Finite Group Databasebeta

← #29697708

Subgroups — #29697708

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#107×D61C211yesmaximal, characteristic
2#21448687C211yesmaximal, characteristic
3#21448684C211yesmaximal, characteristic
4#21446556C211yesmaximal, characteristic, complemented
5#21446554C211yesmaximal, characteristic, complemented
6#21446413C211yesmaximal, characteristic, complemented
7#21446412C211yesmaximal, characteristic, complemented
8#21446410C211yesmaximal, characteristic, complemented
9#21445904C211yesmaximal, characteristic
10#21445904C211yesmaximal, characteristic
11#21445843C211yesmaximal, characteristic
12#21445717C211yesmaximal, characteristic, complemented
13#21445715C211yesmaximal, characteristic, complemented
14#21445594C211yesmaximal, characteristic
15C61×#1921C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519735—21no—
18C2×#14519734—21no—
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24#107×D61C418yescomplemented
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30#16839661C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839649—21no—
34#16839646—21no—
35#16839644—21no—
36#16839643—21no—
37#16839624—21no—
38#16839624—21no—
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×#11758198C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C2×C8×D61—22no—
66C2×C8×D61—22no—
67C2×C8×D61C2×C414yes—
68#14519735—44no—
69#14519734—44no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C4⋊C4×D61—22no—
80C4⋊C4×D61—22no—
81C4⋊C4×D61C2×C414yes—
82C4⋊C4×D61C2×C414yes—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198D411yescharacteristic
86C2×#11758198D411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91#14519711C2×C414yes—
92#14519710C2×C414yes—
93#14519710C2×C414yes—
94#14519708—22no—
95#14519708—22no—
96#14519689—22no—
97#14519688—22no—
98#14519687—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#107×C61C2×C414yes—
119#95×C61—21no—
120#95×C61—21no—
121C23×D61C2×D411yescharacteristic
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—24no—
141C2×C4×D61—24no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×C4×D61C22×C412yes—
147C2×C4×D61#4112yes—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61C22×C412yes—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61C22×C412yes—
162C4⋊C4×C61C22×C412yes—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—42no—
166#11758198—22no—
167#11758198—22no—
168#11758198—42no—
169#11758198—42no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—42no—
177C4⋊Dic61C22×C412yes—
178C4⋊Dic61C22×C412yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—42no—
182#11758196—42no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—42no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418C22×C412yes—
192C2×#92418#4112yes—
193C2×#92418#4112yes—
194C8×D61—28no—
195C23×C61C22×D411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61C4×D412yes—
201C22×D61AGL(1,ℤ/8)12yes—
202C22×D61AGL(1,ℤ/8)12yes—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C23×C411yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61C4×D412yes—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—48no—
253C4×D61—48no—
254C8×C61—22no—
255C8×C61—41no—
256#92418—41no—
257#92418#11714yescomplemented
258#22617—611nomaximal, Sylow
259C22×C61C2×C4×D411yescharacteristic
260C22×C61#54711yescharacteristic
261C22×C61#54711yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C61#57811yescharacteristic
265C22×C61#57811yescharacteristic
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61#44214yes—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61C2×#11711yescharacteristic
280C4×C61#51911yescharacteristic
281C4×C61#55211yescharacteristic
282C4×C61#55211yescharacteristic
283Dic61—22no—
284Dic61—41no—
285Dic61—41no—
286Dic61—42no—
287Dic61—42no—
288Dic61—81no—
289Dic61—81no—
290C22×#107—611no—
291#2344—611no—
292#2344—611no—
293#1943—611no—
294#1943—611no—
295#1921—611no—
296#1921—611no—
297#1921—611no—
298#1921D6111yescharacteristic, complemented
299#1823—611no—
300#1823—611no—
301#1807—611no—
302#1776—611no—
303#1776—611no—
304#1752—611no—
305C2×C61#234411yescharacteristic
306C2×C61C2×#44211yescharacteristic
307C2×C61#295011yescharacteristic
308C2×C61#334311yescharacteristic
309C2×C61#334311yescharacteristic
310C2×C61#334311yescharacteristic
311C2×C61#334311yescharacteristic
312D61—24no—
313C23×C8—611no—
314C2×#117—1221no—
315C2×#117—1221no—
316C22×C4⋊C4—611no—
