Finite Group Databasebeta

← #30620406

Subgroups — #30620406

Order 15616.

1292 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840814C212yesmaximal, complemented
2#21461993C212yesmaximal, complemented
3#21461992C212yesmaximal, complemented
4#21461990C212yesmaximal, complemented
5#21461676C211yesmaximal, characteristic
6#21461676C211yesmaximal, characteristic
7#21461674C211yesmaximal, characteristic
8#21461546C211yesmaximal, characteristic
9#21461546C211yesmaximal, characteristic
10C4×#14519834C211yesmaximal, characteristic
11C2×C4×Q8×D61C211yesmaximal, characteristic
12#16840814—28no—
13C2×#14519844C2212yes—
14C2×#14519844C2212yes—
15C2×C4⋊#92420C2212yes—
16C2×#14519834C2211yescharacteristic
17C2×#14519834C2211yescharacteristic
18C2×#14519834C2211yescharacteristic
19C2×#14519834C2211yescharacteristic
20C2×#14519832C2212yes—
21C2×#14519832C2212yes—
22C2×C4×#92420C2212yes—
23#16840628C2212yes—
24#16840628C2212yes—
25#16840628C2212yes—
26#16840628C2212yes—
27#16840628C2212yes—
28#16840628C2212yes—
29C22×Q8×D61C411yescharacteristic
30C2×Q8×Dic61C2211yescharacteristic
31C2×C4×Dic122C411yescharacteristic
32C2×C4×Dic122C411yescharacteristic
33C2×C4×Dic122C411yescharacteristic
34C2×#14519742C2211yescharacteristic
35C2×#14519742C2211yescharacteristic
36C2×#14519742C2211yescharacteristic
37C4×Q8×D61—28no—
38C2×C4⋊C4×D61C411yescharacteristic
39C2×C4⋊C4×D61C411yescharacteristic
40C2×C4⋊C4×D61C411yescharacteristic
41C2×C42×D61C2211yescharacteristic
42C2×C42×D61C2211yescharacteristic
43C2×C42×D61C2211yescharacteristic
44C2×C4×Q8×C61C411yescharacteristic
45C22×#92420C2312yes—
46C22×#92420C2312yes—
47C22×#92420C2312yes—
48C22×#92420C2312yes—
49#14519844—24no—
50#14519844—24no—
51C4⋊#92420—24no—
52#14519834—22no—
53#14519834—22no—
54#14519834—22no—
55#14519834Q814yescomplemented
56#14519832—24no—
57#14519832—24no—
58C4×#92420—24no—
59C22×Dic122C2×C411yescharacteristic
60C22×Dic122C2×C411yescharacteristic
61C22×Dic122C2×C411yescharacteristic
62C2×Q8×D61—24no—
63C2×Q8×D61—24no—
64C2×Q8×D61—24no—
65C22×C4×D61C2×C411yescharacteristic
66C22×C4×D61C2×C411yescharacteristic
67C22×C4×D61C2×C411yescharacteristic
68C22×C4×D61C2×C411yescharacteristic
69C22×C4×D61C2×C411yescharacteristic
70C22×C4×D61C2×C411yescharacteristic
71C22×C4×D61C2311yescharacteristic
72Q8×Dic61—24no—
73C4×Dic122—24no—
74C4×Dic122—24no—
75C4×Dic122—24no—
76#14519742—24no—
77#14519742—24no—
78#14519742D418yes—
79C2×C4⋊Dic61C2×C411yescharacteristic
80C2×C4⋊Dic61C2×C411yescharacteristic
81C2×C4⋊Dic61C2×C411yescharacteristic
82C2×#11758196C2×C411yescharacteristic
83C2×#11758196C2×C411yescharacteristic
84C2×#11758196C2×C411yescharacteristic
85C2×#11758196C2×C411yescharacteristic
86C2×#11758196C2×C411yescharacteristic
87C2×#11758196C2×C411yescharacteristic
88C4⋊C4×D61—24no—
89C4⋊C4×D61—24no—
90C4⋊C4×D61—24no—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61C2×C411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—24no—
101C42×D61—24no—
102C42×D61—24no—
103C22×Q8×C61C2×C411yescharacteristic
104C4×Q8×C61—24no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C2×C4⋊C4×C61C2×C411yescharacteristic
108C2×C42×C61C2×C411yescharacteristic
109C2×C42×C61C2×C411yescharacteristic
110C2×C42×C61C2×C411yescharacteristic
111C23×D61C22×C411yescharacteristic
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C22×C411yescharacteristic
124C22×Dic61C22×C411yescharacteristic
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C411yescharacteristic
128C22×Dic61C2411yescharacteristic
