Finite Group Databasebeta

← #29696639

Subgroups — #29696639

Order 15616.

1110 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840471C211yesmaximal, characteristic
2C2×#16839890C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
4#21446788C211yesmaximal, characteristic, complemented
5#21446786C211yesmaximal, characteristic, complemented
6#21446778C211yesmaximal, characteristic, complemented
7#21446776C211yesmaximal, characteristic, complemented
8#21446775C211yesmaximal, characteristic, complemented
9#21446765C211yesmaximal, characteristic, complemented
10#21445819C211yesmaximal, characteristic
11#21445817C211yesmaximal, characteristic
12#21445817C211yesmaximal, characteristic
13#21445646C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#1762C211yesmaximal, characteristic
16#16840471—24no—
17#16840471C418yescomplemented
18C2×#14519776C2211yescharacteristic
19C2×#14519769C2211yescharacteristic
20C2×#14519768C2211yescharacteristic
21C2×#14519756—21no—
22C2×#14519754—21no—
23C2×#14519754—21no—
24C2×C4×Dic122—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×D4×Dic61C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C22×#11758196—21no—
29#16839890—24no—
30C2×C4⋊#92418C2211yescharacteristic
31C2×#14519706C2211yescharacteristic
32C2×#14519706C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839660—21no—
38#16839659—21no—
39#16839659—21no—
40#16839632C2211yescharacteristic
41#16839631C2211yescharacteristic
42#16839631C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#97×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—21no—
63C22×Dic122D411yescharacteristic
64C23×Dic61—21no—
65C23×Dic61—21no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776C2×C414yes—
70#14519769—22no—
71#14519769C2×C414yes—
72#14519768—22no—
73#14519768C2×C414yes—
74#14519756—28no—
75#14519754—28no—
76#14519754—28no—
77C4×Dic122—44no—
78C4×Dic122C2×C414yes—
79C4×Dic122D414yes—
80D4×Dic61—22no—
81D4×Dic61—22no—
82D4×Dic61C2×C414yes—
83D4×Dic61C2×C414yes—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61D411yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—28no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196D411yescharacteristic
102C2×#11758196D411yescharacteristic
103C2×C22⋊Dic61—21no—
104C2×C22⋊Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C4⋊#92418—22no—
110#14519706—22no—
111#14519706—22no—
112#14519698—22no—
113#14519698—22no—
114#14519687—22no—
115C8×Dic61—22no—
116#14519684—21no—
117#14519684—21no—
118C22×D4×C61C2311yescharacteristic
119C23×C4×C61—21no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×D4×C61—22no—
123C4×D4×C61C2×C414yes—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130#102×C61—22no—
131#102×C61Q814yescomplemented
132#97×C61—22no—
133#95×C61—21no—
134C24×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C22×C412yes—
139C2×D4×C61C2×Q812yes—
140C2×D4×C61#4112yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—21no—
145C22×C4×C61—81no—
146C22×C4×C61—81no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—24no—
153C22×Dic61—24no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—81no—
159C22×Dic61—81no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122—24no—
167C2×Dic122—81no—
168C2×Dic122—81no—
169C2×Dic122C22⋊C414yes—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×Q812yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C412yes—
181C4⋊C4×C61C2×Q812yes—
182C22⋊C4×C61—24no—
183C42×C61—42no—
184C42×C61C22×C412yes—
185C42×C61C2×D412yes—
186C22⋊Dic61—24no—
187C22⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—24no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—81no—
197C4⋊Dic61C22⋊C414yes—
198C4⋊Dic61C22×C412yes—
199C4⋊Dic61C22×C412yes—
200#11758196—24no—
201#11758196—24no—
202#11758196—24no—
203#11758196—24no—
204#11758196—42no—
205#11758196—42no—
206#11758196—81no—
207#11758196—81no—
208#11758196C22⋊C414yes—
209#11758196C22⋊C414yes—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61C22×C412yes—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C22×D411yescharacteristic
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61C4×Q814yescomplemented
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—84no—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—48no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—84no—
276C2×Dic61C2×C22⋊C412yes—
277C2×Dic61C2×C22⋊C412yes—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280C2×Dic61AGL(1,ℤ/8)12yes—
281C2×Dic61#13712yes—
282Dic122—28no—
283Dic122—84no—
284C8×C61—22no—
285C8×C61—41no—
286#92418—41no—
287#92418—41no—
288#22522—611nomaximal, Sylow
289C22×C61—24no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61C22×C22⋊C411yescharacteristic
297C22×C61#54811yescharacteristic
298C22×C61D4×Q811yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61#57611yescharacteristic
302C22×C61#57711yescharacteristic
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61—81no—
308C4×C61C2×C4×Q811yescharacteristic
309C4×C61#51911yescharacteristic
310C4×C61#54811yescharacteristic
311C4×C61D4×Q811yescharacteristic
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316Dic61—42no—
317Dic61—81no—
318Dic61—81no—
319Dic61#41714yes—
320C2×#554—611no—
321C2×#446—611no—
322Q8×C4⋊C4—611no—
323#1992—611no—
324#1992—611no—
325#1990—611no—
