Finite Group Databasebeta

← #35319400

Subgroups — #35319400

Order 21248.

752 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127675C211yesmaximal, characteristic
2#24915908C211yesmaximal, characteristic
3#24905593C214yesmaximal, complemented
4C2×#413×C83C211yesmaximal, characteristic
5C22×#13050574C411yescharacteristic, complemented
6C22×#13050574C411yescharacteristic, complemented
7C2×#14581751—21no—
8#19127675—21no—
9#19127675—21no—
10#19127675C412yes—
11#19127675C2212yes—
12C2×#14581707—21no—
13C2×#14581707C412yes—
14#19127668—21no—
15#19127668—21no—
16#19127668C412yescomplemented
17#19127668C2212yes—
18#19127665—21no—
19#19127665—21no—
20C22×#41×C83C2211yescharacteristic
21C22×M16×C83—21no—
22#413×C83—21no—
23#413×C83—21no—
24#413×C83C412yescomplemented
25#413×C83C2212yes—
26C2×#101×C83C412yescomplemented
27C2×#100×C83—21no—
28C2×#13050574—21no—
29C2×#13050574—21no—
30C2×#13050574—21no—
31C2×#13050574—41no—
32C2×#13050574—41no—
33C2×#13050574—41no—
34C2×#13050574—41no—
35C2×#13050574C2×C412yescomplemented
36C22×C83⋊C8—21no—
37C22×C83⋊C8C2×C412yescomplemented
38#14581751—22no—
39#14581751—22no—
40#14581751—41no—
41#14581751—41no—
42C2×C4⋊Dic83—41no—
43C2×C22⋊Dic83—22no—
44C2×C4×Dic83—41no—
45#14581707—22no—
46#14581707—22no—
47#14581707—22no—
48#14581707C2×C414yes—
49C22⋊(C83⋊C8)—22no—
50C22⋊(C83⋊C8)—22no—
51C22⋊(C83⋊C8)—22no—
52C22⋊(C83⋊C8)—22no—
53C2×#41×C83—41no—
54C2×#41×C83—41no—
55C2×#41×C83—41no—
56C2×#41×C83—41no—
57C2×#41×C83C2×C411yescharacteristic
58C2×#41×C83C2×C411yescharacteristic
59C2×#41×C83C2×C411yescharacteristic
60C2×#41×C83D411yescharacteristic
61C2×#41×C83D411yescharacteristic
62C2×#41×C83D411yescharacteristic
63C2×#41×C83Q811yescharacteristic
64C2×#41×C83C2311yescharacteristic
65C22×Q8×C83C2×C411yescharacteristic
66C22×D4×C83C2×C411yescharacteristic
67C22×D4×C83D411yescharacteristic
68C22×D4×C83Q811yescharacteristic
69C23×C4×C83C2×C411yescharacteristic
70C23×C4×C83D411yescharacteristic
71C23×C4×C83D411yescharacteristic
72C2×M16×C83—22no—
73C2×M16×C83—22no—
74C2×M16×C83—22no—
75C2×M16×C83—41no—
76C2×M16×C83—41no—
77C2×M16×C83—41no—
78C22×C8×C83—41no—
79#101×C83—22no—
80#101×C83C2×C414yescomplemented
81#100×C83—22no—
82#100×C83—22no—
83C22×Dic83—42no—
84#13050574—42no—
85#13050574—41no—
86#13050574—41no—
87#13050574—41no—
88#13050574—41no—
89#13050574—42no—
90C2×C83⋊C8—22no—
91C2×C83⋊C8—22no—
92C2×C83⋊C8—42no—
93C2×C83⋊C8—42no—
94C4⋊Dic83—42no—
95C4⋊Dic83—42no—
96C22⋊Dic83—24no—
97C22⋊Dic83—42no—
98C4×Dic83—42no—
99C4×Dic83—42no—
100C24×C83C22⋊C411yescharacteristic
101C24×C83C4⋊C411yescharacteristic
102C24×C83C4⋊C411yescharacteristic
103#41×C83—41no—
104#41×C83—41no—
105#41×C83—41no—
106#41×C83—41no—
107#41×C83—41no—
108#41×C83—41no—
109#41×C83—41no—
110#41×C83—41no—
111#41×C83—41no—
112#41×C83—41no—
113#41×C83—41no—
114#41×C83—41no—
115#41×C83—41no—
116#41×C83—41no—
117#41×C83—41no—
118#41×C83—41no—
119C2×Q8×C83—41no—
120C2×Q8×C83—41no—
121C2×Q8×C83C4211yescharacteristic
122C2×Q8×C83C22⋊C411yescharacteristic
123C2×Q8×C83C4⋊C411yescharacteristic
124C2×Q8×C83C22×C411yescharacteristic
