Finite Group Databasebeta

← #29736628

Subgroups — #29736628

Order 15616.

1446 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
1C2×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840110C211yesmaximal, characteristic, complemented
3C2×#16840109C211yesmaximal, characteristic, complemented
4#21454165C211yesmaximal, characteristic
5#21446954C214yesmaximal, complemented
6#21446779C211yesmaximal, characteristic, complemented
7#21446135C214yesmaximal, complemented
8C4×#14519689C211yesmaximal, characteristic, complemented
9C61×#1990C211yesmaximal, characteristic, complemented
10C22×Q8×D61—21no—
11C2×#14519777C2211yescharacteristic
12C2×#14519776—21no—
13C2×#14519776C2212yes—
14C2×#14519775—21no—
15C4⋊Q8×D61—24no—
16C4⋊Q8×D61—24no—
17C2×#14519774C2211yescharacteristic
18C2×#14519752—21no—
19C2×#14519752—21no—
20C2×#14519752C2212yes—
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2212yes—
23C2×C42×D61C2211yescharacteristic
24#16840110—24no—
25#16840109—24no—
26C2×#14519699C2214yes—
27C2×#14519697C2214yes—
28C2×#14519689C2212yes—
29C2×C4×#92418C2211yescharacteristic
30#16839663—21no—
31#16839663—21no—
32#16839663C2212yes—
33#16839524C2214yes—
34#16839516C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×C8×C61C2211yescharacteristic
37C61×#384C2211yescharacteristic
38C61×#340C2214yes—
39C22×Dic122—21no—
40C22×Dic122—22no—
41C2×Q8×D61—28no—
42C2×Q8×D61—44no—
43C2×Q8×D61—44no—
44C22×D244—22no—
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48C22×C4×D61—22no—
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777D414yescomplemented
55#14519776—28no—
56#14519776—24no—
57#14519776—24no—
58#14519775—28no—
59#14519774—22no—
60#14519774—22no—
61#14519752—28no—
62#14519752—44no—
63#14519752—44no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C4⋊C4×D61—28no—
69C4⋊C4×D61—24no—
70C4⋊C4×D61—24no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×#11758198—22no—
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61D411yescharacteristic
76C2×C4×Dic61C2311yescharacteristic
77C42×D61—22no—
78C42×D61—22no—
79#14519699—28no—
80#14519697—28no—
81#14519689—24no—
82C4×#92418—22no—
83C22×Q8×C61—21no—
84C22×C8×C61C2312yes—
85C4⋊Q8×C61—22no—
86C4⋊Q8×C61D414yescomplemented
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2312yes—
91C2×C4⋊C4×C61C2312yes—
92C2×C42×C61C2311yescharacteristic
93C4×C8×C61—22no—
94C23×D61—22no—
95C23×D61C2×D411yescharacteristic
96C2×Q8×C61—24no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2411yescharacteristic
104C22×Dic61—22no—
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108Q8×D61—432no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—24no—
112C2×D244—24no—
113C2×D244—44no—
114C2×D244—44no—
115C2×D244—44no—
116C2×D244—42no—
117C2×D244—42no—
118C2×D244—42no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—42no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—22no—
129C2×C4×D61—28no—
130C2×C4×D61—44no—
131C2×C4×D61—44no—
132C2×C4×D61D814yes—
133C2×C4×D61SD1614yes—
134C2×C4×D61#4114yes—
135C2×Dic122—28no—
136C2×Dic122—24no—
137C2×Dic122—44no—
138C2×Dic122—42no—
139C2×Dic122—42no—
140C2×Dic122—44no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61C2×D414yes—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152#11758198—44no—
153#11758198—44no—
154#11758198—44no—
155C4⋊Dic61—24no—
156C4⋊Dic61—22no—
157C4⋊Dic61C2×D414yes—
158#11758196—28no—
159#11758196—28no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61C22×D411yescharacteristic
