Finite Group Databasebeta

← #29698378

Subgroups — #29698378

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839863C211yesmaximal, characteristic
2C2×SD16×Dic61C211yesmaximal, characteristic
3Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
4#21448729C211yesmaximal, characteristic
5#21446558C211yesmaximal, characteristic, complemented
6#21446553C211yesmaximal, characteristic, complemented
7#21446089C211yesmaximal, characteristic, complemented
8#21446082C211yesmaximal, characteristic
9#21446075C211yesmaximal, characteristic, complemented
10#21446068C211yesmaximal, characteristic
11#21445987C211yesmaximal, characteristic, complemented
12#21445834C211yesmaximal, characteristic, complemented
13#21445819C211yesmaximal, characteristic
14#21445691C211yesmaximal, characteristic, complemented
15C2×C61×#440C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16839863—24no—
21SD16×Dic61—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839661—21no—
34#16839661—21no—
35#16839661C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839637—21no—
40#16839637—21no—
41#16839632—21no—
42#16839632C2211yescharacteristic
43#16839628C2211yescharacteristic
44#16839622—21no—
45C4×C4⋊Dic61—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47C2×#14519684—21no—
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C61×#440C418yescomplemented
58C2×#107×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C2×#97×C61C2211yescharacteristic
62C23×Dic61—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68Q8×Dic61—41no—
69Q8×Dic61—41no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73D4×Dic61—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519711—22no—
95#14519711—41no—
96#14519709—22no—
97#14519709—41no—
98C4⋊#92418—22no—
99#14519701—22no—
100#14519701—22no—
101#14519698—22no—
102#14519698—22no—
103#14519687—22no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C23×C4×C61—21no—
116C2×SD16×C61C2×C414yes—
117C2×SD16×C61D414yes—
118C2×SD16×C61Q814yes—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61C2×C414yescomplemented
123C4×D4×C61—22no—
124C4×D4×C61C2×C414yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#107×C61C2×C414yescomplemented
132#103×C61C2×C414yescomplemented
133#102×C61C2×C414yes—
134#97×C61C2×C414yescomplemented
135C24×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61C22×C412yes—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×Q812yes—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61C22×C412yes—
146C2×D4×C61C2×D412yes—
147C2×D4×C61C2×Q812yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—24no—
152C22×C4×C61—81no—
153C22×C4×C61—81no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61—81no—
166C22×Dic61—81no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171SD16×C61C4⋊C4116yescomplemented
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C22×C412yes—
175C2×C8×C61C22×C412yes—
176C2×C8×C61C2×D412yes—
177C2×C8×C61C2×Q812yes—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C22×C412yes—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—24no—
184C42×C61—24no—
185C42×C61C22×C412yes—
186C42×C61#4112yes—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C4⋊Dic61—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61—42no—
202C4⋊Dic61—81no—
203C4⋊Dic61—81no—
204C4⋊Dic61—81no—
205C4⋊Dic61—81no—
206C4⋊Dic61—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C22×D411yescharacteristic
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61C2×C4⋊C414yes—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61C2×C4⋊C414yes—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—84no—
249C2×C4×C61C4×D412yes—
250C2×C4×C61C4×D412yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
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×Dic61—41no—
280C2×Dic61—84no—
281C2×Dic61—84no—
282C2×Dic61#13712yes—
283C8×C61—22no—
284C8×C61C2×C4⋊C414yes—
285#92418—41no—
286#92418—41no—
287#22678—611nomaximal, Sylow
288C22×C61—24no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61C2×C4×D411yescharacteristic
296C22×C61D4211yescharacteristic
297C22×C61D4×Q811yescharacteristic
298C22×C61C2×#13711yescharacteristic
299C22×C61#57511yescharacteristic
300C22×C61#57611yescharacteristic
301C22×C61#57711yescharacteristic
302C4×C61—24no—
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61C22×C4⋊C411yescharacteristic
308C4×C61#52011yescharacteristic
309C4×C61#54711yescharacteristic
310C4×C61#55411yescharacteristic
311Dic61—22no—
312Dic61—41no—
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317Dic61—81no—
318Dic61—81no—
319C2×#440—611no—
320C2×#440D6111yescharacteristic
321C2×C4×SD16—611no—
322Q8×C4⋊C4—611no—
323#2349—611no—
324#1943—611no—
325#1942—611no—
326#1865—611no—
327#1864—611no—
328#1863—611no—
329#1862—611no—
330#1847—611no—
331#1805—611no—
332#1802—611no—
333#1771—611no—
334C2×C61—44no—
335C2×C61D4×C4⋊C411yescharacteristic
336C2×C61C4×#13711yescharacteristic
337C2×C61#294611yescharacteristic
338C2×C61#326611yescharacteristic
339C2×C61#326811yescharacteristic
340C2×C61#338511yescharacteristic
341C2×C61#338811yescharacteristic
342C22×SD16C2×D6111yescharacteristic
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8C2×D6111yescharacteristic
