Finite Group Databasebeta

← #29587601

Subgroups — #29587601

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519703C211yesmaximal, characteristic
2#21454339C211yesmaximal, characteristic
3C2×#16839744C211yesmaximal, characteristic
4#21445557C211yesmaximal, characteristic, complemented
5C2×#16839523C211yesmaximal, characteristic, complemented
6#21445512C211yesmaximal, characteristic, complemented
7C2×C61×#338C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11C22×C4×Dic61C2211yescharacteristic
12C2×#14519703—22no—
13C2×#14519703—22no—
14C2×#14519703—22no—
15#16839772—21no—
16#16839768—21no—
17#16839764—22no—
18#16839759—22no—
19#16839744—22no—
20#16839744C414yescomplemented
21C2×#14519695—21no—
22C2×#14519693—21no—
23#16839530—22no—
24#16839523—22no—
25#16839516—22no—
26C22×M16×C61C2211yescharacteristic
27C22×M16×C61C2211yescharacteristic
28C22×M16×C61C2211yescharacteristic
29C61×#338—22no—
30C2×#11758222—21no—
31C2×#11758222—21no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222C2×C412yes—
41C2×#11758222D412yes—
42C22×Dic122D411yescharacteristic
43C22×D244D411yescharacteristic
44C22×C22⋊D61—21no—
45C23×Dic61—21no—
46C22×C4×D61—21no—
47C2×#11758194—21no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194C2×C412yes—
53C22×#92418—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58C2×C4⋊Dic61D412yes—
59C2×C22⋊Dic61—21no—
60C2×C4×Dic61—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61D412yes—
67#14519703—28no—
68#14519702—24no—
69#14519699—24no—
70#14519695—24no—
71#14519693—24no—
72C23×C4×C61C2311yescharacteristic
73C2×M16×C61—21no—
74C2×M16×C61—21no—
75C2×M16×C61—21no—
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61—21no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C2×M16×C61—21no—
86C2×M16×C61—21no—
87C2×M16×C61—21no—
88C2×M16×C61—21no—
89C2×M16×C61C2×C412yes—
90C2×M16×C61D412yes—
91C22×C8×C61—21no—
92C22×C8×C61—21no—
93C22×C8×C61D412yes—
94C24×C61#4111yescharacteristic
95C23×D61—21no—
96C22×C4×C61C22×C411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×C22⋊D61—22no—
111C2×C22⋊D61—22no—
112C2×C22⋊D61—41no—
113C2×C22⋊D61—41no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—41no—
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—44no—
120C22×Dic61—41no—
121C22×Dic61—41no—
122#11758222—22no—
123#11758222—22no—
124#11758222—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—41no—
131#11758222—41no—
132#11758222—41no—
133#11758222—41no—
134#11758222—41no—
135#11758222—41no—
136#11758222—41no—
137#11758222—41no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C22⋊C412yes—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C22⋊C412yes—
156M16×C61—22no—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—22no—
160M16×C61—41no—
161M16×C61—41no—
162M16×C61—41no—
163M16×C61—41no—
164M16×C61—41no—
165M16×C61—41no—
166M16×C61—41no—
167M16×C61—41no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—41no—
174C2×C8×C61—41no—
175C2×C8×C61—41no—
176C2×C8×C61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61—41no—
184C4×Dic61—41no—
185C4×Dic61—41no—
186C4×Dic61—41no—
187#11758194—22no—
188#11758194—22no—
189#11758194—41no—
190#11758194—41no—
191C2×#92418—22no—
192C2×#92418—41no—
193C2×#92418—41no—
194C23×C61—42no—
195C23×C61#11711yescharacteristic
196C23×C61#12111yescharacteristic
197C23×C61#12311yescharacteristic
198C23×C61#12311yescharacteristic
199C23×C61#12311yescharacteristic
200C23×C61#12411yescharacteristic
201C23×C61#12611yescharacteristic
202C22×D61—22no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—82no—
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×C4×C61#11711yescharacteristic
208C2×C4×C61C4×D411yescharacteristic
209C2×C4×C61C4×D411yescharacteristic
210C2×C4×C61C4×D411yescharacteristic
211C2×C4×C61C4×D411yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12111yescharacteristic
215C2×C4×C61#12111yescharacteristic
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61#12111yescharacteristic
