Finite Group Databasebeta

← #29707914

Subgroups — #29707914

Order 15616.

1326 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic
2C2×#16839811C211yesmaximal, characteristic, complemented
3#21454336C211yesmaximal, characteristic
4C22×#14519696C211yesmaximal, characteristic, complemented
5#21454263C211yesmaximal, characteristic
6#21447119C218yesmaximal, complemented
7#21445558C211yesmaximal, characteristic, complemented
8C2×C61×#419C211yesmaximal, characteristic, complemented
9C22×#11758222C2211yescharacteristic
10C2×#14519808—24no—
11C2×#14519808—24no—
12C2×#14519808—22no—
13C2×#14519808C2214yes—
14C22×#11758202—21no—
15C22×#11758201—21no—
16C22×#11758194C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C22×#11758194C2211yescharacteristic
19#16839811—24no—
20C2×#14519713C2214yescomplemented
21#16839770—21no—
22#16839766—21no—
23#16839761C2214yes—
24#16839756—22no—
25C2×#14519696—24no—
26C2×#14519696C2214yescomplemented
27#16839730—22no—
28#16839728C2214yes—
29#16839725—22no—
30#16839531C2214yescomplemented
31C23×Q8×C61C2211yescharacteristic
32C22×M16×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C61×#419—22no—
35C61×#419—22no—
36C2×#103×C61C2214yescomplemented
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222D412yes—
44C2×#11758222C2312yes—
45C22×Dic122D411yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61—21no—
49#14519808—216no—
50#14519808—28no—
51#14519808—28no—
52C2×#11758202—28no—
53C2×#11758202—24no—
54C2×#11758201—28no—
55C2×#11758201—24no—
56C2×#11758194—22no—
57C2×#11758194—21no—
58C2×#11758194—21no—
59C2×#11758194—21no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194D414yes—
65C2×#11758194C2312yes—
66C2×#11758194C2312yes—
67C2×#11758194C2312yes—
68C22×#92418—21no—
69C22×#92418—21no—
70C22×#92418—21no—
71#14519713—28no—
72#14519700—24no—
73#14519697—24no—
74#14519696—28no—
75#14519689—28no—
76C22×Q8×C61—22no—
77C22×Q8×C61—21no—
78C22×Q8×C61—21no—
79C22×Q8×C61—21no—
80C22×Q8×C61—22no—
81C22×Q8×C61—21no—
82C22×Q8×C61—21no—
83C22×Q8×C61—22no—
84C22×Q8×C61D414yes—
85C22×Q8×C61C2312yes—
86C23×C4×C61—21no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—22no—
89C2×M16×C61—22no—
90C2×M16×C61D412yes—
91C2×M16×C61D412yes—
92C22×C8×C61C2312yes—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61—21no—
96#117×C61—21no—
97C2×C4⋊C4×C61C2312yes—
98C2×C22⋊C4×C61—21no—
99C2×C42×C61—21no—
100#103×C61—28no—
101C24×C61#4111yescharacteristic
102C23×D61—21no—
103C2×Q8×C61—24no—
104C2×Q8×C61—22no—
105C2×Q8×C61—24no—
106C2×Q8×C61—24no—
107C2×Q8×C61—24no—
108C2×Q8×C61—24no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×Q8×C61—41no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C2×Q8×C61—42no—
118C2×Q8×C61—42no—
119C2×Q8×C61—42no—
120C2×Q8×C61C2×D414yes—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—41no—
125C22×C4×C61—41no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4112yes—
134C22×C4×C61#4111yescharacteristic
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—22no—
138C2×C22⋊D61—22no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—21no—
142#11758222—24no—
143#11758222—24no—
144#11758222—22no—
145#11758222—22no—
146#11758222—22no—
147#11758222—22no—
148#11758222—42no—
149#11758222—42no—
150#11758222—42no—
151#11758222—42no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244C2×D412yes—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—21no—
163C2×Dic122C2×D412yes—
164M16×C61—24no—
165M16×C61—24no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61C2×D414yes—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C22⋊C4×C61—41no—
172C22⋊C4×C61—41no—
173C42×C61—41no—
174C42×C61—41no—
175#11758202—216no—
176#11758201—216no—
177#11758194—24no—
178#11758194—22no—
179#11758194—22no—
