Finite Group Databasebeta

← #29828852

Subgroups — #29828852

Order 15616.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839972C211yesmaximal, characteristic, complemented
2C2×#16839967C211yesmaximal, characteristic, complemented
3C2×#16839924C212yesmaximal
4#21454171C211yesmaximal, characteristic, complemented
5#21446160C212yesmaximal, complemented
6#21446153C212yesmaximal, complemented
7#21446131C212yesmaximal, complemented
8#21445824C211yesmaximal, characteristic, complemented
9#21445819C211yesmaximal, characteristic, complemented
10#21445799C211yesmaximal, characteristic, complemented
11C2×C4×#41×C61C211yesmaximal, characteristic
12C22×#11758202C2212yes—
13C22×#11758200C2212yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519756C2212yes—
16#16839972—24no—
17#16839967—24no—
18#16839924—28no—
19C2×C4⋊#92418C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2212yes—
22C2×#14519700C2212yes—
23C2×#14519700C2212yes—
24C2×#14519698C2212yes—
25C2×C22⋊#92418C2212yes—
26#16839668C2212yes—
27#16839660—21no—
28#16839660—21no—
29#16839632C2211yescharacteristic
30#16839622—21no—
31#16839622—21no—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C22×#41×C61C2211yescharacteristic
36C4×#41×C61—24no—
37C4×#41×C61—24no—
38C2×C4×Q8×C61C2211yescharacteristic
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4×D4×C61—21no—
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61—21no—
45C2×#117×C61C2211yescharacteristic
46C22×C42×C61—21no—
47C22×C42×C61C2211yescharacteristic
48C22×Dic122C2312yes—
49C2×#11758202—24no—
50C2×#11758202—24no—
51C2×#11758202—24no—
52C2×#11758200—24no—
53C2×#11758200—24no—
54C2×#11758200—24no—
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57#14519777—22no—
58#14519777—22no—
59#14519756—24no—
60#14519756—24no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—22no—
67C2×C22⋊Dic61—22no—
68C4⋊#92418—22no—
69C4⋊#92418—22no—
70#14519701—24no—
71#14519700—24no—
72#14519700—24no—
73#14519698—24no—
74C22⋊#92418—24no—
75#14519684—22no—
76#14519684—22no—
77C2×#41×C61—22no—
78C2×#41×C61—22no—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—22no—
82C2×#41×C61D414yes—
83C22×Q8×C61C2311yescharacteristic
84C22×D4×C61—21no—
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88C23×C4×C61—21no—
89C23×C4×C61D411yescharacteristic
90C23×C4×C61D411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C4×Q8×C61—22no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61D414yes—
96C4×D4×C61—28no—
97C4×D4×C61—28no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61D414yes—
102#117×C61—28no—
103#117×C61—22no—
104#117×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C42×C61—28no—
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61—22no—
117C2×C42×C61—22no—
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C24×C61—21no—
123C24×C61C2×D411yescharacteristic
124#41×C61—28no—
125#41×C61—28no—
126#41×C61—28no—
127#41×C61—28no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61#4112yes—
134C2×D4×C61—24no—
135C2×D4×C61—24no—
136C2×D4×C61—24no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61C2×D412yes—
141C2×D4×C61C2×D412yes—
142C2×D4×C61#4112yes—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—21no—
146C22×C4×C61—24no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—24no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—24no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—41no—
161C22×C4×C61—42no—
162C22×C4×C61—42no—
163C22×C4×C61—42no—
164C22×C4×C61—42no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61#4112yes—
172C22×C4×C61C2411yescharacteristic
173C22×Dic61—22no—
174C22×Dic61—22no—
175C2×Dic122—22no—
176C2×Dic122—22no—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C2×D412yes—
186C4⋊C4×C61C2×D412yes—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—24no—
189C22⋊C4×C61—24no—
190C42×C61—24no—
191C42×C61—24no—
192C42×C61—22no—
193C42×C61—24no—
194C42×C61—22no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61C2×D412yes—
198C42×C61C2×D412yes—
199C22⋊Dic61—44no—
200#11758202—48no—
