Finite Group Databasebeta

← #29696203

Subgroups — #29696203

Order 15616.

1132 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
1C2×#16840462C211yesmaximal, characteristic
2M16×D244C212yesmaximal, complemented
3#21446812C212yesmaximal, complemented
4#21446279C212yesmaximal, complemented
5#21446279C212yesmaximal, complemented
6#21445901C211yesmaximal, characteristic, complemented
7#21445898C211yesmaximal, characteristic, complemented
8#21445811C211yesmaximal, characteristic, complemented
9#21445556C211yesmaximal, characteristic, complemented
10#343×D61C211yesmaximal, characteristic, complemented
11C4×#100×C61C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13C2×M16×D61C2212yes—
14#16840462—22no—
15#16840462—22no—
16#16840462—22no—
17#16840462—22no—
18C2×#14519770C412yes—
19C2×#14519769C2211yescharacteristic
20C2×#14519760C412yes—
21C2×#14519750C412yes—
22C2×#14519748C2211yescharacteristic
23C2×#14519748C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×C4×C22⋊D61C412yes—
26C2×#14519735C2211yescharacteristic
27C2×C4⋊C4×D61C2211yescharacteristic
28#16839841—22no—
29#16839835—22no—
30#16839835—22no—
31C8×D244—22no—
32#16839786C2212yes—
33C2×#14519694C2212yes—
34#16839725C2212yes—
35#16839725C2212yes—
36#16839531C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839530C2212yes—
39#16839530C2212yes—
40#16839530C2212yes—
41#16839530C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#100×C61C2212yes—
44C4×M16×C61C2212yes—
45C61×#343C2211yescharacteristic
46C61×#343C2211yescharacteristic
47C22×D244C2×C411yescharacteristic
48C22×C22⋊D61C2×C412yes—
49C22×C22⋊D61C2×C412yes—
50C22×C4×D61C2×C412yes—
51C22×C4×D61C2311yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C2×#11758194C2312yes—
54C2×#11758194C2312yes—
55C2×#11758189—22no—
56C2×#11758189—22no—
57M16×D61—24no—
58M16×D61—24no—
59M16×D61—42no—
60M16×D61—42no—
61C2×C8×D61—22no—
62C2×C8×D61—22no—
63#14519770—24no—
64#14519770—24no—
65#14519769—22no—
66#14519769—22no—
67#14519760—24no—
68#14519760—24no—
69#14519750—24no—
70#14519750—24no—
71#14519748—22no—
72#14519748—22no—
73#14519748—22no—
74#14519748—22no—
75C4×D244D412yes—
76C4×D244D412yes—
77C4×D244D412yes—
78C4×D244D412yes—
79C4×C22⋊D61—22no—
80C4×C22⋊D61—22no—
81C4×C22⋊D61—22no—
82C4×C22⋊D61—22no—
83#14519735—22no—
84#14519735—22no—
85C2×C4⋊Dic61C2×C411yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196C2×C412yes—
89C2×#11758196C2×C412yes—
90C2×#11758196C2×C412yes—
91C4⋊C4×D61—22no—
92C4⋊C4×D61—22no—
93C2×C22⋊Dic61C2×C412yes—
94C2×C22⋊Dic61C2×C412yes—
95C2×#11758198C2×C412yes—
96C2×#11758198C2×C411yescharacteristic
97C2×#11758198C2×C411yescharacteristic
98C2×C4×Dic61C2×C412yes—
99C4⋊#92418—22no—
100C4⋊#92418—22no—
101#14519694—42no—
102#14519689—22no—
103#14519689—22no—
104#14519689—22no—
105#14519689—22no—
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61C2×C412yes—
108C2×M16×C61C2312yes—
109C2×M16×C61C2312yes—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61C2×C412yes—
114C2×C42×C61C2311yescharacteristic
115#100×C61—24no—
116#97×C61—22no—
117C4×C8×C61—22no—
118C24×C61C22×C412yes—
119C23×D61C22×C411yescharacteristic
120C23×D61C22×C411yescharacteristic
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—82no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—22no—
133C2×C22⋊D61—22no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—44no—
136C2×C22⋊D61—82no—
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C412yes—
140C22×Dic61C22×C412yes—
141C2×D244—42no—
142C2×D244C22⋊C412yes—
143C2×D244C22⋊C412yes—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—82no—
153C2×C4×D61C2×D412yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61C2×D412yes—
156C2×C4×D61C2×D412yes—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—22no—
160M16×C61#4114yes—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—22no—
