Finite Group Databasebeta

← #29737023

Subgroups — #29737023

Order 15616.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840471C211yesmaximal, characteristic
2C2×#16840131C211yesmaximal, characteristic, complemented
3C2×#16839967C211yesmaximal, characteristic
4#21451034C211yesmaximal, characteristic, complemented
5#21446969C212yesmaximal, complemented
6#21446959C212yesmaximal, complemented
7#21446776C211yesmaximal, characteristic, complemented
8#21446759C211yesmaximal, characteristic, complemented
9#21446151C212yesmaximal
10#21446151C212yesmaximal
11C61×#1762C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13#16840471—24no—
14#16840471—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519756—21no—
20C2×#14519754—21no—
21C2×#14519754—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24C2×D4×Dic61C2211yescharacteristic
25C22×#11758196—21no—
26#16840131—24no—
27#16839967—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519702C2212yes—
30C2×#14519702C2212yes—
31C2×#14519700C2212yes—
32#16839702—21no—
33#16839702—21no—
34#16839668—21no—
35#16839667C2212yes—
36#16839667C2212yes—
37#16839628—21no—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41#16839516C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C4×C4⋊C4×C61C2211yescharacteristic
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122D411yescharacteristic
49C22×Dic122C2312yes—
50C23×Dic61—21no—
51C23×Dic61—21no—
52C2×#11758200—24no—
53C2×#11758200—28no—
54C22×#92418C2312yes—
55#14519777—22no—
56#14519777—22no—
57#14519776—22no—
58#14519776—22no—
59#14519769—22no—
60#14519769D414yes—
61#14519768—22no—
62#14519768—22no—
63#14519756—28no—
64#14519754—28no—
65#14519754—28no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—28no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C4⋊#92418—22no—
90#14519702—24no—
91#14519702—24no—
92#14519700—24no—
93#14519684—22no—
94#14519684—22no—
95#14519684—22no—
96#14519684—22no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C22×C8×C61C2312yes—
100C4×D4×C61—22no—
101C4×D4×C61D414yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107#102×C61—24no—
108#97×C61—22no—
109#95×C61—21no—
110C24×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61C2×D412yes—
114C2×D4×C61#4114yes—
115C22×C4×C61—21no—
116C22×C4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—82no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—24no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—24no—
126C22×Dic61—22no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61—82no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—24no—
139C2×Dic122—24no—
140C2×Dic122—24no—
141C2×Dic122—41no—
142C2×Dic122—82no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C22⋊C4×C61—24no—
151C42×C61—42no—
152C42×C61C2×D412yes—
153C42×C61C2×D412yes—
154C22⋊Dic61—24no—
155C22⋊Dic61—24no—
156#11758200—48no—
157C4⋊Dic61—24no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—42no—
162C4⋊Dic61—42no—
163C4⋊Dic61C2×D412yes—
164C4⋊Dic61C2×D412yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—22no—
168#11758196—24no—
169#11758196—24no—
170#11758196—22no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C2×#92418—22no—
176C2×#92418—24no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—42no—
182C23×C61C22×D411yescharacteristic
183D4×C61—22no—
184D4×C61—24no—
185D4×C61#12414yes—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—84no—
195C2×C4×C61#12012yes—
196C2×C4×C61#12012yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4112yes—
200C2×C4×C612−1+412yes—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—48no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—44no—
218C2×Dic61—84no—
219C2×Dic61#13712yes—
220Dic122—44no—
221Dic122—42no—
222Dic122—42no—
223Dic122—44no—
224Dic122—84no—
225C8×C61—42no—
226#92418—42no—
227#25945—611nomaximal, Sylow
228C22×C61—24no—
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61C2×#12011yescharacteristic
234C22×C61#54812yes—
235C22×C61C2×#13711yescharacteristic
236C22×C61#57511yescharacteristic
237C22×C61#57712yes—
238C4×C61—22no—
239C4×C61—22no—
240C4×C61—42no—
241C4×C61—82no—
242C4×C61C2×#12411yescharacteristic
243C4×C61#53611yescharacteristic
244C4×C61#54812yes—
245Dic61—22no—
246Dic61—41no—
247Dic61—42no—
248Dic61—41no—
249Dic61—42no—
250Dic61—41no—
251Dic61—82no—
252C2×#554—611no—
253C2×#491—611no—
254C2×#461—611no—
255#2568—611no—
256#2026—612no—
257#2024—612no—
258#1989—611no—
259#1985—611no—
260#1877—612no—
261#1877—612no—
262#1762D6111yescharacteristic
263C2×C61—44no—
264C2×C61#243611yescharacteristic
265C2×C61#300711yescharacteristic
266C2×C61#300911yescharacteristic
267C2×C61#337212yes—
268C2×C61#337612yes—
