Finite Group Databasebeta

← #29707122

Subgroups — #29707122

Order 15616.

1938 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519807C211yesmaximal, characteristic
2C23×#11758202C211yesmaximal, characteristic, complemented
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839758C211yesmaximal, characteristic
5#21448403C211yesmaximal, characteristic, complemented
6#21446919C214yesmaximal, complemented
7#21446913C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C2×C61×#536C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11C2×#14519807—24no—
12C2×#14519807—24no—
13C2×#14519807—24no—
14C22×#11758202—24no—
15C22×#11758202—24no—
16C22×#11758202—24no—
17C22×#11758202C2214yes—
18C22×#11758202C2212yes—
19C22×#11758200C2212yes—
20C22×#11758194C2211yescharacteristic
21C23×#92418C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839771—24no—
24#16839758—24no—
25C2×#14519701C2214yes—
26C2×#14519701C2212yes—
27C2×#14519698C2212yes—
28#16839668C2214yes—
29#16839667—22no—
30#16839623—22no—
31#16839517C2214yes—
32C22×#41×C61C2211yescharacteristic
33C23×Q8×C61C2211yescharacteristic
34C61×#536—24no—
35C61×#536—24no—
36C2×C4⋊Q8×C61C2212yes—
37C2×#124×C61C2212yes—
38C2×#123×C61—22no—
39C2×C22⋊Q8×C61—22no—
40C2×#117×C61C2211yescharacteristic
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—22no—
45C22×Dic122—21no—
46C22×Dic122—22no—
47C22×Dic122D414yes—
48C22×Dic122C2312yes—
49C23×Dic61—21no—
50#14519807—232no—
51C2×#11758202—232no—
52C2×#11758202—28no—
53C2×#11758202—24no—
54C2×#11758202—28no—
55C2×#11758202—24no—
56C2×#11758202—28no—
57C2×#11758202—24no—
58C2×#11758200—24no—
59C2×#11758200—24no—
60C2×#11758200—24no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194D414yes—
64C22×#92418—22no—
65C22×#92418—22no—
66C22×#92418D414yes—
67C22×#92418C2312yes—
68C22×#92418C2312yes—
69#14519736—22no—
70#14519736—22no—
71C2×C4⋊Dic61C2312yes—
72C2×C22⋊Dic61—21no—
73C2×C4×Dic61—21no—
74#14519701—24no—
75#14519701—28no—
76#14519698—24no—
77#14519684—28no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61D414yes—
84C22×Q8×C61—22no—
85C22×Q8×C61—21no—
86C22×Q8×C61—22no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61D414yes—
91C22×Q8×C61C2311yescharacteristic
92C22×Q8×C61C2312yes—
93C22×D4×C61—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C4⋊Q8×C61—24no—
99C4⋊Q8×C61—24no—
100#124×C61—24no—
101#124×C61—24no—
102#123×C61—216no—
103C22⋊Q8×C61—216no—
104#117×C61—22no—
105#117×C61—22no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61D412yes—
108C2×C22⋊C4×C61—24no—
109C2×C22⋊C4×C61D412yes—
110C2×C42×C61C2312yes—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113#41×C61—28no—
114#41×C61—28no—
115#41×C61—44no—
116#41×C61—44no—
117C2×Q8×C61—24no—
118C2×Q8×C61—22no—
119C2×Q8×C61—21no—
120C2×Q8×C61—28no—
121C2×Q8×C61—28no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—21no—
130C2×Q8×C61—44no—
131C2×Q8×C61—42no—
132C2×Q8×C61—44no—
133C2×Q8×C61—42no—
134C2×Q8×C61C2×D414yes—
135C2×Q8×C61C2×D412yes—
136C2×D4×C61—24no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—42no—
143C2×D4×C61—42no—
144C2×D4×C61C2×D412yes—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—21no—
148C22×C4×C61—24no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61C2×D414yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2×D412yes—
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61#4112yes—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—24no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C2×Dic122—28no—
168C2×Dic122—28no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—24no—
174C2×Dic122—22no—
175C2×Dic122—42no—
176C2×Dic122—44no—
177C2×Dic122—44no—
178C2×Dic122—42no—