201#11758200—48no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61—42no—
205C4⋊Dic61—42no—
206C4⋊Dic61—42no—
207C4⋊Dic61—42no—
208#11758196—44no—
209C2×#92418—22no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C23×C61—21no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—21no—
219C23×C61—22no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61#12112yes—
229C23×C61C22×D411yescharacteristic
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61#12114yes—
234D4×C61—216no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61#12114yes—
239C2×C4×C61—216no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—44no—
259C2×C4×C61—48no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—44no—
265C2×C4×C61—48no—
266C2×C4×C61#12012yes—
267C2×C4×C61#12012yes—
268C2×C4×C61#12112yes—
269C2×C4×C61#13512yes—
270C2×C4×C61#13712yes—
271C2×C4×C61#13712yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283C2×Dic61—42no—
284Dic122—42no—
285Dic122—42no—
286Dic122—42no—
287Dic122—42no—
288#92418—42no—
289#92418—42no—
290#32907—611nomaximal, Sylow
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—41no—
296C22×C61—42no—
297C22×C61—41no—
298C22×C61—42no—
299C22×C61—41no—
300C22×C61—42no—
301C22×C61—41no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61C2×#12011yescharacteristic
306C22×C61C2×#12111yescharacteristic
307C22×C61#52511yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C22×C61C2×#13711yescharacteristic
310C22×C61C2×#13711yescharacteristic
311C22×C61#57511yescharacteristic
312C4×C61—22no—
313C4×C61—24no—
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—41no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61#45214yescomplemented
320C4×C61C2×#12111yescharacteristic
321C4×C61C2×#12111yescharacteristic
322C4×C61#52511yescharacteristic
323C4×C61#52511yescharacteristic
324Dic61—82no—
325Dic61—82no—
326C2×C4×#41D6111yescharacteristic
327C2×#462—611no—
328C2×#461—611no—
329C2×#452—612no—
330#2857—611no—
331#1878—612no—
332#1877—612no—
333#1873—612no—
334#1803—611no—
335#1802—611no—
336#1797—611no—
337C2×C61—42no—
338C2×C61—44no—
339C2×C61#284711yescharacteristic
340C2×C61C2×#45211yescharacteristic
341C2×C61#300711yescharacteristic
342C2×C61#303511yescharacteristic
343C2×C61#303911yescharacteristic
344C2×C61#310711yescharacteristic
345C2×C61#311611yescharacteristic
346C22×#41C2×D6111yescharacteristic
347C22×Q16—612no—
348C22×SD16—612no—
349C2×C4⋊Q8—611no—
350C2×C22⋊Q8—612no—
351C4×#41—24no—
352C4×#41—24no—
353C2×C4×Q8C2×D6111yescharacteristic
354C2×C4×Q8C2×D6111yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4—21no—
357C2×C4×D4C2×D6111yescharacteristic
358C2×C4×D4C2×D6111yescharacteristic
359C2×#117—21no—
360C2×#117C2×D6111yescharacteristic
361C22×C42—21no—
362C22×C42C2×D6111yescharacteristic
363#462—1224no—
364#461—1224no—
365#452—1228no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×#103—612no—
369C2×#103—612no—
370C2×#103—612no—
371C2×#102—612no—
372C2×C22⋊C8—612no—
373#391—612no—
374#389—1221no—
375#389—1221no—
376#384—611no—
377#380—1221no—
378#380—1221no—
379#340—612no—
380#340—612no—
381#336—612no—
382C61#3290711yescharacteristic, complemented, Sylow
383C2×#41—22no—
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41D24414yes—
389C22×Q8—612no—
390C22×Q8C22×D6111yescharacteristic
391C22×D4—21no—
392C22×D4C22×D6111yescharacteristic
393C23×C4—21no—
394C23×C4—21no—
395C23×C4—21no—
396C23×C4C22⋊D6111yescharacteristic
397C23×C4C22⋊D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399C2×Q16—1224no—
400C2×Q16—1224no—
401C2×Q16—1224no—
402C2×SD16—1224no—
403C2×SD16—1224no—
404C2×SD16—1224no—
405C22×C8—612no—
406C22×C8—612no—
407C4⋊Q8—1222no—
408C4⋊Q8—1222no—
409C22⋊Q8—1224no—
410C22⋊Q8—1224no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8C22⋊D6114yes—
415C4×D4—28no—
416C4×D4—28no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4C22⋊D6114yes—
421#117—28no—
422#117—22no—
423#117—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1222no—
438C2×C42—28no—
439C2×C42—21no—
440C2×C42—21no—
441C2×C42—21no—
