Finite Group Databasebeta

← #29840946

Subgroups — #29840946

Order 15616.

2858 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×#16840238C211yesmaximal, characteristic
2C2×#16840238C211yesmaximal, characteristic
3C2×C4×Q8×D61C211yesmaximal, characteristic
4#21448738C213yesmaximal, complemented
5#21448737C211yesmaximal, characteristic, complemented
6#21448736C213yesmaximal, complemented
7Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic122C213yesmaximal, complemented
9#21448733C213yesmaximal, complemented
10Q8×#11758196C211yesmaximal, characteristic, complemented
11#21448444C213yesmaximal
12#21448441C213yesmaximal
13#21448300C213yesmaximal
14#21448299C213yesmaximal
15C61×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17#16840238—28no—
18#16840238C4116yescomplemented
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×C4×Dic122C2213yes—
23C2×C4×Dic122C2213yes—
24C2×C4×Dic122C2213yes—
25C2×#14519742C2213yes—
26C2×#14519742C2213yes—
27C2×#14519742C2213yes—
28C4×Q8×D61—28no—
29C2×#14519735C2213yes—
30C2×#14519735C2213yes—
31C2×#14519734C2213yes—
32C2×#14519734C2213yes—
33C2×C4⋊C4×D61C2213yes—
34C2×C42×D61C2213yes—
35#16839661C2213yes—
36#16839661C2213yes—
37#16839660C2213yes—
38#16839660C2213yes—
39#16839659C2216yes—
40#16839659C2213yes—
41#16839659C2216yes—
42#16839659C2213yes—
43#16839658C2213yes—
44#16839658C2213yes—
45#16839657C2213yes—
46#16839657C2213yes—
47#16839636C2211yescharacteristic
48#16839636C2213yes—
49#16839636C2216yes—
50#16839633C2213yes—
51#16839633C2213yes—
52#16839632C2213yes—
53#16839631C2213yes—
54#16839631C2216yes—
55#16839627C2213yes—
56#16839626C2216yes—
57C4⋊C4×Dic61C2216yes—
58C4×#11758196C2213yes—
59C4×#11758196C2213yes—
60C4×C4⋊Dic61C2213yes—
61#16839613C2216yes—
62C2×C4×Q8×C61C2211yescharacteristic
63C2×C4×Q8×C61C2211yescharacteristic
64C2×C4×Q8×C61C2211yescharacteristic
65C61×#389C2213yes—
66C61×#389C2213yes—
67C61×#382C2213yes—
68C4×C4⋊C4×C61C2213yes—
69C22×Dic122C2313yes—
70C2×Q8×D61C2×C418yes—
71C2×Q8×D61D418yes—
72C2×Q8×D61Q818yes—
73C22×C4×D61C2311yescharacteristic
74C22×C4×D61C2313yes—
75C22×C4×D61C2313yes—
76Q8×Dic61—24no—
77Q8×Dic61—24no—
78Q8×Dic61C2×C418yescomplemented
79C4×Dic122—212no—
80C4×Dic122—212no—
81C4×Dic122C2×C4124yescomplemented
82#14519742—212no—
83#14519742—212no—
84#14519742C2×C4124yescomplemented
85#14519735—212no—
86#14519735C2×C4124yes—
87#14519734—212no—
88#14519734C2×C4124yes—
89C2×C4⋊Dic61C2313yes—
90C2×C4⋊Dic61C2313yes—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2313yes—
93C2×C4⋊Dic61C2316yes—
94C2×C4⋊Dic61C2313yes—
95C2×#11758196C2316yes—
96C2×#11758196C2313yes—
97C2×#11758196C2313yes—
98C2×#11758196C2313yes—
99C2×#11758196C2311yescharacteristic
100C2×#11758196C2316yes—