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12311yescharacteristic
222C2×C4×C61#12311yescharacteristic
223C2×C4×C61#12311yescharacteristic
224C2×C4×C61#12311yescharacteristic
225C2×C4×C61#12311yescharacteristic
226C2×C4×C61#12311yescharacteristic
227C2×C4×C61#12311yescharacteristic
228C2×C4×C61#12411yescharacteristic
229C2×C4×C61#12411yescharacteristic
230C2×C4×C61#12411yescharacteristic
231C2×C4×C61#12611yescharacteristic
232C2×C4×C61#12611yescharacteristic
233C2×C4×C61#12611yescharacteristic
234C22⋊D61—42no—
235C22⋊D61—42no—
236C22⋊D61—42no—
237C22⋊D61—42no—
238C2×Dic61—24no—
239C2×Dic61—22no—
240C2×Dic61—42no—
241C2×Dic61—44no—
242C2×Dic61—44no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247C2×Dic61—82no—
248D244—22no—
249D244—22no—
250D244—41no—
251D244—41no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—42no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—42no—
261C8×C61—42no—
262C8×C61—42no—
263#92418—42no—
264#11151—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61#38111yescharacteristic
273C22×C61#38811yescharacteristic
274C22×C61#38811yescharacteristic
275C22×C61#39411yescharacteristic
276C22×C61#39411yescharacteristic
277C22×C61#39711yescharacteristic
278C22×C61#39711yescharacteristic
279C2×D61—42no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61—82no—
283C4×C61#38611yescharacteristic
284C4×C61#38611yescharacteristic
285C4×C61#38811yescharacteristic
286C4×C61#38811yescharacteristic
287C4×C61#39411yescharacteristic
288C4×C61#39411yescharacteristic
289C4×C61#39711yescharacteristic
290C4×C61#39711yescharacteristic
291Dic61—44no—
292Dic61—42no—
293Dic61—82no—
294C22×#104—611no—
295#2896—611no—
296C2×#413—611no—
297#1742—611no—
298C2×#339—611no—
299C2×#338D6111yescharacteristic, complemented
300#1728—611no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61#144011yescharacteristic
308C2×C61#205012yes—
309D61—82no—
310C22×#41—611no—
311C22×M16—611no—
312C22×M16C2×D6111yescharacteristic
313C22×M16C2×D6111yescharacteristic
314C22×M16C2×D6111yescharacteristic
315C2×#117—611no—
316C22×C42—611no—
317C2×#104—1222no—
318C2×#104—1222no—
319C2×#104—1222no—
320#419—1221no—
321#418—1221no—
322#417—1222no—
323#416—1222no—
324#413—614no—
325#413—1222no—
326C2×#101—1221no—
327C2×#100—1221no—
328#343—1222no—
329#339—1222no—
330#338—22no—
331#336—1222no—
332C61#1115111yescharacteristic, complemented, Sylow
333C2×#41—612no—
334C2×#41—612no—
335C2×#41—1221no—
336C2×#41—1221no—
337C2×#41—1221no—
338C2×#41—1221no—
339C2×#41—1221no—
340C2×#41—1221no—
341C2×#41—1221no—
342C2×#41—1221no—
343C2×#41—1221no—
344C2×#41—1221no—
345C22×Q8—611no—
346C22×D4—611no—
347C22×D4—1221no—
348C23×C4—1221no—
349C23×C4—1221no—
350C23×C4C22×D6111yescharacteristic
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—612no—
368C2×M16—1221no—
369C2×M16—1221no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1221no—
373C2×M16C4×D6112yes—
374C2×M16C22⋊D6112yes—
375C22×C8—21no—
376C22×C8—21no—
377C22×C8—1221no—
378C22×C8D24412yes—
379#117—1221no—
380#117—1221no—
381#117—1221no—
382#117—1221no—
383C2×C4⋊C4—612no—
384C2×C22⋊C4—1221no—
385C2×C42—612no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42—1221no—
392#104—1228no—
393#103—1224no—
394#102—1224no—
395#101—1224no—
396#100—1224no—
397C24—1221no—
398C24#1175822611yescharacteristic
399#41—1222no—
400#41—1222no—
401#41—1222no—
402#41—1222no—
403#41—1222no—
404#41—1222no—
405#41—1222no—
406#41—1222no—
407#41—2441no—
408#41—2441no—
409#41—2441no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41—2441no—
415C2×Q8—612no—
416C2×Q8—1221no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×D4—612no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1221no—
425C2×D4—1221no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—2441no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4—2441no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—2444no—
439C22×C4—2441no—
440C22×C4—2441no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2441no—
