Finite Group Databasebeta

← #29665410

Subgroups — #29665410

Order 15616.

674 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#21456744C211yesmaximal, characteristic, complemented
2#21455628C211yesmaximal, characteristic, complemented
3#21455276C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic, complemented
5#21445248C212yesmaximal, complemented
6#21445241C212yesmaximal, complemented
7#21445170C212yesmaximal, complemented
8#21445163C212yesmaximal, complemented
9#21444589C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C2×C61×#332C211yesmaximal, characteristic, complemented
12C2×#14519799—21no—
13#16840312C2211yescharacteristic
14C4×#11758222C2211yescharacteristic
15#16840136C2212yes—
16#16840112C2212yes—
17#16840054—21no—
18#16840049—21no—
19#16840044C2212yes—
20#16840039C2212yes—
21#16839985—21no—
22#16839978—21no—
23#16839973C2211yescharacteristic
24#16839968C2211yescharacteristic
25#16839964—21no—
26#16839964—21no—
27#16839963C2211yescharacteristic
28#16839958C2211yescharacteristic
29#16839901—22no—
30#16839886C2212yes—
31#16839827C2212yes—
32C8×C22⋊D61—22no—
33#16839815C2211yescharacteristic
34#16839814—21no—
35#16839802—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839761—21no—
39#16839759—21no—
40#16839512C2212yes—
41#16839512C2212yes—
42#16839507C2212yes—
43#16839505C2212yes—
44C2×C4⋊Q8×C61C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yes—
48C61×#332C2212yes—
49C2×#11758222—21no—
50C2×#11758222D411yescharacteristic
51#14519799—44no—
52C2×Dic244—21no—
53C2×#11758190—21no—
54C2×#11758190—21no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C2×D488—21no—
58C2×#11758194—21no—
59C2×C8×D61—22no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×Dic122D411yescharacteristic
66C4×Dic122C2311yescharacteristic
67C4×D244—21no—
68C4×D244—21no—
69C4×D244D411yescharacteristic
70C4×D244C2311yescharacteristic
71C4×C22⋊D61—21no—
72C4×C22⋊D61—21no—
73#14519736D411yescharacteristic
74#14519734—21no—
75C42×D61—21no—
76#14519710—22no—
77#14519708—22no—
78C4⋊#92418—21no—
79C4⋊#92418D411yescharacteristic
80C4⋊#92418D411yescharacteristic
81C4⋊#92418C2312yes—
82#14519706—22no—
83#14519702—21no—
84#14519702—21no—
85#14519701—22no—
86#14519701—22no—
87#14519700—22no—
88#14519699—21no—
89#14519699—21no—
90#14519697—22no—
91C22⋊#92418D412yes—
92#14519689—22no—
93C4×#92418C2312yes—
94C8×Dic61—22no—
95C22×Q8×C61—21no—
96C22×C8×C61—21no—
97C22×C8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61C2312yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2311yescharacteristic
108C4⋊C8×C61C2311yescharacteristic
109C4⋊C8×C61C2312yes—
110C2×Q8×C61—22no—
111C2×Q8×C61—41no—
112C2×Q8×C61—81no—
113C2×Q8×C61—81no—
114C22×C4×C61—41no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C2×C22⋊D61—21no—
119C2×C22⋊D61—41no—
120#11758222—22no—
121#11758222—42no—
122#11758222—42no—
123#11758222SD1614yescomplemented
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244C2×D411yescharacteristic
128C2×C4×D61—21no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—41no—
132C2×Dic122—21no—
133C2×Dic122C2×D411yescharacteristic
134C2×C8×C61—22no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—42no—
138C2×C8×C61—81no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61—41no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61#4111yescharacteristic
146C42×C61C2411yescharacteristic
147C22⋊Dic61—21no—
148#11758201—82no—
149#11758201—82no—
150#11758198—21no—
151#11758198—21no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61C2×D411yescharacteristic
157C4⋊Dic61C2×D411yescharacteristic
158#11758196—21no—
159#11758196—21no—
160C4×Dic61—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163#11758194—42no—
164C2×#92418—22no—
165C2×#92418C2×D412yes—
166C2×#92418C2×D412yes—
167Dic244—42no—
168D488—42no—
169#11758190—42no—
170#11758190—42no—
171C8×D61—82no—
172C23×C61#12111yescharacteristic
173C22×D61—21no—
174C22×D61—41no—
175Q8×C61—42no—
176Q8×C61—42no—
177Q8×C61—82no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—82no—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61C2×SD1612yes—
187C2×C4×C61#13512yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C22⋊D61—24no—
192C22⋊D61—82no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—41no—
197D244—21no—
198D244—41no—
199D244—41no—
200D244—41no—
201D244—41no—
202D244—81no—
203D244C2×SD1612yes—
204C4×D61—24no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—81no—
208C4×D61—82no—
209Dic122—21no—
210Dic122—41no—
211Dic122—41no—
212Dic122C2×SD1612yes—
213C8×C61—41no—
214C8×C61—41no—
215C8×C61—42no—
216#92418—22no—
217#92418—22no—
218#92418—42no—
219#19408—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61#46512yes—
223C22×C61C2×#12111yescharacteristic
224C2×D61—22no—
225C2×D61—41no—