164C22⋊C4×C61—24no—
165C42×C61C2×D412yes—
166C42×C61C2×D412yes—
167C22⋊Dic61—22no—
168C22⋊Dic61—22no—
169C22⋊Dic61—24no—
170#11758198—24no—
171#11758198C22⋊C411yescharacteristic
172#11758198C22⋊C411yescharacteristic
173#11758198C22⋊C411yescharacteristic
174#11758198C22⋊C411yescharacteristic
175#11758198C22⋊C412yes—
176#11758198C22⋊C412yes—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61C22⋊C412yes—
180C4⋊Dic61C22⋊C412yes—
181#11758196—22no—
182#11758196—22no—
183#11758196—24no—
184#11758196—24no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—22no—
190#11758194—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193#11758189—44no—
194#11758189—44no—
195C8×D61—44no—
196C8×D61—44no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61C23×C411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—22no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61C2×C22⋊C412yes—
212C22×D61C2×C22⋊C411yescharacteristic
213C22×D61C2×C22⋊C411yescharacteristic
214D4×C61—42no—
215D4×C61—42no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—42no—
220C2×C4×C61—82no—
221C2×C4×C61—82no—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C22×D412yes—
226C2×C4×C61C2×#4112yes—
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C22⋊D61—44no—
230C22⋊D61—44no—
231C22⋊D61—84no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C412yes—
243D244—24no—
244D244—42no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—82no—
254C4×D61—82no—
255C8×C61—24no—
256C8×C61—42no—
257#92418—42no—
258#92418—42no—
259#22482—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—82no—
266C22×C61—82no—
267C22×C61C22×C22⋊C411yescharacteristic
268C22×C61C4×#4111yescharacteristic
269C22×C61#52011yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—22no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61#41312yes—
283C4×C61—22no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—42no—
287C4×C61#54612yes—
288C4×C61#54712yes—
289Dic61—21no—
290Dic61—22no—
291Dic61—21no—
292Dic61—42no—
293Dic61—42no—
294C2×#553—611no—
295D4×M16—612no—
296#1998—612no—
297#1900—612no—
298#1900—612no—
299#1822—611no—
300#1821—611no—
301#1800—611no—
302C4×#100D6111yescharacteristic, complemented
303#1742—611no—
304C2×#343—611no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—82no—
308C2×C61#233911yescharacteristic
309C2×C61C2×#41311yescharacteristic
310C2×C61#288811yescharacteristic
311D61—22no—
312D61—41no—
313D61—42no—
314C22×M16—612no—
315C22×M16—612no—
316#553—1222no—
317#553—1222no—
318#553—1222no—
319#553—1222no—
320C2×#126—612no—
321C2×#125—611no—
322C2×#123—612no—
323C2×#121—611no—
324C2×#121—611no—
325C2×#121—612no—
326C2×C4×D4—612no—
327C2×C4×D4—611no—
328C2×C4×D4C2×D6111yescharacteristic
329C2×#117—611no—
330C22×C4⋊C4—611no—
331#436—1222no—
332#435—1222no—
333#435—1222no—
334C8×D4—1222no—
335#423—612no—
336C2×#100—612no—
337C2×#100C2×D6112yes—
338#407—612no—
339#407—612no—
340C4×M16C2×D6112yes—
341#343—612no—
342#343—612no—
343#343—612no—
344#343—612no—
345#343—611no—
346#343—611no—
347#343C2×D6111yescharacteristic
348#343C2×D6111yescharacteristic
349C61#2248211yescharacteristic, complemented, Sylow
350C22×D4—612no—
351C22×D4—612no—
352C22×D4—611no—
353C22×D4C22×D6111yescharacteristic
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—612no—
357C23×C4C4×D6112yes—
358C2×M16—612no—
359C2×M16—612no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16—1224no—
363C2×M16—1224no—
364C2×M16—2442no—
365C2×M16—2442no—
366C2×M16C22×D6112yes—
367C2×M16C22×D6112yes—
368C22×C8—1222no—
369C22×C8—1222no—
370#126—1224no—
371#126—1224no—
372#125—1222no—
373#125—1222no—
374#123—1224no—
375#123—1224no—
376#121—1224no—
377#121—1224no—
378#121—1222no—
379#121—1222no—
380#121—1222no—
381#121—1222no—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—612no—
385C4×D4—612no—
386C4×D4—612no—