445C22×C4C2×C4×D6111yescharacteristic
446C22×C4C2×D24411yescharacteristic
447C22×C4#1175822211yescharacteristic
448C22×C4#1175822211yescharacteristic
449C22×C4#1175822211yescharacteristic
450C22×C4#1175822211yescharacteristic
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4#1175822411yescharacteristic
456C22×C4#1175822411yescharacteristic
457C22×C4#1175822611yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—1222no—
472M16—1222no—
473M16—2441no—
474M16—2441no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—1222no—
485C2×C8—2441no—
486C2×C8—2441no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C42—1222no—
492C42—1222no—
493C42—2441no—
494C42—2441no—
495C42—2441no—
496C42—2441no—
497C23—42no—
498C23—1222no—
499C23—4882no—
500C23—2441no—
501C23—2441no—
502C23#1451973511yescharacteristic
503C23#1451975211yescharacteristic
504C23#1451976211yescharacteristic
505C23#1451976211yescharacteristic
506C23#1451976211yescharacteristic
507C23#1451976611yescharacteristic
508C23#1451977211yescharacteristic
509Q8—1222no—
510Q8—1222no—
511Q8—2441no—
512Q8—2441no—
513D4—1222no—
514D4—1222no—
515D4—2442no—
516D4—2442no—
517D4—2442no—
518D4—2442no—
519D4—2441no—
520D4—2441no—
521C2×C4—1224no—
522C2×C4—1222no—
523C2×C4—4882no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2444no—
527C2×C4—2444no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4C22⋊C4×D6111yescharacteristic
536C2×C4#1451973411yescharacteristic
537C2×C4#1451973711yescharacteristic
538C2×C4C4×C22⋊D6111yescharacteristic
539C2×C4#1451973911yescharacteristic
540C2×C4C4×D24411yescharacteristic
541C2×C4#1451974511yescharacteristic
542C2×C4#1451974711yescharacteristic
543C2×C4#1451974811yescharacteristic
544C2×C4#1451974911yescharacteristic
545C2×C4#1451975011yescharacteristic
546C2×C4#1451975011yescharacteristic
547C2×C4#1451975011yescharacteristic
548C2×C4#1451975111yescharacteristic
549C2×C4#1451975211yescharacteristic
550C2×C4#1451975911yescharacteristic
551C2×C4#1451975911yescharacteristic
552C2×C4#1451975911yescharacteristic
553C2×C4#1451976011yescharacteristic
554C2×C4#1451976111yescharacteristic
555C2×C4#1451976211yescharacteristic
556C2×C4#1451976211yescharacteristic
557C2×C4#1451976311yescharacteristic
558C2×C4#1451976311yescharacteristic
559C2×C4#1451976511yescharacteristic
560C2×C4#1451977111yescharacteristic
561C2×C4#1451977211yescharacteristic
562C2×C4#1451977211yescharacteristic
563C8—42no—
564C8—42no—
565C8—42no—
566C8—2442no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—4882no—
575C22—4882no—
576C22—4882no—
577C22—2442no—
578C22#1683962411yescharacteristic
579C22#1683965211yescharacteristic
580C22#1683965411yescharacteristic
581C22#1683967911yescharacteristic
582C22#1683968011yescharacteristic
583C22#1683968811yescharacteristic
584C22#1683969111yescharacteristic
585C4—4882no—
586C4—2444no—
587C4—2442no—
588C4#1683963811yescharacteristic
589C4#1683964011yescharacteristic
590C4#1683965111yescharacteristic
591C4#1683965311yescharacteristic
592C4#1683967711yescharacteristic
593C4#1683967811yescharacteristic
594C4#1683968911yescharacteristic
595C4#1683969111yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—4882no—
603C2#2144406711yescharacteristic
604C2#2144710712yes—
subgroupG/H
centerC2×C4#14519765
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#338C2
socleC22×C61#394
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenterC2×#338D61
hypocenterC61#11151
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#338

Complements

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

NH ≅ G/Ntype
4. #21445557C2semidirect
5. C2×#16839523C2semidirect
6. #21445512C2semidirect
7. C2×C61×#338C2semidirect
20. #16839744C4semidirect
299. C2×#338D61semidirect
332. C61#11151semidirect

Maximal subgroups

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

subgroupindex
C22×#145197032
#214543392
C2×#168397442
#214455572
C2×#168395232
#214455122
C2×C61×#3382
#1115161