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61#12412yes—
180C22×D61C2×D812yes—
181C22×D61C2×SD1612yes—
182Q8×C61—48no—
183C2×C4×C61—24no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61#12012yes—
190C2×C4×C61#12012yes—
191C2×C4×C61C2×D812yes—
192C2×C4×C61C2×SD1612yes—
193C2×C4×C61AGL(1,ℤ/8)12yes—
194C2×C4×C61#13712yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4112yes—
198C2×C4×C612−1+412yes—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—24no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61#12412yes—
209C2×Dic61C2×D812yes—
210C2×Dic61C2×SD1612yes—
211D244—44no—
212D244—44no—
213D244—88no—
214D244—84no—
215D244—84no—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—28no—
219C4×D61—28no—
220C4×D61—416no—
221Dic122—416no—
222Dic122—48no—
223C8×C61—42no—
224#92418—42no—
225#25903—611nomaximal, Sylow
226C22×C61C2×#12011yescharacteristic
227C22×C61C2×#12411yescharacteristic
228C22×C61#53611yescharacteristic
229C22×C61C22×D811yescharacteristic
230C22×C61C22×SD1611yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61C2×#13711yescharacteristic
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C2×D61#48614yescomplemented
247C4×C61—44no—
248C4×C61—82no—
249C4×C61#44714yes—
250C4×C61#44814yes—
251C4×C61#54812yes—
252C4×C61#54812yes—
253Dic61—22no—
254Dic61—22no—
255Dic61—22no—
256Dic61—44no—
257C22×C4⋊Q8—611no—
258C2×#486—611no—
259C2×#486—611no—
260#2856—611no—
261#2023—614no—
262#1990—611no—
263#1990D6111yescharacteristic, complemented
264#1874—614no—
265C4×C22⋊C8—611no—
266C2×C61#243611yescharacteristic
267C2×C61C2×#44711yescharacteristic
268C2×C61C2×#44811yescharacteristic
269C2×C61C2×#48611yescharacteristic
270C2×C61#314111yescharacteristic
271C2×C61#322611yescharacteristic
272C2×C61#322711yescharacteristic
273D61—24no—
274D61—162no—
275C23×Q8—1221no—
276C2×C4⋊Q8—612no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8—1221no—
279C2×C4⋊Q8—1221no—
280C2×C4⋊Q8—1224no—
281C2×C4⋊Q8—1224no—
282C2×C4⋊Q8C2×D6111yescharacteristic
283C2×#127—611no—
284C2×#121—612no—
285C2×#121—1221no—
286C2×#121—1221no—
287C22×C4⋊C4—612no—
288C22×C4⋊C4—1221no—
289C22×C42—611no—
290#486—1224no—
291#486—1224no—
292C2×#102—614no—
293C2×#102—614no—
294C2×C22⋊C8—612no—
295C2×C4×C8—611no—
296C2×C4×C8C2×D6111yescharacteristic
297#390—612no—
298#390—1221no—
299#390—1221no—
300#384C2×D6111yescharacteristic
301#340—614no—
302#340C2×D6114yes—
303#336—612no—
304C61#2590311yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C22×Q8—1222no—
307C22×Q8—1228no—
308C22×Q8—1221no—
309C22×Q8—2444no—
310C22×Q8—2444no—
311C22×D4—612no—
312C22×D4—1222no—
313C22×D4—1221no—
314C22×D4—1221no—
315C23×C4—611no—
316C23×C4—611no—
317C23×C4—611no—
318C23×C4—1222no—
319C22×C8—612no—
320C22×C8C22×D6112yes—
321C4⋊Q8—22no—
322C4⋊Q8—614no—
323C4⋊Q8—1224no—
324C4⋊Q8—1228no—
325C4⋊Q8—1228no—
326C4⋊Q8—1224no—
327C4⋊Q8—1222no—
328C4⋊Q8D24414yescomplemented
329#127—1222no—
330#127—1222no—
331#121—1228no—
332#121—2444no—
333#121—2444no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1224no—
339C2×C4⋊C4—1228no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1224no—
344C2×C4⋊C4C22×D6112yes—
345C2×C4⋊C4C22×D6112yes—
346C2×C22⋊C4—1222no—
347C2×C22⋊C4—1222no—
348C2×C22⋊C4—1222no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—1222no—
353C2×C42—1222no—
354C2×C42C22×D6111yescharacteristic
355#102—1228no—
356#102—1228no—
357C22⋊C8—1224no—