125C2×D4×C83—41no—
126C2×D4×C83—41no—
127C2×D4×C83—41no—
128C2×D4×C83—41no—
129C2×D4×C83—41no—
130C2×D4×C83—41no—
131C2×D4×C83C4211yescharacteristic
132C2×D4×C83C22⋊C411yescharacteristic
133C2×D4×C83C22⋊C411yescharacteristic
134C2×D4×C83C22⋊C411yescharacteristic
135C2×D4×C83C4⋊C411yescharacteristic
136C2×D4×C83C4⋊C411yescharacteristic
137C2×D4×C83C4⋊C411yescharacteristic
138C2×D4×C83C4⋊C411yescharacteristic
139C2×D4×C83C4⋊C411yescharacteristic
140C2×D4×C83C22×C411yescharacteristic
141C2×D4×C83C2×D411yescharacteristic
142C2×D4×C83C2×Q811yescharacteristic
143C22×C4×C83—21no—
144C22×C4×C83—21no—
145C22×C4×C83—21no—
146C22×C4×C83—21no—
147C22×C4×C83—21no—
148C22×C4×C83—21no—
149C22×C4×C83—21no—
150C22×C4×C83—21no—
151C22×C4×C83—21no—
152C22×C4×C83—21no—
153C22×C4×C83—21no—
154C22×C4×C83—21no—
155C22×C4×C83C4211yescharacteristic
156C22×C4×C83C4211yescharacteristic
157C22×C4×C83C22⋊C411yescharacteristic
158C22×C4×C83C22⋊C411yescharacteristic
159C22×C4×C83C22⋊C411yescharacteristic
160C22×C4×C83C22⋊C411yescharacteristic
161C22×C4×C83C22⋊C411yescharacteristic
162C22×C4×C83C22⋊C411yescharacteristic
163C22×C4×C83C22⋊C411yescharacteristic
164C22×C4×C83C4⋊C411yescharacteristic
165C22×C4×C83C4⋊C411yescharacteristic
166C22×C4×C83C4⋊C411yescharacteristic
167C22×C4×C83C4⋊C411yescharacteristic
168C22×C4×C83C22×C411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171M16×C83—24no—
172M16×C83—42no—
173M16×C83—42no—
174M16×C83—42no—
175C2×C8×C83—42no—
176C2×C8×C83—42no—
177C2×C8×C83—42no—
178C23×C83—21no—
179C23×C83—21no—
180C23×C83—21no—
181C23×C83—21no—
182C23×C83—21no—
183C23×C83—21no—
184C23×C83—81no—
185C23×C83—81no—
186C23×C83—81no—
187C23×C83#9511yescharacteristic
188C23×C83#9511yescharacteristic
189C23×C83#9511yescharacteristic
190C23×C83#9511yescharacteristic
191C23×C83C2×C22⋊C411yescharacteristic
192C23×C83C2×C4⋊C411yescharacteristic
193C23×C83C2×C4⋊C411yescharacteristic
194Q8×C83—41no—
195Q8×C83—41no—
196Q8×C83—41no—
197Q8×C83—41no—
198D4×C83—41no—
199D4×C83—41no—
200D4×C83—41no—
201D4×C83—41no—
202D4×C83—41no—
203D4×C83—41no—
204D4×C83—41no—
205D4×C83—41no—
206D4×C83—41no—
207D4×C83—41no—
208D4×C83—41no—
209D4×C83—41no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83#9511yescharacteristic
241C2×C4×C83#9511yescharacteristic
242C2×C4×C83#9511yescharacteristic
243C2×C4×C83#9511yescharacteristic
244C2×C4×C83#9911yescharacteristic
245C2×C4×C83#9911yescharacteristic
246C2×C4×C83#9911yescharacteristic
247C2×C4×C83#9911yescharacteristic
248C2×C4×C83C2×C4211yescharacteristic
249C2×C4×C83C2×C22⋊C411yescharacteristic
250C2×C4×C83C2×C22⋊C411yescharacteristic
251C2×C4×C83C2×C4⋊C411yescharacteristic
252C2×Dic83—44no—
253C2×Dic83—84no—
254C8×C83—44no—
255C83⋊C8—42no—
256C83⋊C8—42no—
257C22×C83—21no—
258C22×C83—21no—
259C22×C83—21no—
260C22×C83—21no—
261C22×C83—21no—
262C22×C83—21no—
263C22×C83—41no—
264C22×C83—41no—
265C22×C83—41no—
266C22×C83—41no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—81no—
273C22×C83—81no—
274C22×C83—81no—
275C22×C83—81no—
276C22×C83#34211yescharacteristic