179C2×Dic122C2×D414yes—
180C4⋊C4×C61—216no—
181C4⋊C4×C61—24no—
182C22⋊C4×C61—216no—
183C22⋊C4×C61—24no—
184C42×C61—22no—
185C42×C61—22no—
186C22⋊Dic61—42no—
187#11758202—232no—
188#11758202—216no—
189#11758200—216no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4×Dic61—42no—
193#11758194—28no—
194C2×#92418—28no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418C2×D414yes—
200C2×#92418C2×D414yes—
201C23×C61—24no—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—44no—
206C23×C61#12012yes—
207C23×C61#12112yes—
208C23×C61#12712yes—
209C23×C61C22×D411yescharacteristic
210Q8×C61—24no—
211Q8×C61—22no—
212Q8×C61—24no—
213Q8×C61—22no—
214Q8×C61—42no—
215Q8×C61—41no—
216Q8×C61—42no—
217Q8×C61—41no—
218Q8×C61—48no—
219D4×C61—22no—
220D4×C61—22no—
221D4×C61—48no—
222D4×C61—41no—
223D4×C61—41no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—28no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—48no—
233C2×C4×C61—48no—
234C2×C4×C61#12014yes—
235C2×C4×C61#12014yes—
236C2×C4×C61#12012yes—
237C2×C4×C61#12114yes—
238C2×C4×C61#12114yes—
239C2×C4×C61#12112yes—
240C2×C4×C61#12712yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×Dic61—24no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—48no—
251C2×Dic61—42no—
252C2×Dic61—44no—
253Dic122—24no—
254Dic122—24no—
255Dic122—42no—
256Dic122—42no—
257Dic122—48no—
258#92418—24no—
259#92418—24no—
260#23425—611nomaximal, Sylow
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—24no—
264C22×C61—42no—
265C22×C61—44no—
266C22×C61—82no—
267C22×C61—82no—
268C22×C61#39214yes—
269C22×C61C2×#12011yescharacteristic
270C22×C61C2×#12111yescharacteristic
271C22×C61C2×#12711yescharacteristic
272C22×C61#57814yes—
273C4×C61—42no—
274C4×C61—44no—
275C4×C61—44no—
276C4×C61#39214yes—
277C4×C61C2×#12011yescharacteristic
278C4×C61C2×#12011yescharacteristic
279C4×C61C2×#12111yescharacteristic
280C4×C61C2×#12111yescharacteristic
281Dic61—44no—
282Dic61—82no—
283C22×#137—611no—
284C23×Q16—611no—
285C2×#536D6111yescharacteristic
286C2×#419—611no—
287C2×#416—611no—
288#2310—611no—
289#2016—614no—
290#2015—614no—
291#1729—611no—
292C2×C61—24no—
293C2×C61—82no—
294C2×C61C2×#39211yescharacteristic
295C2×C61#301112yes—
296C2×C61#304412yes—
297C2×C61#315612yes—
298C22×#41C2×D6111yescharacteristic
299C23×Q8—611no—
300C23×Q8C2×D6111yescharacteristic
301C2×#137—1224no—
302C2×#137—1224no—
303C2×#137—1224no—
304C22×Q16—614no—
305C22×Q16—612no—
306C22×Q16—1224no—
307C22×Q16—1224no—
308C22×Q16—1224no—
309C22×SD16—612no—
310C22×M16—611no—
311C23×C8—611no—
312#536—24no—
313#536—24no—
314C2×C4⋊Q8C2×D6112yes—
315C2×#124C2×D6112yes—
316C2×#123—22no—
317C2×C22⋊Q8—22no—
318C2×#117—611no—
319C2×#117C2×D6111yescharacteristic
320#419—1224no—
321#416—1224no—
322C2×#103—612no—
323C2×#103—614no—
324C2×#102—612no—
325#391—614no—
326#391—1222no—
327#381—1222no—
328#336—614no—
329C61#2342511yescharacteristic, complemented, Sylow
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41C22⋊D6114yes—
336C22×Q8—21no—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—614no—
343C22×Q8—612no—
344C22×Q8—1222no—
345C22×Q8—1221no—
346C22×Q8—1222no—
347C22×Q8—1222no—
348C22×Q8—1222no—
349C22×Q8—1222no—
350C22×Q8C22⋊D6114yes—
351C22×Q8C22×D6112yes—
352C22×Q8C22×D6111yescharacteristic
353C22×D4—21no—
354C22×D4C22×D6111yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—1221no—
358C23×C4C22×D6111yescharacteristic
359#137—12232no—
360C2×Q16—1228no—
361C2×Q16—1224no—
362C2×Q16—1228no—
363C2×Q16—1224no—
364C2×Q16—12232no—
365C2×Q16—1228no—
366C2×Q16—1224no—
367C2×SD16—1224no—
368C2×SD16—1224no—
369C2×SD16—1224no—
370C2×M16—614no—
371C2×M16—1222no—
372C2×M16—1222no—
373C22×C8—614no—
374C22×C8—612no—
