Finite Group Databasebeta

← #29736811

Subgroups — #29736811

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456208C211yesmaximal, characteristic, complemented
2#21455313C211yesmaximal, characteristic
3#21454192C211yesmaximal, characteristic, complemented
4#21446977C212yesmaximal, complemented
5#21446972C212yesmaximal, complemented
6#21446813C211yesmaximal, characteristic, complemented
7#21446797C211yesmaximal, characteristic, complemented
8#21446195C212yesmaximal
9#21446188C212yesmaximal, complemented
10C4×#14519704C211yesmaximal, characteristic, complemented
11C61×#3504C211yesmaximal, characteristic, complemented
12C2×#14519807C2211yescharacteristic
13C2×#14519802C2211yescharacteristic
14C2×#14519765C2211yescharacteristic
15#16840124—22no—
16#16840119—21no—
17#16840119—21no—
18#16839984—22no—
19#16839977—22no—
20#16839786C2211yescharacteristic
21C2×#14519704C2212yes—
22#16839770C2212yes—
23#16839770C2211yescharacteristic
24#16839766C2212yes—
25#16839766C2211yescharacteristic
26C4×#11758194C2211yescharacteristic
27#16839667C2212yes—
28#16839663C2212yes—
29#16839652—24no—
30C42×Dic61C2211yescharacteristic
31#16839531C2212yes—
32#16839531C2212yes—
33#16839521C2212yes—
34#16839521C2212yes—
35#16839521C2212yes—
36C61×#557—22no—
37C61×#554—22no—
38C61×#554—24no—
39C61×#544—22no—
40C61×#544C2212yes—
41C61×#533—21no—
42C61×#533C2211yescharacteristic
43C2×#125×C61—21no—
44C2×#125×C61C2211yescharacteristic
45C61×#520C2212yes—
46C4×#41×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49#14519807—24no—
50#14519802—24no—
51C2×#11758202—21no—
52C2×#11758200—21no—
53C2×#11758201—21no—
54C2×#11758199—21no—
55C2×#11758194C2312yes—
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2311yescharacteristic
58#14519765—22no—
59#14519765—21no—
60#14519765—21no—
61C2×#11758198—22no—
62C2×#11758198—24no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61C2312yes—
65C4⋊#92418—22no—
66#14519704—24no—
67#14519700—22no—
68#14519700—22no—
69#14519700—22no—
70#14519697—22no—
71#14519697—22no—
72#14519697—22no—
73#14519688—21no—
74C4×#92418—21no—
75#14519684—24no—
76C2×#41×C61C2311yescharacteristic
77C4⋊Q8×C61—24no—
78C4⋊Q8×C61—22no—
79C4⋊Q8×C61—22no—
80C4⋊Q8×C61D412yes—
81#125×C61—24no—
82#125×C61—24no—
83#125×C61—22no—
84#125×C61—22no—
85#125×C61—21no—
86#125×C61—21no—
87#125×C61—22no—
88#125×C61D411yescharacteristic
89#125×C61D411yescharacteristic
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—24no—
92C22⋊Q8×C61—24no—
93C22⋊Q8×C61—22no—
94C22⋊Q8×C61—24no—
95C4×Q8×C61—24no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61D412yes—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—24no—
102C4×D4×C61D412yes—
103#117×C61—22no—
104#117×C61—21no—
105#117×C61C2312yes—
106#117×C61C2311yescharacteristic
107C2×C4⋊C4×C61—24no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61D414yes—
110C2×C4⋊C4×C61C2312yes—
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C23×D61—21no—
114#41×C61—22no—
115#41×C61#4114yes—
116C2×Q8×C61—24no—
117C2×Q8×C61—22no—
118C2×Q8×C61C2×D411yescharacteristic
119C2×D4×C61—21no—
120C2×D4×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61C2×D414yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—22no—
128C22×Dic61—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—22no—
132C2×D244—22no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—21no—
155C22⋊C4×C61—22no—
156C42×C61—24no—
157C42×C61—22no—
