Finite Group Databasebeta

← #29730122

Subgroups — #29730122

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
2C2×#16839827C211yesmaximal, characteristic, complemented
3#21450814C211yesmaximal, characteristic, complemented
4#21450735C211yesmaximal, characteristic
5#21447244C212yesmaximal, complemented
6#21447003C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446246C212yesmaximal, complemented
9#21445845C212yesmaximal, complemented
10C4×#14519697C211yesmaximal, characteristic, complemented
11C61×#1947C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519765—21no—
15C2×#14519752—21no—
16C2×#14519752C2212yes—
17C2×C4×D244C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16840043—24no—
20#16839827—24no—
21C2×#14519711C2212yes—
22C2×#14519699C2212yes—
23C2×#14519697C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519689C2212yes—
26C2×C4×#92418C2211yescharacteristic
27#16839689—22no—
28#16839679—22no—
29#16839668—21no—
30#16839619—21no—
31#16839619C2212yes—
32#16839619C2212yes—
33#16839619C2212yes—
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2212yes—
37#16839524C2212yes—
38C2×#107×C61C2212yes—
39C2×C4⋊C8×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122—21no—
44C22×D244—22no—
45C22×D244D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61C2312yes—
49C22×#92418C2311yescharacteristic
50C22×#92418C2311yescharacteristic
51C2×C8×D61—24no—
52C2×C8×D61—24no—
53C2×C8×D61—42no—
54#14519777—22no—
55#14519777—22no—
56#14519765—44no—
57#14519752—24no—
58#14519752—42no—
59#14519752—44no—
60C4×D244—22no—
61C4×D244—22no—
62C4×D244—22no—
63C4×D244—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×#11758198—22no—
73C2×#11758198D412yes—
74#14519711—24no—
75#14519699—24no—
76#14519697—22no—
77#14519697—22no—
78#14519689—24no—
79C4×#92418—22no—
80#14519684—22no—
81#14519684—22no—
82#14519684—22no—
83C22×C8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C42×C61—21no—
90C2×C42×C61C2311yescharacteristic
91#107×C61—42no—
92C4⋊C8×C61D414yescomplemented
93#95×C61—21no—
94C23×D61—21no—
95C23×D61C2×D412yes—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—22no—
104C22×Dic61C2×D412yes—
105C2×D244—22no—
106C2×D244—22no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—44no—
113C2×D244—44no—
114C2×D244—82no—
115C2×D244D814yes—
116C2×D244#4112yes—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×Dic122—42no—
127C2×Dic122—42no—
128C2×Dic122—42no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61C2×D414yes—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—81no—
140C42×C61—42no—
141C42×C61C2×D412yes—
142C42×C61#4112yes—
143#11758198—24no—
144#11758198—44no—
145#11758198—44no—
146#11758198—44no—
147#11758198—44no—
148#11758198—42no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—42no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418#4112yes—
161C8×D61—48no—
162C23×C61C22×D411yescharacteristic
163C22×D61—22no—
164C22×D61—22no—
165C22×D61—22no—
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61C2×D814yes—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61#12112yes—
189C2×C4×C61#12112yes—
190C2×C4×C61C2×D812yes—
191C2×C4×C61#13512yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61H(5,𝔽2)11yescharacteristic
197C2×C4×C612−1+411yescharacteristic
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207D244—22no—
208D244—22no—
209D244—28no—
210D244—41no—
211D244—41no—
212D244—88no—
213C4×D61—42no—
214C4×D61—42no—
215C4×D61—42no—
216C4×D61—42no—
217C4×D61—84no—
218Dic122—84no—
219C8×C61—24no—
220#92418—22no—
221#92418—41no—
222#25338—611nomaximal, Sylow
223C22×C61C2×#12111yescharacteristic
224C22×C61#54611yescharacteristic
225C22×C61#54711yescharacteristic
226C22×C61C22×D811yescharacteristic
227C22×C61C2×#13511yescharacteristic
228C22×C61#57711yescharacteristic
229C22×C61#57811yescharacteristic
230C2×D61—24no—
231C2×D61—24no—
232C2×D61—24no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—42no—
236C2×D61—42no—
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—81no—
246C4×C61—82no—
247C4×C61—81no—
248C4×C61#46614yes—
249C4×C61#52911yescharacteristic
250C4×C61#54211yescharacteristic
251C4×C61#54611yescharacteristic
252C4×C61#54811yescharacteristic
253Dic61—42no—
254Dic61—82no—
255Dic61—81no—
256C2×#474—611no—
257C2×C8×D4—611no—
258#2547—611no—
259#2537—611no—
260#2075—612no—
261#2033—612no—
262#1990—611no—
263#1947D6111yescharacteristic, complemented
264#1893—612no—
265#1808—612no—
266C4×#102—611no—
267C2×C61#242711yescharacteristic
268C2×C61C2×#46611yescharacteristic
269C2×C61#306311yescharacteristic
270C2×C61#329411yescharacteristic
271C2×C61#330211yescharacteristic
272C2×C61#333411yescharacteristic
273C2×C61#334011yescharacteristic
274D61—44no—
275D61—82no—
276C23×C8—612no—
277C2×C4⋊Q8—611no—
278C2×#124—1221no—
279C2×#121—612no—
280C2×#121—1221no—