317C22×C4⋊C4—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×#107—611no—
321C2×#107—618no—
322C2×#107C2×D6111yescharacteristic
323C2×#106—611no—
324C2×#106—611no—
325C2×C22⋊C8—611no—
326C2×#97—611no—
327C2×#97—611no—
328#389—611no—
329#389—611no—
330#389—611no—
331#389C2×D6111yescharacteristic
332#387—1221no—
333#387—1221no—
334#387—1221no—
335#387—1221no—
336#381—1221no—
337#381—1221no—
338#380—611no—
339#380—611no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—611no—
342C4×C4⋊C4—611no—
343C4×C4⋊C4C2×D6111yescharacteristic
344C4×C22⋊C4—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340C2×D6111yescharacteristic
353#340C2×D6111yescharacteristic
354#340C2×D6111yescharacteristic
355#336—611no—
356#336C2×D6111yescharacteristic
357C61#2261711yescharacteristic, complemented, Sylow
358C23×C4—611no—
359C23×C4—1221no—
360C23×C4—1221no—
361C22×C8—611no—
362C22×C8—614no—
363C22×C8—611no—
364C22×C8—1222no—
365C22×C8—1222no—
366C22×C8C22×D6111yescharacteristic
367C22×C8C22×D6111yescharacteristic
368#117—2444no—
369#117—2444no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—614no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C42—21no—
393C2×C42—611no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398#107—614no—
399#107—614no—
400#107—614no—
401#107C4×D6114yes—
402#106—1222no—
403#106—1222no—
404C22⋊C8—1222no—
405#97—1222no—
406#97—1222no—
407#95—21no—
408#95—21no—
409#95—611no—
410#95—611no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419C24—611no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—611no—
425C22×C4—612no—
426C22×C4—612no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—1224no—
430C22×C4—1224no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4—2442no—
443C22×C4D4×D6111yescharacteristic
444C22×C4D4×D6111yescharacteristic
445C22×C4C23×D6111yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—612no—
452C2×C8—612no—
453C2×C8—612no—
454C2×C8—1228no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8C2×C4×D6112yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—1224no—
469C4⋊C4—1224no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4C2×C4×D6112yes—
480C4⋊C4C2×C4×D6112yes—
481C22⋊C4—1222no—
482C22⋊C4—1222no—
483C22⋊C4—2442no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—1221no—
489C42—1221no—
490C42—2442no—
491C42—2442no—
492C42—2442no—
493C23—612no—
494C23—612no—
495C23—612no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23C2×D4×D6111yescharacteristic
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—612no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—2448no—
535C2×C4—2448no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4C22×C4×D6111yescharacteristic
547C2×C4#41×D6111yescharacteristic
548C2×C4#41×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C2×C4#1451982411yescharacteristic
551C2×C4#1451982511yescharacteristic
552C8—22no—
553C8—41no—
554C8—614no—
555C8—2441no—
556C22—614no—
557C22—1222no—
558C22—1222no—
559C22—1222no—
560C22—1222no—
561C22—1222no—
562C22—1222no—
563C22C4×D4×D6111yescharacteristic
564C22#1684041011yescharacteristic
565C22#1684041411yescharacteristic
566C22AGL(1,ℤ/8)×D6111yescharacteristic
567C22AGL(1,ℤ/8)×D6111yescharacteristic
568C22#1684057511yescharacteristic
569C22#1684057611yescharacteristic
570C4—42no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—1222no—
577C4—4881no—
578C4—4881no—
579C4—2442no—
580C4—2442no—
581C4—2441no—
582C4—2441no—
583C4#117×D6111yescharacteristic
584C4#1684023811yescharacteristic
585C4#1684044411yescharacteristic
586C4#1684044811yescharacteristic
587C2—1224no—
588C2#2144868511yescharacteristic
589C2#442×D6111yescharacteristic
590C2#2145468211yescharacteristic
591C2#2145825211yescharacteristic
592C2#2145825611yescharacteristic
593C2#2145826211yescharacteristic
594C2#2145826611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1921C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1921D61
hypocenterC61#22617
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #21446556C2semidirect
5. #21446554C2semidirect
6. #21446413C2semidirect
7. #21446412C2semidirect
8. #21446410C2semidirect
12. #21445717C2semidirect
13. #21445715C2semidirect
15. C61×#1921C2semidirect
24. #107×D61C4semidirect
257. #92418#117semidirect
298. #1921D61semidirect
357. C61#22617semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D612
#214486872
#214486842
#214465562
#214465542
#214464132
#214464122
#214464102
#214459042
#214458432
#214457172
#214457152
#214455942
C61×#19212
#2261761