158C42×C61—21no—
159C42×C61—22no—
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D412yes—
163#11758202—42no—
164#11758201—42no—
165#11758200—42no—
166#11758199—42no—
167#11758198—48no—
168#11758198—42no—
169#11758198—42no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172C4×Dic61—44no—
173#11758194—22no—
174#11758194—22no—
175#11758194—22no—
176#11758194—22no—
177C2×#92418—22no—
178C2×#92418—21no—
179C2×#92418—21no—
180C23×C61—21no—
181C23×C61C22×D411yescharacteristic
182C22×D61—44no—
183C22×D61—41no—
184C22×D61—42no—
185Q8×C61—21no—
186Q8×C61—82no—
187Q8×C61—84no—
188Q8×C61#12412yes—
189D4×C61—24no—
190D4×C61—21no—
191D4×C61#12412yes—
192C2×C4×C61—21no—
193C2×C4×C61—24no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—42no—
199C2×C4×C61—44no—
200C2×C4×C61—82no—
201C2×C4×C61—84no—
202C2×C4×C61#12012yes—
203C2×C4×C61#12012yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4112yes—
207C2×C4×C612−1+412yes—
208C2×Dic61—24no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213D244—41no—
214D244—41no—
215D244—42no—
216Dic122—41no—
217Dic122—41no—
218Dic122—42no—
219#92418—41no—
220#92418—41no—
221#92418—42no—
222#25925—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—42no—
226C22×C61C2×#12011yescharacteristic
227C22×C61#54812yes—
228C2×D61—84no—
229C2×D61—81no—
230C2×D61—82no—
231C4×C61—22no—
232C4×C61—22no—
233C4×C61—22no—
234C4×C61—44no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61C2×#12411yescharacteristic
239C4×C61#53611yescharacteristic
240C4×C61#54812yes—
241Dic61—44no—
242Dic61—81no—
243#3504D6111yescharacteristic, complemented
244#3140—611no—
245#3040—611no—
246#2861—611no—
247#2028—612no—
248#2027—612no—
249#1998—611no—
250#1994—611no—
251#1883—612no—
252#1882—612no—
253C4×#104—611no—
254C2×C61—22no—
255C2×C61—42no—
256C2×C61#243611yescharacteristic
257C2×C61#301812yes—
258D61—161no—
259C2×#137—611no—
260C2×AGL(1,ℤ/8)—611no—
261#557—22no—
262#554—22no—
263#554—24no—
264#544—22no—
265#544C2×D6112yes—
266#533—21no—
267#533C2×D6111yescharacteristic
268C2×#125—21no—
269C2×#125C2×D6111yescharacteristic
270C2×#124—611no—
271#520C2×D6112yes—
272C4×#41C2×D6111yescharacteristic
273#489—1222no—
274#488—1221no—
275#488—1221no—
276#464—1222no—
277#463—1222no—
278#423—611no—
279C2×#104—612no—
280#419—612no—
281#419—611no—
282#418—612no—
283#418—611no—
284C4×M16—611no—
285#391—612no—
286#390—612no—
287#388—1224no—
288C43—611no—
289#343—612no—
290#343—612no—
291#339—612no—
292#339—612no—
293#339—612no—
294C61#2592511yescharacteristic, complemented, Sylow
295C2×#41C22×D6111yescharacteristic
296C22×Q8—611no—
297C22×D4—611no—
298#137—1224no—
299AGL(1,ℤ/8)—1224no—
300C2×Q16—1221no—
301C2×SD16—1221no—
302C2×SD16—1221no—
303C2×D8—1221no—
304C2×M16—611no—
305C2×M16—611no—
306C2×M16—612no—
307C4⋊Q8—24no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8C22⋊D6112yes—
311#125—24no—
312#125—24no—
313#125—22no—
314#125—22no—
315#125—21no—
316#125—21no—
317#125—22no—
318#125C22⋊D6111yescharacteristic
319#125C22⋊D6111yescharacteristic
320#124—1222no—
321#124—1221no—
322#124—1221no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—22no—
327C22⋊Q8—24no—
328C4×Q8—24no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8C22⋊D6112yes—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—24no—
335C4×D4C22⋊D6112yes—
336#117—22no—
337#117—21no—
338#117C22×D6111yescharacteristic
