Finite Group Databasebeta

← #29721580

Subgroups — #29721580

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839987C212yesmaximal
2C2×#16839969C211yesmaximal, characteristic, complemented
3C2×C4×#11758190C211yesmaximal, characteristic
4#21450815C211yesmaximal, characteristic
5#21448731C211yesmaximal, characteristic, complemented
6#21447198C212yesmaximal, complemented
7#21447056C212yesmaximal
8#21446537C211yesmaximal, characteristic, complemented
9#21446157C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic
11C61×#1956C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13C22×#11758190C2211yescharacteristic
14C2×#14519766C2212yes—
15C2×C4×Dic122C2211yescharacteristic
16C2×#14519742C2212yes—
17C2×C4×D244C2211yescharacteristic
18#16839987—28no—
19#16839969—24no—
20C4×#11758190—24no—
21C2×#14519709C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519706C2212yes—
24C2×#14519702C2211yescharacteristic
25C2×#14519701C2212yes—
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2212yes—
28#16839688—22no—
29#16839669C2212yes—
30#16839658C2212yes—
31#16839657—22no—
32#16839631—21no—
33#16839619—21no—
34#16839619—21no—
35C4⋊C4×Dic61—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839524C2212yes—
38#16839516C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×#103×C61C2212yes—
41C2×#103×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C22×Dic122C2311yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758202—24no—
48C2×#11758202—24no—
49C2×#11758202—24no—
50C2×#11758190—22no—
51C2×#11758190—22no—
52C2×#11758190D414yes—
53C22×#92418C2312yes—
54#14519766—24no—
55#14519766—42no—
56C4×Dic122—22no—
57C4×Dic122D414yes—
58#14519742—24no—
59#14519742—42no—
60C4×D244—22no—
61C4×D244—22no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×#11758196—21no—
67C2×#11758196—22no—
68C2×#11758196—22no—
69C2×#11758198—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61C2312yes—
75#14519709—22no—
76C4⋊#92418—22no—
77#14519706—24no—
78#14519702—22no—
79#14519701—24no—
80#14519699—22no—
81#14519697—24no—
82#14519684—22no—
83#14519684—22no—
84#14519684—22no—
85#14519684D412yes—
86C22×Q8×C61C2312yes—
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C4⋊Q8×C61—22no—
90C4⋊Q8×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61C2312yes—
96C2×C42×C61C2311yescharacteristic
97#103×C61—24no—
98#103×C61—42no—
99C4×C8×C61—22no—
100C23×D61—21no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C22×Dic61C2×D412yes—
116C22×Dic61C2×D412yes—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244C2×D412yes—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122C2×D412yes—
130C2×Dic122C2×D412yes—
131C2×Dic122#4112yes—
132C2×C8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61C2×D412yes—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—42no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61—81no—
144C42×C61C2×D412yes—
145C42×C61#4112yes—
146#11758202—48no—
147#11758198—44no—
148#11758198—44no—
149#11758198—42no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61C2×D412yes—
156#11758196—24no—
157#11758196—44no—
158#11758196—82no—
159C4×Dic61—24no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—44no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166#11758190—28no—
167C23×C61C22×D411yescharacteristic
168C22×D61—22no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175Q8×C61—42no—
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—42no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61#12112yes—
188C2×C4×C61#12112yes—
189C2×C4×C61#13512yes—
190C2×C4×C61#13712yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212D244—22no—
213D244—22no—
214D244—41no—
215D244—41no—
216C4×D61—84no—
217Dic122—22no—
218Dic122—22no—
219Dic122—22no—
220Dic122#12114yes—
221C8×C61—22no—
222C8×C61—41no—
223#92418—42no—
224#24729—611nomaximal, Sylow
225C22×C61C2×#12111yescharacteristic
226C22×C61D4211yescharacteristic
227C22×C61#54811yescharacteristic
228C22×C61C2×#13511yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C22×C61#57711yescharacteristic
231C22×C61#57811yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—22no—
240C4×C61—82no—
241C4×C61—82no—
242C4×C61#46414yescomplemented
243C4×C61C2×#12111yescharacteristic
244C4×C61#53711yescharacteristic
245C4×C61#54711yescharacteristic
246C4×C61#55311yescharacteristic
247Dic61—24no—
248Dic61—42no—
249Dic61—42no—
250Dic61—42no—
251Dic61—81no—
252C2×#464—612no—
253C2×#462—611no—
254C2×C4×SD16—611no—
255#2547—611no—
256Q8×C4⋊C4—611no—
257#2066—612no—
258#2042—612no—
259#1956D6111yescharacteristic, complemented
260#1939—611no—
261#1878—612no—
262#1802—611no—
263C2×C61#241611yescharacteristic
264C2×C61C2×#46411yescharacteristic
265C2×C61#304611yescharacteristic
266C2×C61#327611yescharacteristic