180#11758194—22no—
181#11758194—22no—
182#11758194—22no—
183#11758194—42no—
184#11758194—42no—
185#11758194C2×D414yes—
186C2×#92418—24no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—42no—
191C2×#92418—42no—
192C23×C61—42no—
193C23×C61#12111yescharacteristic
194C23×C61#12111yescharacteristic
195C23×C61#12312yes—
196C23×C61#12412yes—
197C23×C61C2×#4111yescharacteristic
198C22×D61—22no—
199C22×D61—42no—
200C22×D61—82no—
201Q8×C61—24no—
202Q8×C61—24no—
203Q8×C61—42no—
204Q8×C61—42no—
205Q8×C61—44no—
206Q8×C61—44no—
207C2×C4×C61—24no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—44no—
211C2×C4×C61—44no—
212C2×C4×C61—42no—
213C2×C4×C61—82no—
214C2×C4×C61#12012yes—
215C2×C4×C61#12012yes—
216C2×C4×C61#12112yes—
217C2×C4×C61#12112yes—
218C2×C4×C61#12112yes—
219C2×C4×C61#12112yes—
220C2×C4×C61#12111yescharacteristic
221C2×C4×C61#12111yescharacteristic
222C2×C4×C61#12312yes—
223C2×C4×C61#12312yes—
224C2×C4×C61#12312yes—
225C2×C4×C61#12412yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—42no—
233C22⋊D61—42no—
234C22⋊D61—44no—
235C2×Dic61—22no—
236C2×Dic61—42no—
237D244—22no—
238D244—22no—
239D244—42no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61—44no—
243Dic122—22no—
244Dic122—22no—
245Dic122—42no—
246C8×C61—24no—
247#92418—24no—
248#92418—42no—
249#92418—42no—
250#23532—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61#39412yes—
257C22×C61#39412yes—
258C22×C61C2×#12111yescharacteristic
259C22×C61C2×#12311yescharacteristic
260C22×C61C2×#12411yescharacteristic
261C2×D61—42no—
262C2×D61—44no—
263C2×D61—84no—
264C4×C61—44no—
265C4×C61—82no—
266C4×C61#39412yes—
267C4×C61#39412yes—
268C4×C61C2×#12011yescharacteristic
269C4×C61C2×#12111yescharacteristic
270C4×C61C2×#12111yescharacteristic
271C4×C61C2×#12311yescharacteristic
272Dic61—42no—
273C22×#137—611no—
274C2×#430—611no—
275#2896—611no—
276C2×#419D6111yescharacteristic, complemented
277C22×#101—611no—
278#2879—611no—
279#2052—618no—
280#1742—611no—
281C2×C61—22no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—22no—
285C2×C61C2×#39411yescharacteristic
286C2×C61#306912yes—
287D61—82no—
288C22×#41—611no—
289C23×Q8C2×D6111yescharacteristic
290C2×#137—614no—
291C2×#137—1224no—
292C2×#137—1222no—
293C2×#137—1224no—
294C22×Q16—1221no—
295C22×SD16—1221no—
296C22×M16—611no—
297C22×M16—611no—
298C22×M16—611no—
299C22×M16C2×D6111yescharacteristic
300C2×#117C2×D6111yescharacteristic
301#430—1224no—
302C2×#108—614no—
303#419—22no—
304#419—22no—
305#419—1221no—
306#418—1221no—
307#417—614no—
308#416—1222no—
309C2×#103C2×D6114yescomplemented
310C2×#101—614no—
311C2×#101—1224no—
312#408—614no—
313#408—1222no—
314#407—1222no—
315#343—614no—
316C61#2353211yescharacteristic, complemented, Sylow
317C2×#41—612no—
318C2×#41—612no—
319C2×#41—1221no—
320C2×#41—1221no—
321C2×#41—1222no—
322C2×#41—1222no—
323C2×#41—1222no—
324C2×#41—1222no—
325C22×Q8—21no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—21no—
329C22×Q8—21no—
330C22×Q8—22no—
331C22×Q8—21no—
332C22×Q8—21no—
333C22×Q8—611no—
334C22×Q8D24414yes—
335C22×Q8C22×D6112yes—
336C22×D4—611no—
337C22×D4—1221no—
338C23×C4—21no—
339C23×C4—1221no—
340C23×C4C22×D6111yescharacteristic
341#137—1228no—
342#137—12216no—
343#137—1228no—
344C2×Q16—1224no—
345C2×Q16—1228no—
346C2×SD16—1224no—
347C2×SD16—1228no—
348C2×M16—22no—
349C2×M16—22no—
350C2×M16—614no—
351C2×M16—612no—
352C2×M16—612no—
353C2×M16—612no—
354C2×M16—1222no—
355C2×M16—1221no—
356C2×M16—1221no—
357C2×M16—1221no—
358C2×M16—1222no—
359C2×M16—1222no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16C22⋊D6112yes—
363C2×M16C22⋊D6112yes—
364C22×C8—1221no—
365C22×C8—1221no—
366C22×C8—1221no—
367C22×C8C22×D6112yes—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—21no—