101C2×#11758196C2313yes—
102C2×#11758196C2313yes—
103C2×#11758196C2313yes—
104C2×#11758196C2313yes—
105C4⋊C4×D61—212no—
106C2×#11758198C2311yescharacteristic
107C2×#11758198C2313yes—
108C2×#11758198C2311yescharacteristic
109C2×#11758198C2313yes—
110C2×C4×Dic61C2313yes—
111C2×C4×Dic61C2313yes—
112C2×C4×Dic61C2313yes—
113C2×C4×Dic61C2313yes—
114C2×C4×Dic61C2316yes—
115C2×C4×Dic61C2313yes—
116C2×C4×Dic61C2316yes—
117C42×D61—212no—
118#14519684C2313yes—
119#14519684C2313yes—
120#14519684C2311yescharacteristic
121#14519684C2313yes—
122#14519684C2313yes—
123#14519684C2313yes—
124#14519684C2316yes—
125C22×Q8×C61C2311yescharacteristic
126C4×Q8×C61—24no—
127C4×Q8×C61—24no—
128C4×Q8×C61C2×C418yescomplemented
129C2×C4⋊C4×C61C2313yes—
130C2×C4⋊C4×C61C2313yes—
131C2×C4⋊C4×C61C2313yes—
132C2×C4⋊C4×C61C2313yes—
133C2×C4⋊C4×C61C2313yes—
134C2×C42×C61C2313yes—
135C2×C42×C61C2313yes—
136C2×C42×C61C2313yes—
137#95×C61C2311yescharacteristic
138#95×C61C2313yes—
139#95×C61C2316yes—
140C23×D61C2411yescharacteristic
141C2×Q8×C61C22×C414yes—
142C2×Q8×C61C2×D414yes—
143C2×Q8×C61C2×Q814yes—
144C22×C4×C61C2413yes—
145C22×C4×C61C2413yes—
146C22×C4×C61C2411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×C4×C61C2413yes—
149C22×C4×C61C2411yescharacteristic
150C22×C4×C61C2413yes—
151C22×Dic61C2413yes—
152C22×Dic61C2411yescharacteristic
153C22×Dic61C2413yes—
154C22×Dic61C2411yescharacteristic
155C22×Dic61C2413yes—
156C22×Dic61C2411yescharacteristic
157C22×Dic61C2413yes—
158Q8×D61C4⋊C4164yescomplemented
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—26no—
163C2×C4×D61—43no—
164C2×C4×D61—43no—
165C2×C4×D61C22×C4112yes—
166C2×C4×D61C2×D4112yes—
167C2×C4×D61C2×Q8112yes—
168C2×Dic122C22×C4112yes—
169C2×Dic122C2×D4112yes—
170C2×Dic122C2×Q8112yes—
171C4⋊C4×C61—26no—
172C4⋊C4×C61—26no—
173C4⋊C4×C61—26no—
174C4⋊C4×C61—43no—
175C4⋊C4×C61C22×C4112yes—
176C42×C61—26no—
177C42×C61—26no—
178C42×C61C22×C4112yes—
179#11758198—22no—
180#11758198—26no—
181#11758198C22×C414yes—
182#11758198C22×C4112yes—
183C4⋊Dic61—26no—
184C4⋊Dic61—26no—
185C4⋊Dic61—22no—
186C4⋊Dic61—43no—
187C4⋊Dic61—46no—
188C4⋊Dic61C22×C4112yes—
189#11758196—22no—
190#11758196—212no—
191#11758196—26no—
192#11758196—26no—
193#11758196—26no—
194#11758196—46no—
195#11758196—43no—
196#11758196—43no—
197#11758196C22×C4112yes—
198#11758196C22×C4112yes—
199C4×Dic61—26no—
200C4×Dic61—26no—
201C4×Dic61—26no—
202C4×Dic61—26no—
203C4×Dic61—212no—
204C4×Dic61C22×C4112yes—
205C4×Dic61C22×C4124yes—
206C23×C61C2511yescharacteristic
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61C23×C412yes—
212C22×D61C22×D412yes—
213C22×D61C22×Q812yes—
214Q8×C61C2×C4⋊C4116yes—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—23no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—23no—
