Finite Group Databasebeta

← #29779574

Subgroups — #29779574

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454166C211yesmaximal, characteristic, complemented
2#21448308C211yesmaximal, characteristic
3#21447227C211yesmaximal, characteristic, complemented
4#21447224C211yesmaximal, characteristic, complemented
5#21446952C211yesmaximal, characteristic, complemented
6#21446514C212yesmaximal, complemented
7#21446506C212yesmaximal, complemented
8#21445912C212yesmaximal, complemented
9#21445758C212yesmaximal
10#21445580C211yesmaximal, characteristic, complemented
11C61×#2024C211yesmaximal, characteristic, complemented
12C22×#11758226C2211yescharacteristic
13C2×#14519753—21no—
14C2×#14519735C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16C2×#14519701C2212yes—
17C2×#14519699C2212yes—
18C2×#14519697C2212yes—
19C2×#14519689C2212yes—
20C2×#14519687C2212yes—
21#16839669—21no—
22#16839669C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839663C2211yescharacteristic
25#16839658—21no—
26#16839658C2212yes—
27#16839640—21no—
28#16839640—21no—
29#16839632C2211yescharacteristic
30#16839619—21no—
31#16839619C2212yes—
32#16839609—21no—
33#16839609—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×#103×C61C2212yes—
40C61×#391C2211yescharacteristic
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C2×#11758226—22no—
44C2×#11758226—22no—
45C2×#11758226—24no—
46C2×#11758226—42no—
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2312yes—
54#14519753—44no—
55#14519735—24no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—22no—
61C2×#11758198—21no—
62C2×#11758198—22no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61C2311yescharacteristic
71C42×D61—42no—
72#14519701—24no—
73#14519699—24no—
74#14519697—42no—
75#14519689—24no—
76#14519687—42no—
77#14519684—21no—
78#14519684—21no—
79#14519684—22no—
80#14519684—22no—
81C22×Q8×C61—21no—
82C22×Q8×C61C2311yescharacteristic
83C22×C8×C61C2312yes—
84C4⋊Q8×C61—42no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C42×C61C2311yescharacteristic
89#103×C61—42no—
90#95×C61—21no—
91#95×C61—21no—
92C23×D61—21no—
93C23×D61C2×D411yescharacteristic
94C2×Q8×C61—21no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C2×Q8×C61#4112yes—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61C2×D411yescharacteristic
110C22×Dic61C2×D411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112#11758226—48no—
113#11758226—48no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—24no—
117C2×D244—21no—
118C2×D244—42no—
119C2×D244—42no—
120C2×D244#4112yes—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—22no—
124C2×C4×D61—24no—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61#4114yes—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—42no—
141C42×C61—22no—
142#11758198—42no—
143#11758198—42no—
144#11758198—44no—
145#11758198—42no—
146#11758198—42no—
147#11758198—42no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61—42no—
153#11758196—44no—
154C4×Dic61—22no—
155C4×Dic61—41no—
156C4×Dic61—41no—
157C4×Dic61—42no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C23×C61C22×D411yescharacteristic
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—22no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61#12412yes—
176Q8×C61—24no—
177Q8×C61—42no—
178Q8×C61—84no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61AGL(1,ℤ/8)12yes—
193C2×C4×C61#13712yes—
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4112yes—
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)12yes—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61#12412yes—
213D244—24no—
214D244—48no—
215D244—42no—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—48no—
219C4×D61—44no—
220C4×D61—44no—
221C8×C61—42no—
222#92418—42no—
223#29113—611nomaximal, Sylow
224C22×C61#52511yescharacteristic
225C22×C61C2×#12411yescharacteristic
226C22×C61#53411yescharacteristic
227C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
228C22×C61C2×#13711yescharacteristic
229C22×C61#57511yescharacteristic
230C22×C61#57511yescharacteristic
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—41no—
235C2×D61—41no—
236C2×D61—41no—
237C2×D61—41no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—42no—
248C4×C61—81no—
249C4×C61—82no—
250C4×C61#55112yes—
251C4×C61#55112yes—
252Dic61—22no—
253Dic61—41no—
254Dic61—41no—
255Dic61—82no—
256Dic61—81no—
257#2857—611no—
258#2297—611no—
259#2072—611no—
260#2071—611no—
261#2024D6111yescharacteristic, complemented
262#2023—611no—
263#1936—612no—
264#1935—612no—
265#1825—612no—
266#1788—612no—
267#1749—611no—
268C2×C61#256111yescharacteristic
269C2×C61#311511yescharacteristic
270C2×C61#311611yescharacteristic
271C2×C61#313511yescharacteristic
272C2×C61#314111yescharacteristic
273C2×C61#319411yescharacteristic
