—
571C4—732no—
572C4—1461no—
573C4—1461no—
574C4C2×#1455667011yescharacteristic
575C4#41×#6440811yescharacteristic
576C2—42no—
577C2—2922no—
578C2—732no—
579C2—732no—
580C2#2262638511yescharacteristic
581C2#2262669111yescharacteristic
582C2#2262669411yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
FittingC4×SD16×C73C4
socleC22×C73C2×C4×D4
prime radicalC2×D4×D73C23
cosocleC2×C4×D73C24
hypercenterC4×SD16#64408
hypocenterC73#21133
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #22626415C2semidirect
3. #22626010C2semidirect
4. #22625993C2semidirect
7. C4×#14556659C2semidirect
8. #22625703C2semidirect
10. #22625649C2semidirect
37. #17988678C4semidirect
38. C4×#11924017C4semidirect
39. SD16×Dic73C4semidirect
40. C4×#11924015C4semidirect
41. #17988674C4semidirect
42. C4×#11924016C4semidirect
43. #17988672C4semidirect
48. C4×SD16×C73C4semidirect
84. Q8×Dic73C2×C4semidirect
86. C4×Dic146C2×C4semidirect
88. #14556556C2×C4semidirect
100. #14556523C2×C4semidirect
101. #14556522C2×C4semidirect
102. #14556522C2×C4semidirect
103. #14556516C2×C4semidirect
104. #14556515C2×C4semidirect
105. #14556514C2×C4semidirect
109. C4×#10596831C2×C4semidirect
110. C8×Dic73C2×C4semidirect
111. C8×Dic73C2×C4semidirect
113. C4×Q8×C73C2×C4semidirect
115. #106×C73C2×C4semidirect
116. #103×C73C2×C4semidirect
118. C4×C8×C73C2×C4semidirect
303. C73#21133semidirect
315. C4×SD16#64408semidirect
358. C4×Q8C2×#64408semidirect
385. #106C2×#64408semidirect
390. #103C2×#64408semidirect
401. C4×C8C2×#64408semidirect

Maximal subgroups

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

subgroupindex
#226264152
#226263702
#226260102
#226259932
#226259242
#226258962
C4×#145566592
#226257032
#226257012
#226256492
C4×SD16×D732
#2113373