Finite Group Databasebeta

← #29730118

Subgroups — #29730118

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840043C211yesmaximal, characteristic, complemented
2#21450811C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic
4#21447242C212yesmaximal, complemented
5#21447002C212yesmaximal, complemented
6#21446780C211yesmaximal, characteristic, complemented
7#21446578C211yesmaximal, characteristic, complemented
8#21446247C212yesmaximal, complemented
9#21445848C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C8×D4×C61C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519764—21no—
14C2×#14519751C2212yes—
15C2×#14519748—21no—
16C2×C4×D244C2211yescharacteristic
17#16840043—24no—
18C2×#14519711C2212yes—
19C2×#14519711C2212yes—
20C2×#14519699C2211yescharacteristic
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2212yes—
23#16839687—22no—
24#16839678—22no—
25#16839667—21no—
26#16839632C2211yescharacteristic
27#16839622C2212yes—
28#16839622C2212yes—
29#16839622C2212yes—
30#16839619—21no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839525C2211yescharacteristic
33#16839525C2211yescharacteristic
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37C23×C8×C61C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C8×D4×C61—24no—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×C22⋊C8×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C22⋊D61—22no—
46C22×C4×D61—21no—
47#14519777—22no—
48#14519777—22no—
49#14519764—44no—
50#14519751—24no—
51#14519751—42no—
52#14519748—44no—
53C4×D244—22no—
54C4×D244—22no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×#11758198—21no—
67C2×#11758198—22no—
68C2×C4×Dic61—21no—
69#14519711—24no—
70#14519711—42no—
71#14519699—22no—
72#14519699—24no—
73#14519699—22no—
74#14519684—21no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78C22×D4×C61D411yescharacteristic
79C23×C4×C61C2312yes—
80C22×C8×C61—24no—
81C22×C8×C61—24no—
82C22×C8×C61—42no—
83C22×C8×C61C2311yescharacteristic
84C22×C8×C61C2312yes—
85C22×C8×C61C2311yescharacteristic
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C2×C4⋊C4×C61D411yescharacteristic
89C2×C22⋊C4×C61D412yes—
90C2×C42×C61C2311yescharacteristic
91C4⋊C8×C61D414yescomplemented
92C22⋊C8×C61—24no—
93C4×C8×C61—22no—
94C24×C61C2×D412yes—
95C23×D61—21no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61D814yes—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C2×C22⋊D61—44no—
110C2×C22⋊D61—44no—
111C2×C22⋊D61—82no—
112C22×Dic61—21no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244#4112yes—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×Dic122—42no—
127C2×Dic122—42no—
128C2×Dic122—42no—
129C2×C8×C61—21no—
130C2×C8×C61—22no—
131C2×C8×C61—21no—
132C2×C8×C61—22no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—21no—
136C2×C8×C61—48no—
137C2×C8×C61C2×D414yes—
138C2×C8×C61#4112yes—
139C4⋊C4×C61—22no—
140C22⋊C4×C61—24no—
141C42×C61C2×D412yes—
142C42×C61#4112yes—
143C22⋊Dic61—44no—
144C22⋊Dic61—44no—
145C22⋊Dic61—44no—
146#11758198—44no—
147#11758198—42no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—42no—
159C4⋊Dic61—42no—
160C4⋊Dic61—81no—
161C4×Dic61—42no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61C2×D814yes—
169C23×C61C22×D411yescharacteristic
170C22×D61—22no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177D4×C61—28no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
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×C61C2×D812yes—
190C2×C4×C61#13512yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)11yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C22⋊D61—88no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213D244—22no—
214D244—22no—
215D244—41no—
216D244—41no—
217C4×D61—84no—
218Dic122—84no—
219C8×C61—22no—
220C8×C61—24no—
221C8×C61—41no—
222#25338—611nomaximal, Sylow
223C22×C61—24no—
224C22×C61—24no—
225C22×C61—24no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61C2×#12111yescharacteristic
231C22×C61#54611yescharacteristic
232C22×C61#54711yescharacteristic
233C22×C61C22×D811yescharacteristic
234C22×C61C2×#13511yescharacteristic
235C22×C61#57711yescharacteristic
236C22×C61#57811yescharacteristic
237C2×D61—42no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—81no—
244C4×C61—22no—
245C4×C61—42no—
246C4×C61#46614yes—
247C4×C61#52911yescharacteristic
248C4×C61#54211yescharacteristic
249C4×C61#54611yescharacteristic
250C4×C61#54811yescharacteristic
251Dic61—81no—
252Dic61—81no—
253Dic61—82no—
254Dic61—82no—
255Dic61—81no—
256C2×#474—611no—
257C2×C8×D4D6111yescharacteristic, complemented
258#2547—611no—
259#2537—611no—
260#2075—612no—
261#2033—612no—
262#1990—611no—
263#1947—611no—
264#1893—612no—
265#1808—612no—
266C4×#102—611no—
267C2×C61—44no—
268C2×C61#242711yescharacteristic
269C2×C61C2×#46611yescharacteristic
270C2×C61#306311yescharacteristic
271C2×C61#329411yescharacteristic
272C2×C61#330211yescharacteristic
273C2×C61#333411yescharacteristic
