Finite Group Databasebeta

← #29663853

Subgroups — #29663853

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456394C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic, complemented
3C2×Q8×#92418C211yesmaximal, characteristic, complemented
4#21445265C212yesmaximal, complemented
5#21445261C212yesmaximal, complemented
6#21445211C212yesmaximal, complemented
7#21445177C212yesmaximal, complemented
8#21444577C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic, complemented
10#21444496C211yesmaximal, characteristic, complemented
11C61×#3037C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13#16840312C2211yescharacteristic
14#16840165C2211yescharacteristic
15#16840138C2211yescharacteristic
16#16840110C2211yescharacteristic
17#16840108C2211yescharacteristic
18#16839996—21no—
19#16839996—21no—
20#16839984—22no—
21#16839972C2212yes—
22#16839962C2212yes—
23Q8×#92418—21no—
24Q8×#92418—21no—
25Q8×#92418C2212yes—
26Q8×#92418C2212yes—
27C4×#11758199—21no—
28C4×#11758199—21no—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418—21no—
31C2×C4⋊#92418C2211yescharacteristic
32#16839761—21no—
33C2×C4×#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yes—
36#16839514C2212yes—
37#16839509C2212yes—
38#16839508C2212yes—
39#16839506C2212yes—
40#16839505C2212yes—
41C2×#137×C61—21no—
42C61×#532C2211yescharacteristic
43C2×C4×Q8×C61C2211yescharacteristic
44C61×#464—22no—
45C61×#462C2212yes—
46C61×#460C2212yes—
47C61×#423C2211yescharacteristic
48C61×#417—21no—
49C2×#11758222—21no—
50#14519808—44no—
51C2×#11758202—22no—
52C2×#11758201—22no—
53C2×#11758199—21no—
54C2×#11758199—21no—
55C2×#11758194—21no—
56C22×#92418—21no—
57C22×#92418D411yescharacteristic
58C22×#92418D411yescharacteristic
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519710—21no—
67#14519710—21no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418—22no—
71C4⋊#92418—22no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519700—22no—
75#14519698—21no—
76#14519698—21no—
77#14519697—22no—
78#14519697—21no—
79#14519697—21no—
80C4×#92418—22no—
81C4×#92418—22no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C2×#41×C61—21no—
87C22×Q8×C61D411yescharacteristic
88#137×C61—44no—
89C2×Q16×C61—22no—
90C2×SD16×C61—22no—
91C2×M16×C61—21no—
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61C2311yescharacteristic
94#124×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61D412yes—
100C4×Q8×C61C2312yes—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#103×C61—22no—
110#102×C61—22no—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—21no—
116C2×Q8×C61D814yescomplemented
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—41no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—41no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C2×C22⋊D61—41no—
129#11758222—42no—
130#11758222—42no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×C4×D61—41no—
135C2×Dic122—21no—
136Q16×C61—44no—
137SD16×C61—44no—
138M16×C61—42no—
139C2×C8×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61C2×D412yes—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61—22no—
149C42×C61—22no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758202—44no—
155#11758201—44no—
156#11758199—42no—
157#11758199—42no—
158#11758198—41no—
159#11758198—41no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162#11758194—42no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—41no—
167C2×#92418—42no—
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C23×C61—41no—
173C23×C61#12111yescharacteristic
174C22×D61—41no—
175Q8×C61—22no—
176Q8×C61—24no—
177Q8×C61—41no—
178Q8×C61—41no—
179Q8×C61C2×D814yes—
180D4×C61—41no—
181D4×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184D4×C61—81no—
185D4×C61—82no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—82no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61C2×D812yes—
199C2×C4×C61#13512yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C22⋊D61—82no—
204C2×Dic61—41no—
205D244—41no—
206D244—41no—
207D244—41no—
208D244—41no—
209D244—81no—
210C4×D61—82no—
211Dic122—41no—
212Dic122—41no—
213C8×C61—42no—
214#92418—21no—
215#92418—21no—
216#92418—21no—
217#92418—42no—
218#92418—42no—
219#92418#13712yescomplemented
220#19261—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61#46612yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—24no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—81no—
237C4×C61#46612yes—
238C4×C61C22×D811yescharacteristic
239C4×C61C2×#13511yescharacteristic
240C4×C61C2×#13711yescharacteristic
241C4×C61#57511yescharacteristic
242Dic61—81no—
243#3169—611no—
244#3037—611no—
245#3037D6111yescharacteristic, complemented
246C2×C8×Q8—611no—
247#1682—612no—
248#1682—612no—
249#1674—612no—
250#1668—612no—