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—21no—
374C2×C42—21no—
375#108—1228no—
376#103—28no—
377#103—1224no—
378#102—1224no—
379#101—1228no—
380C22⋊C8—1228no—
381C24—1221no—
382C24#1175822411yescharacteristic
383#41—1222no—
384#41—1222no—
385#41—1222no—
386#41—1222no—
387#41—1224no—
388#41—1224no—
389#41—2442no—
390#41—2442no—
391#41—2442no—
392#41—2442no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—612no—
411C2×Q8—1221no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×Q8C2×D24414yes—
415C2×D4—612no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—1221no—
420C2×D4—1222no—
421C2×D4—2442no—
422C2×D4—2442no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—1221no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4C2×D24412yes—
435C22×C4#1175822212yes—
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6112yes—
439C22×C4#1175822412yes—
440C22×C4#1175822411yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C23×D6111yescharacteristic
444Q16—12216no—
445SD16—12216no—
446M16—24no—
447M16—24no—
448M16—614no—
449M16—1222no—
450M16—1224no—
451M16—1222no—
452M16—1222no—
453M16—1222no—
454M16—1222no—
455M16—2442no—
456M16—2442no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—1222no—
460C2×C8—1224no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—2442no—
464C2×C8—2442no—
465C2×C8C2×C22⋊D6114yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C42—41no—
471C42—41no—
472C23—42no—
473C23—1222no—
474C23—4882no—
475C23—2442no—
476C23#1451974811yescharacteristic
477C23#1451974811yescharacteristic
478C23#1451976112yes—
479C23#1451976312yes—
480C23C2×#1175822411yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—44no—
487Q8—1222no—
488Q8—1222no—
489Q8—2442no—
490D4—1222no—
491D4—1222no—
492D4—2442no—
493D4—2442no—
494D4—2444no—
495D4—2442no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—1222no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2444no—
508C2×C4#1451974712yes—
509C2×C4#1451974712yes—
510C2×C4#1451974811yescharacteristic
511C2×C4#1451974811yescharacteristic
512C2×C4#1451974912yes—
513C2×C4#1451974912yes—
514C2×C4#1451975112yes—
515C2×C4#1451975112yes—
516C2×C4#1451975912yes—
517C2×C4#1451975912yes—
518C2×C4#1451976112yes—
519C2×C4#1451976312yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C2×D4×D6111yescharacteristic
522C2×C4C22×C22⋊D6111yescharacteristic
523C2×C4C22×D24411yescharacteristic
524C2×C4C2×#1175822411yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C8—24no—
527C8—1224no—
528C8—2442no—
529C8—2442no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—4884no—
536C22—2442no—
537C22—2444no—
538C22#1683967812yes—
539C22#1683967812yes—
540C22C2×#1451974811yescharacteristic
541C22C2×#1451976111yescharacteristic
542C22C2×#1451976311yescharacteristic
543C4—44no—
544C4—82no—
545C4—2442no—
546C4#1683967812yes—
547C4#1683967812yes—
548C4C2×#1451974711yescharacteristic
549C4C2×#1451974911yescharacteristic
550C4C2×#1451975111yescharacteristic
551C4C2×#1451975911yescharacteristic
552C2—22no—
553C2—21no—
554C2—21no—
555C2—22no—
556C2—4882no—
557C2C2×#1683967811yescharacteristic
558C2#2145555212yes—
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
hypercenterC2×#419D61
hypocenterC61#23532
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×#16839811C2semidirect
4. C22×#14519696C2semidirect
6. #21447119C2semidirect
7. #21445558C2semidirect
8. C2×C61×#419C2semidirect
20. C2×#14519713C22semidirect
26. C2×#14519696C22semidirect
30. #16839531C22semidirect
36. C2×#103×C61C22semidirect
276. C2×#419D61semidirect
309. C2×#103C2×D61semidirect
316. C61#23532semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168398112
#214543362
C22×#145196962
#214542632
#214471192
#214455582
C2×C61×#4192
#2353261