Finite Group Databasebeta

← #29779572

Subgroups — #29779572

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#21454172C211yesmaximal, characteristic, complemented
2#21448306C211yesmaximal, characteristic
3#21447228C211yesmaximal, characteristic, complemented
4#21446961C211yesmaximal, characteristic, complemented
5#21446955C211yesmaximal, characteristic, complemented
6#21446518C212yesmaximal, complemented
7#21446510C212yesmaximal, complemented
8#21445913C212yesmaximal, complemented
9#21445756C212yesmaximal
10#21445580C211yesmaximal, characteristic, complemented
11C61×#2071C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14C2×#14519757—21no—
15C2×#14519735C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17C2×#14519702C2212yes—
18C2×#14519700C2212yes—
19C2×#14519698C2212yes—
20C2×#14519689C2212yes—
21C2×#14519687C2212yes—
22#16839668C2211yescharacteristic
23#16839667—21no—
24#16839667C2211yescharacteristic
25#16839665C2211yescharacteristic
26#16839661—21no—
27#16839661C2212yes—
28#16839638—21no—
29#16839638—21no—
30#16839619—21no—
31#16839619C2212yes—
32C22⋊C4×Dic61—21no—
33C22⋊C4×Dic61—21no—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2211yescharacteristic
38C2×#102×C61C2212yes—
39C61×#390C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C61×#340C2212yes—
42C61×#336C2211yescharacteristic
43C2×#11758224—22no—
44C2×#11758224—24no—
45C2×#11758224—22no—
46C2×#11758224—42no—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C22×C22⋊D61—21no—
50C23×Dic61—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55#14519777—42no—
56#14519757—44no—
57#14519735—24no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—21no—
64C2×#11758198—21no—
65C2×#11758198—22no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C42×D61—42no—
71#14519702—42no—
72#14519700—24no—
73#14519698—42no—
74#14519689—24no—
75#14519687—42no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684—21no—
80#14519684—22no—
81#14519684—22no—
82C22×D4×C61C2311yescharacteristic
83C22×C8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2312yes—
88C2×C22⋊C4×C61—21no—
89C2×C22⋊C4×C61—21no—
90C2×C42×C61C2311yescharacteristic
91#102×C61—24no—
92C24×C61—21no—
93C23×D61C2×D411yescharacteristic
94C2×D4×C61—22no—
95C2×D4×C61—21no—
96C2×D4×C61—22no—
97C2×D4×C61#4112yes—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2411yescharacteristic
103C2×C22⋊D61—24no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—42no—
106C22×Dic61—24no—
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61—21no—
110C22×Dic61—42no—
111C22×Dic61—42no—
112C22×Dic61C2×D411yescharacteristic
113C22×Dic61C2×D411yescharacteristic
114C22×Dic61C2×D411yescharacteristic
115#11758224—48no—
116#11758224—48no—
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61—41no—
126C2×C4×D61#4114yes—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—21no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×Dic122#4112yes—
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—
141C4⋊C4×C61—42no—
142C22⋊C4×C61—42no—
143C22⋊C4×C61—42no—
144C42×C61—22no—
145C22⋊Dic61—42no—
146C22⋊Dic61—42no—
147#11758198—44no—
148#11758198—42no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—44no—
153C4⋊Dic61—42no—
154C4×Dic61—22no—
155C4×Dic61—41no—
156C4×Dic61—41no—
157C4×Dic61—42no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C23×C61—22no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61C22×D411yescharacteristic
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—21no—
175C22×D61#12412yes—
176D4×C61—24no—
177D4×C61—42no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61AGL(1,ℤ/8)12yes—
189C2×C4×C61#13712yes—
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4112yes—
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61H(5,𝔽2)12yes—
194C22⋊D61—48no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—22no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—48no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61#12412yes—
214C4×D61—24no—
215C4×D61—24no—
216C4×D61—44no—
217C4×D61—44no—
218Dic122—24no—
219Dic122—42no—
220Dic122—84no—
221C8×C61—42no—
222#92418—42no—
223#29113—611nomaximal, Sylow
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61#52511yescharacteristic
232C22×C61C2×#12411yescharacteristic
233C22×C61#53411yescharacteristic
234C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
235C22×C61C2×#13711yescharacteristic
236C22×C61#57511yescharacteristic
237C22×C61#57511yescharacteristic
238C2×D61—22no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—82no—
248C4×C61—81no—
249C4×C61#55112yes—
250C4×C61#55112yes—
251Dic61—22no—
252Dic61—42no—
