Finite Group Databasebeta

← #30602781

Subgroups — #30602781

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840704C211yesmaximal, characteristic
2C2×#16840704C211yesmaximal, characteristic
3C2×#16840639C214yesmaximal
4C4×C22⋊C4×D61C211yesmaximal, characteristic
5C22×C4⋊#6923—21no—
6C22×C4⋊#6923C412yescomplemented
7C22×C22⋊#6923C2212yes—
8C22×C22⋊#6923C2212yes—
9#16840704—21no—
10#16840704—21no—
11#16840704—21no—
12#16840704—21no—
13#16840704—21no—
14#16840704—21no—
15#16840704C412yescomplemented
16#16840704C412yescomplemented
17C2×(C2×#6923)⋊C4—21no—
18C2×(C2×#6923)⋊C4C412yescomplemented
19#16840639—28no—
20C23×C4×D61C2211yescharacteristic
21C2×C22⋊C4×D61C2212yes—
22C2×C42×D61—21no—
23C22⋊C4×Dic61C412yescomplemented
24C4×#11758198—21no—
25C4×C22⋊Dic61C411yescharacteristic, complemented
26#16839609—21no—
27#95×D61—21no—
28C4×C22⋊C4×C61C411yescharacteristic, complemented
29C23×#6923—22no—
30C23×#6923C2×C414yes—
31C2×C4⋊#6923—22no—
32C2×C4⋊#6923—22no—
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923—22no—
35C2×C4⋊#6923—44no—
36C2×C4⋊#6923—44no—
37C2×C22⋊#6923—24no—
38C2×C22⋊#6923—24no—
39C2×C22⋊#6923—22no—
40C2×C22⋊#6923—22no—
41C2×C22⋊#6923—22no—
42C2×C22⋊#6923—24no—
43C2×C22⋊#6923—24no—
44C2×C22⋊#6923C2×C414yes—
45(C2×#6923)⋊C4—22no—
46(C2×#6923)⋊C4—22no—
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—22no—
49C24×D61C2311yescharacteristic
50C23×Dic61C2×C411yescharacteristic
51C22×C4×D61—22no—
52C22×C4×D61—44no—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61Q811yescharacteristic
64C22×C4×D61Q811yescharacteristic
65C2×C22⋊Dic61C2×C412yes—
66C2×#11758198—22no—
67C22⋊C4×D61—22no—
68C22⋊C4×D61—22no—
69C22⋊C4×D61D414yes—
70C22⋊C4×D61Q814yes—
71C2×C4×Dic61—21no—
72C2×C4×Dic61—22no—
73C42×D61—44no—
74#14519684—21no—
75#14519684—22no—
76C23×C4×C61C2×C411yescharacteristic
77C2×C22⋊C4×C61C2×C412yes—
78C2×C42×C61—21no—
79#95×C61—21no—
80C24×C61C22×C411yescharacteristic
81C23×D61—42no—
82C23×D61—42no—
83C23×D61C22×C411yescharacteristic
84C23×D61C22×C411yescharacteristic
85C23×D61C2×D411yescharacteristic
86C23×D61C2×D411yescharacteristic
87C23×D61C2×D411yescharacteristic
88C23×D61C2×D412yes—
89C23×D61C2×Q811yescharacteristic
90C23×D61C2×Q812yes—
91C23×D61#4112yes—
92C23×D61#4112yes—
93C22×#6923—24no—
94C22×#6923—24no—
95C22×#6923—28no—
96C22×#6923—44no—
97C22×#6923—44no—
98C22×#6923—42no—
99C22×#6923—42no—
100C22×#6923—44no—
101C22×#6923—44no—
102C22×C4×C61—22no—
103C22×C4×C61—42no—
104C22×C4×C61C4211yescharacteristic
105C22×C4×C61C4211yescharacteristic
106C22×C4×C61C22⋊C411yescharacteristic
107C22×C4×C61C22⋊C411yescharacteristic
108C22×C4×C61C4⋊C411yescharacteristic
109C22×C4×C61C4⋊C411yescharacteristic
110C22×Dic61—22no—
111C22×Dic61—42no—
112C22×Dic61C4211yescharacteristic
113C22×Dic61C4211yescharacteristic
114C22×Dic61C22⋊C411yescharacteristic
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C4⋊C411yescharacteristic
117C22×Dic61C4⋊C411yescharacteristic
118C2×C4×D61—24no—
119C2×C4×D61—24no—
120C2×C4×D61—24no—
121C2×C4×D61—24no—
