Finite Group Databasebeta

← #29736148

Subgroups — #29736148

Order 15616.

1076 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×#16840468C211yesmaximal, characteristic
2C2×#16840172C211yesmaximal, characteristic, complemented
3C2×#16840053C211yesmaximal, characteristic
4#21450807C211yesmaximal, characteristic, complemented
5#21447156C212yesmaximal, complemented
6#21447151C212yesmaximal, complemented
7#21446758C211yesmaximal, characteristic, complemented
8#21446081C212yesmaximal
9#21446067C212yesmaximal
10#21445647C211yesmaximal, characteristic
11C61×#1989C211yesmaximal, characteristic, complemented
12#16840468—24no—
13#16840468—24no—
14C2×#14519777—21no—
15C2×#14519777C2211yescharacteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519757—21no—
19C2×#14519755—21no—
20C2×#14519755—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519739C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25C2×C4⋊C4×D61—21no—
26#16840172—24no—
27#16840053—24no—
28C2×#14519710C2212yes—
29C2×#14519709C2212yes—
30C2×#14519708C2212yes—
31C2×#14519699C2212yes—
32C2×#14519697C2212yes—
33C2×#14519688C2211yescharacteristic
34#16839694—21no—
35#16839694—21no—
36#16839668—21no—
37#16839661C2212yes—
38#16839661C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41#16839516C2212yes—
42C2×#125×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×C4⋊C4×C61C2211yescharacteristic
45C61×#340C2212yes—
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122—21no—
49C22×Dic122D411yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×#92418C2312yes—
55#14519777—22no—
56#14519777—22no—
57#14519777—44no—
58#14519776—22no—
59#14519776—22no—
60#14519768—22no—
61#14519768—22no—
62#14519757—28no—
63#14519755—28no—
64#14519755—28no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C4×D244—22no—
68C4×D244D414yes—
69#14519739—22no—
70#14519739—22no—
71#14519739—22no—
72#14519739—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C4⋊C4×D61—28no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—24no—
91#14519709—24no—
92#14519708—24no—
93#14519699—24no—
94#14519697Q818yescomplemented
95#14519688—22no—
96#14519684—22no—
97#14519684—22no—
98C22×C8×C61C2312yes—
99#125×C61—22no—
100#125×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2312yes—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—22no—
109#95×C61—21no—
110C23×D61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×D244—21no—
124C2×D244—22no—
125C2×D244C2×D412yes—
126C2×D244C2×Q814yes—
127C2×C4×D61—24no—
128C2×C4×D61—24no—
129C2×C4×D61—24no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—82no—
133C2×C4×D61—82no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—24no—
137C2×Dic122—41no—
138C2×Dic122—42no—
139C2×Dic122—42no—
140C2×Dic122—42no—
141C2×Dic122—82no—
142C2×C8×C61—22no—
143C2×C8×C61—24no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×Q814yes—
152C42×C61—42no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155#11758198—24no—
156#11758198—24no—
157#11758198—24no—
158C4⋊Dic61—24no—
159C4⋊Dic61—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—44no—
164C4⋊Dic61—42no—
165C4⋊Dic61—44no—
166C4⋊Dic61C2×D412yes—
167#11758196—24no—
168#11758196—24no—
169#11758196—22no—
170#11758196—22no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418C2×Q814yes—
178C23×C61C22×D411yescharacteristic
179C22×D61—22no—
180C22×D61—24no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61#12012yes—
196C2×C4×C61#12012yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×Q812yes—
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—21no—
208C2×Dic61—21no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61AGL(1,ℤ/8)12yes—
216D244—22no—
217D244—24no—
218D244C4⋊Q814yescomplemented
219C4×D61—48no—
220C4×D61—84no—
221C4×D61—84no—
222Dic122—44no—
223Dic122—84no—
224Dic122—84no—
225C8×C61—42no—
226#92418—24no—
227#25851—611nomaximal, Sylow
228C22×C61C2×#12011yescharacteristic
229C22×C61D4×Q812yes—
230C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C61#57511yescharacteristic
232C22×C61#57712yes—
233C2×D61—24no—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—42no—
237C2×D61—42no—
238C4×C61—22no—
239C4×C61—22no—
240C4×C61—44no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61C2×C4⋊Q811yescharacteristic
244C4×C61#53611yescharacteristic
245C4×C61D4×Q812yes—
246Dic61—22no—
247Dic61—41no—
248Dic61—41no—
249Dic61—42no—
250Dic61—41no—
251Dic61—82no—
252C2×#554—611no—
253C2×#501—611no—
254C2×#476—611no—
255#2546—611no—
256#2059—612no—
257#2058—612no—
258#1989D6111yescharacteristic, complemented
259#1985—611no—
260#1864—612no—
261#1862—612no—
262#1762—611no—
263C2×C61#243411yescharacteristic
