Finite Group Databasebeta

← #29737511

Subgroups — #29737511

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840055C211yesmaximal, characteristic, complemented
2C2×#16839769C211yesmaximal, characteristic
3#21449513C211yesmaximal, characteristic
4Q8×#11758198C211yesmaximal, characteristic
5#21447370C211yesmaximal, characteristic, complemented
6#21447213C211yesmaximal, characteristic, complemented
7#21447201C211yesmaximal, characteristic, complemented
8#21446959C211yesmaximal, characteristic, complemented
9#21446087C211yesmaximal, characteristic, complemented
10#21446085C211yesmaximal, characteristic, complemented
11#21446021C211yesmaximal, characteristic
12#21446000C211yesmaximal, characteristic
13#21445851C211yesmaximal, characteristic
14#21445752C211yesmaximal, characteristic
15C61×#1805C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×#14519776C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21#16840055—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27#16839769—24no—
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35#16839686—21no—
36#16839675—21no—
37#16839668—21no—
38#16839668C2211yescharacteristic
39#16839667—21no—
40#16839667C2211yescharacteristic
41#16839666—21no—
42#16839660C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839644—21no—
45#16839619—21no—
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48C4×#11758198—21no—
49C4×#11758198C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#382C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—21no—
63C22×Dic122C2311yescharacteristic
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—41no—
68C2×Q8×D61—41no—
69C2×Q8×D61Q814yescomplemented
70C22×C4×D61—21no—
71C22×C4×D61C2311yescharacteristic
72C2×#11758189—22no—
73C2×#11758189—22no—
74C2×#11758189—41no—
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77#14519776—22no—
78#14519776—41no—
79Q8×Dic61—22no—
80Q8×Dic61—41no—
81#14519735—22no—
82#14519735—41no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D411yescharacteristic
94C2×#11758198D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98#14519711—22no—
99#14519710—22no—
100#14519708—41no—
101#14519708—41no—
102#14519706—22no—
103#14519702—22no—
104#14519702—41no—
105#14519701—22no—
106#14519701D414yes—
107#14519700—22no—
108#14519700—41no—
109#14519689—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4×Q8×C61—22no—
124C4×Q8×C61—41no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—41no—
132#103×C61—22no—
133#103×C61—22no—
134#95×C61—21no—
135#95×C61—21no—
136C23×D61C2×D411yescharacteristic
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×Q812yes—
142C2×Q8×C61#4112yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157Q8×D61—28no—
158Q8×D61—44no—
159Q8×D61—44no—
160Q8×D61—44no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61C2×Q812yes—
169C2×C4×D61#4112yes—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—24no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122—81no—
181C2×Dic122C2×Q812yes—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—42no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—42no—
198#11758198—22no—
199#11758198—41no—
200#11758198—42no—
201#11758198—42no—
202#11758198—81no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209C4⋊Dic61C2×D412yes—
210#11758196—42no—
211#11758196—42no—
212#11758196—81no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C2×D412yes—
223#11758189—44no—
224C23×C61C22×D411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61C22⋊Q812yes—
232Q8×C61—22no—
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61C22⋊Q814yes—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61C22⋊Q812yes—
280C2×Dic61#13712yes—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—48no—
286Dic122—22no—
287Dic122—22no—
288Dic122—48no—
289Dic122—41no—
290Dic122—41no—
291Dic122—84no—
292C8×C61—41no—
293C8×C61—41no—
294#92418—22no—
295#92418—41no—
296#25979—611nomaximal, Sylow
297C22×C61C2×C22⋊Q811yescharacteristic
298C22×C61D4211yescharacteristic
299C22×C61#54611yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57711yescharacteristic
303C22×C61#57811yescharacteristic
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—24no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61—81no—
317C4×C61C2×C22⋊Q811yescharacteristic
318C4×C61#54311yescharacteristic
319C4×C61#54611yescharacteristic
320C4×C61#55111yescharacteristic
321Dic61—22no—
322Dic61—22no—
323Dic61—42no—
324Dic61—41no—
325Dic61—42no—
326Dic61—81no—
327Dic61—81no—
328C2×#477—611no—
329C2×#419—611no—
330#2439—611no—
331Q8×C22⋊C4—611no—
332#2097—611no—
333#2068—611no—
334#2066—611no—
335#2024—611no—
336#1865—611no—
337#1865—611no—
338#1854—611no—
339#1851—611no—
340#1809—611no—
341#1805D6111yescharacteristic, complemented
342#1787—611no—
343C2×C61#243711yescharacteristic
344C2×C61#307811yescharacteristic
345C2×C61#308211yescharacteristic
346C2×C61#327011yescharacteristic
347C2×C61#327511yescharacteristic
348C2×C61#331311yescharacteristic
349C2×C61#332911yescharacteristic
350D61—42no—
351C23×Q8—611no—
352C22×M16—611no—