122C2×C4×D61—24no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—44no—
133C2×C4×D61—44no—
134C2×C4×D61C22⋊C411yescharacteristic
135C2×C4×D61C22⋊C411yescharacteristic
136C2×C4×D61C22⋊C411yescharacteristic
137C2×C4×D61C22⋊C411yescharacteristic
138C2×C4×D61C22⋊C411yescharacteristic
139C2×C4×D61C22⋊C411yescharacteristic
140C2×C4×D61C22⋊C411yescharacteristic
141C2×C4×D61C22⋊C411yescharacteristic
142C2×C4×D61C4⋊C411yescharacteristic
143C2×C4×D61C4⋊C411yescharacteristic
144C2×C4×D61C4⋊C411yescharacteristic
145C2×C4×D61C4⋊C411yescharacteristic
146C2×C4×D61C4⋊C411yescharacteristic
147C2×C4×D61C4⋊C411yescharacteristic
148C2×C4×D61C4⋊C411yescharacteristic
149C2×C4×D61C4⋊C411yescharacteristic
150C22⋊#6923—28no—
151C22⋊#6923—44no—
152C22⋊#6923—44no—
153C22⋊#6923—42no—
154C22⋊#6923—42no—
155C4⋊#6923—44no—
156C4⋊#6923—44no—
157C4⋊#6923—44no—
158C4⋊#6923—44no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61C4⋊C414yescomplemented
161C42×C61—42no—
162C22⋊Dic61—22no—
163C22⋊Dic61C4⋊C414yescomplemented
164#11758198—24no—
165#11758198—24no—
166C4×Dic61—44no—
167C4×Dic61—42no—
168C23×C61—41no—
169C23×C61—41no—
170C23×C61C2×C4211yescharacteristic
171C23×C61C2×C22⋊C411yescharacteristic
172C23×C61C2×C4⋊C411yescharacteristic
173C23×C61C2×C4⋊C412yes—
174C23×C61#11712yes—
175C22×D61—22no—
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—44no—
182C22×D61—44no—
183C22×D61—44no—
184C22×D61—44no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61C2×C22⋊C411yescharacteristic
192C22×D61C2×C22⋊C411yescharacteristic
193C22×D61C2×C4⋊C411yescharacteristic
194C22×D61C2×C4⋊C411yescharacteristic
195C22×D61C4×D412yes—
196C22×D61C4×D412yes—
197C22×D61C4×Q812yes—
198C22×D61C4×Q812yes—
199C22×D61#12112yes—
200C22×D61#12112yes—
201C22×D61C22⋊Q812yes—
202C22×D61C22⋊Q812yes—
203C22×D61#12411yescharacteristic
204C22×D61#12411yescharacteristic
205C22×D61#12512yes—
206C22×D61#12711yescharacteristic
207C22×D61C4⋊Q811yescharacteristic
208C22×D61C4⋊Q812yes—
209C2×#6923—44no—
210C2×#6923—44no—
211C2×#6923—44no—
212C2×#6923—44no—
213C2×#6923—44no—
214C2×#6923—44no—
215C2×#6923—84no—
216C2×#6923—84no—
217C2×#6923—88no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—24no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61#9511yescharacteristic
228C2×C4×C61#9511yescharacteristic
229C2×C4×C61#9511yescharacteristic
230C2×C4×C61#9511yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61#9511yescharacteristic
241C2×Dic61#9511yescharacteristic
242C2×Dic61#9511yescharacteristic
243C2×Dic61#9511yescharacteristic
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—48no—
260C4×D61—48no—
261C4×D61#9914yes—
262#10854—611nomaximal, Sylow
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61C2×#9511yescharacteristic
272C22×C61C4×C4⋊C412yes—
273C22×C61#38012yes—
274C22×C61#38211yescharacteristic
275C22×C61#38411yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—24no—
283C2×D61—24no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—24no—
287C2×D61—22no—
288C2×D61—24no—
289C2×D61—22no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61#38912yes—