129Q8×D61—232no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61M1614yes—
150C2×C4×D61C2×Q812yes—
151C2×C4×D61#4112yes—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122C22⋊C414yes—
160C2×Dic122M1614yes—
161C2×#92420—22no—
162C2×#92420—22no—
163C2×#92420—22no—
164C2×#92420—22no—
165C2×#92420—22no—
166C2×#92420—22no—
167C2×#92420—22no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420C2×Q814yescomplemented
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C22⋊C414yes—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182#11758196—22no—
183#11758196—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196C22⋊C414yes—
187#11758196C22⋊C414yes—
188C4×Dic61—22no—
189C4×Dic61—22no—
190C4×Dic61—22no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61C2×D414yes—
195C4×Dic61C2×D414yes—
196C4×Dic61C2×D414yes—
197C23×C61C23×C411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61C4×Q812yes—
203C22×D61C2×M1612yes—
204C22×D61#13112yes—
205Q8×C61—28no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C4×Q812yes—
225C2×C4×C61C2×M1612yes—
226C2×C4×C61C2×M1612yes—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C412yes—
241C2×Dic61C2×C22⋊C412yes—
242C2×Dic61C2×M1612yes—
243C2×Dic61C2×M1612yes—
244C2×Dic61C2×M1612yes—
245C2×Dic61C22×D412yes—
246C2×Dic61C22×Q811yescharacteristic
247C2×Dic61C2×#4111yescharacteristic
248C2×Dic612−1+412yes—
249C4×D61—28no—
250C4×D61—28no—
251C4×D61—24no—
252C4×D61—24no—
253C4×D61—28no—
254C4×D61—44no—
255C4×D61—44no—
256C4×D61—44no—
257Dic122—28no—
258Dic122—28no—
259Dic122—28no—
260#92420—24no—
261#92420—42no—
262#92420—42no—
263#92420—42no—
264#22510—611nomaximal, Sylow
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61C2×C4×Q811yescharacteristic
267C22×C61#51911yescharacteristic
268C22×C61C22×M1611yescharacteristic
269C22×C61C22×M1611yescharacteristic
270C22×C61C2×#13111yescharacteristic
271C22×C61#56811yescharacteristic
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61#44614yescomplemented
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—41no—
286Dic61—22no—
287Dic61—22no—
288Dic61—22no—
289Dic61—41no—
290Dic61—41no—
291Dic61#40714yes—
292Dic61#54812yes—
293Dic61D4×Q812yes—
294C22×C4×Q8—611no—
295C2×#446—612no—
296#1985—612no—
297#1985—612no—
298#1984—612no—
299#1797—611no—
300#1797—611no—
301#1797—611no—
302#1749—611no—
303#1749—611no—
304C4×C22⋊C8—611no—
305C2×C61Q8×C22⋊C411yescharacteristic
306C2×C61C2×#40711yescharacteristic
307C2×C61#288011yescharacteristic
308C2×C61C2×#44611yescharacteristic
309C2×C61Q8×M1611yescharacteristic
310C2×C61#298311yescharacteristic
311C2×C61#298411yescharacteristic
312D61—24no—
313C23×Q8—611no—
314C2×C4×Q8—611no—
315C2×C4×Q8—611no—
316C2×C4×Q8—611no—
317C2×C4×Q8—611no—
318C2×C4×Q8—611no—
319C2×C4×Q8—611no—
320C2×C4×Q8—611no—
321C2×C4×Q8—1228no—
322C2×C4×Q8#692311yescharacteristic
323C22×C4⋊C4—611no—
324C22×C4⋊C4—611no—
325C22×C4⋊C4—611no—
326C22×C42—611no—
327C22×C42—611no—
328C22×C42—611no—
329#446—1228no—
330C2×C4⋊C8—612no—
331C2×C4⋊C8—612no—
332C2×C4⋊C8—612no—
333C2×C22⋊C8—611no—
334C2×C22⋊C8—611no—
335C2×C22⋊C8—611no—
336C2×C22⋊C8—611no—
337C2×#97—612no—
338C2×#97—612no—
339C2×C4×C8—612no—
340#336—612no—
341#336—612no—
342#336—612no—
343#336—612no—
344#336—612no—
345#336—612no—
346C61#2251011yescharacteristic, complemented, Sylow
347C22×Q8—611no—
348C22×Q8—611no—
349C22×Q8—611no—
350C22×Q8—1224no—
351C22×Q8—1224no—
352C22×Q8—1224no—
353C22×Q8C2×#692311yescharacteristic
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—611no—
361C22×C8—612no—
362C22×C8—612no—
363C22×C8—612no—
364C22×C8—612no—
