Finite Group Databasebeta

← #29665505

Subgroups — #29665505

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456940C211yesmaximal, characteristic, complemented
2#21455274C211yesmaximal, characteristic
3#21454930C211yesmaximal, characteristic, complemented
4#21445265C212yesmaximal, complemented
5#21445258C212yesmaximal, complemented
6#21445211C212yesmaximal, complemented
7#21445206C212yesmaximal, complemented
8#21444595C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic, complemented
10C2×#16839508C211yesmaximal, characteristic, complemented
11C61×#3039C211yesmaximal, characteristic, complemented
12C2×#14519797—21no—
13#16840312C2211yescharacteristic
14#16840165C2212yes—
15#16840110C2212yes—
16#16840075—22no—
17#16840039—21no—
18#16840039—21no—
19#16839984—21no—
20#16839977—21no—
21#16839972C2211yescharacteristic
22#16839967C2211yescharacteristic
23#16839962C2211yescharacteristic
24#16839957C2211yescharacteristic
25Q8×#92418C2212yes—
26D4×#92418—22no—
27D4×#92418C2212yes—
28#16839815—21no—
29#16839815C2211yescharacteristic
30C2×C4⋊#92418—21no—
31C2×C4⋊#92418C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839756—21no—
34#16839514C2212yes—
35#16839514C2212yes—
36#16839509C2212yes—
37#16839508C2212yes—
38#16839508C2212yes—
39#16839505C2212yes—
40C2×#137×C61—21no—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C4×#41×C61C2211yescharacteristic
43C61×#462—21no—
44C61×#462C2212yes—
45C61×#461—21no—
46C61×#461C2212yes—
47C61×#423C2211yescharacteristic
48C61×#419—21no—
49C2×#11758222—21no—
50#14519797—44no—
51C2×#11758202—21no—
52C2×#11758200—21no—
53C2×#11758201—21no—
54C2×#11758199—21no—
55C22×#92418—21no—
56C22×#92418—21no—
57C22×#92418—22no—
58#14519777C2311yescharacteristic
59#14519774C2311yescharacteristic
60#14519765—21no—
61#14519751—21no—
62#14519751—21no—
63C4×D244—21no—
64C4×D244—21no—
65#14519710—22no—
66#14519710—22no—
67C4⋊#92418—21no—
68C4⋊#92418—21no—
69C4⋊#92418—22no—
70C4⋊#92418D411yescharacteristic
71C4⋊#92418D411yescharacteristic
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75#14519700—21no—
76#14519700—21no—
77#14519697—21no—
78#14519697—21no—
79#14519697—22no—
80#14519697—22no—
81C22⋊#92418—22no—
82C22⋊#92418D412yes—
83C4×#92418—22no—
84C4×#92418C2312yes—
85C2×#41×C61D411yescharacteristic
86C22×Q8×C61—21no—
87#137×C61—44no—
88C2×Q16×C61—22no—
89C2×SD16×C61—22no—
90C2×M16×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61C2312yes—
95C4×Q8×C61D411yescharacteristic
96C4×Q8×C61C2311yescharacteristic
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61D411yescharacteristic
100C4×D4×C61C2311yescharacteristic
101#117×C61—21no—
102#117×C61D411yescharacteristic
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#103×C61—22no—
110#103×C61—22no—
111#41×C61—22no—
112#41×C61D814yes—
113C2×Q8×C61—22no—
114C2×Q8×C61—41no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C2×Q8×C61C2×D411yescharacteristic
118C2×D4×C61—21no—
119C2×D4×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C2×C22⋊D61—41no—
128#11758222—42no—
129#11758222—42no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×C4×D61—41no—
134C2×Dic122—21no—
135Q16×C61—44no—
136SD16×C61—44no—
137M16×C61—42no—
138C2×C8×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61C2×D411yescharacteristic
146C4⋊C4×C61C2×D411yescharacteristic
147C22⋊C4×C61—21no—
148C22⋊C4×C61—21no—
149C22⋊C4×C61—21no—