221C2×C4×C61—23no—
222C2×C4×C61—23no—
223C2×C4×C61—21no—
224C2×C4×C61—23no—
225C2×C4×C61—23no—
226C2×C4×C61—21no—
227C2×C4×C61—23no—
228C2×C4×C61—23no—
229C2×C4×C61—21no—
230C2×C4×C61C23×C416yes—
231C2×C4×C61C23×C412yes—
232C2×C4×C61C23×C416yes—
233C2×C4×C61C22×D416yes—
234C2×C4×C61C22×Q816yes—
235C2×C4×C612−1+412yes—
236C2×Dic61—21no—
237C2×Dic61—23no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—23no—
242C2×Dic61—21no—
243C2×Dic61—23no—
244C2×Dic61—23no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—23no—
248C2×Dic61—23no—
249C2×Dic61—23no—
250C2×Dic61—23no—
251C2×Dic61C23×C412yes—
252C2×Dic61C23×C416yes—
253C2×Dic61C23×C416yes—
254C2×Dic61C22×D416yes—
255C2×Dic61C22×Q816yes—
256C2×Dic612−1+412yes—
257C4×D61—28no—
258C4×D61—412no—
259C4×D61C2×C4⋊C4148yes—
260Dic122C2×C4⋊C4148yes—
261#34249—611nomaximal, Sylow
262C22×C61C24×C411yescharacteristic
263C22×C61C23×D411yescharacteristic
264C22×C61C23×Q811yescharacteristic
265C22×C61C2×2−1+411yescharacteristic
266C22×C61C2×2−1+411yescharacteristic
267C22×C61#58511yescharacteristic
268C22×C61#58511yescharacteristic
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61C22×C4⋊C414yes—
276C4×C61—22no—
277C4×C61—26no—
278C4×C61—22no—
279C4×C61—43no—
280C4×C61—43no—
281C4×C61—41no—
282C4×C61C22×C4⋊C4112yes—
283Dic61—26no—
284Dic61—22no—
285Dic61—22no—
286Dic61—41no—
287Dic61—43no—
288Dic61—43no—
289Dic61C22×C4⋊C4112yes—
290C2×#519—611no—
291C2×#519—611no—
292C22×C4×Q8—611no—
293#2351—613no—
294#2351—613no—
295#2351—611no—
296#2351D6111yescharacteristic, complemented
297Q8×C4⋊C4—613no—
298Q8×C4⋊C4—613no—
299Q8×C4⋊C4—611no—
300Q8×C4⋊C4—611no—
301#2317—613no—
302#2317—613no—
303#2296—613no—
304#2296—613no—
305C2×C61C23×C4⋊C411yescharacteristic
306C2×C61C4×2−1+411yescharacteristic
307C2×C61C4×2−1+411yescharacteristic
308C2×C61#348611yescharacteristic
309C2×C61#348711yescharacteristic
310C2×C61#350211yescharacteristic
311C2×C61#350211yescharacteristic
312D61—24no—
313C23×Q8—611no—
314#519—6116no—
315#519—1228no—
316C2×C4×Q8—611no—
317C2×C4×Q8—611no—
318C2×C4×Q8—611no—
319C2×C4×Q8—613no—
320C2×C4×Q8—613no—
321C2×C4×Q8—613no—
322C2×C4×Q8—613no—
323C2×C4×Q8—613no—
324C2×C4×Q8—613no—
325C2×C4×Q8—1228no—
326C2×C4×Q8C2×D6111yescharacteristic
327C2×C4×Q8C2×D6111yescharacteristic
328C2×C4×Q8C2×D6111yescharacteristic
329C2×#117—613no—
330C2×#117—613no—
331C2×#117—613no—
332C2×#117—613no—
333C22×C4⋊C4—613no—
334C22×C42—613no—
335#389—613no—
336#389—613no—
337#389—613no—
338#389—613no—
339#389—613no—
340#389—613no—
341#389—613no—
342#389—613no—
343#389—616no—
344#389—616no—
345#389—613no—
346#389—613no—
347#389C2×D6113yes—
348#389C2×D6113yes—
349#385—613no—
350#385—613no—
351#385—616no—
352#385—613no—
