Finite Group Databasebeta

← #29656306

Subgroups — #29656306

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456193C211yesmaximal, characteristic, complemented
2#21455949C211yesmaximal, characteristic, complemented
3#21455918C211yesmaximal, characteristic, complemented
4#21445019C214yesmaximal, complemented
5#21445012C214yesmaximal, complemented
6#21444533C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic, complemented
9C2×C61×#550C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840126—22no—
12#16840108C2212yescomplemented
13#16840108C2212yescomplemented
14#16840097—21no—
15#16840092—21no—
16#16840047C2212yescomplemented
17#16840042C2212yescomplemented
18#16839972C2212yescomplemented
19#16839967C2212yescomplemented
20#16839918—21no—
21#16839913—21no—
22#16839867—22no—
23#16839862—22no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839767—22no—
27C4×#11758194C2211yescharacteristic
28#16839506C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C22×#41×C61—21no—
32C61×#550—22no—
33C61×#550—22no—
34C61×#550—22no—
35C61×#550C2214yes—
36C2×#127×C61—21no—
37C2×#127×C61C2211yescharacteristic
38C2×#121×C61—22no—
39C2×#121×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758202—22no—
44C2×#11758200—22no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519708—22no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—22no—
59#14519700—22no—
60#14519700—22no—
61#14519698—24no—
62#14519698—24no—
63#14519698—22no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688—21no—
68#14519688—21no—
69C4×#92418C2312yes—
70C2×#41×C61—24no—
71C2×#41×C61—24no—
72C2×#41×C61—48no—
73C22×Q8×C61D411yescharacteristic
74C22×D4×C61—22no—
75C22×D4×C61—22no—
76C22×D4×C61—21no—
77C22×D4×C61—21no—
78C22×D4×C61D411yescharacteristic
79C22×D4×C61D412yes—
80C23×C4×C61—22no—
81C23×C4×C61—21no—
82#127×C61—24no—
83#127×C61—24no—
84#127×C61—21no—
85#127×C61—21no—
86#127×C61D412yes—
87#127×C61C2312yes—
88#121×C61—24no—
89#121×C61—28no—
90#121×C61—28no—
91#121×C61—24no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61D412yes—
95C4×Q8×C61C2312yes—
96C4×D4×C61—24no—
97C4×D4×C61—24no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61D412yes—
101C4×D4×C61C2312yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—22no—
110C24×C61—21no—
111#41×C61—416no—
112#41×C61—48no—
113#41×C61—48no—
114C2×Q8×C61—24no—
115C2×Q8×C61—21no—
116C2×Q8×C61C2×D412yes—
117C2×D4×C61—28no—
118C2×D4×C61—28no—
119C2×D4×C61—21no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—42no—
126C2×D4×C61—44no—
127C2×D4×C61—48no—
128C2×D4×C61—44no—
129C2×D4×C61—44no—
130C2×D4×C61—84no—
131C2×D4×C61C2×D412yes—
132C2×D4×C61C2×D414yes—
133C22×C4×C61—21no—
134C22×C4×C61—28no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—48no—
140C22×C4×C61—44no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×Dic122—21no—
145C2×Dic122—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2411yescharacteristic
161C22⋊Dic61—41no—
162#11758202—44no—
163#11758200—44no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168#11758196—41no—
169#11758196—41no—
170C4×Dic61—41no—
171#11758194—42no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—24no—
177C23×C61—44no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—44no—
181C23×C61—44no—
182C23×C61—42no—
183C23×C61#12011yescharacteristic
184Q8×C61—24no—
185Q8×C61—22no—
186Q8×C61AGL(1,ℤ/8)14yes—
187D4×C61—22no—
188D4×C61—22no—
189D4×C61—44no—
190D4×C61—42no—
191D4×C61—44no—
192D4×C61—416no—
193D4×C61—44no—
194D4×C61—44no—
195D4×C61—44no—
196D4×C61—82no—
197D4×C61—88no—
198C2×C4×C61—24no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—416no—
206C2×C4×C61—44no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—44no—
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×Dic61—41no—
217C2×Dic61—41no—
218Dic122—42no—
219Dic122—42no—
220#92418—42no—
221#92418—42no—
222#92418—42no—
223#18550—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—44no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—42no—
232C22×C61—88no—
233C22×C61—84no—
234C22×C61#45512yes—
235C22×C61C2×#12011yescharacteristic
236C4×C61—24no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61C2×AGL(1,ℤ/8)12yes—
244C4×C61#57512yes—
245Dic61—81no—
246Dic61—81no—
247C2×#550D6111yescharacteristic
248#3138—611no—
249#3113—611no—
250#3110—611no—
251#1642—614no—
252#1641—614no—
253#1555—611no—
254#1491—611no—
255#1490—611no—
256C2×C61—22no—
257C2×C61—82no—
258C2×C61—84no—
259C2×C61#300311yescharacteristic
260C2×C61#300511yescharacteristic
261C2×C61C2×#45511yescharacteristic
262C22×#41—21no—
263#550—22no—
264#550—22no—