150C42×C61—21no—
151C42×C61—21no—
152C42×C61—21no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61#4111yescharacteristic
156C42×C61C2411yescharacteristic
157#11758202—42no—
158#11758201—42no—
159#11758200—42no—
160#11758199—42no—
161#11758198—41no—
162#11758198—41no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C2×#92418—22no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—42no—
169C2×#92418—82no—
170C2×#92418—81no—
171C2×#92418C2×D412yes—
172C2×#92418C2×D412yes—
173C23×C61—21no—
174C23×C61#12111yescharacteristic
175C22×D61—41no—
176Q8×C61—21no—
177Q8×C61—42no—
178Q8×C61—42no—
179Q8×C61—82no—
180Q8×C61C2×D812yes—
181D4×C61—24no—
182D4×C61—21no—
183D4×C61C2×D812yes—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—24no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—81no—
195C2×C4×C61—82no—
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61C2×D812yes—
200C2×C4×C61#13512yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C22⋊D61—82no—
205C2×Dic61—41no—
206D244—41no—
207D244—41no—
208D244—41no—
209D244—41no—
210D244—81no—
211C4×D61—82no—
212Dic122—41no—
213Dic122—41no—
214C8×C61—42no—
215#92418—22no—
216#92418—22no—
217#92418—41no—
218#92418—41no—
219#92418—42no—
220#19416—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—22no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61#46612yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—22no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—41no—
236C4×C61—41no—
237C4×C61—82no—
238C4×C61#46612yes—
239C4×C61C22×D811yescharacteristic
240C4×C61C2×#13511yescharacteristic
241C4×C61#57711yescharacteristic
242C4×C61#57811yescharacteristic
243Dic61—81no—
244#3226—611no—
245#3039D6111yescharacteristic, complemented
246#3035—611no—
247#2988—611no—
248#1682—612no—
249#1681—612no—
250#1674—612no—
251#1673—612no—
252#1571—611no—
253#1547—611no—
254C2×#332—611no—
255C2×C61—22no—
256C2×C61—42no—
257C2×C61#304611yescharacteristic
258C2×C61C2×#46611yescharacteristic
259C2×C61#306911yescharacteristic
260D61—161no—
261C2×#137—21no—
262C2×#135—1221no—
263#532—611no—
264C2×C4⋊Q8C2×D6111yescharacteristic
265C4×#41C2×D6111yescharacteristic
266#500—612no—
267#486—612no—
268#480—1222no—
269#474—1221no—
270#474—1221no—
271#464—1221no—
272#463—1221no—
273#462—21no—
274#462—611no—
275#462C2×D6112yes—
276#461—21no—
277#461—611no—
278#461C2×D6112yes—
279#460—611no—
280#459—611no—
281C8×Q8—612no—
282C8×D4—612no—
283C8×D4—1222no—
284#431—611no—
285#431—1221no—
286#423C2×D6111yescharacteristic
287C2×C4⋊C8—611no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—1221no—
290#419—21no—
291#416—1221no—
292#335—612no—
293#335—612no—
294#332—612no—
295#332—612no—
296#332—612no—
297#331—612no—
298C61#1941611yescharacteristic, complemented, Sylow
299C2×#41—1221no—
300C2×#41C22⋊D6111yescharacteristic
301C22×Q8—21no—
302#137—44no—
303#135—2444no—
304C2×Q16—22no—
305C2×Q16—1221no—
306C2×SD16—22no—
307C2×SD16—1221no—
308C2×SD16—1221no—
309C2×D8—1221no—
310C2×M16—21no—
311C22×C8—1222no—
312C22×C8—1221no—
313C22×C8—1221no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—611no—
318C4⋊Q8C22×D6112yes—
319#127—611no—
320#124—1221no—
321#121—1221no—
322#121—1221no—
323C4×Q8C22⋊D6111yescharacteristic
324C4×Q8C22×D6111yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4C22⋊D6111yescharacteristic
330C4×D4C22×D6111yescharacteristic
331#117—21no—
