Finite Group Databasebeta

← #29702312

Subgroups — #29702312

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839984C211yesmaximal, characteristic
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic
4#21448888C211yesmaximal, characteristic
5#21446628C211yesmaximal, characteristic
6#21446585C211yesmaximal, characteristic, complemented
7#21446501C211yesmaximal, characteristic
8#21446494C211yesmaximal, characteristic, complemented
9#21446174C211yesmaximal, characteristic
10#21446160C211yesmaximal, characteristic, complemented
11#21446086C211yesmaximal, characteristic, complemented
12#21446065C211yesmaximal, characteristic
13C4⋊C4×#92418C211yesmaximal, characteristic, complemented
14C4×#14519701C211yesmaximal, characteristic, complemented
15C61×#2351C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16839984C418yescomplemented
22C4×#11758202—24no—
23C4×#11758201—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×C4×#92418C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839652—21no—
41#16839651—21no—
42#16839633—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44#16839609—21no—
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839517C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C61×#389—21no—
55C61×#389—21no—
56C61×#389C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#382—21no—
59C61×#382C2211yescharacteristic
60C4×C4⋊C4×C61—21no—
61C4×C4⋊C4×C61C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202C2×C414yescomplemented
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519765—22no—
76#14519765C2×C414yes—
77C4×Dic122—22no—
78C4×Dic122—41no—
79C4×D244—22no—
80C4×D244—41no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91#14519710—22no—
92#14519708—22no—
93C4⋊#92418C2×C414yescomplemented
94#14519701—22no—
95#14519701D414yes—
96#14519700—22no—
97#14519700—41no—
98#14519700C2×C414yescomplemented
99#14519697—22no—
100#14519697—41no—
101#14519697C2×C414yes—
102C4×#92418—22no—
103C4×#92418—22no—
104C4×#92418—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108C22×Q8×C61C2311yescharacteristic
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C4×Q8×C61—22no—
112C4×Q8×C61—41no—
113C4×Q8×C61—41no—
114C4×Q8×C61C2×C414yescomplemented
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131#95×C61—21no—
132#95×C61—21no—
133#95×C61—21no—
134#95×C61—21no—
135#95×C61D411yescharacteristic
136#95×C61D411yescharacteristic
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C412yes—
142C2×Q8×C61C2×D412yes—
143C2×Q8×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—81no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61C22×C412yes—
186C42×C61—24no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—42no—
192C42×C61—42no—
193C42×C61C22×C412yes—
194C42×C61#4112yes—
195#11758202—28no—
196#11758201—28no—
197#11758198—24no—
198#11758198—24no—
199#11758198—42no—
200#11758198—42no—
201#11758198—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D412yes—
206#11758196—42no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C22×C412yes—
219C2×#92418C22×C412yes—
220C2×#92418C2×D412yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61C4×D414yes—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
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—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61C4×D412yes—
260C2×C4×C61C4×D412yes—
261C2×C4×C61#13512yes—
262C2×C4×C61#13512yes—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278D244—22no—
279D244—22no—
280D244—41no—
281D244—41no—
282C4×D61—84no—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287#92418—22no—
288#92418—22no—
289#92418—22no—
290#92418—41no—
291#23000—611nomaximal, Sylow
292C22×C61C2×C4×D411yescharacteristic
293C22×C61D4211yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61C2×#13511yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57811yescharacteristic
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C4×C61—24no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61C2×C4×D411yescharacteristic
317C4×C61#52011yescharacteristic
318C4×C61#54711yescharacteristic
319C4×C61#55311yescharacteristic
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323C2×#464—611no—
324C2×C4×Q16—611no—
325C2×C4×SD16—611no—
326#2367—611no—
327#2351D6111yescharacteristic, complemented
328#1958—611no—
329#1949—611no—
330#1934—611no—
331#1933—611no—
332#1880—611no—
333#1878—611no—
334#1865—611no—
335#1862—611no—
336C8×C4⋊C4—611no—
337C4×#103—611no—
338C2×C61#236111yescharacteristic
339C2×C61C4×#13511yescharacteristic
340C2×C61#294711yescharacteristic
341C2×C61#328211yescharacteristic
342C2×C61#328511yescharacteristic
343C2×C61#336411yescharacteristic
344C2×C61#336511yescharacteristic
345D61—82no—
346C22×Q16—611no—
347C22×SD16—611no—
348C2×#124—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8C2×D6111yescharacteristic
351C2×C4×Q8C2×D6111yescharacteristic