339#117C22×D6112yes—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4C22⋊D6114yes—
343C2×C4⋊C4C22×D6112yes—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1224no—
346C2×C42—21no—
347C2×C42—612no—
348C2×C42—1222no—
349C2×C42C22×D6111yescharacteristic
350C4⋊C8—1222no—
351#104—1224no—
352#103—1222no—
353#103—1222no—
354#103—1222no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#97—1221no—
359C4×C8—1221no—
360#95—1224no—
361C24—1221no—
362#41—22no—
363#41#1175822614yes—
364C2×Q8—24no—
365C2×Q8—22no—
366C2×Q8—1221no—
367C2×Q8—1221no—
368C2×Q8—1222no—
369C2×Q8—1222no—
370C2×Q8C2×C22⋊D6111yescharacteristic
371C2×D4—21no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—1222no—
375C2×D4—1222no—
376C2×D4C2×C22⋊D6111yescharacteristic
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—1222no—
381C22×C4—1221no—
382C22×C4C2×C22⋊D6114yes—
383C22×C4C2×C22⋊D6112yes—
384C22×C4C23×D6111yescharacteristic
385Q16—2442no—
386SD16—2442no—
387SD16—2442no—
388D8—2442no—
389M16—1222no—
390M16—1222no—
391M16—1222no—
392M16—1222no—
393C2×C8—1222no—
394C2×C8—1221no—
395C2×C8—1221no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4C2×C22⋊D6112yes—
409C4⋊C4C2×C22⋊D6112yes—
410C4⋊C4C2×C22⋊D6112yes—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C22⋊C4—2448no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C42—24no—
419C42—21no—
420C42—22no—
421C42—22no—
422C42—1222no—
423C42—1222no—
424C42—2444no—
425C42C2×C22⋊D6111yescharacteristic
426C42C2×C22⋊D6111yescharacteristic
427C42C2×C22⋊D6112yes—
428C23—21no—
429C23—2444no—
430C23—2441no—
431C23—2442no—
432C23C22×C22⋊D6111yescharacteristic
433Q8—21no—
434Q8—82no—
435Q8—84no—
436Q8—2441no—
437Q8—2441no—
438Q8—2442no—
439Q8#1451976612yes—
440D4—24no—
441D4—21no—
442D4—2441no—
443D4—2441no—
444D4—2442no—
445D4#1451976612yes—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—82no—
455C2×C4—84no—
456C2×C4—1224no—
457C2×C4—2441no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4#1451974612yes—
462C2×C4#1451974612yes—
463C2×C4C22×C22⋊D6111yescharacteristic
464C2×C4C22×C22⋊D6111yescharacteristic
465C2×C4C2×#1175822612yes—
466C2×C4#1451982512yes—
467C8—2441no—
468C8—2441no—
469C8—2442no—
470C22—22no—
471C22—22no—
472C22—42no—
473C22—4884no—
474C22—4881no—
475C22—4882no—
476C22C2×#1451974611yescharacteristic
477C22#1684042312yes—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—44no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—4881no—
486C4—2444no—
487C4C2×#1451976611yescharacteristic
488C4#1684033611yescharacteristic
489C4#1684042312yes—
490C2—22no—
491C2—42no—
492C2—9761no—
493C2#2144947311yescharacteristic
494C2#2145516812yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3504C2
socleC22×C61C2×#120
prime radicalC2×#14519802C22
hypercenter#3504D61
hypocenterC61#25925
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3504

Complements

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

NH ≅ G/Ntype
1. #21456208C2semidirect
3. #21454192C2semidirect
4. #21446977C2semidirect
5. #21446972C2semidirect
6. #21446813C2semidirect
7. #21446797C2semidirect
9. #21446188C2semidirect
10. C4×#14519704C2semidirect
11. C61×#3504C2semidirect
243. #3504D61semidirect
294. C61#25925semidirect

Maximal subgroups

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

subgroupindex
#214562082
#214553132
#214541922
#214469772
#214469722
#214468132
#214467972
#214461952
#214461882
C4×#145197042
C61×#35042
#2592561