269C22×SD16—612no—
270#554—1224no—
271#554—1224no—
272C2×C4⋊Q8—611no—
273C2×C4⋊Q8—611no—
274C2×#125—611no—
275C2×#125—611no—
276C2×C22⋊Q8—1221no—
277C2×C22⋊Q8—1221no—
278C2×C22⋊Q8—1221no—
279C2×C4×Q8—611no—
280C2×C4×D4—611no—
281C2×C4×D4—611no—
282C2×C4×D4C2×D6111yescharacteristic
283C22×C4⋊C4—1221no—
284#491—1224no—
285#461—1224no—
286C2×C4⋊C8—611no—
287C2×#103—612no—
288C2×#103—612no—
289C2×#103—612no—
290C2×#102C2×D6112yes—
291C2×#97C2×D6111yescharacteristic
292#400—1221no—
293#400—1221no—
294#391—612no—
295#391—612no—
296#391—1221no—
297#382—1221no—
298C4×C4⋊C4C2×D6111yescharacteristic
299#340—612no—
300#340—612no—
301#340C2×D6112yes—
302#336—612no—
303#336—612no—
304C61#2594511yescharacteristic, complemented, Sylow
305C22×Q8—611no—
306C22×Q8—612no—
307C22×Q8—1221no—
308C22×D4C22×D6111yescharacteristic
309C23×C4—21no—
310C23×C4—1221no—
311C23×C4—1221no—
312C2×SD16—1224no—
313C2×SD16—1228no—
314C22×C8—612no—
315C22×C8C22×D6112yes—
316C4⋊Q8—1222no—
317C4⋊Q8—1222no—
318C4⋊Q8—1222no—
319C4⋊Q8—1222no—
320#125—614no—
321#125—1222no—
322#125—1222no—
323#125—1222no—
324C22⋊Q8—1228no—
325C22⋊Q8—1228no—
326C22⋊Q8—1228no—
327C4×Q8—1222no—
328C4×Q8—1222no—
329C4×D4—22no—
330C4×D4—1222no—
331C4×D4—1222no—
332C4×D4—1222no—
333C4×D4—1222no—
334C4×D4D24414yes—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1228no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4C22×D6112yes—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C42—21no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42C22×D6111yescharacteristic
357C4⋊C8—1222no—
358#103—1224no—
359#103—1224no—
360#103—1224no—
361#102—24no—
362#97—22no—
363#95—21no—
364#95—1222no—
365#95—1222no—
366#95—1222no—
367#95—1222no—
368C24—21no—
369C2×Q8—1222no—
370C2×Q8—1224no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1224no—
375C2×Q8—1224no—
376C2×Q8—4882no—
377C2×Q8—2441no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4C2×D24412yes—
381C2×D4#1175822214yes—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—82no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—1224no—
391C22×C4—1224no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4—4882no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4D4×D6111yescharacteristic
399C22×C4D4×D6111yescharacteristic
400C22×C4C23×D6111yescharacteristic
401SD16—2448no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—1222no—
406C2×C8—1224no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—612no—
411C4⋊C4—612no—
412C4⋊C4—1224no—
413C4⋊C4—1224no—
414C4⋊C4—1224no—
415C4⋊C4—1221no—
416C4⋊C4—1221no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—1224no—
421C4⋊C4—1224no—
422C4⋊C4—2442no—
423C4⋊C4—2442no—
424C4⋊C4C2×D24412yes—
425C22⋊C4—24no—
426C22⋊C4—1224no—
427C22⋊C4—1224no—
428C42—42no—
429C42—1221no—
430C42—1221no—
431C42—1221no—
432C42—1221no—
433C42C2×D24412yes—
434C42D4×D6112yes—
435C23—22no—
436C23—24no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23C2×D4×D6111yescharacteristic
441Q8—4884no—
442Q8—2444no—
443Q8—2442no—
444Q8—2442no—
445Q8—2444no—
446D4—22no—
447D4—24no—
448D4#1451976514yes—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—84no—
458C2×C4—612no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—4884no—
471C2×C4—2448no—
472C2×C4—2442no—
473C2×C4—2444no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2442no—
477C2×C4#1451974712yes—
478C2×C4#1451974712yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C22×D24411yescharacteristic
481C2×C4C2×#1175822212yes—
482C2×C4#1451982412yes—
483C8—42no—
484C8—2442no—
485C22—24no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22C2×#1451974711yescharacteristic
491C22#1684042112yes—
492C22#137×D6111yescharacteristic
493C22#1684055811yescharacteristic
494C22#1684057212yes—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—82no—
499C4—1222no—
500C4—4882no—
501C4—2441no—
502C4—2442no—
503C4—2441no—
504C4—2441no—
505C4—2442no—
506C4C2×#1451976511yescharacteristic
507C4#1684033811yescharacteristic
508C4#1684042112yes—
509C2—44no—
510C2#2144947711yescharacteristic
511C2#2145508011yescharacteristic
512C2#2145509711yescharacteristic
513C2#2145860112yes—
514C2#2145864712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1762D61
hypocenterC61#25945
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16840131C2semidirect
4. #21451034C2semidirect
5. #21446969C2semidirect
6. #21446959C2semidirect
7. #21446776C2semidirect
8. #21446759C2semidirect
304. C61#25945semidirect

Maximal subgroups

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

subgroupindex
C2×#168404712
C2×#168401312
C2×#168399672
#214510342
#214469692
#214469592
#214467762
#214467592
#214461512
C61×#17622
#2594561