358C4×C8—22no—
359C4×C8—1222no—
360C24—611no—
361C24—1222no—
362C2×Q8—24no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—1228no—
366C2×Q8—1224no—
367C2×Q8—2444no—
368C2×Q8—2444no—
369C2×Q8—24432no—
370C2×Q8—2442no—
371C2×Q8—2442no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—1224no—
375C2×D4—1224no—
376C2×D4—2444no—
377C2×D4—2444no—
378C2×D4—2444no—
379C2×D4—2442no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4—2442no—
384C2×D4—2442no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—614no—
391C22×C4—614no—
392C22×C4—614no—
393C22×C4—1228no—
394C22×C4—1222no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—2444no—
403C22×C4—2444no—
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—614no—
419C4⋊C4—1228no—
420C4⋊C4—1228no—
421C4⋊C4—1222no—
422C4⋊C4—1224no—
423C4⋊C4C2×D24414yes—
424C22⋊C4—2444no—
425C22⋊C4—2444no—
426C22⋊C4—2444no—
427C42—1221no—
428C42—1221no—
429C42—1222no—
430C42—1222no—
431C42C2×D24412yes—
432C42D4×D6112yes—
433C23—612no—
434C23—612no—
435C23—612no—
436C23—1221no—
437C23—1221no—
438C23—1221no—
439C23—1221no—
440C23—2442no—
441C23—2442no—
442C23—2442no—
443C23—2442no—
444C23—2442no—
445C23—2442no—
446C23—2442no—
447C23C2×D4×D6111yescharacteristic
448Q8—48no—
449Q8—24416no—
450Q8—2448no—
451D4—4888no—
452D4—4884no—
453D4—4884no—
454D4—2444no—
455D4—2444no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—612no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1221no—
470C2×C4—1221no—
471C2×C4—1228no—
472C2×C4—1228no—
473C2×C4—1224no—
474C2×C4—1224no—
475C2×C4—1224no—
476C2×C4—24416no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4#1451974712yes—
480C2×C4#1451974712yes—
481C2×C4D8×D6112yes—
482C2×C4SD16×D6112yes—
483C2×C4#1451980412yes—
484C2×C4#1451980912yes—
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C22×D24411yescharacteristic
487C2×C4#41×D6112yes—
488C2×C4#1451982512yes—
489C8—42no—
490C8—2442no—
491C22—614no—
492C22—1222no—
493C22—1222no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22—1222no—
498C22—4882no—
499C22—4882no—
500C22—4882no—
501C22—4882no—
502C22—4882no—
503C22—4882no—
504C22—4882no—
505C22C2×#1451974711yescharacteristic
506C22#124×D6111yescharacteristic
507C22#1684033711yescharacteristic
508C22C2×D8×D6111yescharacteristic
509C22C2×SD16×D6111yescharacteristic
510C22C2×#1451980411yescharacteristic
511C22C2×#1451980911yescharacteristic
512C4—44no—
513C4—82no—
514C4—1222no—
515C4—1222no—
516C4—1222no—
517C4—2444no—
518C4#1683989714yes—
519C4#1683990214yes—
520C4#1684042212yes—
521C4#1684042412yes—
522C2—1224no—
523C2—9762no—
524C2#2144947611yescharacteristic
525C2C2×#1683989711yescharacteristic
526C2C2×#1683990211yescharacteristic
527C2#486×D6111yescharacteristic
528C2#2145621411yescharacteristic
529C2#2145694111yescharacteristic
530C2#2145694811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radical#21454165C2
hypercenter#1990D61
hypocenterC61#25903
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
2. C2×#16840110C2semidirect
3. C2×#16840109C2semidirect
5. #21446954C2semidirect
6. #21446779C2semidirect
7. #21446135C2semidirect
8. C4×#14519689C2semidirect
9. C61×#1990C2semidirect
54. #14519777D4semidirect
86. C4⋊Q8×C61D4semidirect
246. C2×D61#486semidirect
263. #1990D61semidirect
304. C61#25903semidirect
328. C4⋊Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168401102
C2×#168401092
#214541652
#214469542
#214467792
#214461352
C4×#145196892
C61×#19902
#2590361