277C22×C83#34211yescharacteristic
278C22×C83C2×#9511yescharacteristic
279C4×C83—21no—
280C4×C83—21no—
281C4×C83—21no—
282C4×C83—21no—
283C4×C83—21no—
284C4×C83—21no—
285C4×C83#34211yescharacteristic
286C4×C83#34211yescharacteristic
287C4×C83C2×#9911yescharacteristic
288C4×C83C2×#9911yescharacteristic
289Dic83—84no—
290#12589—831nomaximal, Sylow
291C2×C83—21no—
292C2×C83—21no—
293C2×C83—21no—
294C2×C83—41no—
295C2×C83—41no—
296C2×C83—41no—
297C2×C83—41no—
298C2×C83—41no—
299C2×C83—41no—
300C2×C83C2×#34211yescharacteristic
301C2×#413D8311yescharacteristic
302C2×#410—831no—
303#2879—831no—
304#1395—834no—
305C83#1258911yescharacteristic, complemented, Sylow
306C22×#41C2×D8311yescharacteristic
307C22×M16—21no—
308C22×M16—831no—
309C22×M16—831no—
310C2×#117—1661no—
311#413—21no—
312#413—21no—
313#413Dic8312yescomplemented
314#413C2×D8312yes—
315C2×#101Dic8312yescomplemented
316C2×#100—21no—
317#410—832no—
318#410—832no—
319#410—1661no—
320#410—1661no—
321C2×#99—832no—
322C2×#99—1661no—
323#408—832no—
324#408—832no—
325#408—1661no—
326#408—1661no—
327#407—1661no—
328#407—1661no—
329C2×#41—41no—
330C2×#41—41no—
331C2×#41—41no—
332C2×#41—41no—
333C2×#41Dic16611yescharacteristic
334C2×#41C4×D8311yescharacteristic
335C2×#41C4×D8311yescharacteristic
336C2×#41D33211yescharacteristic
337C2×#41C2×Dic8311yescharacteristic
338C2×#41C22⋊D8311yescharacteristic
339C2×#41C22⋊D8311yescharacteristic
340C2×#41C22×D8311yescharacteristic
341C22×Q8C2×Dic8311yescharacteristic
342C22×D4Dic16611yescharacteristic
343C22×D4C4×D8311yescharacteristic
344C22×D4C22⋊D8311yescharacteristic
345C23×C4C4×D8311yescharacteristic
346C23×C4D33211yescharacteristic
347C23×C4C22⋊D8311yescharacteristic
348C2×M16—22no—
349C2×M16—22no—
350C2×M16—22no—
351C2×M16—41no—
352C2×M16—41no—
353C2×M16—41no—
354C2×M16—3321no—
355C2×M16—3321no—
356C2×M16—3321no—
357C2×M16—3321no—
358C2×M16—832no—
359C2×M16—1661no—
360C2×M16—1661no—
361C2×M16—1661no—
362C22×C8—41no—
363C22×C8—832no—
364C22×C8—1661no—
365#117—3321no—
366#117—3321no—
367#117—1662no—
368#117—1662no—
369C2×C4⋊C4—3321no—
370C2×C22⋊C4—1662no—
371C2×C42—3321no—
372#101—22no—
373#101C2×Dic8314yescomplemented
374#100—22no—
375#100—22no—
376#99—834no—
377#99—1662no—
378#99—1662no—
379#99—1662no—
380C22⋊C8—1662no—
381C22⋊C8—1662no—
382C22⋊C8—1662no—
383C22⋊C8—1662no—
384C24(C2×D83)⋊C411yescharacteristic
385C24Dic83⋊C411yescharacteristic
386C24Dic83⋊C411yescharacteristic
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8C4×Dic8311yescharacteristic
406C2×Q8C22⋊Dic8311yescharacteristic
407C2×Q8C4⋊Dic8311yescharacteristic
408C2×Q8C22×Dic8311yescharacteristic
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C4×Dic8311yescharacteristic
416C2×D4(C2×D83)⋊C411yescharacteristic
417C2×D4(C2×D83)⋊C411yescharacteristic
418C2×D4C22⋊Dic8311yescharacteristic
419C2×D4Dic83⋊C411yescharacteristic
420C2×D4Dic83⋊C411yescharacteristic
421C2×D4Dic83⋊C411yescharacteristic
422C2×D4Dic83⋊C411yescharacteristic
423C2×D4C4⋊Dic8311yescharacteristic
424C2×D4C2×C4×D8311yescharacteristic
425C2×D4C2×C22⋊D8311yescharacteristic