353#385—611no—
354#384—616no—
355#384—613no—
356#384—613no—
357#382—613no—
358#382—616no—
359#382C2×D6113yes—
360C4×C4⋊C4—616no—
361C4×C4⋊C4—613no—
362C4×C4⋊C4—613no—
363C4×C4⋊C4—613no—
364C4×C4⋊C4—616no—
365C4×C4⋊C4C2×D6113yes—
366C61#3424911yescharacteristic, complemented, Sylow
367C22×Q8—618no—
368C22×Q8—618no—
369C22×Q8—618no—
370C22×Q8—613no—
371C22×Q8C22×D6111yescharacteristic
372C23×C4—613no—
373C23×C4—611no—
374C23×C4—613no—
375C4×Q8—24no—
376C4×Q8—24no—
377C4×Q8—618no—
378C4×Q8—6124no—
379C4×Q8—6124no—
380C4×Q8—12212no—
381C4×Q8—12212no—
382C4×Q8—12212no—
383C4×Q8—12212no—
384C4×Q8—1224no—
385C4×Q8—1224no—
386C4×Q8C4×D6118yescomplemented
387#117—6124no—
388#117—6124no—
389#117—12212no—
390#117—12212no—
391C2×C4⋊C4—613no—
392C2×C4⋊C4—613no—
393C2×C4⋊C4—613no—
394C2×C4⋊C4—616no—
395C2×C4⋊C4—613no—
396C2×C4⋊C4—613no—
397C2×C4⋊C4—613no—
398C2×C4⋊C4—613no—
399C2×C4⋊C4—613no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—613no—
403C2×C4⋊C4—613no—
404C2×C4⋊C4—616no—
405C2×C4⋊C4—616no—
406C2×C4⋊C4—613no—
407C2×C4⋊C4—12212no—
408C2×C4⋊C4C22×D6113yes—
409C2×C4⋊C4C22×D6113yes—
410C2×C4⋊C4C22×D6113yes—
411C2×C4⋊C4C22×D6113yes—
412C2×C4⋊C4C22×D6113yes—
413C2×C22⋊C4—613no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4—611no—
416C2×C22⋊C4—613no—
417C2×C42—616no—
418C2×C42—613no—
419C2×C42—616no—
420C2×C42—613no—
421C2×C42—613no—
422C2×C42—613no—
423C2×C42—613no—
424C2×C42—12212no—
425C2×C42C22×D6113yes—
426C2×C42C22×D6113yes—
427C2×C42C22×D6113yes—
428#95—611no—
429#95—613no—
430#95—613no—
431#95—613no—
432#95—616no—
433#95—613no—
434#95—613no—
435#95C22×D6116yes—
436#95C22×D6111yescharacteristic
437#95C22×D6113yes—
438C24—611no—
439C2×Q8—6164no—
440C2×Q8—6112no—
441C2×Q8—6112no—
442C2×Q8—6112no—
443C2×Q8C2×C4×D6114yes—
444C2×Q8D4×D6114yes—
445C2×Q8Q8×D6114yes—
446C22×C4—613no—
447C22×C4—6112no—
448C22×C4—6112no—
449C22×C4—6112no—
450C22×C4—611no—
451C22×C4—613no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—613no—
455C22×C4—613no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1226no—
459C22×C4—1222no—
460C22×C4—2443no—
461C22×C4—2443no—
462C22×C4C23×D6113yes—
463C22×C4C23×D6113yes—
464C22×C4C23×D6111yescharacteristic
465C22×C4C23×D6113yes—
466C22×C4C23×D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468C22×C4C23×D6113yes—
469C4⋊C4—26no—
470C4⋊C4—26no—
471C4⋊C4—26no—
472C4⋊C4—43no—
473C4⋊C4—6112no—
474C4⋊C4—6112no—
475C4⋊C4—6112no—
476C4⋊C4—12212no—
477C4⋊C4—1226no—
478C4⋊C4—1226no—
479C4⋊C4—1226no—
480C4⋊C4—1222no—
481C4⋊C4—1226no—
482C4⋊C4—1226no—
483C4⋊C4—1222no—
484C4⋊C4—2443no—
485C4⋊C4—2443no—
486C4⋊C4—2446no—
487C4⋊C4—2446no—
488C4⋊C4—2443no—
489C4⋊C4C2×C4×D61112yes—
490C22⋊C4—6112no—
491C22⋊C4—614no—
492C22⋊C4—1222no—
493C22⋊C4—1226no—
494C42—26no—
