Finite Group Databasebeta

← #29663855

Subgroups — #29663855

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#21455293C211yesmaximal, characteristic, complemented
3#21455287C211yesmaximal, characteristic, complemented
4C2×#16839886C211yesmaximal, characteristic, complemented
5#21445266C212yesmaximal, complemented
6#21445262C212yesmaximal, complemented
7#21445212C212yesmaximal, complemented
8#21445178C212yesmaximal, complemented
9#21444577C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519807—21no—
14#16840312C2211yescharacteristic
15C2×C4×Dic122C2211yescharacteristic
16#16840165C2211yescharacteristic
17#16840138C2211yescharacteristic
18#16840110C2211yescharacteristic
19#16840108C2211yescharacteristic
20#16839996—21no—
21#16839996—21no—
22#16839985—22no—
23#16839981—22no—
24#16839973C2212yes—
25#16839969C2212yes—
26#16839963C2212yes—
27#16839959C2212yes—
28#16839886—21no—
29#16839886—21no—
30#16839886C2212yes—
31#16839886C2212yes—
32C4×#11758199—21no—
33C4×#11758199—21no—
34#16839786C2211yescharacteristic
35C2×#14519706—21no—
36#16839764—21no—
37#16839763—21no—
38C2×C4×#92418C2211yescharacteristic
39C2×C8×Dic61—21no—
40#16839514C2212yes—
41#16839514C2212yes—
42#16839507C2212yes—
43#16839505C2212yes—
44C61×#532C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yes—
48C61×#331C2212yes—
49C2×#11758222—21no—
50C22×Dic122D411yescharacteristic
51#14519810—44no—
52#14519807—44no—
53C2×Dic244—22no—
54C2×#11758202—22no—
55C2×#11758190—22no—
56C2×#11758200—22no—
57C2×#11758199—21no—
58C2×#11758199—21no—
59C2×#11758194—21no—
60C22×#92418D411yescharacteristic
61C22×#92418D411yescharacteristic
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×Dic122D412yes—
70C4×Dic122C2312yes—
71C4×D244—21no—
72C4×D244—21no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×#11758196—21no—
75C2×C4×Dic61—21no—
76#14519710—21no—
77#14519710—21no—
78C4⋊#92418—21no—
79C4⋊#92418C2312yes—
80#14519706—22no—
81#14519706—22no—
82#14519702—22no—
83#14519701—22no—
84#14519699—22no—
85#14519698—22no—
86#14519698—21no—
87#14519698—21no—
88#14519697—21no—
89#14519697—21no—
90C4×#92418D411yescharacteristic
91C4×#92418D411yescharacteristic
92C4×#92418C2311yescharacteristic
93C4×#92418C2311yescharacteristic
94C8×Dic61—22no—
95C8×Dic61—22no—
96C2×#41×C61—21no—
97C2×M16×C61—21no—
98C22×C8×C61—21no—
99C4⋊Q8×C61C2311yescharacteristic
100#127×C61C2311yescharacteristic
101#124×C61—21no—
102#121×C61—21no—
103#121×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61—21no—
109C4⋊C8×C61C2312yes—
110C4⋊C8×C61C2312yes—
111#41×C61—42no—
112#41×C61—42no—
113C2×Q8×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C2×C22⋊D61—41no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125#11758222—42no—
126#11758222—42no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132C2×Dic122—22no—
133C2×Dic122—21no—
134C2×Dic122D814yescomplemented
135C2×Dic122C2×D412yes—
136M16×C61—42no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—42no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147C42×C61C2411yescharacteristic
148#11758202—44no—
149#11758200—44no—
150#11758199—42no—
151#11758199—42no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61C2×D412yes—
158#11758196—22no—
159#11758196—22no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162#11758194—42no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—41no—
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169C2×#92418C2×D411yescharacteristic
170Dic244—44no—
171#11758190—44no—
172C23×C61—41no—
173C23×C61#12111yescharacteristic
174C22×D61—41no—
175Q8×C61—41no—
176Q8×C61—41no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—81no—
182D4×C61—82no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—82no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61C2×D812yes—
192C2×C4×C61#13512yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C22⋊D61—82no—
197C2×Dic61—22no—
198C2×Dic61—24no—
199C2×Dic61—22no—
200C2×Dic61—42no—
201C2×Dic61—41no—
202D244—41no—
203D244—41no—
204D244—41no—
205D244—41no—
206D244—81no—
207C4×D61—82no—
208Dic122—22no—
209Dic122—24no—
210Dic122—41no—
211Dic122—41no—
212Dic122C2×D814yes—
213C8×C61—42no—
214C8×C61—42no—
215#92418—21no—
216#92418—21no—
217#92418—21no—
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—21no—
233C4×C61—21no—
234C4×C61—81no—
235C4×C61#46612yes—
236C4×C61C22×D811yescharacteristic
237C4×C61C2×#13511yescharacteristic
238C4×C61C2×#13711yescharacteristic
239C4×C61#57511yescharacteristic
240Dic61—24no—
241Dic61—42no—
242Dic61—81no—
243#3169—611no—
244#3037—611no—
245#3037—611no—
