Finite Group Databasebeta

← #29698383

Subgroups — #29698383

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×#16839864C211yesmaximal, characteristic
2C2×#16839864C211yesmaximal, characteristic
3C2×C4×#11758190C211yesmaximal, characteristic
4C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
5#21448730C211yesmaximal, characteristic
6#21446557C211yesmaximal, characteristic, complemented
7#21446552C211yesmaximal, characteristic, complemented
8#21446090C211yesmaximal, characteristic, complemented
9#21446083C211yesmaximal, characteristic
10#21446076C211yesmaximal, characteristic, complemented
11#21446069C211yesmaximal, characteristic
12#21445987C211yesmaximal, characteristic, complemented
13#21445835C211yesmaximal, characteristic, complemented
14#21445820C211yesmaximal, characteristic
15C61×#1771C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16839864—24no—
24#16839864C418yescomplemented
25C4×#11758190—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519711C2211yescharacteristic
28C2×#14519711C2211yescharacteristic
29C2×#14519709C2211yescharacteristic
30C2×#14519709C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839659—21no—
42#16839659—21no—
43#16839633—21no—
44#16839633—21no—
45#16839632C2211yescharacteristic
46#16839631—21no—
47#16839628C2211yescharacteristic
48#16839621—21no—
49C4⋊C4×Dic61—21no—
50#95×D61—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54#16839525C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190C2×C414yes—
68C2×#11758190D414yes—
69C2×#11758190Q814yes—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73C4×Dic122—41no—
74C4×Dic122—41no—
75C4×Dic122C2×C414yescomplemented
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244—41no—
80C4×D244—41no—
81C4×D244C2×C414yes—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103#14519711—22no—
104#14519711—41no—
105#14519711C2×C414yescomplemented
106#14519709—22no—
107#14519709—41no—
108#14519702—22no—
109#14519702—22no—
110#14519702C2×C414yescomplemented
111#14519699—22no—
112#14519699—22no—
113#14519699C2×C414yes—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C4⋊C4×C61D411yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130#97×C61—22no—
131#97×C61C2×C414yescomplemented
132C4×C8×C61—22no—
133#95×C61D411yescharacteristic
134#95×C61D411yescharacteristic
135C23×D61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×D244C2×D412yes—
155C2×D244C2×Q812yes—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—81no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C22×C412yes—
169C2×Dic122C2×D412yes—
170C2×Dic122C2×Q812yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C22×C412yes—
180C2×C8×C61C22×C412yes—
181C2×C8×C61C2×D412yes—
182C2×C8×C61C2×Q812yes—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61C22×C412yes—
190C42×C61#4112yes—
191#11758198—24no—
192#11758198—42no—
193#11758198—42no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—42no—
205C4⋊Dic61—81no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C412yes—
208#11758196—24no—
209#11758196—42no—
210#11758196—42no—
211#11758196—81no—
212#11758196—81no—
213#11758196—81no—
214#11758196—81no—
215C4×Dic61—24no—
216C4×Dic61—42no—
217C4×Dic61—42no—
218#11758190C4⋊C4116yescomplemented
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61C4×D412yes—
243C2×C4×C61C4×D412yes—
244C2×C4×C61#13712yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272D244—22no—
273D244—22no—
274D244—22no—
275D244C2×C4⋊C414yes—
276C4×D61—84no—
277C4×D61—84no—
278C4×D61—84no—
279Dic122—22no—
280Dic122—22no—
281Dic122—22no—
282Dic122C2×C4⋊C414yes—
283C8×C61—22no—
284C8×C61—41no—
285C8×C61—41no—
286C8×C61C2×C4⋊C414yes—
287#22678—611nomaximal, Sylow
288C22×C61C2×C4×D411yescharacteristic
289C22×C61D4211yescharacteristic
290C22×C61D4×Q811yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61#57511yescharacteristic
293C22×C61#57611yescharacteristic
294C22×C61#57711yescharacteristic
295C2×D61—24no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61C22×C4⋊C411yescharacteristic
309C4×C61#52011yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61#55411yescharacteristic
312Dic61—24no—
313Dic61—42no—
314Dic61—42no—
315Dic61—81no—
316Dic61—81no—
317Dic61—81no—
318Dic61—81no—
319C2×#440—611no—
320C2×#440—611no—
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#1771D6111yescharacteristic, complemented
334C2×C61D4×C4⋊C411yescharacteristic
335C2×C61C4×#13711yescharacteristic
336C2×C61#294611yescharacteristic
337C2×C61#326611yescharacteristic
338C2×C61#326811yescharacteristic
339C2×C61#338511yescharacteristic
340C2×C61#338811yescharacteristic
341D61—44no—
342C22×SD16—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×D4—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349#440—618no—
350#440—1224no—
351C4×SD16—1224no—
352C2×#107—611no—
353C2×#107—611no—
354C2×#107—611no—
355C2×#106—611no—
356C2×#106—611no—
357C2×C4⋊C8C2×D6111yescharacteristic