253Dic61—41no—
254Dic61—41no—
255Dic61—81no—
256Dic61—82no—
257#2857—611no—
258#2297—611no—
259#2072—611no—
260#2071D6111yescharacteristic, complemented
261#2024—611no—
262#2023—611no—
263#1936—612no—
264#1935—612no—
265#1825—612no—
266#1788—612no—
267#1749—611no—
268C2×C61—82no—
269C2×C61#256111yescharacteristic
270C2×C61#311511yescharacteristic
271C2×C61#311611yescharacteristic
272C2×C61#313511yescharacteristic
273C2×C61#314111yescharacteristic
274C2×C61#319411yescharacteristic
275C2×C61#319611yescharacteristic
276D61—42no—
277C22×#41—611no—
278C2×C4⋊Q8—611no—
279C2×C22⋊Q8—1221no—
280C2×#117—611no—
281C22×C42—611no—
282C2×#103—612no—
283C2×#103—612no—
284C2×#102—612no—
285C2×#102C2×D6112yes—
286C2×C22⋊C8—612no—
287C2×#97—612no—
288#391—611no—
289#391—611no—
290#391—1221no—
291#390—611no—
292#390C2×D6111yescharacteristic
293#389—612no—
294#389—1221no—
295#386—1221no—
296#386—1221no—
297#384C2×D6111yescharacteristic
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—611no—
314C22×Q8—1221no—
315C22×D4—1221no—
316C22×D4C22×D6111yescharacteristic
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—1221no—
321C22×C8—612no—
322C22×C8C22×D6112yes—
323C4⋊Q8—2442no—
324C22⋊Q8—2444no—
325#117—1224no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C4⋊C4C22×D6112yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
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—1224no—
347#103—2442no—
348#102—24no—
349#102—2442no—
350C22⋊C8—1224no—
351#97—2442no—
352#95—1221no—
353#95—1221no—
354#95—1222no—
355#95—1222no—
356#95—1221no—
357#95—1221no—
358C24—21no—
359C24—611no—
360#41—2448no—
361#41—2448no—
362C2×Q8—612no—
363C2×Q8—1221no—
364C2×Q8—1222no—
365C2×Q8—1222no—
366C2×Q8—2442no—
367C2×Q8—2442no—
368C2×Q8—2442no—
369C2×D4—22no—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—1224no—
373C2×D4—2442no—
374C2×D4—2442no—
375C2×D4#1175822412yes—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—611no—
379C22×C4—614no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1222no—
385C22×C4—1224no—
386C22×C4—1221no—
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—42no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—2442no—
415C4⋊C4—2444no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—2442no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C42—22no—
423C42—1222no—
424C42—2441no—
425C42—2441no—
426C42—2442no—
427C23—22no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—612no—
435C23—1221no—
436C23—1221no—
437C23—1221no—
438C23—1221no—
439C23—1221no—
440C23—1221no—
441C23C2×D4×D6111yescharacteristic
442Q8—1224no—
443Q8—4884no—
444Q8—2442no—
445D4—24no—
446D4—42no—
447D4—2448no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—612no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1224no—
462C2×C4—1224no—
463C2×C4—1222no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—2444no—
471C2×C4—2444no—
472C2×C4—2448no—
473C2×C4—2441no—
474C2×C4—2442no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4#1451980112yes—
482C2×C4#1451980612yes—
483C2×C4#41×D6112yes—
484C2×C4#41×D6111yescharacteristic
485C2×C4C2×#1175822411yescharacteristic
486C2×C4#1451982312yes—
487C8—42no—
488C8—2442no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—1222no—
497C22—1222no—
498C22—1222no—
499C22—2441no—
500C22—2441no—
501C22—2441no—
502C22—2441no—
503C22#1684027211yescharacteristic
504C22#124×D6111yescharacteristic
505C22#1684032311yescharacteristic
506C22C2×#1451980111yescharacteristic
507C22C2×#1451980611yescharacteristic
508C22#1684055811yescharacteristic
509C22#1684056111yescharacteristic
510C4—22no—
511C4—22no—
512C4—81no—
513C4—82no—
514C4—1222no—
515C4—4881no—
516C4—4882no—
517C4—2441no—
518C4—2442no—
519C4—2441no—
520C4#1684043512yes—
521C4#1684043712yes—
522C2—82no—
523C2—2442no—
524C2#2145097111yescharacteristic
525C2#2145596711yescharacteristic
526C2#2145597811yescharacteristic
527C2#2145616111yescharacteristic
528C2#2145620911yescharacteristic
529C2#2145663211yescharacteristic
530C2#2145664811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#2071D61
hypocenterC61#29113
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
1. #21454172C2semidirect
3. #21447228C2semidirect
4. #21446961C2semidirect
5. #21446955C2semidirect
6. #21446518C2semidirect
7. #21446510C2semidirect
8. #21445913C2semidirect
10. #21445580C2semidirect
11. C61×#2071C2semidirect
260. #2071D61semidirect
308. C61#29113semidirect

Maximal subgroups

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

subgroupindex
#214541722
#214483062
#214472282
#214469612
#214469552
#214465182
#214465102
#214459132
#214457562
#214455802
C61×#20712
#2911361