Finite Group Databasebeta

← #28559607

Subgroups — #28559607

Order 15104.

670 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434125C211yesmaximal, characteristic
2Q8×#11745659C211yesmaximal, characteristic, complemented
3#21432835C211yesmaximal, characteristic
4#21422989C211yesmaximal, characteristic, complemented
5#21422988C211yesmaximal, characteristic, complemented
6#21422940C211yesmaximal, characteristic, complemented
7#21422939C211yesmaximal, characteristic, complemented
8#21422932C211yesmaximal, characteristic, complemented
9#21422931C211yesmaximal, characteristic, complemented
10#21422893C211yesmaximal, characteristic, complemented
11#21422865C211yesmaximal, characteristic, complemented
12#21422545C211yesmaximal, characteristic
13Q8⋊C8×D59C211yesmaximal, characteristic, complemented
14C8×#11745659C211yesmaximal, characteristic, complemented
15C8×SD16×C59C211yesmaximal, characteristic, complemented
16C2×#14273733—21no—
17#16837949C2211yescharacteristic
18C4×Q8×D59C2211yescharacteristic
19#16837722C411yescharacteristic
20#16837719C411yescharacteristic
21#16837686C411yescharacteristic
22#16837685C411yescharacteristic
23#16837601C411yescharacteristic
24#16837600C411yescharacteristic
25#16837542C411yescharacteristic
26#16837539C411yescharacteristic
27#16837532—21no—
28#16837532C2211yescharacteristic
29#16837529—21no—
30#16837529C2211yescharacteristic
31Q8×C59⋊C8C2211yescharacteristic
32C4×#11745660C2211yescharacteristic
33C4×#11745671C2211yescharacteristic
34C4×#11745670C2211yescharacteristic
35#16837481C2211yescharacteristic
36#16837480C2211yescharacteristic
37#16837476—21no—
38#16837473—21no—
39#16837467C2211yescharacteristic
40#16837453C2211yescharacteristic
41#16837424—21no—
42#16837424C2211yescharacteristic
43C4⋊C8×D59C2211yescharacteristic
44#16837367—21no—
45C4×#11745659—21no—
46C4×#11745659C2211yescharacteristic
47C4×C8×D59C2211yescharacteristic
48#16837153C2211yescharacteristic
49#16837152C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52Q8⋊(C59⋊C8)C2211yescharacteristic
53#16837132C2211yescharacteristic
54#16837131C2211yescharacteristic
55D4⋊(C59⋊C8)C2211yescharacteristic
56#16837129C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837123C2211yescharacteristic
59C8×C59⋊C8C2211yescharacteristic
60C8×Q8×C59C2211yescharacteristic
61C4×SD16×C59C2211yescharacteristic
62C8×D4×C59C2211yescharacteristic
63C59×#334C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685—21no—
68C2×Q8×D59C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×#11745670C2×C411yescharacteristic
71C2×#11745671C2×C411yescharacteristic
72#14273733—44no—
73C2×#11745664—21no—
74C2×#11745659—21no—
75C2×#11745659—21no—
76C2×#11745659D411yescharacteristic
77C2×#11745659D411yescharacteristic
78C2×C8×D59—21no—
79C2×C8×D59—21no—
80C2×C8×D59—21no—
81#14273726C2×C411yescharacteristic
82#14273723C2×C411yescharacteristic
83#14273713—21no—
84#14273702C2×C411yescharacteristic
85#14273700—21no—
86#14273698C2×C411yescharacteristic
87Q8×Dic59C2×C411yescharacteristic
88C4×Dic118—21no—
89C4×Dic118C2311yescharacteristic
90#14273692—21no—
91#14273692C2×C411yescharacteristic
92C4×D236C2311yescharacteristic
93C4×C22⋊D59—21no—
94C4⋊C4×D59—21no—
95C4⋊C4×D59C2×C411yescharacteristic
96C42×D59—21no—
97C42×D59C2311yescharacteristic
98#14273659C2×C411yescharacteristic
99#14273658C2×C411yescharacteristic
100#14273658C2×C411yescharacteristic
101C4⋊(C59⋊C8)—21no—
102C4⋊(C59⋊C8)C2311yescharacteristic
103C4⋊(C59⋊C8)C2311yescharacteristic
104Dic59⋊C8—21no—
105Dic59⋊C8—21no—
106Dic59⋊C8—21no—
107Dic59⋊C8—21no—
108Dic59⋊C8C2×C411yescharacteristic