265#550—22no—
266#550C2×D6114yes—
267#533—611no—
268C2×#127—21no—
269C2×#127C2×D6111yescharacteristic
270C2×#121—22no—
271C2×#121—21no—
272C2×C4×Q8C2×D6111yescharacteristic
273C2×C4×D4—21no—
274C2×C4×D4C2×D6111yescharacteristic
275#490—1222no—
276#486—612no—
277#486—612no—
278#484—1221no—
279#483—1221no—
280#475—612no—
281#474—612no—
282#462—612no—
283#461—612no—
284#451—1221no—
285#450—1221no—
286#441—1222no—
287#440—1222no—
288#432—611no—
289#432—611no—
290#418—1222no—
291C4×M16—611no—
292#331—614no—
293Q8⋊C8—614no—
294D4⋊C8—614no—
295C61#1855011yescharacteristic, complemented, Sylow
296C2×#41—24no—
297C2×#41—24no—
298C2×#41—48no—
299C22×Q8C22⋊D6111yescharacteristic
300C22×D4—22no—
301C22×D4—22no—
302C22×D4—21no—
303C22×D4—21no—
304C22×D4C22⋊D6111yescharacteristic
305C22×D4C22⋊D6112yes—
306C23×C4—22no—
307C23×C4—21no—
308C2×Q16—1222no—
309C2×SD16—1222no—
310C2×M16—1221no—
311C4⋊Q8—611no—
312C4⋊Q8—611no—
313#127—24no—
314#127—24no—
315#127—21no—
316#127—21no—
317#127C22⋊D6112yes—
318#127C22×D6112yes—
319#125—1221no—
320C22⋊Q8—1221no—
321C22⋊Q8—1221no—
322#121—24no—
323#121—28no—
324#121—28no—
325#121—24no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—1221no—
329C4×Q8—1221no—
330C4×Q8C22⋊D6112yes—
331C4×Q8C22×D6112yes—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4C22⋊D6112yes—
337C4×D4C22×D6112yes—
338#117—1221no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4C22⋊D6111yescharacteristic
341C2×C4⋊C4C22⋊D6111yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C42—21no—
345C2×C42C22×D6111yescharacteristic
346#107—1222no—
347#106—1222no—
348C4⋊C8—612no—
349C4⋊C8—612no—
350#103—1222no—
351#103—1222no—
352#103—1222no—
353#102—1224no—
354#102—1224no—
355#102—1222no—
356C22⋊C8—1221no—
357C22⋊C8—1221no—
358#97—1221no—
359#97—1221no—
360#97—1221no—
361C4×C8—612no—
362C24—22no—
363C24—21no—
364#41—416no—
365#41—48no—
366#41—48no—
367C2×Q8—24no—
368C2×Q8—21no—
369C2×Q8—1221no—
370C2×Q8—1221no—
371C2×Q8C2×C22⋊D6112yes—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—84no—
386C2×D4C2×C22⋊D6112yes—
387C2×D4C2×C22⋊D6114yes—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399Q16—2444no—
400SD16—2444no—
401M16—2442no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1222no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4C2×C22⋊D6112yes—
416C4⋊C4C2×C22⋊D6112yes—
417C22⋊C4—28no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—2441no—
421C42—22no—
422C42—22no—
423C42—2441no—
424C42C2×C22⋊D6111yescharacteristic
425C42C2×C22⋊D6111yescharacteristic
426C42C2×C22⋊D6111yescharacteristic
427C42C23×D6111yescharacteristic
428C23—24no—
429C23—22no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23#1451974611yescharacteristic
437Q8—24no—
438Q8—22no—
439Q8—2442no—
440Q8—2442no—
441Q8#1451980314yes—
442D4—22no—
443D4—22no—
444D4—44no—
445D4—42no—
446D4—44no—
447D4—416no—
448D4—44no—
449D4—44no—
450D4—44no—
451D4—82no—
452D4—88no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—416no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4#1451974611yescharacteristic
468C2×C4#1451974611yescharacteristic
469C2×C4#1451974611yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C2×C4C22×C22⋊D6111yescharacteristic
472C2×C4C22×C22⋊D6111yescharacteristic
473C8—2442no—
474C8—2442no—
475C8—2442no—
476C22—21no—
477C22—21no—
478C22—44no—
479C22—42no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—88no—
485C22—84no—
486C22#1683994012yes—
487C22C2×#1451974611yescharacteristic
488C4—21no—
489C4—21no—
490C4—24no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—4881no—
496C4—4881no—
497C4C2×#1451980312yes—
498C4#1684056012yes—
499C2—22no—
500C2—84no—
501C2—82no—
502C2#2145504611yescharacteristic
503C2#2145506111yescharacteristic
504C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#550C2
socleC22×C61C2×#120
hypercenterC2×#550D61
hypocenterC61#18550
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #21456193C2semidirect
2. #21455949C2semidirect
3. #21455918C2semidirect
4. #21445019C2semidirect
5. #21445012C2semidirect
6. #21444533C2semidirect
7. #21444266C2semidirect
8. #21444262C2semidirect
12. #16840108C22semidirect
13. #16840108C22semidirect
16. #16840047C22semidirect
17. #16840042C22semidirect
18. #16839972C22semidirect
19. #16839967C22semidirect
28. #16839506C22semidirect
29. Q8⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
295. C61#18550semidirect

Maximal subgroups

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

subgroupindex
#214561932
#214559492
#214559182
#214450192
#214450122
#214445332
#214442662
#214442622
C2×C61×#5502
#1855061