Finite Group Databasebeta

← #29707112

Subgroups — #29707112

Order 15616.

2098 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
1C22×#14519803C211yesmaximal, characteristic
2C23×#11758200C211yesmaximal, characteristic, complemented
3C2×#16840338C211yesmaximal, characteristic, complemented
4C2×#16839770C211yesmaximal, characteristic, complemented
5C2×#16839756C211yesmaximal, characteristic, complemented
6#21446901C214yesmaximal, complemented
7#21446895C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C61×#2309C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×Dic122C2211yescharacteristic
12C2×#14519803—24no—
13C2×#14519803—24no—
14C2×#14519803—24no—
15C22×#11758200—24no—
16C22×#11758200—24no—
17C22×#11758200—24no—
18C22×#11758200C2214yes—
19C22×#11758200C2212yes—
20C22×#11758199C2212yes—
21C22×#11758194C2211yescharacteristic
22C23×#92418C2211yescharacteristic
23#16840338—24no—
24#16840338—24no—
25C2×#14519777C2212yescomplemented
26C2×#14519765C2212yescomplemented
27C2×#14519761—22no—
28C2×#14519757—22no—
29#16839770—24no—
30#16839756—24no—
31C2×#14519700C2212yescomplemented
32C2×#14519700C2214yescomplemented
33C2×#14519697C2212yescomplemented
34#16839517C2214yescomplemented
35C23×D4×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#390—22no—
39C61×#390C2214yes—
40C61×#379—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222D414yes—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122—22no—
50C22×Dic122—22no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122D414yes—
54C22×Dic122C2312yes—
55C22×Dic122C2311yescharacteristic
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—21no—
58C23×Dic61—21no—
59C22×C4×D61—21no—
60#14519803—232no—
61C2×#11758200—232no—
62C2×#11758200—28no—
63C2×#11758200—24no—
64C2×#11758200—28no—
65C2×#11758200—24no—
66C2×#11758200—28no—
67C2×#11758200—24no—
68C2×#11758199—24no—
69C2×#11758199—24no—
70C2×#11758199—24no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194D414yes—
74C22×#92418—22no—
75C22×#92418—22no—
76C22×#92418D414yes—
77C22×#92418C2312yes—
78C22×#92418C2312yes—
79#14519777—24no—
80#14519777—24no—
81#14519765—24no—
82#14519765—24no—
83#14519761—216no—
84#14519757—216no—
85C2×C4⋊Dic61—24no—
86C2×#11758198—24no—
87#14519700—24no—
88#14519700—28no—
89#14519697—24no—
90C25×C61—21no—
91C22×D4×C61—22no—
92C22×D4×C61—22no—
93C22×D4×C61—22no—
94C22×D4×C61—22no—
95C22×D4×C61—21no—
96C22×D4×C61—22no—
97C22×D4×C61D414yes—
98C22×D4×C61C2312yes—
99C23×C4×C61C2311yescharacteristic
100#117×C61—22no—
101#117×C61—22no—
102#117×C61—22no—
103#117×C61—22no—
104C2×C4⋊C4×C61D412yes—
105C2×C4⋊C4×C61C2312yes—
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—28no—
108C2×C22⋊C4×C61D412yes—
109C2×C42×C61—21no—
110C2×C42×C61C2312yes—
111C24×C61—22no—
112C24×C61—24no—
113C24×C61—42no—
114C24×C61—42no—
115C24×C61—84no—
116C24×C61C2×D411yescharacteristic
117C23×D61—21no—
118C2×D4×C61—28no—
119C2×D4×C61—28no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—42no—
127C2×D4×C61—44no—
128C2×D4×C61—44no—
129C2×D4×C61—42no—
130C2×D4×C61C2×D414yes—
131C22×C4×C61—22no—
132C22×C4×C61C2×D414yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61#4112yes—
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C22×Dic61—24no—
145C22×Dic61—21no—
146C22×Dic61—22no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149#11758222—28no—
150#11758222—28no—
151#11758222—44no—
152#11758222—44no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244C2×D412yes—
159C2×C4×D61—24no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×Dic122—28no—
163C2×Dic122—28no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—22no—
171C2×Dic122—21no—
172C2×Dic122—24no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122—44no—
176C2×Dic122—44no—
177C2×Dic122—42no—
178C2×Dic122—42no—
179C2×Dic122C2×D414yes—
180C2×Dic122C2×D412yes—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—88no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61—42no—
190#11758200—232no—
191#11758200—216no—
192#11758199—216no—
193#11758198—216no—