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#97C2×D6111yescharacteristic
365C2×#97C2×D6111yescharacteristic
366C2×C4×C8C2×D6111yescharacteristic
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—1221no—
380C4×C4⋊C4C2×D6111yescharacteristic
381C2×#95—1221no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#336C2×D6111yescharacteristic
387#336C2×D6111yescharacteristic
388C61#2267811yescharacteristic, complemented, Sylow
389C22×Q8—611no—
390C22×D4—611no—
391C23×C4—1221no—
392C23×C4—1221no—
393C23×C4—1221no—
394C2×SD16—614no—
395C2×SD16—614no—
396C2×SD16—614no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C4×Q8—614no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—2441no—
406C4×Q8—2441no—
407C4×D4—614no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—2441no—
412C4×D4—2441no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
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⋊C4D24411yescharacteristic
429C2×C4⋊C4D24411yescharacteristic
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439#107—614no—
440#107—1222no—
441#107—2441no—
442#106—1222no—
443#106—2441no—
444C4⋊C8—22no—
445#103—614no—
446#103—1222no—
447#103—1222no—
448#102—614no—
449#102—1222no—
450#102—1222no—
451#97—22no—
452#97C4×D6114yescomplemented
453C4×C8—22no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95D24411yescharacteristic
461#95D24411yescharacteristic
462C24—1221no—
463C2×Q8—612no—
464C2×Q8—612no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×D4—612no—
470C2×D4—612no—
471C2×D4—612no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1224no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—4881no—
484C22×C4—4881no—
485C22×C4—4881no—
486C22×C4—4881no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4C2×D24411yescharacteristic
492C22×C4C2×D24411yescharacteristic
493C22×C4C2×D24411yescharacteristic
494C22×C4C2×D24411yescharacteristic
495C22×C4C2×D24411yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16—6116no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8C2×C4×D6112yes—
508C2×C8C2×C4×D6112yes—
509C2×C8D4×D6112yes—
510C2×C8Q8×D6112yes—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1224no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—4881no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—4881no—
530C4⋊C4—4881no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C22⋊C4—1224no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—1224no—
542C42—2442no—
543C42—2442no—
544C42C2×C4×D6112yes—
545C42#1175822212yes—
546C23—1222no—
547C23—1222no—
548C23—1222no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23C22×D24411yescharacteristic
554Q8—614no—
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558D4—614no—
559D4—1222no—
560D4—1222no—
561D4—1222no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—4884no—
583C2×C4—4884no—
584C2×C4—4884no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
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×C4C4×D24412yes—
602C2×C4C4×D24412yes—
603C2×C4#1451981012yes—
604C2×C4C22×C4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×Q8×D6111yescharacteristic
607C2×C4C2×#1175822211yescharacteristic
608C2×C4#1451982311yescharacteristic
609C2×C4#1451982411yescharacteristic
610C8—22no—
611C8—41no—
612C8—41no—
613C8C4⋊C4×D6114yes—
614C22—1224no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22C2×C4×D24411yescharacteristic
622C22D4×D24411yescharacteristic
623C22Q8×D24411yescharacteristic
624C22C2×#1451981011yescharacteristic
625C22#1684056211yescharacteristic
626C22#1684056711yescharacteristic
627C22#1684057411yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—1224no—
635C4—4881no—
636C4—4881no—
637C4—4881no—
638C4—4881no—
639C4—2442no—
640C4—2442no—
641C4C2×C4⋊C4×D6111yescharacteristic
642C4#1684024211yescharacteristic
643C4#1684041711yescharacteristic
644C4#1684047211yescharacteristic
645C2—2444no—
646C2C4⋊C4×D24411yescharacteristic
647C2C4×#1451981011yescharacteristic
648C2#2145465311yescharacteristic
649C2#2145733411yescharacteristic
650C2#2145736411yescharacteristic
651C2#2145873111yescharacteristic
652C2#2145876611yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1771C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1771D61
hypocenterC61#22678
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic122C2semidirect
6. #21446557C2semidirect
7. #21446552C2semidirect
8. #21446090C2semidirect
10. #21446076C2semidirect
12. #21445987C2semidirect
13. #21445835C2semidirect
15. C61×#1771C2semidirect
24. #16839864C4semidirect
75. C4×Dic122C2×C4semidirect
105. #14519711C2×C4semidirect
110. #14519702C2×C4semidirect
131. #97×C61C2×C4semidirect
218. #11758190C4⋊C4semidirect
333. #1771D61semidirect
388. C61#22678semidirect
452. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
C2×C4×#117581902
C4⋊C4×Dic1222
#214487302
#214465572
#214465522
#214460902
#214460832
#214460762
#214460692
#214459872
#214458352
#214458202
C61×#17712
#2267861