352C2×C4×Q8C2×D6111yescharacteristic
353C2×C4×D4—611no—
354#464—618no—
355C4×Q16—1224no—
356C4×SD16—1224no—
357C2×#107—611no—
358C2×#106—611no—
359C2×C4⋊C8—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×C4×C8—611no—
369C2×C4×C8—611no—
370C2×C4×C8—611no—
371#391—611no—
372#390—611no—
373#389—21no—
374#389—21no—
375#389C2×D6111yescharacteristic
376#389C2×D6111yescharacteristic
377#388—1221no—
378#388—1221no—
379#385—1221no—
380#382—21no—
381#382C2×D6111yescharacteristic
382C4×C4⋊C4—21no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4C2×D6111yescharacteristic
385C4×C22⋊C4—1221no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#336—611no—
391#336—611no—
392C61#2300011yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×Q8C22×D6111yescharacteristic
395C22×D4—611no—
396C23×C4—1221no—
397C2×Q16—614no—
398C2×Q16—1222no—
399C2×Q16—1222no—
400C2×SD16—614no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8—611no—
407#124—614no—
408#124—1222no—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—41no—
413C4×Q8—41no—
414C4×Q8—1222no—
415C4×Q8—2441no—
416C4×Q8C4×D6114yescomplemented
417C4×D4—1222no—
418C4×D4—2441no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22⋊D6111yescharacteristic
428C2×C4⋊C4C22⋊D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#106—1222no—
447C4⋊C8—614no—
448#103—614no—
449#103—614no—
450#103—1222no—
451#103—1222no—
452#103—2441no—
453#102—614no—
454#102—1222no—
455#102—2441no—
456C4×C8—1222no—
457C4×C8—1222no—
458C4×C8—1222no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95C22⋊D6111yescharacteristic
467#95C22⋊D6111yescharacteristic
468C24—1221no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—612no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8C2×C4×D6112yes—
479C2×Q8D4×D6112yes—
480C2×Q8#1175822612yes—
481C2×D4—612no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4881no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—1228no—
505SD16—1228no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—612no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C4⋊C4—21no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—612no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2442no—
535C4⋊C4C2×C4×D6112yes—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—4881no—
539C22⋊C4—2442no—
540C22⋊C4—2442no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—24no—
546C42—42no—
547C42—42no—
548C42—4881no—
549C42—2442no—
550C42C2×C4×D6112yes—
551C42#1175822212yes—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23C22×C22⋊D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—1222no—
564Q8—1222no—
565Q8—2441no—
566Q8—2441no—
567Q8#1451973914yes—
568D4—1222no—
569D4—1222no—
570D4—2441no—
571D4—2441no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—4884no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4C4×C22⋊D6112yes—
609C2×C4C4×C22⋊D6112yes—
610C2×C4#1451979712yes—
611C2×C4#1451979712yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4C2×#1175822211yescharacteristic
615C2×C4C2×#1175822611yescharacteristic
616C2×C4#1451982211yescharacteristic
617C2×C4#1451982511yescharacteristic
618C8—1222no—
619C8—1222no—
620C8—1222no—
621C8—2441no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22C2×C4×C22⋊D6111yescharacteristic
630C22D4×C22⋊D6111yescharacteristic
631C22#1684042311yescharacteristic
632C22C2×#1451979711yescharacteristic
633C22C2×#1451979711yescharacteristic
634C22#1684057011yescharacteristic
635C22#1684057711yescharacteristic
636C4—22no—
637C4—22no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—81no—
645C4—81no—
646C4—4881no—
647C4—2442no—
648C4—2442no—
649C4C2×#1451973911yescharacteristic
650C4#1684024311yescharacteristic
651C4#1684041311yescharacteristic
652C4#1684046011yescharacteristic
653C2—4882no—
654C2#2144883511yescharacteristic
655C2C4×#1451979711yescharacteristic
656C2#2145466211yescharacteristic
657C2#2145752711yescharacteristic
658C2#2145755611yescharacteristic
659C2#2145850411yescharacteristic
660C2#2145851511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2351C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#2351D61
hypocenterC61#23000
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758202C2semidirect
6. #21446585C2semidirect
8. #21446494C2semidirect
10. #21446160C2semidirect
11. #21446086C2semidirect
13. C4⋊C4×#92418C2semidirect
14. C4×#14519701C2semidirect
15. C61×#2351C2semidirect
21. #16839984C4semidirect
67. C2×#11758202C2×C4semidirect
93. C4⋊#92418C2×C4semidirect
98. #14519700C2×C4semidirect
114. C4×Q8×C61C2×C4semidirect
327. #2351D61semidirect
392. C61#23000semidirect
416. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399842
C2×C4×#117582022
C2×C4×#117582012
#214488882
#214466282
#214465852
#214465012
#214464942
#214461742
#214461602
#214460862
#214460652
C4⋊C4×#924182
C4×#145197012
C61×#23512
#2300061