353C2×C4⋊Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×Q8C2×D6111yescharacteristic
356C2×#117—611no—
357#477—1224no—
358C2×#107—611no—
359C2×#107—611no—
360C2×#106—611no—
361C2×#106—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8C2×D6111yescharacteristic
364#419—1224no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103C2×D6111yescharacteristic
372C2×#103C2×D6111yescharacteristic
373C2×C22⋊C8—611no—
374#396—1221no—
375#393—1221no—
376#391—611no—
377#391—611no—
378#391—1221no—
379#391—1221no—
380#391—1221no—
381#389—611no—
382#389—611no—
383#387—1221no—
384#382C2×D6111yescharacteristic
385#380—611no—
386#380—611no—
387#380—1221no—
388C4×C22⋊C4—611no—
389C4×C22⋊C4—1221no—
390#340—611no—
391#340—611no—
392#340C2×D6111yescharacteristic
393#340C2×D6111yescharacteristic
394#336—611no—
395#336—611no—
396C61#2597911yescharacteristic, complemented, Sylow
397C22×Q8—614no—
398C22×Q8—611no—
399C22×Q8—1221no—
400C22×Q8—1221no—
401C22×Q8—1222no—
402C22×Q8—1222no—
403C22×Q8—1222no—
404C22×Q8—2441no—
405C22×Q8—2441no—
406C22×Q8C22×D6111yescharacteristic
407C23×C4—611no—
408C23×C4—1221no—
409C2×M16—1222no—
410C2×M16—1222no—
411C2×M16—2441no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—1222no—
417C4⋊Q8—2441no—
418C4×Q8—22no—
419C4×Q8—41no—
420C4×Q8—1222no—
421C4×Q8—2441no—
422#117—1222no—
423#117—2441no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
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×C4⋊C4C22×D6111yescharacteristic
435C2×C22⋊C4—611no—
436C2×C22⋊C4—611no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C42—21no—
441C2×C42—611no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#107—1222no—
447#106—2441no—
448#106—2441no—
449C4⋊C8—41no—
450C4⋊C8—1222no—
451#103—22no—
452#103—22no—
453#103—614no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—2441no—
458#103—2441no—
459C22⋊C8—1222no—
460#95—21no—
461#95—21no—
462#95—611no—
463#95—611no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—611no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—612no—
477C2×Q8—1224no—
478C2×Q8—1221no—
479C2×Q8—1228no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—4881no—
485C2×Q8—2442no—
486C2×Q8—2442no—
487C2×Q8—2442no—
488C2×Q8—2442no—
489C2×Q8—2444no—
490C2×Q8—2444no—
491C2×Q8—2444no—
492C2×Q8#1175822212yes—
493C2×Q8D4×D6112yes—
494C2×Q8Q8×D6112yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—612no—
502C22×C4—612no—
503C22×C4—611no—
504C22×C4—1224no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518M16—2444no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—612no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—612no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—4881no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C22⋊C4—1222no—
549C22⋊C4—4881no—
550C22⋊C4—2441no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—1221no—
557C42—1221no—
558C42—2442no—
559C42—2442no—
560C23—612no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23C2×D4×D6111yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—1222no—
572Q8—1222no—
573Q8—4884no—
574Q8—2448no—
575Q8—2441no—
576Q8—2441no—
577Q8#1451975514yes—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—612no—
597C2×C4—612no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—2448no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4C2×D4×D6111yescharacteristic
623C2×C4C2×Q8×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822211yescharacteristic
626C2×C4#1451982211yescharacteristic
627C2×C4#1451982311yescharacteristic
628C8—41no—
629C8—41no—
630C8—1222no—
631C8—2441no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22C22⋊Q8×D6111yescharacteristic
640C22#1684039611yescharacteristic
641C22#1684040211yescharacteristic
642C22#137×D6111yescharacteristic
643C22#1684055911yescharacteristic
644C22#1684057111yescharacteristic
645C22#1684057611yescharacteristic
646C4—24no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—1222no—
653C4—1222no—
654C4—4881no—
655C4—4881no—
656C4—2442no—
657C4—2441no—
658C4—2442no—
659C4C2×#1451975511yescharacteristic
660C4#1684038711yescharacteristic
661C4#1684040111yescharacteristic
662C4#1684043611yescharacteristic
663C2—2442no—
664C2#2144948711yescharacteristic
665C2#2145562211yescharacteristic
666C2#2145566011yescharacteristic
667C2#2145738411yescharacteristic
668C2#2145744511yescharacteristic
669C2#2145788111yescharacteristic
670C2#2145810911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1805D61
hypocenterC61#25979
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×#16840055C2semidirect
5. #21447370C2semidirect
6. #21447213C2semidirect
7. #21447201C2semidirect
8. #21446959C2semidirect
9. #21446087C2semidirect
10. #21446085C2semidirect
15. C61×#1805C2semidirect
69. C2×Q8×D61Q8semidirect
341. #1805D61semidirect
396. C61#25979semidirect

Maximal subgroups

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

subgroupindex
C2×#168400552
C2×#168397692
#214495132
Q8×#117581982
#214473702
#214472132
#214472012
#214469592
#214460872
#214460852
#214460212
#214460002
#214458512
#214457522
C61×#18052
#2597961