326#1989—611no—
327#1989—611no—
328#1987—611no—
329#1802—611no—
330#1802—611no—
331#1802—611no—
332#1762—611no—
333#1762D6111yescharacteristic
334C4×#102—611no—
335C2×C61—44no—
336C2×C61Q8×C22⋊C411yescharacteristic
337C2×C61C2×#41711yescharacteristic
338C2×C61#289711yescharacteristic
339C2×C61#336611yescharacteristic
340C2×C61#337211yescharacteristic
341C2×C61#338411yescharacteristic
342C2×C61#338511yescharacteristic
343#554—618no—
344#554—1224no—
345C2×C4⋊Q8—611no—
346C2×#125—611no—
347C2×#125—611no—
348C2×C22⋊Q8—1221no—
349C2×C22⋊Q8—1221no—
350C2×C22⋊Q8—1221no—
351C2×C4×Q8—611no—
352C2×C4×Q8—1221no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4C2×D6111yescharacteristic
356C22×C4⋊C4—1221no—
357#446—1224no—
358C2×C4⋊C8—611no—
359C2×C4⋊C8—611no—
360C2×C4⋊C8—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102C2×D6111yescharacteristic
364C2×#102C2×D6111yescharacteristic
365C2×#97—611no—
366C2×#97C2×D6111yescharacteristic
367C2×C4×C8—611no—
368#389—1221no—
369#389—1221no—
370#389—1221no—
371#384—611no—
372#384—611no—
373#384—611no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4C2×D6111yescharacteristic
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389C61#2252211yescharacteristic, complemented, Sylow
390C22×Q8—611no—
391C22×Q8—1221no—
392C22×D4C22×D6111yescharacteristic
393C23×C4—21no—
394C23×C4—1221no—
395C23×C4—1221no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400C4⋊Q8—614no—
401C4⋊Q8—1222no—
402#125—614no—
403#125—614no—
404#125—1222no—
405#125—1222no—
406C22⋊Q8—1228no—
407C22⋊Q8—1228no—
408C22⋊Q8—1228no—
409C4×Q8—614no—
410C4×Q8—614no—
411C4×Q8—2444no—
412C4×D4—22no—
413C4×D4—614no—
414C4×D4—614no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4C4×D6114yes—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1228no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—21no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452#102—22no—
453#102—1222no—
454#102—1222no—
455#102Dic12214yescomplemented
456#97—22no—
457#97—1222no—
458C4×C8—1222no—
459#95—21no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C2×Q8—614no—
464C2×Q8—1222no—
465C2×Q8—1222no—
466C2×Q8—1224no—
467C2×Q8—4881no—
468C2×Q8—4881no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×Dic12212yes—
473C2×D4C2×C4×D6112yes—
474C2×D4#1175822212yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—4881no—
492C22×C4—4881no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8C2×Dic12212yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—614no—
515C4⋊C4—614no—
516C4⋊C4—614no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—1224no—
520C4⋊C4—1224no—
521C4⋊C4—1224no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1224no—
525C4⋊C4—1224no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—4881no—
531C4⋊C4—4881no—
532C4⋊C4—4881no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4C2×Dic12212yes—
537C4⋊C4C2×C4×D6112yes—
538C22⋊C4—24no—
539C22⋊C4—1224no—
540C22⋊C4—1224no—
541C42—42no—
542C42—612no—
543C42—612no—
544C42—612no—
545C42—612no—
546C42—2442no—
547C42—2442no—
548C42C2×C4×D6112yes—
549C42D4×D6112yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×D4×D6111yescharacteristic
558Q8—1228no—
559Q8—4884no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4C4×Dic12214yescomplemented
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—84no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—4884no—
593C2×C4—2448no—
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×C4—2441no—
603C2×C4—2441no—
604C2×C4C22⋊C4×D6112yes—
605C2×C4C22⋊C4×D6112yes—
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4C22×Dic12211yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4#1451982411yescharacteristic
611C2×C4#1451982411yescharacteristic
612C8—22no—
613C8—41no—
614C8—2441no—
615C8—2441no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×C22⋊C4×D6111yescharacteristic
624C22#1684042111yescharacteristic
625C22D4×Dic12211yescharacteristic
626C22AGL(1,ℤ/8)×D6111yescharacteristic
627C22#137×D6111yescharacteristic
628C22#1684056611yescharacteristic
629C22#1684057211yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—614no—
636C4—1222no—
637C4—1222no—
638C4—1222no—
639C4—4881no—
640C4—4881no—
641C4—2442no—
642C4—2442no—
643C4C2×C4×Dic12211yescharacteristic
644C4#1684023911yescharacteristic
645C4#1684042111yescharacteristic
646C4D4×Dic12211yescharacteristic
647C2—44no—
648C2C22⋊C4×Dic12211yescharacteristic
649C2#417×D6111yescharacteristic
650C2#2145434411yescharacteristic
651C2#2145852711yescharacteristic
652C2#2145860111yescharacteristic
653C2#2145871911yescharacteristic
654C2#2145873011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#22522
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16839890C2semidirect
3. C4⋊C4×Dic122C2semidirect
4. #21446788C2semidirect
5. #21446786C2semidirect
6. #21446778C2semidirect
7. #21446776C2semidirect
8. #21446775C2semidirect
9. #21446765C2semidirect
17. #16840471C4semidirect
131. #102×C61Q8semidirect
233. D4×C61C4×Q8semidirect
389. C61#22522semidirect
455. #102Dic122semidirect
563. D4C4×Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168404712
C2×#168398902
C4⋊C4×Dic1222
#214467882
#214467862
#214467782
#214467762
#214467752
#214467652
#214458192
#214458172
#214456462
#102×Dic612
C61×#17622
#2252261