246C2×C8×Q8—611no—
247#1682—612no—
248#1682—612no—
249#1674—612no—
250#1668—612no—
251#1567—611no—
252#1547—611no—
253#1547D6111yescharacteristic, complemented
254C2×C61—22no—
255C2×C61—161no—
256C2×C61#303711yescharacteristic
257C2×C61C2×#46611yescharacteristic
258C2×C61#306711yescharacteristic
259D61—161no—
260C2×#137—1221no—
261C2×#137—1221no—
262#532—611no—
263#532C2×D6111yescharacteristic
264C2×C4×Q8—611no—
265#500—611no—
266#493—611no—
267#486—611no—
268#486—611no—
269#466—1221no—
270#466—1221no—
271#464—1222no—
272#464—1222no—
273#462—612no—
274#462—612no—
275#460—612no—
276#460—612no—
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—1221no—
286C2×C4⋊C8C2×D6111yescharacteristic
287#417—1221no—
288#417—1221no—
289C2×C4×C8—611no—
290C2×C4×C8—1221no—
291#335—612no—
292#335—612no—
293#332—612no—
294#332C2×D6112yes—
295#331—612no—
296#331C2×D6112yes—
297C61#1926111yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41—1221no—
300C22×Q8—611no—
301#137—2444no—
302#137—2444no—
303C2×Q16—1222no—
304C2×Q16—1222no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×D8—1221no—
308C2×D8—1221no—
309C2×M16—21no—
310C2×M16—1221no—
311C22×C8—21no—
312C22×C8—611no—
313C22×C8—611no—
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—612no—
325C4×Q8—612no—
326C4×Q8—1221no—
327C4×Q8—1221no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1221no—
331C4×D4—1221no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—1221no—
334C2×C42—1221no—
335C2×C42C22×D6111yescharacteristic
336#107—1221no—
337#107—1221no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—612no—
341C4⋊C8—1222no—
342C4⋊C8—1222no—
343C4⋊C8—1221no—
344C4⋊C8C22×D6112yes—
345C4⋊C8C22×D6112yes—
346#103—1222no—
347#103—1222no—
348#102—1221no—
349#102—1221no—
350#102—1222no—
351#102—1222no—
352#102—1221no—
353#102—1221no—
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—612no—
366C2×Q8—614no—
367C2×Q8—1221no—
368C2×Q8—1222no—
369C2×Q8—1221no—
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—41no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—2441no—
382C22×C4#1175822211yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4D4×D6111yescharacteristic
385Q16—2444no—
386Q16—2444no—
387SD16—2444no—
388SD16—2444no—
389D8—2442no—
390D8—2442no—
391M16—42no—
392M16—2442no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—42no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—611no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—2441no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—612no—
406C4⋊C4—1222no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C42—1222no—
416C42—1222no—
417C42#1175822211yescharacteristic
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—41no—
422C23—2441no—
423C23#1451975011yescharacteristic
424Q8—41no—
425Q8—41no—
426Q8—614no—
427Q8—1224no—
428Q8—1222no—
429Q8—2441no—
430Q8—2441no—
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—21no—
444C2×C4—21no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—1224no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—4882no—
452C2×C4—2441no—
453C2×C4—2442no—
454C2×C4#1451975011yescharacteristic
455C2×C4#1451975011yescharacteristic
456C2×C4#1451975011yescharacteristic
457C2×C4D8×D6112yes—
458C2×C4#1451979812yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C2×#1175822211yescharacteristic
462C8—42no—
463C8—42no—
464C8—612no—
465C8—1221no—
466C8—1221no—
467C8—1221no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22#1683999712yes—
478C22C2×#1451975011yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—1224no—
484C4—4881no—
485C4—2442no—
486C4#1683999712yes—
487C4C2×D8×D6111yescharacteristic
488C4C2×#1451979811yescharacteristic
489C4#137×D6111yescharacteristic
490C4#1684055811yescharacteristic
491C2—22no—
492C2—161no—
493C2—9761no—
494C2#2145529011yescharacteristic
495C2C2×#1683999711yescharacteristic
496C2#2145553811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C2×#121
prime radical#21456394C2
hypercenter#1547D61
hypocenterC61#19261
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #21455293C2semidirect
3. #21455287C2semidirect
4. C2×#16839886C2semidirect
5. #21445266C2semidirect
6. #21445262C2semidirect
7. #21445212C2semidirect
8. #21445178C2semidirect
9. #21444577C2semidirect
10. #21444494C2semidirect
11. C61×#1547C2semidirect
134. C2×Dic122D8semidirect
219. #92418#137semidirect
253. #1547D61semidirect
297. C61#19261semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214552932
#214552872
C2×#168398862
#214452662
#214452622
#214452122
#214451782
#214445772
#214444942
C61×#15472
#1926161