297C2×D61#38912yes—
298C2×D61#39011yescharacteristic
299C2×D61#39011yescharacteristic
300C2×D61#39111yescharacteristic
301C2×D61#39111yescharacteristic
302C2×D61C2×#9912yes—
303C2×D61#41012yes—
304#6923—84no—
305#6923—84no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—44no—
310C4×C61#34212yescomplemented
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—44no—
315Dic61#34212yescomplemented
316C2×#380—611no—
317C2×#380—611no—
318C2×C4×C22⋊C4—611no—
319C2×#342—614no—
320C2×C61—22no—
321C2×C61—22no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61C4⋊#9511yescharacteristic
325C2×C61C2×#34211yescharacteristic
326C2×C61#174011yescharacteristic
327D61—21no—
328D61—22no—
329D61—22no—
330D61—21no—
331D61#196412yes—
332D61#199612yes—
333C24×C4—611no—
334C22×C4⋊C4—612no—
335C22×C4⋊C4—1221no—
336C22×C22⋊C4—612no—
337C22×C22⋊C4—612no—
338C22×C22⋊C4—612no—
339C22×C42—1221no—
340#380—612no—
341#380—612no—
342#380—1221no—
343#380—1221no—
344#380—1221no—
345#380—1221no—
346#380—1221no—
347#380—1221no—
348C4×C22⋊C4—611no—
349C4×C22⋊C4—612no—
350C4×C22⋊C4—1221no—
351C4×C22⋊C4—1221no—
352C4×C22⋊C4#692311yescharacteristic, complemented
353C2×#95—612no—
354C2×#95—1221no—
355C2×#95—1221no—
356#342—1228no—
357C61#1085411yescharacteristic, complemented, Sylow
358C25—611no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—614no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—2444no—
376C23×C4C2×#692311yescharacteristic
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—2444no—
382C2×C4⋊C4—2444no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—614no—
385C2×C22⋊C4—614no—
386C2×C22⋊C4—614no—
387C2×C22⋊C4—1222no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1224no—
393C2×C22⋊C4—1224no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4C2×#692312yes—
398C2×C42—21no—
399C2×C42—1222no—
400C2×C42—1221no—
401C2×C42—2444no—
402#95—21no—
403#95—1222no—
404#95—1222no—
405#95—1222no—
406#95—1222no—
407#95—1222no—
408#95—1221no—
409C24—611no—
410C24—611no—
411C24—611no—
412C24—611no—
413C24—611no—
414C24—612no—
415C24—612no—
416C24—611no—
417C24—612no—
418C24—612no—
419C24—2442no—
420C24—2442no—
421C24C22×#692311yescharacteristic
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—1222no—
452C22×C4—1228no—
453C22×C4—1224no—
454C22×C4—1224no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2444no—
465C22×C4—2444no—
466C22×C4—2442no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2444no—
470C22×C4—2444no—
471C22×C4—2444no—
472C22×C4—2444no—
473C22×C4C4×#692311yescharacteristic
474C22×C4C4×#692311yescharacteristic
475C22×C4C4⋊#692311yescharacteristic
476C22×C4C4⋊#692311yescharacteristic
477C22×C4C22⋊#692311yescharacteristic
478C22×C4C22⋊#692311yescharacteristic
479C4⋊C4—2444no—
480C4⋊C4—2444no—
481C4⋊C4—2444no—
482C4⋊C4—2444no—
483C22⋊C4—22no—
484C22⋊C4—614no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—1222no—
488C22⋊C4—1228no—
489C22⋊C4—2444no—
490C22⋊C4—2444no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C22⋊C4C4⋊#692314yescomplemented
494C42—42no—
495C42—2442no—