346C2×C4×D4—611no—
347C2×C4×D4—611no—
348C2×C4×D4C2×D6111yescharacteristic
349#440—1224no—
350#440Dic6118yescomplemented
351C4×SD16—1224no—
352C2×#107—611no—
353C2×#107—611no—
354C2×#107C2×D6111yescharacteristic
355C2×#106—611no—
356C2×#106—611no—
357C2×C4⋊C8—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103C2×D6111yescharacteristic
361C2×#102—611no—
362C2×#102—611no—
363C2×#102C2×D6111yescharacteristic
364C2×#97—611no—
365C2×#97C2×D6111yescharacteristic
366C2×C4×C8—611no—
367#389—611no—
368#389—611no—
369#389—611no—
370#389—611no—
371#389—1221no—
372#389—1221no—
373#385—1221no—
374#385—1221no—
375#384—611no—
376#384—1221no—
377#382—611no—
378#380—1221no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—1221no—
381C2×#95—1221no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#336—611no—
387#336—611no—
388C61#2267811yescharacteristic, complemented, Sylow
389C22×Q8C22×D6111yescharacteristic
390C22×D4C22×D6111yescharacteristic
391C23×C4—21no—
392C23×C4—1221no—
393C23×C4—1221no—
394C2×SD16Dic12214yes—
395C2×SD16D24414yes—
396C2×SD16C2×Dic6114yes—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C4×Q8—22no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—2441no—
405C4×Q8—2441no—
406C4×Q8C2×Dic6114yescomplemented
407C4×D4—22no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—2441no—
411C4×D4—2441no—
412C4×D4C2×Dic6114yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#107—1222no—
440#107—2441no—
441#107C2×Dic6114yescomplemented
442#106—1222no—
443#106—2441no—
444C4⋊C8—1222no—
445#103—1222no—
446#103—1222no—
447#103C2×Dic6114yescomplemented
448#102—1222no—
449#102—1222no—
450#102C2×Dic6114yes—
451#97—1222no—
452#97C2×Dic6114yescomplemented
453C4×C8—1222no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×Dic12212yes—
467C2×Q8C2×D24412yes—
468C2×Q8C22×Dic6112yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×Dic12212yes—
473C2×D4C2×D24412yes—
474C2×D4C22×Dic6112yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—4881no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16C4⋊Dic61116yescomplemented
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8C2×Dic12212yes—
508C2×C8C2×D24412yes—
509C2×C8C22×Dic6112yes—
510C2×C8C22×Dic6112yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—24no—
514C4⋊C4—1222no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4881no—
524C4⋊C4—4881no—
525C4⋊C4—4881no—
526C4⋊C4—4881no—
527C4⋊C4—4881no—
528C4⋊C4—2441no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4C22×Dic6112yes—
533C4⋊C4C22×Dic6112yes—
534C22⋊C4—24no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C42—24no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42—1221no—
542C42—2442no—
543C42—2442no—
544C42#1175822412yes—
545C42C22×Dic6112yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D4×D6111yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C2×C4⋊Dic6114yes—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C2×C4⋊Dic6114yes—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—612no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—4884no—
589C2×C4—4884no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4D4×Dic6112yes—
603C2×C4D4×Dic6112yes—
604C2×C4C23×Dic6111yescharacteristic
605C2×C4C22×D24411yescharacteristic
606C2×C4C22×Dic12211yescharacteristic
607C2×C4C2×#1175822411yescharacteristic
608C2×C4#1451982311yescharacteristic
609C2×C4#1451982411yescharacteristic
610C8—22no—
611C8—2441no—
612C8—2441no—
613C8C2×C4⋊Dic6114yes—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×D4×Dic6111yescharacteristic
622C22D4×D24411yescharacteristic
623C22D4×Dic12211yescharacteristic
624C22#137×D6111yescharacteristic
625C22#1684056111yescharacteristic
626C22#1684056611yescharacteristic
627C22#1684057211yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—1222no—
634C4—4881no—
635C4—4881no—
636C4—4881no—
637C4—4881no—
638C4—2441no—
639C4—2441no—
640C4—2441no—
641C4C22×C4⋊Dic6111yescharacteristic
642C4#1684024511yescharacteristic
643C4#1684041611yescharacteristic
644C4#1684046911yescharacteristic
645C2—44no—
646C2D4×C4⋊Dic6111yescharacteristic
647C2#137×Dic6111yescharacteristic
648C2#2145465211yescharacteristic
649C2#2145733111yescharacteristic
650C2#2145736111yescharacteristic
651C2#2145873011yescharacteristic
652C2#2145876511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#440D61
hypocenterC61#22678
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. Q8×C4⋊Dic61C2semidirect
5. #21446558C2semidirect
6. #21446553C2semidirect
7. #21446089C2semidirect
9. #21446075C2semidirect
11. #21445987C2semidirect
12. #21445834C2semidirect
14. #21445691C2semidirect
57. C61×#440C4semidirect
122. C4×Q8×C61C2×C4semidirect
131. #107×C61C2×C4semidirect
132. #103×C61C2×C4semidirect
134. #97×C61C2×C4semidirect
171. SD16×C61C4⋊C4semidirect
350. #440Dic61semidirect
388. C61#22678semidirect
406. C4×Q8C2×Dic61semidirect
441. #107C2×Dic61semidirect
447. #103C2×Dic61semidirect
452. #97C2×Dic61semidirect
498. SD16C4⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398632
C2×SD16×Dic612
Q8×C4⋊Dic612
#214487292
#214465582
#214465532
#214460892
#214460822
#214460752
#214460682
#214459872
#214458342
#214458192
#214456912
C2×C61×#4402
#2267861