375C22×C8—612no—
376C22×C8—1222no—
377C22×C8—1222no—
378C4⋊Q8—24no—
379C4⋊Q8—24no—
380#124—24no—
381#124—24no—
382#123—216no—
383C22⋊Q8—216no—
384#117—22no—
385#117—22no—
386#117—1222no—
387#117—1222no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4C22⋊D6112yes—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4C22⋊D6112yes—
394C2×C42—1221no—
395C2×C42C22×D6112yes—
396#103—1224no—
397#103—1228no—
398#102—1224no—
399#95—1228no—
400C24—21no—
401C24C2×C22⋊D6111yescharacteristic
402#41—28no—
403#41—28no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—28no—
418C2×Q8—28no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—614no—
424C2×Q8—1222no—
425C2×Q8—1224no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1228no—
431C2×Q8—1228no—
432C2×Q8—2442no—
433C2×Q8—2444no—
434C2×Q8—2444no—
435C2×Q8—2442no—
436C2×Q8C2×C22⋊D6114yes—
437C2×Q8C2×C22⋊D6112yes—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4C2×C22⋊D6112yes—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—1222no—
456C22×C4—1224no—
457C22×C4—1222no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4D4×D6112yes—
461C22×C4D4×D6112yes—
462C22×C4D4×D6112yes—
463C22×C4#1175822412yes—
464C22×C4C2×C22⋊D6114yes—
465C22×C4C2×C22⋊D6112yes—
466C22×C4C2×C22⋊D6112yes—
467C22×C4C2×C22⋊D6111yescharacteristic
468C22×C4C23×D6111yescharacteristic
469Q16—12232no—
470Q16—12216no—
471SD16—12216no—
472M16—1228no—
473C2×C8—614no—
474C2×C8—614no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1228no—
480C4⋊C4—216no—
481C4⋊C4—24no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C22⋊C4—216no—
485C22⋊C4—24no—
486C22⋊C4—2442no—
487C42—22no—
488C42—22no—
489C42—2442no—
490C23—24no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—44no—
495C23#1451974612yes—
496C23#1451974812yes—
497C23#1451977312yes—
498C23C22×C22⋊D6111yescharacteristic
499Q8—24no—
500Q8—22no—
501Q8—24no—
502Q8—22no—
503Q8—42no—
504Q8—41no—
505Q8—42no—
506Q8—41no—
507Q8—48no—
508Q8—1224no—
509Q8—1224no—
510Q8—2442no—
511Q8—2442no—
512Q8—2448no—
513D4—22no—
514D4—22no—
515D4—48no—
516D4—41no—
517D4—41no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—1224no—
529C2×C4—2442no—
530C2×C4—2444no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2448no—
534C2×C4#1451974514yes—
535C2×C4#1451974514yes—
536C2×C4#1451974612yes—
537C2×C4#1451974812yes—
538C2×C4#1451974914yes—
539C2×C4#1451974914yes—
540C2×C4#1451977312yes—
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C8—1224no—
548C8—1224no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—44no—
553C22—42no—
554C22—82no—
555C22—82no—
556C22#1683967014yes—
557C22C2×#1451974611yescharacteristic
558C22C2×#1451974811yescharacteristic
559C22C2×#1451977311yescharacteristic
560C22#1684057714yes—
561C4—44no—
562C4—42no—
563C4—44no—
564C4—4882no—
565C4—2444no—
566C4#1683967014yes—
567C4C2×#1451974511yescharacteristic
568C4C2×#1451974511yescharacteristic
569C4C2×#1451974911yescharacteristic
570C4C2×#1451974911yescharacteristic
571C2—24no—
572C2—82no—
573C2C2×#1683967011yescharacteristic
574C2#2145511412yes—
575C2#2145534512yes—
576C2#2145631012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#536C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenterC2×#536D61
hypocenterC61#23425
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C23×#11758202C2semidirect
3. C2×#16839771C2semidirect
5. #21448403C2semidirect
6. #21446919C2semidirect
7. #21446913C2semidirect
8. #21445516C2semidirect
329. C61#23425semidirect

Maximal subgroups

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

subgroupindex
C22×#145198072
C23×#117582022
C2×#168397712
C2×#168397582
#214484032
#214469192
#214469132
#214455162
C2×C61×#5362
#2342561