365C4×Q8—24no—
366C4×Q8—618no—
367C4×Q8—1224no—
368C4×Q8—1224no—
369C4×Q8—1224no—
370C4×Q8—1224no—
371C4×Q8—1224no—
372C4×Q8—1224no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—1224no—
383C2×C4⋊C4—1224no—
384C2×C4⋊C4—1224no—
385C2×C4⋊C4C2×#692311yescharacteristic
386C2×C4⋊C4C2×#692311yescharacteristic
387C2×C4⋊C4C2×#692311yescharacteristic
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—1224no—
398C2×C42—1224no—
399C2×C42—1224no—
400C2×C42C2×#692311yescharacteristic
401C2×C42C2×#692311yescharacteristic
402C2×C42C2×#692311yescharacteristic
403C4⋊C8—1224no—
404C4⋊C8—1224no—
405C4⋊C8—1224no—
406C22⋊C8—614no—
407C22⋊C8—1222no—
408C22⋊C8—1222no—
409C22⋊C8—1222no—
410#97—1224no—
411#97—1224no—
412C4×C8—1224no—
413C24—611no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—614no—
418C2×Q8—614no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8—1222no—
425C2×Q8—12232no—
426C2×Q8—1222no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—614no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—2441no—
454C22×C4—2441no—
455C22×C4—2441no—
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C22×C4C22×#692311yescharacteristic
459C22×C4C22×#692311yescharacteristic
460C22×C4C22×#692311yescharacteristic
461C22×C4C22×#692311yescharacteristic
462C22×C4C22×#692311yescharacteristic
463C2×C8—614no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—614no—
478C4⋊C4—614no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4C22⋊#692314yes—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—614no—
491C42—614no—
492C42—614no—
493C42—1222no—
494C42—1222no—
495C42—1222no—
496C42—1222no—
497C42—1222no—
498C42—1222no—
499C23—612no—
500C23—612no—
501C23—612no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23—1221no—
506C23C23×#692311yescharacteristic
507Q8—28no—
508Q8—1228no—
509Q8—1228no—
510Q8—1228no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1228no—
539C2×C4—1228no—
540C2×C4—1228no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4C2×C22⋊#692312yes—
557C2×C4C2×C22⋊#692312yes—
558C2×C4C2×C22⋊#692312yes—
559C2×C4Q8×#692312yes—
560C2×C4D61⋊M1612yes—
561C2×C4D61⋊M1612yes—
562C8—1224no—
563C8—2442no—
564C8—2442no—
565C8—2442no—
566C22—614no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22—1222no—
572C22—1222no—
573C22C22×C22⋊#692311yescharacteristic
574C22C2×Q8×#692311yescharacteristic
575C22#1684083011yescharacteristic
576C22C2×D61⋊M1611yescharacteristic
577C22C2×D61⋊M1611yescharacteristic
578C22D61⋊#13111yescharacteristic
579C22#1684084111yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—612no—
588C4—612no—
589C4—614no—
590C4—1222no—
591C4—1222no—
592C4—1222no—
593C4—2441no—
594C4—2441no—
595C2—1224no—
596C2Q8×C22⋊#692311yescharacteristic
597C2D61⋊#40711yescharacteristic
598C2#2146240911yescharacteristic
599C2D61⋊#44611yescharacteristic
600C2#2146262111yescharacteristic
601C2#2146271611yescharacteristic
602C2#2146272011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C4×Q8×C61C4
prime radicalC23×D61C22×C4
cosocleC22×Dic61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#22510
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840814C2semidirect
2. #21461993C2semidirect
3. #21461992C2semidirect
4. #21461990C2semidirect
55. #14519834Q8semidirect
172. C2×#92420C2×Q8semidirect
278. C2×D61#446semidirect
346. C61#22510semidirect

Maximal subgroups

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

subgroupindex
C2×#168408142
#214619932
#214619922
#214619902
#214616762
#214616742
#214615462
C4×#145198342
C2×C4×Q8×D612
#2251061