442C2×C42—22no—
443C2×C42—22no—
444C2×C42C22⋊D6111yescharacteristic
445C2×C42C22⋊D6111yescharacteristic
446C2×C42C22×D6111yescharacteristic
447C2×C42C22×D6111yescharacteristic
448C4⋊C8—1222no—
449C4⋊C8—1222no—
450#103—1224no—
451#103—1224no—
452#103—1224no—
453#102—1224no—
454C22⋊C8—1224no—
455#95—1222no—
456#95—1222no—
457C24—21no—
458C24C2×C22⋊D6111yescharacteristic
459#41—28no—
460#41—28no—
461#41—28no—
462#41—28no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—1222no—
467C2×Q8—1222no—
468C2×Q8—1222no—
469C2×Q8—1222no—
470C2×Q8—1222no—
471C2×Q8—1222no—
472C2×Q8C2×D24412yes—
473C2×Q8#1175822212yes—
474C2×Q8C2×C22⋊D6112yes—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—24no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4C2×D24412yes—
482C2×D4#1175822212yes—
483C2×D4C2×C22⋊D6112yes—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—41no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4C2×D24412yes—
509C22×C4#1175822212yes—
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C23×D6111yescharacteristic
516Q16—2448no—
517SD16—2448no—
518C2×C8—1222no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C2×C8—1222no—
523C2×C8—1222no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—24no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2444no—
534C4⋊C4—2442no—
535C4⋊C4C2×C22⋊D6112yes—
536C4⋊C4C2×C22⋊D6112yes—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—2444no—
541C42—24no—
542C42—24no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—24no—
547C42—22no—
548C42C2×C22⋊D6112yes—
549C42C2×C22⋊D6112yes—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—21no—
554C23—22no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23#1451975112yes—
564C23C22×C22⋊D6111yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—22no—
568Q8—2442no—
569Q8—2442no—
570Q8—2442no—
571Q8—2442no—
572Q8#1451975114yes—
573D4—216no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4#1451975114yes—
578C2×C4—216no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4#1451974612yes—
612C2×C4#1451974612yes—
613C2×C4#1451975112yes—
614C2×C4#1451979712yes—
615C2×C4#1451980712yes—
616C2×C4#1451980712yes—
617C2×C4C22×C22⋊D6111yescharacteristic
618C2×C4C22×C22⋊D6111yescharacteristic
619C2×C4C22×D24411yescharacteristic
620C2×C4C2×#1175822211yescharacteristic
621C2×C4C2×#1175822211yescharacteristic
622C2×C4C2×#1175822211yescharacteristic
623C8—2442no—
624C8—2442no—
625C22—22no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—41no—
630C22—42no—
631C22—41no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22C2×#1451974611yescharacteristic
640C22C2×#1451975111yescharacteristic
641C22#1684027311yescharacteristic
642C22C2×#1451979711yescharacteristic
643C22C2×#1451980711yescharacteristic
644C22C2×#1451980711yescharacteristic
645C22#1684056011yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—4882no—
654C4—4882no—
655C4#1683992414yescomplemented
656C4C2×#1451975111yescharacteristic
657C4C2×#1451975111yescharacteristic
658C4#1684027311yescharacteristic
659C4#1684027311yescharacteristic
660C2—42no—
661C2—44no—
662C2#2145410111yescharacteristic
663C2C2×#1683992411yescharacteristic
664C2#2145507911yescharacteristic
665C2#2145527411yescharacteristic
666C2#2145530611yescharacteristic
667C2#2145589011yescharacteristic
668C2#2145598111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×#41×C61C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenterC2×C4×#41D61
hypocenterC61#32907
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16839972C2semidirect
2. C2×#16839967C2semidirect
4. #21454171C2semidirect
5. #21446160C2semidirect
6. #21446153C2semidirect
7. #21446131C2semidirect
8. #21445824C2semidirect
9. #21445819C2semidirect
10. #21445799C2semidirect
319. C4×C61#452semidirect
382. C61#32907semidirect
655. C4#16839924semidirect

Maximal subgroups

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

subgroupindex
C2×#168399722
C2×#168399672
C2×#168399242
#214541712
#214461602
#214461532
#214461312
#214458242
#214458192
#214457992
C2×C4×#41×C612
#3290761