194C4⋊Dic61—216no—
195#11758194—28no—
196C2×#92418—28no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C2×#92418C2×D414yes—
202C2×#92418C2×D414yes—
203C23×C61—24no—
204C23×C61—24no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61—48no—
208C23×C61—48no—
209C23×C61—88no—
210C23×C61—88no—
211C23×C61—88no—
212C23×C61#12012yes—
213C23×C61#12112yes—
214C23×C61#12712yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—44no—
220D4×C61—24no—
221D4×C61—24no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—48no—
225C2×C4×C61—24no—
226C2×C4×C61—28no—
227C2×C4×C61—42no—
228C2×C4×C61—44no—
229C2×C4×C61#12014yes—
230C2×C4×C61#12014yes—
231C2×C4×C61#12012yes—
232C2×C4×C61#12114yes—
233C2×C4×C61#12114yes—
234C2×C4×C61#12112yes—
235C2×C4×C61#12712yes—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—48no—
243C2×Dic61—22no—
244C2×Dic61—24no—
245C2×Dic61—42no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—41no—
249C2×Dic61—48no—
250D244—22no—
251D244—22no—
252D244—41no—
253D244—41no—
254C4×D61—48no—
255Dic122—24no—
256Dic122—22no—
257Dic122—24no—
258Dic122—22no—
259Dic122—42no—
260Dic122—41no—
261Dic122—42no—
262Dic122—41no—
263Dic122—48no—
264#92418—24no—
265#92418—24no—
266#23424—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—44no—
271C22×C61—48no—
272C22×C61—816no—
273C22×C61—84no—
274C22×C61—84no—
275C22×C61#39214yes—
276C22×C61C2×#12011yescharacteristic
277C22×C61C2×#12111yescharacteristic
278C22×C61C2×#12711yescharacteristic
279C22×C61#57714yes—
280C2×D61—42no—
281C2×D61—44no—
282C2×D61—82no—
283C2×D61—82no—
284C4×C61—44no—
285C4×C61—82no—
286C4×C61#39214yes—
287C4×C61C2×#12011yescharacteristic
288C4×C61C2×#12011yescharacteristic
289C4×C61C2×#12111yescharacteristic
290C4×C61C2×#12111yescharacteristic
291Dic61—44no—
292Dic61—42no—
293C22×AGL(1,ℤ/8)—611no—
294C23×SD16—611no—
295C2×#536—611no—
296C2×#419—611no—
297C2×#416—611no—
298#2309D6111yescharacteristic, complemented
299#2013—614no—
300#2012—614no—
301#1729—611no—
302C2×C61—24no—
303C2×C61—84no—
304C2×C61C2×#39211yescharacteristic
305C2×C61#301012yes—
306C2×C61#304212yes—
307C2×C61#315412yes—
308D61—82no—
309C22×#41—611no—
310C23×Q8—611no—
311C23×D4C2×D6111yescharacteristic
312C2×AGL(1,ℤ/8)—1224no—
313C2×AGL(1,ℤ/8)—1224no—
314C2×AGL(1,ℤ/8)—1224no—
315C22×SD16—614no—
316C22×SD16—612no—
317C22×SD16—1224no—
318C22×SD16—1224no—
319C22×SD16—1224no—
320C22×D8—612no—
321C22×M16—611no—
322C23×C8—611no—
323#536—1224no—
324#536—1224no—
325C2×C4⋊Q8—612no—
326C2×#124—612no—
327C2×#123—1222no—
328C2×C22⋊Q8—1222no—
329C2×#117C2×D6111yescharacteristic
330C2×#117C2×D6111yescharacteristic
331#419—1224no—
332#416—1224no—
333C2×#103—612no—
334C2×#103—614no—
335C2×#102—612no—
336#390—22no—
337#390C2×D6114yes—
338#379—22no—
339#336—614no—
340C61#2342411yescharacteristic, complemented, Sylow
341C25—21no—
342C2×#41—614no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—1222no—
347C2×#41—1222no—
348C22×Q8—614no—
349C22×Q8—612no—
350C22×Q8—611no—
351C22×Q8—1222no—
352C22×Q8—1221no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8—1222no—
356C22×Q8—1222no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—21no—
363C22×D4—611no—
364C22×D4—1221no—
365C22×D4D24414yes—
366C22×D4C22×D6112yes—
367C23×C4—1221no—
368C23×C4—1221no—
369C23×C4C22×D6111yescharacteristic
370AGL(1,ℤ/8)—12232no—
371C2×SD16—1228no—
372C2×SD16—1224no—
373C2×SD16—1228no—
374C2×SD16—1224no—
375C2×SD16—12232no—
376C2×SD16—1228no—
377C2×SD16—1224no—
378C2×D8—1224no—
379C2×D8—1224no—
380C2×D8—1224no—
381C2×M16—614no—
382C2×M16—1222no—
383C2×M16—1222no—
384C22×C8—614no—
385C22×C8—612no—
386C22×C8—612no—
387C22×C8—1222no—
388C22×C8—1222no—
389C4⋊Q8—1224no—
390C4⋊Q8—1224no—
391#124—1224no—
392#124—1224no—
393#123—12216no—
394C22⋊Q8—12216no—
395#117—22no—
396#117—22no—
397#117—22no—
398#117—22no—
399C2×C4⋊C4—1224no—
400C2×C4⋊C4C22⋊D6112yes—
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—28no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4C22⋊D6112yes—
406C2×C42—21no—
407C2×C42C22×D6112yes—
408#103—1224no—
409#103—1228no—
410#102—1224no—