274C2×C61#319611yescharacteristic
275D61—42no—
276D61—82no—
277C22×#41—611no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×C22⋊Q8—1221no—
280C2×#117—611no—
281C22×C42—611no—
282C2×#103—612no—
283C2×#103C2×D6112yes—
284C2×#102—612no—
285C2×#102—612no—
286C2×C22⋊C8—612no—
287C2×#97—612no—
288#391—611no—
289#391—1221no—
290#391C2×D6111yescharacteristic
291#390—611no—
292#390—611no—
293#389—612no—
294#389—1221no—
295#386—1221no—
296#386—1221no—
297#384—611no—
298#380—612no—
299#380—1221no—
300C4×C22⋊C4—1221no—
301C4×C22⋊C4—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306#336—611no—
307#336C2×D6111yescharacteristic
308C61#2911311yescharacteristic, complemented, Sylow
309C2×#41—1224no—
310C2×#41—1222no—
311C2×#41—1222no—
312C2×#41—2442no—
313C22×Q8—21no—
314C22×Q8C22×D6111yescharacteristic
315C22×D4—611no—
316C22×D4—1221no—
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—1221no—
321C22×C8—612no—
322C22×C8C22×D6112yes—
323C4⋊Q8—42no—
324C22⋊Q8—2444no—
325#117—1224no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C4⋊C4C22×D6112yes—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C42—611no—
341C2×C42—611no—
342C2×C42—611no—
343C2×C42—1221no—
344C2×C42—2442no—
345C2×C42C22×D6111yescharacteristic
346#103—42no—
347#103—1224no—
348#102—1224no—
349#102—2442no—
350C22⋊C8—1224no—
351#97—2442no—
352#95—21no—
353#95—21no—
354#95—1221no—
355#95—1221no—
356#95—1222no—
357#95—1222no—
358C24—611no—
359C24—1221no—
360#41—2448no—
361#41—2448no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8#1175822612yes—
369C2×D4—612no—
370C2×D4—1224no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1222no—
374C2×D4—2442no—
375C2×D4—2442no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—611no—
380C22×C4—614no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—1224no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1221no—
390C22×C4—1222no—
391C22×C4—2441no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4—2441no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4D4×D6111yescharacteristic
398C22×C4D4×D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—1222no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—2442no—
414C4⋊C4—2444no—
415C4⋊C4—2442no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C22⋊C4—2442no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C42—22no—
423C42—1222no—
424C42—2441no—
425C42—2441no—
426C42—2442no—
427C23—612no—
428C23—1221no—
429C23—1221no—
430C23—1221no—
431C23—1221no—
432C23—1221no—
433C23—1221no—
434C23—1222no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23C2×D4×D6111yescharacteristic
442Q8—24no—
443Q8—42no—
444Q8—84no—
445D4—1224no—
446D4—2448no—
447D4—2442no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—612no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—2444no—
473C2×C4—2444no—
474C2×C4—2448no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4#1451980412yes—
482C2×C4#1451980912yes—
483C2×C4#41×D6112yes—
484C2×C4#41×D6111yescharacteristic
485C2×C4C2×#1175822611yescharacteristic
486C2×C4#1451982212yes—
487C8—42no—
488C8—2442no—
489C22—1222no—
490C22—1222no—
491C22—1222no—
492C22—4881no—
493C22—4881no—
494C22—4881no—
495C22—4881no—
496C22—2442no—
497C22—2442no—
498C22—2442no—
499C22—2441no—
500C22—2441no—
501C22—2441no—
502C22—2441no—
503C22#1684027511yescharacteristic
504C22#124×D6111yescharacteristic
505C22#1684032211yescharacteristic
506C22C2×#1451980411yescharacteristic
507C22C2×#1451980911yescharacteristic
508C22#1684055811yescharacteristic
509C22#1684056111yescharacteristic
510C4—22no—
511C4—22no—
512C4—42no—
513C4—81no—
514C4—82no—
515C4—1222no—
516C4—4881no—
517C4—4882no—
518C4—2441no—
519C4—2441no—
520C4#1684043912yes—
521C4#1684044212yes—
522C2—4882no—
523C2—2442no—
524C2#2145097311yescharacteristic
525C2#2145597311yescharacteristic
526C2#2145598411yescharacteristic
527C2#2145616111yescharacteristic
528C2#2145621411yescharacteristic
529C2#2145663611yescharacteristic
530C2#2145665211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2024C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
hypercenter#2024D61
hypocenterC61#29113
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2024

Complements

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

NH ≅ G/Ntype
1. #21454166C2semidirect
3. #21447227C2semidirect
4. #21447224C2semidirect
5. #21446952C2semidirect
6. #21446514C2semidirect
7. #21446506C2semidirect
8. #21445912C2semidirect
10. #21445580C2semidirect
11. C61×#2024C2semidirect
261. #2024D61semidirect
308. C61#29113semidirect

Maximal subgroups

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

subgroupindex
#214541662
#214483082
#214472272
#214472242
#214469522
#214465142
#214465062
#214459122
#214457582
#214455802
C61×#20242
#2911361