281C2×C4×D4—611no—
282C2×C4×D4—611no—
283#474—1224no—
284C8×D4—1224no—
285C2×#107—612no—
286C2×#107C2×D6112yes—
287C2×C4⋊C8C2×D6111yescharacteristic
288C2×#102—612no—
289C2×#102—611no—
290C2×#102—611no—
291C2×C22⋊C8—612no—
292C2×C4×C8—611no—
293#397—1222no—
294#394—1222no—
295#391—1221no—
296#384C2×D6111yescharacteristic
297#380—612no—
298#380—612no—
299#380—612no—
300#380—1221no—
301C4×C4⋊C4C2×D6111yescharacteristic
302#340—611no—
303#340—612no—
304#340—611no—
305#340—612no—
306#340C2×D6112yes—
307C61#2533811yescharacteristic, complemented, Sylow
308C22×Q8—1221no—
309C22×D4—611no—
310C22×D4—611no—
311C22×D4—1222no—
312C23×C4—612no—
313C23×C4—1221no—
314C22×C8—611no—
315C22×C8—611no—
316C22×C8—1224no—
317C22×C8—1224no—
318C22×C8—2442no—
319C22×C8C22×D6112yes—
320C4⋊Q8—1222no—
321C4⋊Q8—1222no—
322#124—2444no—
323#121—1224no—
324#121—2442no—
325#121—2444no—
326C4×D4—1222no—
327C4×D4—1222no—
328C4×D4—1222no—
329C4×D4—1222no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4C22×D6112yes—
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C22⋊C4—612no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1221no—
345C2×C42—21no—
346C2×C42C22×D6111yescharacteristic
347#107—42no—
348#107—1224no—
349C4⋊C8D24414yescomplemented
350#102—1222no—
351#102—1222no—
352#102—1224no—
353C22⋊C8—1224no—
354C4×C8—1222no—
355#95—21no—
356#95—1222no—
357#95—1222no—
358#95—1222no—
359C24—612no—
360C24—1221no—
361C2×Q8—2442no—
362C2×Q8—2442no—
363C2×Q8—2442no—
364C2×D4—612no—
365C2×D4—614no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—4882no—
374C2×D4—2444no—
375C2×D4—2444no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—612no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1221no—
387C22×C4—1222no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4D4×D6111yescharacteristic
392C22×C4D4×D6111yescharacteristic
393C22×C4C23×D6111yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—612no—
397C2×C8—1221no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—2448no—
403C2×C8C2×D24414yes—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—42no—
411C4⋊C4—81no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4—1222no—
417C4⋊C4—2442no—
418C22⋊C4—1224no—
419C22⋊C4—2444no—
420C22⋊C4—2444no—
421C22⋊C4—2444no—
422C22⋊C4—2444no—
423C22⋊C4—2442no—
424C42—42no—
425C42C2×D24412yes—
426C42#1175822612yes—
427C23—614no—
428C23—1222no—
429C23—1222no—
430C23—1222no—
431C23—1222no—
432C23—1222no—
433C23—1222no—
434C23—1222no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C2×D4×D6111yescharacteristic
442Q8—4884no—
443D4—1228no—
444D4—1222no—
445D4—1222no—
446D4—4888no—
447D4—2441no—
448D4—2441no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—4884no—
464C2×C4—2441no—
465C2×C4—2442no—
466C2×C4—2441no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4#1451975212yes—
475C2×C4#1451975212yes—
476C2×C4D8×D6112yes—
477C2×C4#1451980012yes—
478C2×C4C22×D24411yescharacteristic
479C2×C4#41×D6111yescharacteristic
480C2×C4#41×D6111yescharacteristic
481C2×C4C2×#1175822611yescharacteristic
482C2×C4#1451982211yescharacteristic
483C2×C4#1451982511yescharacteristic
484C8—24no—
485C8—1222no—
486C8—2441no—
487C22—1224no—
488C22—1224no—
489C22—1224no—
490C22—4881no—
491C22—4881no—
492C22—4881no—
493C22—4881no—
494C22—2442no—
495C22—2442no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22—2442no—
500C22—2442no—
501C22C2×#1451975211yescharacteristic
502C22#1684040911yescharacteristic
503C22#1684041411yescharacteristic
504C22C2×D8×D6111yescharacteristic
505C22C2×#1451980011yescharacteristic
506C22#1684057211yescharacteristic
507C22#1684057811yescharacteristic
508C4—22no—
509C4—81no—
510C4—82no—
511C4—81no—
512C4—4881no—
513C4—4882no—
514C4—2442no—
515C4#1683999914yes—
516C4#1684029711yescharacteristic
517C4#1684038611yescharacteristic
518C4#1684040711yescharacteristic
519C4#1684042411yescharacteristic
520C2—4882no—
521C2—2444no—
522C2#2144939411yescharacteristic
523C2C2×#1683999911yescharacteristic
524C2#2145551111yescharacteristic
525C2#2145763511yescharacteristic
526C2#2145773911yescharacteristic
527C2#2145816511yescharacteristic
528C2#2145823511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radical#21450735C2
hypercenter#1947D61
hypocenterC61#25338
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. C2×#16840043C2semidirect
2. C2×#16839827C2semidirect
3. #21450814C2semidirect
5. #21447244C2semidirect
6. #21447003C2semidirect
7. #21446779C2semidirect
8. #21446246C2semidirect
9. #21445845C2semidirect
10. C4×#14519697C2semidirect
11. C61×#1947C2semidirect
92. C4⋊C8×C61D4semidirect
263. #1947D61semidirect
307. C61#25338semidirect
349. C4⋊C8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
C2×#168398272
#214508142
#214507352
#214472442
#214470032
#214467792
#214462462
#214458452
C4×#145196972
C61×#19472
#2533861