411C24—22no—
412C24—24no—
413C24—42no—
414C24—42no—
415C24—84no—
416C24—1221no—
417C24C2×C22⋊D6111yescharacteristic
418#41—1228no—
419#41—1228no—
420#41—2444no—
421#41—2444no—
422C2×Q8—614no—
423C2×Q8—612no—
424C2×Q8—1222no—
425C2×Q8—1221no—
426C2×Q8—1224no—
427C2×Q8—1222no—
428C2×Q8—1221no—
429C2×Q8—1222no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1221no—
433C2×Q8—1222no—
434C2×Q8—1221no—
435C2×Q8—1228no—
436C2×Q8—1228no—
437C2×Q8—2444no—
438C2×Q8—2444no—
439C2×Q8—2442no—
440C2×Q8—2442no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—28no—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—612no—
454C2×D4—1224no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—2442no—
461C2×D4—2442no—
462C2×D4C2×D24414yes—
463C22×C4—22no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1224no—
467C22×C4—1224no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4C2×D24412yes—
473C22×C4#1175822212yes—
474C22×C4D4×D6112yes—
475C22×C4D4×D6112yes—
476C22×C4D4×D6112yes—
477C22×C4D4×D6112yes—
478C22×C4C2×C22⋊D6114yes—
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C23×D6111yescharacteristic
481SD16—12232no—
482SD16—12216no—
483D8—12216no—
484M16—1228no—
485C2×C8—614no—
486C2×C8—614no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—1228no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—12216no—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—88no—
499C22⋊C4—12216no—
500C42—22no—
501C42—22no—
502C42—42no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—48no—
508C23—48no—
509C23—88no—
510C23—88no—
511C23—88no—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23—2444no—
516C23#1451974512yes—
517C23#1451975112yes—
518C23#1451977312yes—
519C23C22×C22⋊D6111yescharacteristic
520Q8—1224no—
521Q8—1222no—
522Q8—1224no—
523Q8—1222no—
524Q8—2442no—
525Q8—2441no—
526Q8—2442no—
527Q8—2441no—
528Q8—2448no—
529D4—24no—
530D4—24no—
531D4—42no—
532D4—42no—
533D4—48no—
534D4—1222no—
535D4—1222no—
536D4—2448no—
537D4—2441no—
538D4—2441no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—1224no—
544C2×C4—1222no—
545C2×C4—2441no—
546C2×C4—2442no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2448no—
550C2×C4—2448no—
551C2×C4#1451974512yes—
552C2×C4#1451974714yes—
553C2×C4#1451974714yes—
554C2×C4#1451975014yes—
555C2×C4#1451975014yes—
556C2×C4#1451975112yes—
557C2×C4#1451977312yes—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×D24411yescharacteristic
563C2×C4C2×#1175822211yescharacteristic
564C8—1224no—
565C8—1224no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—44no—
570C22—48no—
571C22—816no—
572C22—84no—
573C22—84no—
574C22—4882no—
575C22—4882no—
576C22—2442no—
577C22—2444no—
578C22#1683967114yes—
579C22C2×#1451974511yescharacteristic
580C22C2×#1451975111yescharacteristic
581C22C2×#1451977311yescharacteristic
582C22#1684057314yes—
583C4—44no—
584C4—82no—
585C4—2442no—
586C4—2444no—
587C4#1683967114yes—
588C4C2×#1451974711yescharacteristic
589C4C2×#1451974711yescharacteristic
590C4C2×#1451975011yescharacteristic
591C4C2×#1451975011yescharacteristic
592C2—24no—
593C2—84no—
594C2—4882no—
595C2C2×#1683967111yescharacteristic
596C2#2145510312yes—
597C2#2145533112yes—
598C2#2145629112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2309C2
socleC23×C61C22×D4
prime radicalC22×#14519803C2
hypercenter#2309D61
hypocenterC61#23424
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. C23×#11758200C2semidirect
3. C2×#16840338C2semidirect
4. C2×#16839770C2semidirect
5. C2×#16839756C2semidirect
6. #21446901C2semidirect
7. #21446895C2semidirect
8. #21445516C2semidirect
9. C61×#2309C2semidirect
25. C2×#14519777C22semidirect
26. C2×#14519765C22semidirect
31. C2×#14519700C22semidirect
32. C2×#14519700C22semidirect
33. C2×#14519697C22semidirect
34. #16839517C22semidirect
298. #2309D61semidirect
340. C61#23424semidirect

Maximal subgroups

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

subgroupindex
C22×#145198032
C23×#117582002
C2×#168403382
C2×#168397702
C2×#168397562
#214469012
#214468952
#214455162
C61×#23092
#2342461