387C4×D4—612no—
388C4×D4—1222no—
389C4×D4—1222no—
390C4×D4—1222no—
391C4×D4—1222no—
392#117—1222no—
393#117—1222no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C22⋊C4—611no—
404C2×C22⋊C4—611no—
405C2×C22⋊C4—612no—
406C2×C22⋊C4—612no—
407C2×C22⋊C4—612no—
408C2×C22⋊C4C4×D6112yes—
409C2×C42—612no—
410C2×C42C22×D6111yescharacteristic
411C4⋊C8—1222no—
412C4⋊C8—1222no—
413#100—24no—
414#100—2442no—
415C22⋊C8—1222no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418C22⋊C8—1222no—
419#97—22no—
420C4×C8—22no—
421C24—611no—
422C24—611no—
423C24C2×C4×D6112yes—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—612no—
429C2×D4—612no—
430C2×D4—1224no—
431C2×D4—1222no—
432C2×D4—1222no—
433C2×D4—4882no—
434C2×D4—2442no—
435C2×D4—2444no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—82no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—4882no—
456C22×C4C2×C4×D6112yes—
457C22×C4C2×C4×D6111yescharacteristic
458C22×C4C2×C4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460M16—22no—
461M16—22no—
462M16—22no—
463M16—1222no—
464M16—1222no—
465M16—1222no—
466M16—1222no—
467M16—2444no—
468M16—2444no—
469M16#1175822214yes—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—2444no—
475C2×C8—2444no—
476C4⋊C4—22no—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—1224no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1224no—
485C22⋊C4—24no—
486C22⋊C4—611no—
487C22⋊C4—611no—
488C22⋊C4—611no—
489C22⋊C4—611no—
490C22⋊C4—612no—
491C22⋊C4—612no—
492C22⋊C4—1224no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1224no—
496C42—1222no—
497C42—1222no—
498C42D4×D6112yes—
499C42D4×D6112yes—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—42no—
504C23—42no—
505C23—611no—
506C23—612no—
507C23—611no—
508C23—1222no—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23C22×C4×D6111yescharacteristic
517D4—42no—
518D4—42no—
519D4—1224no—
520D4—4884no—
521D4—2442no—
522D4—2444no—
523D4—2444no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—612no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—4882no—
548C2×C4—4882no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4C22⋊C4×D6112yes—
552C2×C4C22⋊C4×D6111yescharacteristic
553C2×C4C22⋊C4×D6111yescharacteristic
554C2×C4C2×D4×D6112yes—
555C2×C4C2×#1175822212yes—
556C2×C4#1451982311yescharacteristic
557C2×C4#1451982411yescharacteristic
558C8—24no—
559C8—42no—
560C8—2442no—
561C8—2442no—
562C22—21no—
563C22—21no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—612no—
570C22—1221no—
571C22—1221no—
572C22—1222no—
573C22—1221no—
574C22—1222no—
575C22—1221no—
576C22—1221no—
577C22—1222no—
578C22—4881no—
579C22—4881no—
580C22—2442no—
581C22—2442no—
582C22C2×C22⋊C4×D6111yescharacteristic
583C22C4×#1175822211yescharacteristic
584C22#1684024211yescharacteristic
585C4—22no—
586C4—21no—
587C4—21no—
588C4—42no—
589C4—1221no—
590C4—1221no—
591C4—1222no—
592C4—2442no—
593C4—2442no—
594C4#1684040812yes—
595C4#1684041712yes—
596C2—21no—
597C2—21no—
598C2—82no—
599C2—1222no—
600C2—2442no—
601C2—2441no—
602C2#2144865011yescharacteristic
603C2#413×D6111yescharacteristic
604C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#100×C61C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840462C2
cosocleC22×C4×C61C24
hypercenterC4×#100D61
hypocenterC61#22482
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. M16×D244C2semidirect
3. #21446812C2semidirect
4. #21446279C2semidirect
5. #21446279C2semidirect
6. #21445901C2semidirect
7. #21445898C2semidirect
8. #21445811C2semidirect
9. #21445556C2semidirect
10. #343×D61C2semidirect
11. C4×#100×C61C2semidirect
302. C4×#100D61semidirect
349. C61#22482semidirect

Maximal subgroups

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

subgroupindex
C2×#168404622
M16×D2442
#214468122
#214462792
#214459012
#214458982
#214458112
#214455562
#343×D612
C4×#100×C612
#2248261