109Dic59⋊C8C2×C411yescharacteristic
110#14273652C2×C411yescharacteristic
111#14273651C2×C411yescharacteristic
112#14273650C2×C411yescharacteristic
113#14273649C2×C411yescharacteristic
114#14273648C2×C411yescharacteristic
115#14273647C2×C411yescharacteristic
116C22⋊(C59⋊C8)—21no—
117(C2×D59)⋊C8—21no—
118(C2×D59)⋊C8—21no—
119(C2×D59)⋊C8C2×C411yescharacteristic
120(C2×D59)⋊C8C2×C411yescharacteristic
121#14273637—21no—
122#14273637—21no—
123#14273637D411yescharacteristic
124#14273637D411yescharacteristic
125C4×C59⋊C8—21no—
126C4×C59⋊C8C2311yescharacteristic
127C4×C59⋊C8C2311yescharacteristic
128C4×C59⋊C8C2311yescharacteristic
129C8×Dic59—21no—
130C2×SD16×C59C2×C411yescharacteristic
131C22×C8×C59—21no—
132C4×Q8×C59C2311yescharacteristic
133C4×D4×C59C2311yescharacteristic
134#106×C59C2×C411yescharacteristic
135C4⋊C8×C59—21no—
136C4⋊C8×C59C2311yescharacteristic
137C4⋊C8×C59C2311yescharacteristic
138#103×C59C2×C411yescharacteristic
139#102×C59C2×C411yescharacteristic
140C22⋊C8×C59—21no—
141C4×C8×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C4×C8×C59C2311yescharacteristic
145C2×Q8×C59C22×C411yescharacteristic
146C2×D4×C59C22×C411yescharacteristic
147C22×C4×C59—21no—
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150C2×C22⋊D59—21no—
151Q8×D59—22no—
152Q8×D59—41no—
153#11745685—42no—
154#11745685—42no—
155C2×D236—21no—
156C2×D236C22×C411yescharacteristic
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×C4×D59C22×C411yescharacteristic
161C2×C4×D59C2×D411yescharacteristic
162C2×C4×D59#4111yescharacteristic
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118C22×C411yescharacteristic
166SD16×C59—22no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—42no—
171C2×C8×C59C22×C411yescharacteristic
172C2×C8×C59C22×C411yescharacteristic
173C2×C8×C59C2×D411yescharacteristic
174C2×C8×C59C2×D411yescharacteristic
175C4⋊C4×C59—21no—
176C4⋊C4×C59C22×C411yescharacteristic
177C4⋊C4×C59C22×C411yescharacteristic
178C22⋊C4×C59—21no—
179C42×C59—21no—
180C42×C59C2411yescharacteristic
181C22⋊Dic59—21no—
182#11745671—22no—
183#11745670—41no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59C22×C411yescharacteristic
188C4⋊Dic59C22×C411yescharacteristic
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192Dic59⋊C4—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59C22×C411yescharacteristic
197C4×Dic59C2×D411yescharacteristic
198C4×Dic59#4111yescharacteristic
199#11745664—42no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C22×C411yescharacteristic
204C2×C59⋊C8C22×C411yescharacteristic
205C2×C59⋊C8C2×D411yescharacteristic
206C2×C59⋊C8C2×D411yescharacteristic
207#11745660—41no—
208#11745659—22no—
209#11745659—42no—
210#11745659—42no—
211#11745659SD1614yescomplemented
212C8×D59—42no—
213C8×D59—42no—
214C8×D59—42no—
215C23×C59—21no—
216C22×D59—21no—
217C22×D59C4×D411yescharacteristic
218Q8×C59—21no—
219Q8×C59#13112yescomplemented
220D4×C59—21no—
221D4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C22⋊D59—42no—
231C22⋊D59—42no—
232C22⋊D59—81no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59C4×D411yescharacteristic
237C2×Dic59C4×D411yescharacteristic
238C2×Dic59C4×D411yescharacteristic
239D236—21no—
240D236—21no—
241D236—42no—
242C4×D59—21no—
243C4×D59—21no—
244C4×D59—41no—
245C4×D59—42no—
246C4×D59—41no—
247C4×D59—41no—
248C4×D59—81no—
249C4×D59C2×SD1612yes—
250C4×D59#13512yes—
251Dic118—21no—
252Dic118—21no—
253Dic118—42no—
254Dic118—42no—
255C8×C59—22no—
256C8×C59—21no—
257C8×C59—21no—
258C8×C59—21no—
259C8×C59—41no—
260C8×C59—41no—
261C8×C59C2×SD1612yes—
262C59⋊C8—21no—
263C59⋊C8—21no—