274C2×C61#334011yescharacteristic
275D61—82no—
276C23×C8C2×D6112yes—
277C2×C4⋊Q8—611no—
278C2×#124—1221no—
279C2×#121—612no—
280C2×#121—1221no—
281C2×C4×D4—611no—
282C2×C4×D4C2×D6111yescharacteristic
283#474—1224no—
284C8×D4—24no—
285C2×#107—612no—
286C2×#107—612no—
287C2×C4⋊C8C2×D6111yescharacteristic
288C2×#102—611no—
289C2×#102—612no—
290C2×#102—611no—
291C2×C22⋊C8C2×D6112yes—
292C2×C4×C8C2×D6111yescharacteristic
293#397—1222no—
294#394—1222no—
295#391—1221no—
296#384—611no—
297#380—612no—
298#380—612no—
299#380—612no—
300#380—1221no—
301C4×C4⋊C4—611no—
302#340—611no—
303#340—612no—
304#340—611no—
305#340—612no—
306#340—612no—
307C61#2533811yescharacteristic, complemented, Sylow
308C22×Q8—1221no—
309C22×D4—611no—
310C22×D4—1222no—
311C22×D4D24411yescharacteristic
312C23×C4—1221no—
313C23×C4C22×D6112yes—
314C22×C8—24no—
315C22×C8—24no—
316C22×C8—42no—
317C22×C8C22×D6111yescharacteristic
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6112yes—
320C4⋊Q8—1222no—
321C4⋊Q8—1222no—
322#124—2444no—
323#121—1224no—
324#121—2442no—
325#121—2444no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—1222no—
329C4×D4—1222no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—612no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4D24411yescharacteristic
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4D24412yes—
345C2×C42—1221no—
346C2×C42C22×D6111yescharacteristic
347#107—1224no—
348#107—2442no—
349C4⋊C8C22⋊D6114yescomplemented
350#102—1222no—
351#102—1222no—
352#102—1224no—
353C22⋊C8—24no—
354C4×C8—22no—
355#95—1221no—
356#95—1222no—
357#95—1222no—
358#95—1222no—
359C24—1221no—
360C24C2×D24412yes—
361C2×Q8—2442no—
362C2×Q8—2442no—
363C2×Q8—2442no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—612no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—1221no—
372C2×D4—4882no—
373C2×D4—2444no—
374C2×D4—2444no—
375C2×D4D48814yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4C2×D24412yes—
391C22×C4C2×D24411yescharacteristic
392C22×C4C2×D24411yescharacteristic
393C22×C4C23×D6111yescharacteristic
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—48no—
402C2×C8#1175822412yes—
403C2×C8C2×C22⋊D6114yes—
404C4⋊C4—22no—
405C4⋊C4—1222no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—1221no—
413C4⋊C4—1222no—
414C4⋊C4—1221no—
415C4⋊C4—4881no—
416C4⋊C4—2442no—
417C4⋊C4—2442no—
418C22⋊C4—24no—
419C22⋊C4—2444no—
420C22⋊C4—2444no—
421C22⋊C4—2444no—
422C22⋊C4—2444no—
423C22⋊C4—2442no—
424C42—2442no—
425C42#1175822212yes—
426C42C2×C22⋊D6112yes—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—1222no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23C2×D48814yes—
441C23C22×D24411yescharacteristic
442Q8—4884no—
443D4—28no—
444D4—1222no—
445D4—1222no—
446D4—4888no—
447D4—2441no—
448D4—2441no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—4884no—
459C2×C4—2441no—
460C2×C4—2442no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2441no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4#1451975112yes—
475C2×C4#1451975112yes—
476C2×C4C2×D48812yes—
477C2×C4#1451979912yes—
478C2×C4C22×C22⋊D6111yescharacteristic
479C2×C4#41×D6111yescharacteristic
480C2×C4C2×#1175822411yescharacteristic
481C2×C4C2×#1175822211yescharacteristic
482C2×C4#1451982211yescharacteristic
483C2×C4#1451982411yescharacteristic
484C8—22no—
485C8—24no—
486C8—41no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22—2442no—
499C22—2442no—
500C22—2442no—
501C22C2×#1451975111yescharacteristic
502C22#1684040711yescharacteristic
503C22#1684041611yescharacteristic
504C22C22×D48811yescharacteristic
505C22C2×#1451979911yescharacteristic
506C22#1684057411yescharacteristic
507C22#1684057911yescharacteristic
508C4—22no—
509C4—42no—
510C4—4881no—
511C4—4882no—
512C4—4881no—
513C4—4882no—
514C4—4881no—
515C4#1683999814yes—
516C4#1684029411yescharacteristic
517C4#1684037711yescharacteristic
518C4#1684040611yescharacteristic
519C4#1684041911yescharacteristic
520C2—44no—
521C2—4882no—
522C2#2144939011yescharacteristic
523C2C2×#1683999811yescharacteristic
524C2#2145551011yescharacteristic
525C2#2145763311yescharacteristic
526C2#2145773711yescharacteristic
527C2#2145816711yescharacteristic
528C2#2145823311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×D4×C61C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC2×C8×D4D61
hypocenterC61#25338
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840043C2semidirect
2. #21450811C2semidirect
4. #21447242C2semidirect
5. #21447002C2semidirect
6. #21446780C2semidirect
7. #21446578C2semidirect
8. #21446247C2semidirect
9. #21445848C2semidirect
10. C4×#14519699C2semidirect
11. C2×C8×D4×C61C2semidirect
91. C4⋊C8×C61D4semidirect
257. C2×C8×D4D61semidirect
307. C61#25338semidirect
349. C4⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
#214508112
#214507322
#214472422
#214470022
#214467802
#214465782
#214462472
#214458482
C4×#145196992
C2×C8×D4×C612
#2533861