267C2×C61#328411yescharacteristic
268C2×C61#336411yescharacteristic
269C2×C61#336911yescharacteristic
270D61—82no—
271C22×Q16—612no—
272C22×SD16—611no—
273C2×C4⋊Q8C2×D6111yescharacteristic
274C2×#124—612no—
275C2×C4×Q8—612no—
276C2×C4×Q8—611no—
277C2×C4×D4—611no—
278#464—1228no—
279#462—1224no—
280C4×SD16—1224no—
281C2×#106—611no—
282C2×C4⋊C8—612no—
283C2×C4⋊C8—611no—
284C2×#103—612no—
285C2×#103—611no—
286C2×#103C2×D6112yes—
287C2×#103C2×D6112yes—
288C2×#102—611no—
289C2×#102—612no—
290C2×C4×C8C2×D6111yescharacteristic
291#397—1222no—
292#391—612no—
293#389—612no—
294#389—1222no—
295#384—1221no—
296#384C2×D6111yescharacteristic
297#380—1221no—
298#380—1221no—
299C4×C4⋊C4—611no—
300C4×C4⋊C4—1221no—
301#340—612no—
302#336—612no—
303C61#2472911yescharacteristic, complemented, Sylow
304C22×Q8—611no—
305C22×Q8C22×D6112yes—
306C22×D4—611no—
307C23×C4—1221no—
308C2×Q16—1224no—
309C2×Q16—1224no—
310C2×Q16—1224no—
311C2×SD16—614no—
312C2×SD16—1222no—
313C2×SD16—1222no—
314C22×C8—612no—
315C22×C8C22×D6111yescharacteristic
316C22×C8C22×D6111yescharacteristic
317C4⋊Q8—22no—
318C4⋊Q8—22no—
319#124—1224no—
320#124—2442no—
321C4×Q8—614no—
322C4×Q8—1222no—
323C4×Q8—1224no—
324C4×Q8—2442no—
325C4×D4—1222no—
326C4×D4—1222no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4C22×D6112yes—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1221no—
342C2×C42—612no—
343C2×C42—1221no—
344C2×C42—1222no—
345C2×C42C22×D6111yescharacteristic
346#106—1222no—
347C4⋊C8—1224no—
348C4⋊C8—1222no—
349#103—24no—
350#103—42no—
351#103—1224no—
352#103—1222no—
353#102—1224no—
354#102—1222no—
355C4×C8—22no—
356#95—612no—
357#95—1222no—
358#95—1222no—
359#95—1222no—
360C24—1221no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—612no—
368C2×Q8—612no—
369C2×Q8—612no—
370C2×Q8—1221no—
371C2×Q8—1221no—
372C2×Q8—1221no—
373C2×D4—612no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—1221no—
378C2×D4—1221no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4C23×D6111yescharacteristic
392Q16—2448no—
393SD16—1228no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8D4×D6112yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—81no—
409C4⋊C4—612no—
410C4⋊C4—1222no—
411C4⋊C4—1221no—
412C4⋊C4—1224no—
413C4⋊C4—1222no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4—4882no—
417C4⋊C4—2444no—
418C22⋊C4—2444no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C42—1222no—
422C42—1222no—
423C42—1224no—
424C42—2444no—
425C42#1175822212yes—
426C42D4×D6112yes—
427C23—1222no—
428C23—2441no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2441no—
433C23—2441no—
434C23C2×D4×D6111yescharacteristic
435Q8—42no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—614no—
440Q8—1222no—
441Q8—1222no—
442Q8—1222no—
443D4—1222no—
444D4—1222no—
445D4—2441no—
446D4—2441no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—4884no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4#1451975012yes—
472C2×C4#1451975012yes—
473C2×C4#1451979812yes—
474C2×C4#1451980912yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4#41×D6111yescharacteristic
478C2×C4C2×#1175822211yescharacteristic
479C2×C4#1451982211yescharacteristic
480C2×C4#1451982411yescharacteristic
481C8—22no—
482C8—41no—
483C8—2442no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—2442no—
489C22—2442no—
490C22—2442no—
491C22C2×#1451975011yescharacteristic
492C22#1684039711yescharacteristic
493C22#1684042011yescharacteristic
494C22C2×#1451979811yescharacteristic
495C22C2×#1451980911yescharacteristic
496C22#1684056811yescharacteristic
497C22#1684057511yescharacteristic
498C4—22no—
499C4—82no—
500C4—82no—
501C4—1224no—
502C4—4881no—
503C4—2442no—
504C4—2442no—
505C4—2442no—
506C4#1683998714yescomplemented
507C4#121×D6111yescharacteristic
508C4#1684034111yescharacteristic
509C4#1684041711yescharacteristic
510C4#1684046211yescharacteristic
511C2—4882no—
512C2#2144926211yescharacteristic
513C2C2×#1683998711yescharacteristic
514C2#2145537411yescharacteristic
515C2#2145746411yescharacteristic
516C2#2145754811yescharacteristic
517C2#2145850911yescharacteristic
518C2#2145856411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1956C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1956D61
hypocenterC61#24729
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
2. C2×#16839969C2semidirect
5. #21448731C2semidirect
6. #21447198C2semidirect
8. #21446537C2semidirect
9. #21446157C2semidirect
11. C61×#1956C2semidirect
242. C4×C61#464semidirect
259. #1956D61semidirect
303. C61#24729semidirect
506. C4#16839987semidirect

Maximal subgroups

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

subgroupindex
C2×#168399872
C2×#168399692
C2×C4×#117581902
#214508152
#214487312
#214471982
#214470562
#214465372
#214461572
#214458162
C61×#19562
#2472961