496C42—2444no—
497C23—41no—
498C23—41no—
499C23—611no—
500C23—612no—
501C23—612no—
502C23—611no—
503C23—611no—
504C23—611no—
505C23—611no—
506C23—611no—
507C23—611no—
508C23—611no—
509C23—612no—
510C23—612no—
511C23—612no—
512C23—612no—
513C23—612no—
514C23—612no—
515C23—612no—
516C23—612no—
517C23—1222no—
518C23—1222no—
519C23—1222no—
520C23—1222no—
521C23—1222no—
522C23—1222no—
523C23—2441no—
524C23—2441no—
525C23—2441no—
526C23—2441no—
527C23—2441no—
528C23—2441no—
529C23—2444no—
530C23—2444no—
531C23—2444no—
532C23—2444no—
533C23C2×C4×#692311yescharacteristic
534C23C2×C22⋊#692311yescharacteristic
535C23C2×C4⋊#692311yescharacteristic
536C23C2×C4⋊#692312yes—
537C23#1451985612yes—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—611no—
548C2×C4—614no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1224no—
565C2×C4—4884no—
566C2×C4—4884no—
567C2×C4—4888no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2448no—
575C2×C4—2448no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2444no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4(C2×#6923)⋊C411yescharacteristic
588C2×C4(C2×#6923)⋊C411yescharacteristic
589C2×C4(C2×#6923)⋊C411yescharacteristic
590C2×C4(C2×#6923)⋊C411yescharacteristic
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—42no—
598C22—42no—
599C22—611no—
600C22—611no—
601C22—612no—
602C22—612no—
603C22—611no—
604C22—612no—
605C22—612no—
606C22—611no—
607C22—1222no—
608C22—1222no—
609C22—1224no—
610C22—1224no—
611C22—1224no—
612C22—1222no—
613C22—1222no—
614C22—1224no—
615C22—1221no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—1221no—
620C22—1221no—
621C22—2442no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22—2442no—
627C22C2×(C2×#6923)⋊C411yescharacteristic
628C22C4×C4⋊#692312yes—
629C22#1684070412yes—
630C22#1684070611yescharacteristic
631C22#1684070811yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—44no—
636C4—612no—
637C4—1221no—
638C4—1221no—
639C4—1221no—
640C4—4884no—
641C4—4884no—
642C4—2444no—
643C4#1684063912yescomplemented
644C2—22no—
645C2—21no—
646C2—21no—
647C2—22no—
648C2—612no—
649C2—612no—
650C2—1221no—
651C2—1222no—
652C2—1222no—
653C2—1221no—
654C2#2146099711yescharacteristic
655C2C2×#1684063911yescharacteristic
656C2#2146151711yescharacteristic
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC4×C22⋊C4×C61C4
socleC22×C61C2×#95
prime radical, cosocleC24×D61C23
hypercenterC4×C22⋊C4#6923
hypocenterC61#10854
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊#6923C4semidirect
15. #16840704C4semidirect
16. #16840704C4semidirect
18. C2×(C2×#6923)⋊C4C4semidirect
23. C22⋊C4×Dic61C4semidirect
25. C4×C22⋊Dic61C4semidirect
28. C4×C22⋊C4×C61C4semidirect
160. C22⋊C4×C61C4⋊C4semidirect
163. C22⋊Dic61C4⋊C4semidirect
310. C4×C61#342semidirect
315. Dic61#342semidirect
352. C4×C22⋊C4#6923semidirect
357. C61#10854semidirect
493. C22⋊C4C4⋊#6923semidirect
643. C4#16840639semidirect

Maximal subgroups

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

subgroupindex
C2×#168407042
C2×#168406392
C4×C22⋊C4×D612
#1085461