264C2×C61#300611yescharacteristic
265C2×C61#300911yescharacteristic
266C2×C61#338412yes—
267C2×C61#338812yes—
268D61—44no—
269#554—1224no—
270#554—1224no—
271C2×C4⋊Q8—611no—
272C2×C4⋊Q8—611no—
273C2×C4⋊Q8—1221no—
274C2×#125—611no—
275C2×#125C2×D6111yescharacteristic
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×D4—611no—
283C22×C4⋊C4—1221no—
284#501—1224no—
285#476—1224no—
286C2×#107—612no—
287C2×#106—612no—
288C2×#106—612no—
289C2×C4⋊C8C2×D6111yescharacteristic
290C2×#102—612no—
291C2×#102—612no—
292C2×#97—611no—
293#398—1221no—
294#398—1221no—
295#391—1221no—
296#389—612no—
297#389—612no—
298C4×C4⋊C4C2×D6111yescharacteristic
299#340—612no—
300#340C2×D6112yes—
301#340C2×D6112yes—
302#336—612no—
303#336—612no—
304C61#2585111yescharacteristic, complemented, Sylow
305C22×Q8—611no—
306C22×Q8—1221no—
307C22×Q8—1221no—
308C22×D4—611no—
309C23×C4—1221no—
310C23×C4—1221no—
311C23×C4—1221no—
312C22×C8—612no—
313C22×C8C22×D6112yes—
314C4⋊Q8—1222no—
315C4⋊Q8—1222no—
316C4⋊Q8—1222no—
317C4⋊Q8—1222no—
318C4⋊Q8—2444no—
319#125—22no—
320#125—1222no—
321#125—1222no—
322#125D24414yes—
323C22⋊Q8—1228no—
324C22⋊Q8—1228no—
325C22⋊Q8—1228no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×D4—614no—
329C4×D4—1222no—
330C4×D4—1222no—
331C4×D4—1222no—
332C4×D4—1222no—
333C4×D4—1222no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—1228no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6112yes—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C42—21no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42C22×D6111yescharacteristic
358#107—1224no—
359#106—1224no—
360#106—1224no—
361C4⋊C8—22no—
362#102—618no—
363#102—1224no—
364#97—1222no—
365#95—21no—
366#95—1222no—
367#95—1222no—
368C24—1221no—
369C2×Q8—1222no—
370C2×Q8—1224no—
371C2×Q8—1222no—
372C2×Q8—4882no—
373C2×Q8—2441no—
374C2×Q8—2442no—
375C2×Q8—2442no—
376C2×Q8—2442no—
377C2×D4—612no—
378C2×D4—614no—
379C2×D4—1221no—
380C2×D4—1222no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—1224no—
389C22×C4—1224no—
390C22×C4—1221no—
391C22×C4—1224no—
392C22×C4—1222no—
393C22×C4—4882no—
394C22×C4—4882no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4D4×D6111yescharacteristic
398C22×C4D4×D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—614no—
403C2×C8—1222no—
404C2×C8—1222no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—612no—
411C4⋊C4—1224no—
412C4⋊C4—1224no—
413C4⋊C4—1224no—
414C4⋊C4—1221no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—2444no—
421C4⋊C4—2444no—
422C4⋊C4—2442no—
423C4⋊C4C2×D24412yes—
424C4⋊C4C2×D24412yes—
425C4⋊C4Q8×D6114yes—
426C22⋊C4—1224no—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C42—42no—
430C42—1221no—
431C42—1221no—
432C42—1221no—
433C42—1221no—
434C42C2×D24412yes—
435C42D4×D6112yes—
436C23—1222no—
437C23—1224no—
438C23—2441no—
439C23—2441no—
440C23—2442no—
441C23C2×D4×D6111yescharacteristic
442Q8—4884no—
443Q8—4884no—
444Q8—2444no—
445D4—614no—
446D4—1222no—
447D4—1224no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—612no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1222no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—4884no—
471C2×C4—4884no—
472C2×C4—2448no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4#1451974712yes—
478C2×C4#1451974712yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C22×D24411yescharacteristic
481C2×C4C2×Q8×D6112yes—
482C2×C4#1451982412yes—
483C8—42no—
484C8—1224no—
485C22—1224no—
486C22—2444no—
487C22—2444no—
488C22—2442no—
489C22—2442no—
490C22C2×#1451974711yescharacteristic
491C22#1684042512yes—
492C22AGL(1,ℤ/8)×D6111yescharacteristic
493C22#1684056111yescharacteristic
494C22#1684056912yes—
495C4—22no—
496C4—22no—
497C4—44no—
498C4—42no—
499C4—42no—
500C4—1222no—
501C4—4882no—
502C4—2441no—
503C4—2441no—
504C4—2441no—
505C4—2442no—
506C4C4⋊Q8×D6111yescharacteristic
507C4#1684033211yescharacteristic
508C4Q8×D24412yes—
509C2—2444no—
510C2#2144946211yescharacteristic
511C2#2145507211yescharacteristic
512C2#2145509911yescharacteristic
513C2#2145872112yes—
514C2#2145876712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1989D61
hypocenterC61#25851
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#16840172C2semidirect
4. #21450807C2semidirect
5. #21447156C2semidirect
6. #21447151C2semidirect
7. #21446758C2semidirect
11. C61×#1989C2semidirect
94. #14519697Q8semidirect
218. D244C4⋊Q8semidirect
258. #1989D61semidirect
304. C61#25851semidirect

Maximal subgroups

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

subgroupindex
C2×#168404682
C2×#168401722
C2×#168400532
#214508072
#214471562
#214471512
#214467582
#214460812
#214460672
#214456472
C61×#19892
#2585161