426C2×D4C2×Dic16611yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—3322no—
440C22×C4C4×Dic8311yescharacteristic
441C22×C4C4×Dic8311yescharacteristic
442C22×C4(C2×D83)⋊C411yescharacteristic
443C22×C4(C2×D83)⋊C411yescharacteristic
444C22×C4(C2×D83)⋊C411yescharacteristic
445C22×C4(C2×D83)⋊C411yescharacteristic
446C22×C4(C2×D83)⋊C411yescharacteristic
447C22×C4C22⋊Dic8311yescharacteristic
448C22×C4C22⋊Dic8311yescharacteristic
449C22×C4Dic83⋊C411yescharacteristic
450C22×C4Dic83⋊C411yescharacteristic
451C22×C4C4⋊Dic8311yescharacteristic
452C22×C4C4⋊Dic8311yescharacteristic
453C22×C4C2×C4×D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×D33211yescharacteristic
456M16—24no—
457M16—42no—
458M16—42no—
459M16—42no—
460M16—3322no—
461M16—3321no—
462M16—3321no—
463M16—3321no—
464M16—3321no—
465M16—3322no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—3322no—
470C2×C8—3322no—
471C2×C8—1662no—
472C2×C8—1662no—
473C4⋊C4—3322no—
474C4⋊C4—3322no—
475C22⋊C4—3322no—
476C22⋊C4—1664no—
477C42—3322no—
478C42—3322no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23C83⋊#9511yescharacteristic
489C23C83⋊#9511yescharacteristic
490C23C83⋊#9511yescharacteristic
491C23C83⋊#9511yescharacteristic
492C23C2×(C2×D83)⋊C411yescharacteristic
493C23C2×Dic83⋊C411yescharacteristic
494C23C2×Dic83⋊C411yescharacteristic
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—3324no—
542C2×C4—6644no—
543C2×C4C83⋊#9511yescharacteristic
544C2×C4C83⋊#9511yescharacteristic
545C2×C4C83⋊#9511yescharacteristic
546C2×C4C83⋊#9511yescharacteristic
547C2×C4#1458170611yescharacteristic
548C2×C4#1458170611yescharacteristic
549C2×C4#1458170611yescharacteristic
550C2×C4#1458170611yescharacteristic
551C2×C4C2×C4×Dic8311yescharacteristic
552C2×C4C2×(C2×D83)⋊C411yescharacteristic
553C2×C4C2×C22⋊Dic8311yescharacteristic
554C2×C4C2×C4⋊Dic8311yescharacteristic
555C8—44no—
556C8—3322no—
557C8—3322no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#1912746711yescharacteristic
578C22#1912746711yescharacteristic
579C22C2×C83⋊#9511yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—6644no—
587C4#1912746711yescharacteristic
588C4#1912746711yescharacteristic
589C4C2×#1458170611yescharacteristic
590C4C2×#1458170611yescharacteristic
591C2—21no—
592C2—21no—
593C2—21no—
594C2—41no—
595C2—41no—
596C2—41no—
597C2—41no—
598C2—41no—
599C2—41no—
600C2C2×#1912746711yescharacteristic
subgroupG/H
centerC4C2×#14581706
commutatorC2×C4×C83C2×C42
FrattiniC2×#41C22×D83
FittingC2×#413×C83C2
socleC2×C83C2×#342
prime radicalC22×#41×C83C22
cosocleC2×#41×C83C23
hypercenterC2×#413D83
hypocenterC83#12589
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
3. #24905593C2semidirect
5. C22×#13050574C4semidirect
6. C22×#13050574C4semidirect
16. #19127668C4semidirect
24. #413×C83C4semidirect
26. C2×#101×C83C4semidirect
35. C2×#13050574C2×C4semidirect
37. C22×C83⋊C8C2×C4semidirect
80. #101×C83C2×C4semidirect
305. C83#12589semidirect
313. #413Dic83semidirect
315. C2×#101Dic83semidirect
373. #101C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191276752
#249159082
#249055932
C2×#413×C832
#1258983