495C42—26no—
496C42—6124no—
497C42—6112no—
498C42—1226no—
499C42—1226no—
500C42—12212no—
501C42—1226no—
502C42—1226no—
503C42C2×C4×D61112yes—
504C23—612no—
505C23—612no—
506C23—612no—
507C23—1221no—
508C23—1221no—
509C23—1221no—
510C23—1221no—
511C23C24×D6111yescharacteristic
512Q8—6148no—
513Q8C4⋊C4×D61116yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—23no—
518C2×C4—23no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—23no—
525C2×C4—21no—
526C2×C4—23no—
527C2×C4—21no—
528C2×C4—23no—
529C2×C4—616no—
530C2×C4—616no—
531C2×C4—616no—
532C2×C4—6148no—
533C2×C4—616no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—1228no—
537C2×C4—1221no—
538C2×C4—1223no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1223no—
543C2×C4—1221no—
544C2×C4—1223no—
545C2×C4—1223no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1223no—
549C2×C4—1223no—
550C2×C4—1223no—
551C2×C4—1223no—
552C2×C4—24412no—
553C2×C4C22×C4×D6116yes—
554C2×C4C22×C4×D6112yes—
555C2×C4C22×C4×D6116yes—
556C2×C4C2×D4×D6116yes—
557C2×C4C2×Q8×D6116yes—
558C2×C4#1451982412yes—
559C22—614no—
560C22—1222no—
561C22—1222no—
562C22—1222no—
563C22—1222no—
564C22—1222no—
565C22—1222no—
566C22C23×C4×D6111yescharacteristic
567C22C22×D4×D6111yescharacteristic
568C22C22×Q8×D6111yescharacteristic
569C222−1+4×D6111yescharacteristic
570C22C2×#1451982411yescharacteristic
571C22#1684059811yescharacteristic
572C22#1684059911yescharacteristic
573C4—22no—
574C4—22no—
575C4—26no—
576C4—43no—
577C4—41no—
578C4—43no—
579C4—6112no—
580C4—1222no—
581C4—1226no—
582C4—1222no—
583C4—2441no—
584C4—2443no—
585C4—2443no—
586C4C2×C4⋊C4×D61112yes—
587C2—1224no—
588C2C22×C4⋊C4×D6111yescharacteristic
589C2#2145902811yescharacteristic
590C2C4×#1451982411yescharacteristic
591C2#2145946511yescharacteristic
592C2#2145947611yescharacteristic
593C2#2145962311yescharacteristic
594C2#2145962611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC61×#2351C2
socle, cosocleC23×C61C25
prime radicalC23×D61C24
hypercenter#2351D61
hypocenterC61#34249
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #21448738C2semidirect
5. #21448737C2semidirect
6. #21448736C2semidirect
7. Q8×C4⋊Dic61C2semidirect
8. C4⋊C4×Dic122C2semidirect
9. #21448733C2semidirect
10. Q8×#11758196C2semidirect
15. C61×#2351C2semidirect
18. #16840238C4semidirect
78. Q8×Dic61C2×C4semidirect
81. C4×Dic122C2×C4semidirect
84. #14519742C2×C4semidirect
128. C4×Q8×C61C2×C4semidirect
158. Q8×D61C4⋊C4semidirect
296. #2351D61semidirect
366. C61#34249semidirect
386. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402382
C2×C4×Q8×D612
#214487382
#214487372
#214487362
Q8×C4⋊Dic612
C4⋊C4×Dic1222
#214487332
Q8×#117581962
#214484442
#214484412
#214483002
#214482992
C61×#23512
#3424961