264C59⋊C8—21no—
265C59⋊C8—41no—
266C59⋊C8—41no—
267C59⋊C8—41no—
268C59⋊C8C2×SD1612yes—
269#17997—591nomaximal, Sylow
270C22×C59—41no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59C2×C4×D411yescharacteristic
274C2×D59—21no—
275C2×D59—21no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59C4×SD1612yes—
280C4×C59—22no—
281C4×C59—21no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59C2×#13111yescharacteristic
285C4×C59#56811yescharacteristic
286C4×C59C22×SD1611yescharacteristic
287C4×C59C2×#13511yescharacteristic
288Dic59—21no—
289Dic59—21no—
290Dic59—41no—
291Dic59—41no—
292Dic59—41no—
293Dic59C4×SD1612yes—
294#3103—591no—
295Q8×M16—591no—
296#2935—591no—
297#1590—591no—
298#1590—591no—
299#1583—591no—
300#1583—591no—
301#1582—591no—
302#1582—591no—
303C8×SD16—591no—
304C8×SD16D5911yescharacteristic, complemented
305#1570—591no—
306#1492—591no—
307C2×Q8⋊C8—591no—
308C8×M16—591no—
309C2×C59—81no—
310C2×C59#293311yescharacteristic
311C2×C59C2×C4×SD1611yescharacteristic
312C2×C59#295611yescharacteristic
313D59—22no—
314D59—81no—
315C2×#131—1181no—
316#532—591no—
317C2×C4×Q8—591no—
318#481—591no—
319#481—591no—
320#475—591no—
321#475—591no—
322#460—591no—
323#460—591no—
324#448—591no—
325#448—591no—
326#446—591no—
327#446—591no—
328#446—1181no—
329#446—1181no—
330C8×Q8—591no—
331C8×Q8C2×D5911yescharacteristic
332C4×SD16—591no—
333C4×SD16—591no—
334C4×SD16—591no—
335C4×SD16C2×D5911yescharacteristic
336#436—591no—
337#436—591no—
338#435—1181no—
339#435—1181no—
340C8×D4—591no—
341C8×D4C2×D5911yescharacteristic
342#431—591no—
343#423—591no—
344#423—1181no—
345C2×C4⋊C8—591no—
346#408—1181no—
347C4×M16—591no—
348C4×M16—1181no—
349C2×C4×C8—591no—
350#334—591no—
351#334—591no—
352#334—591no—
353#334C2×D5911yescharacteristic
354Q8⋊C8—591no—
355Q8⋊C8—591no—
356Q8⋊C8—591no—
357Q8⋊C8C2×D5911yescharacteristic
358D4⋊C8—591no—
359D4⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8C2×D5911yescharacteristic
362#322—591no—
363#322—591no—
364C82—591no—
365C82C2×D5911yescharacteristic
366C59#1799711yescharacteristic, complemented, Sylow
367C2×#41—1181no—
368C22×Q8—591no—
369C2×SD16—591no—
370C2×SD16—591no—
371C2×SD16—591no—
372C2×SD16C4×D5911yescharacteristic
373#131—2364no—
374C2×M16—591no—
375C2×M16—591no—
376C2×M16—1181no—
377C2×M16—1181no—
378C2×M16—1181no—
379C22×C8—21no—
380C22×C8—1181no—
381C22×C8—1181no—
382C22×C8—1181no—
383C4⋊Q8—591no—
384#127—591no—
385#124—1181no—
386#121—591no—
387#121—591no—
388#121—1181no—
389C4×Q8—591no—
390C4×Q8—591no—
391C4×Q8—591no—
392C4×Q8—1181no—
393C4×Q8—1181no—
394C4×Q8C22×D5911yescharacteristic
395C4×D4—591no—
396C4×D4—1181no—
397C4×D4C22×D5911yescharacteristic
398C2×C4⋊C4—591no—
399C2×C4⋊C4—1181no—
400C2×C42—591no—
401C2×C42—1181no—
402#106—591no—
403#106—591no—
404#106—591no—
405#106C4×D5911yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—591no—
408C4⋊C8—591no—
409C4⋊C8—591no—
410C4⋊C8—591no—
411C4⋊C8—1181no—
412C4⋊C8—1181no—
413C4⋊C8—1181no—
414C4⋊C8—1181no—
415C4⋊C8—1181no—
416C4⋊C8C22×D5911yescharacteristic
417C4⋊C8C22×D5911yescharacteristic
418#103—591no—
419#103—591no—
420#103—591no—
421#103C4×D5911yescharacteristic
422#102—591no—
423#102—591no—
424#102—591no—
425#102C4×D5911yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—591no—
428C22⋊C8—591no—
429C22⋊C8—1181no—
430C22⋊C8—1181no—
431C22⋊C8—1181no—
432#97—591no—
433#97—591no—
434#97—1181no—
435#97—1181no—
436C4×C8—21no—
437C4×C8—591no—
438C4×C8—591no—
439C4×C8—591no—
440C4×C8—1181no—
441C4×C8—1181no—
442C4×C8C22×D5911yescharacteristic
443C4×C8C22×D5911yescharacteristic
444C4×C8C22×D5911yescharacteristic