251#1567—611no—
252#1547—611no—
253#1547—611no—
254C2×C61—22no—
255C2×C61—161no—
256C2×C61#303711yescharacteristic
257C2×C61C2×#46611yescharacteristic
258C2×C61#306711yescharacteristic
259D61—161no—
260C2×#137—21no—
261C2×#137—1221no—
262#532—611no—
263#532C2×D6111yescharacteristic
264C2×C4×Q8C2×D6111yescharacteristic
265#500—611no—
266#493—611no—
267#486—611no—
268#486—611no—
269#466—1221no—
270#466—1221no—
271#464—22no—
272#464—1222no—
273#462—612no—
274#462C2×D6112yes—
275#460—612no—
276#460C2×D6112yes—
277C8×Q8—612no—
278C8×Q8—612no—
279C8×Q8—1221no—
280C8×Q8—1221no—
281C4×D8—1221no—
282C4×D8—1221no—
283#423—611no—
284#423C2×D6111yescharacteristic
285C2×C4⋊C8—611no—
286C2×C4⋊C8—1221no—
287#417—21no—
288#417—1221no—
289C2×C4×C8—611no—
290C2×C4×C8—1221no—
291#335—612no—
292#335—612no—
293#332—612no—
294#332—612no—
295#331—612no—
296#331—612no—
297C61#1926111yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41—1221no—
300C22×Q8C22⋊D6111yescharacteristic
301#137—44no—
302#137—2444no—
303C2×Q16—22no—
304C2×Q16—1222no—
305C2×SD16—22no—
306C2×SD16—1222no—
307C2×D8—1221no—
308C2×D8—1221no—
309C2×M16—21no—
310C2×M16—1221no—
311C22×C8—611no—
312C22×C8—611no—
313C22×C8—1221no—
314C4⋊Q8—611no—
315C4⋊Q8C22×D6111yescharacteristic
316#127—611no—
317#127C22×D6111yescharacteristic
318#124—21no—
319#124—1221no—
320#121—21no—
321#121—21no—
322#121—1221no—
323#121—1221no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8C22⋊D6112yes—
327C4×Q8C22×D6112yes—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1221no—
331C4×D4—1221no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C22⋊D6111yescharacteristic
334C2×C42—21no—
335C2×C42C22×D6111yescharacteristic
336#107—1221no—
337#107—1221no—
338C4⋊C8—21no—
339C4⋊C8—612no—
340C4⋊C8—612no—
341C4⋊C8—1222no—
342C4⋊C8—1222no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8C22×D6112yes—
346#103—22no—
347#103—1222no—
348#102—22no—
349#102—1221no—
350#102—1221no—
351#102—1221no—
352#102—1221no—
353#102—1222no—
354C4×C8—611no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—1222no—
359C4×C8—1222no—
360#41—42no—
361#41—42no—
362#41—2442no—
363#41—2442no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—1221no—
368C2×Q8#1175819914yescomplemented
369C2×Q8C2×C22⋊D6112yes—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—1221no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—2441no—
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—2444no—
387SD16—44no—
388SD16—2444no—
389D8—2442no—
390D8—2442no—
391M16—42no—
392M16—2442no—
393C2×C8—22no—
394C2×C8—611no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—2441no—
402C2×C8—2442no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4C2×C22⋊D6112yes—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C42—22no—
416C42—22no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—41no—
422C23—2441no—
423C23#1451974911yescharacteristic
424Q8—24no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—2441no—
429Q8—2441no—
430Q8C2×#1175819914yes—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—81no—
436D4—82no—
437D4—4881no—
438D4—4882no—
439D4—2441no—
440D4—2441no—
441D4—2441no—
442D4—2441no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—4882no—
453C2×C4—2441no—
454C2×C4#1451974911yescharacteristic
455C2×C4#1451974911yescharacteristic
456C2×C4#1451974911yescharacteristic
457C2×C4C2×#1175819912yes—
458C2×C4#1451979712yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C2×#1175822411yescharacteristic
462C8—42no—
463C8—612no—
464C8—1221no—
465C8—1221no—
466C8—1221no—
467C8—2442no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22#1683999612yes—
478C22C2×#1451974911yescharacteristic
479C4—24no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—4881no—
486C4#1683999612yes—
487C4C22×#1175819911yescharacteristic
488C4C2×#1451979711yescharacteristic
489C4#137×D6111yescharacteristic
490C4#1684056111yescharacteristic
491C2—22no—
492C2—161no—
493C2—9761no—
494C2#2145528811yescharacteristic
495C2C2×#1683999611yescharacteristic
496C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3037C2
socleC22×C61C2×#121
prime radical#21456394C2
hypercenter#3037D61
hypocenterC61#19261
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. #21455291C2semidirect
3. C2×Q8×#92418C2semidirect
4. #21445265C2semidirect
5. #21445261C2semidirect
6. #21445211C2semidirect
7. #21445177C2semidirect
8. #21444577C2semidirect
9. #21444497C2semidirect
10. #21444496C2semidirect
11. C61×#3037C2semidirect
116. C2×Q8×C61D8semidirect
219. #92418#137semidirect
245. #3037D61semidirect
297. C61#19261semidirect
368. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214552912
C2×Q8×#924182
#214452652
#214452612
#214452112
#214451772
#214445772
#214444972
#214444962
C61×#30372
#1926161