332#117C22⋊D6111yescharacteristic
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C42—21no—
336C2×C42C22×D6111yescharacteristic
337#107—1222no—
338#107—1222no—
339C4⋊C8—21no—
340C4⋊C8—611no—
341C4⋊C8—611no—
342C4⋊C8—612no—
343C4⋊C8—611no—
344C4⋊C8—611no—
345C4⋊C8—1222no—
346C4⋊C8—1221no—
347C4⋊C8—1221no—
348C4⋊C8C22×D6112yes—
349#103—22no—
350#103—22no—
351#103—1221no—
352#103—1221no—
353#102—1222no—
354#102—1222no—
355#102—1221no—
356#102—1221no—
357C22⋊C8—612no—
358C22⋊C8—1222no—
359C4×C8—612no—
360C4×C8—1222no—
361#41—22no—
362#41—2442no—
363#41—2442no—
364#41#1175819914yes—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—1221no—
370C2×Q8C2×C22⋊D6111yescharacteristic
371C2×D4—21no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—2441no—
376C2×D4C2×C22⋊D6111yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—2441no—
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822411yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385Q16—44no—
386Q16—2442no—
387SD16—44no—
388SD16—2442no—
389SD16—2442no—
390D8—2442no—
391M16—42no—
392C2×C8—22no—
393C2×C8—612no—
394C2×C8—612no—
395C2×C8—1222no—
396C2×C8—1221no—
397C2×C8—1221no—
398C2×C8—4882no—
399C2×C8—4881no—
400C2×C8—2442no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4C2×C22⋊D6111yescharacteristic
410C4⋊C4C2×C22⋊D6111yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42D4×D6111yescharacteristic
420C42#1175822411yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C23×D6111yescharacteristic
423C23—21no—
424C23—2441no—
425C23#1451974911yescharacteristic
426Q8—21no—
427Q8—42no—
428Q8—42no—
429Q8—82no—
430Q8—2441no—
431Q8—2441no—
432Q8C2×#1175819912yes—
433D4—24no—
434D4—21no—
435D4—4881no—
436D4—4882no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441D4C2×#1175819912yes—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—4882no—
455C2×C4—2441no—
456C2×C4#1451974911yescharacteristic
457C2×C4#1451974911yescharacteristic
458C2×C4#1451974911yescharacteristic
459C2×C4C2×#1175819912yes—
460C2×C4#1451979712yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C22×C22⋊D6111yescharacteristic
463C2×C4C2×#1175822411yescharacteristic
464C8—42no—
465C8—1222no—
466C8—1222no—
467C8—2441no—
468C8—2441no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—4881no—
476C22—4881no—
477C22—4881no—
478C22#1683999612yes—
479C22C2×#1451974911yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—82no—
487C4—4881no—
488C4#1683999612yes—
489C4C22×#1175819911yescharacteristic
490C4C2×#1451979711yescharacteristic
491C4#1684057111yescharacteristic
492C4#1684057811yescharacteristic
493C2—22no—
494C2—42no—
495C2—9761no—
496C2#2145537311yescharacteristic
497C2C2×#1683999611yescharacteristic
498C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3039C2
socleC22×C61C2×#121
prime radical#21455274C2
hypercenter#3039D61
hypocenterC61#19416
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
1. #21456940C2semidirect
3. #21454930C2semidirect
4. #21445265C2semidirect
5. #21445258C2semidirect
6. #21445211C2semidirect
7. #21445206C2semidirect
8. #21444595C2semidirect
9. #21444497C2semidirect
10. C2×#16839508C2semidirect
11. C61×#3039C2semidirect
245. #3039D61semidirect
298. C61#19416semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214552742
#214549302
#214452652
#214452582
#214452112
#214452062
#214445952
#214444972
C2×#168395082
C61×#30392
#1941661