445#41—2362no—
446#41—2362no—
447C2×Q8—591no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1182no—
451C2×Q8—2361no—
452C2×Q8C2×C4×D5911yescharacteristic
453C2×D4—591no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4C2×C4×D5911yescharacteristic
459C22×C4—21no—
460C22×C4—591no—
461C22×C4—591no—
462C22×C4—591no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466SD16—22no—
467SD16—1182no—
468SD16—2361no—
469SD16—2361no—
470M16—594no—
471M16—1182no—
472M16—2362no—
473M16—2362no—
474M16—2362no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—42no—
479C2×C8—591no—
480C2×C8—591no—
481C2×C8—591no—
482C2×C8—591no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—2362no—
487C2×C8—2362no—
488C2×C8—2362no—
489C2×C8C2×C4×D5911yescharacteristic
490C2×C8C2×C4×D5911yescharacteristic
491C2×C8D4×D5911yescharacteristic
492C2×C8D4×D5911yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—591no—
495C4⋊C4—591no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4C2×C4×D5911yescharacteristic
502C4⋊C4C2×C4×D5911yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1181no—
505C22⋊C4—1181no—
506C22⋊C4—1181no—
507C42—21no—
508C42—591no—
509C42—591no—
510C42—591no—
511C42—1181no—
512C42—1181no—
513C42—1181no—
514C42C23×D5911yescharacteristic
515C23—21no—
516C23—591no—
517C23—1181no—
518Q8—21no—
519Q8—1181no—
520Q8—1181no—
521Q8—2362no—
522Q8—2362no—
523Q8#1427373312yescomplemented
524D4—21no—
525D4—21no—
526D4—1181no—
527D4—1181no—
528D4—4721no—
529D4—2362no—
530D4—2362no—
531D4—2362no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—591no—
538C2×C4—592no—
539C2×C4—592no—
540C2×C4—591no—
541C2×C4—591no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—4721no—
548C2×C4—2361no—
549C2×C4—2362no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4C22×C4×D5911yescharacteristic
553C2×C4C2×D4×D5911yescharacteristic
554C2×C4#41×D5911yescharacteristic
555C8—22no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—592no—
562C8—1181no—
563C8—1181no—
564C8—1181no—
565C8—2361no—
566C8—2361no—
567C8—2361no—
568C8SD16×D5912yes—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—592no—
573C22—1181no—
574C22—1181no—
575C22—2361no—
576C22—2361no—
577C22—2361no—
578C22C4×D4×D5911yescharacteristic
579C4—21no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—592no—
584C4—1181no—
585C4—1181no—
586C4—2361no—
587C4—2361no—
588C4—2361no—
589C4C2×#1427373311yescharacteristic
590C4#1683816711yescharacteristic
591C4C2×SD16×D5911yescharacteristic
592C4#135×D5911yescharacteristic
593C2—81no—
594C2—1182no—
595C2—4721no—
596C2#2143281611yescharacteristic
597C2C4×SD16×D5911yescharacteristic
598C2#2143300111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC8×SD16×C59C2
socleC22×C59C2×C4×D4
prime radical#16837949C22
cosocleC42×C59C24
hypercenterC8×SD16D59
hypocenterC59#17997
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. Q8×#11745659C2semidirect
4. #21422989C2semidirect
5. #21422988C2semidirect
6. #21422940C2semidirect
7. #21422939C2semidirect
8. #21422932C2semidirect
9. #21422931C2semidirect
10. #21422893C2semidirect
11. #21422865C2semidirect
13. Q8⋊C8×D59C2semidirect
14. C8×#11745659C2semidirect
15. C8×SD16×C59C2semidirect
211. #11745659SD16semidirect
219. Q8×C59#131semidirect
304. C8×SD16D59semidirect
366. C59#17997semidirect
523. Q8#14273733semidirect

Maximal subgroups

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

subgroupindex
#214341252
Q8×#117456592
#214328352
#214229892
#214229882
#214229402
#214229392
#214229322
#214229312
#214228932
#214228652
#214225452
Q8⋊C8×D592
C8×#117456592
C8×SD16×C592
#1799759