226C2×D61—41no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61—81no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—82no—
234C4×C61#46512yes—
235C4×C61C22×SD1611yescharacteristic
236C4×C61C2×#13511yescharacteristic
237C4×C61#57611yescharacteristic
238C4×C61#57711yescharacteristic
239Dic61—22no—
240Dic61—41no—
241Dic61—41no—
242Dic61—81no—
243#3206—611no—
244#3079—611no—
245#3035—611no—
246#2988—611no—
247#1680—612no—
248#1679—612no—
249#1666—612no—
250#1665—612no—
251#1570—611no—
252#1547—611no—
253C2×#332D6111yescharacteristic, complemented
254C2×C61—22no—
255C2×C61#304511yescharacteristic
256C2×C61C2×#46511yescharacteristic
257C2×C61#306911yescharacteristic
258D61—42no—
259D61—161no—
260C2×#135—1221no—
261#532—611no—
262C2×C4⋊Q8C2×D6111yescharacteristic
263C4×#41—611no—
264#492—612no—
265#487—612no—
266#477—1221no—
267#476—1221no—
268#475—612no—
269#474—612no—
270#464—1221no—
271#463—1221no—
272#462—611no—
273#461—611no—
274#461—1221no—
275#461—1221no—
276#460—611no—
277#459—611no—
278#448—1222no—
279C8×Q8—612no—
280C8×D4—612no—
281C8×D4—1222no—
282#431—611no—
283#431—1221no—
284#428—1221no—
285#423—611no—
286C2×C4⋊C8—1221no—
287C2×C4⋊C8C2×D6111yescharacteristic
288C2×C4⋊C8C2×D6111yescharacteristic
289#417—1221no—
290#416—1221no—
291#334—612no—
292#334—612no—
293#332—612no—
294#332C2×D6112yes—
295#332C2×D6112yes—
296#331—612no—
297C61#1940811yescharacteristic, complemented, Sylow
298C2×#41—611no—
299C2×#41—1221no—
300C22×Q8—21no—
301#135—2444no—
302C2×Q16—1221no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×D8—1221no—
308C2×M16—1221no—
309C22×C8—21no—
310C22×C8—21no—
311C22×C8—1222no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—611no—
316C4⋊Q8C22×D6112yes—
317#127—611no—
318#124—1221no—
319#121—1221no—
320#121—1221no—
321C4×Q8—611no—
322C4×Q8—611no—
323C4×D4—611no—
324C4×D4—611no—
325C4×D4—1221no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329#117—611no—
330#117—1221no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C42—1221no—
334C2×C42C22×D6111yescharacteristic
335#107—1222no—
336#106—1222no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—611no—
340C4⋊C8—611no—
341C4⋊C8—612no—
342C4⋊C8—1222no—
343C4⋊C8—1221no—
344C4⋊C8C22×D6112yes—
345C4⋊C8C22×D6111yescharacteristic
346C4⋊C8C22×D6111yescharacteristic
347#103—1222no—
348#103—1221no—
349#103—1221no—
350#103—1222no—
351#103—1222no—
352#102—1222no—
353#102—1221no—
354#102—1221no—
355C22⋊C8—612no—
356C22⋊C8—1222no—
357C4×C8—612no—
358C4×C8—1222no—
359#41—614no—
360#41—1222no—
361#41—2442no—
362#41—2442no—
363C2×Q8—22no—
364C2×Q8—41no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×Q8—611no—
368C2×Q8—1221no—
369C2×D4—611no—
370C2×D4—1221no—
371C2×D4—1221no—
372C2×D4—1221no—
373C2×D4—1221no—
374C2×D4—2441no—
375C22×C4—41no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—2441no—
380C22×C4#1175822211yescharacteristic
381C22×C4D4×D6111yescharacteristic
382C22×C4D4×D6111yescharacteristic
383Q16—2442no—
384SD16—4882no—
385SD16—4882no—
386SD16—2442no—
387SD16—2442no—
388D8—2442no—
389M16—2442no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—42no—
394C2×C8—81no—
395C2×C8—612no—
396C2×C8—612no—
397C2×C8—1222no—
398C2×C8—4882no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—611no—
404C4⋊C4—611no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C22⋊C4—1221no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—1221no—
415C42—1221no—
416C42—1221no—
417C42#1175822211yescharacteristic
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—1221no—
422C23—2441no—
423C23#1451975011yescharacteristic
424Q8—42no—
425Q8—42no—
426Q8—82no—
427Q8—612no—
428Q8—1221no—
429Q8—2441no—
430Q8—2441no—
431D4—612no—
432D4—1224no—
433D4—1221no—
434D4—4881no—
435D4—4882no—
436D4—2441no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—1221no—
446C2×C4—1224no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—4881no—
450C2×C4—4882no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4#1451975011yescharacteristic
455C2×C4#1451975011yescharacteristic
456C2×C4#1451975011yescharacteristic
457C2×C4SD16×D6112yes—
458C2×C4#1451980012yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C2×#1175822211yescharacteristic
462C8—41no—
463C8—41no—
464C8—42no—
465C8—1222no—
466C8—1222no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—1222no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22—2441no—
475C22—2441no—
476C22#1683999412yes—
477C22C2×#1451975011yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—82no—
482C4—1222no—
483C4—4881no—
484C4—2441no—
485C4—2441no—
486C4#1683999412yes—
487C4C2×SD16×D6111yescharacteristic
488C4C2×#1451980011yescharacteristic
489C4#1684056311yescharacteristic
490C4#1684056811yescharacteristic
491C2—22no—
492C2—9761no—
493C2—2442no—
494C2#2145536311yescharacteristic
495C2C2×#1683999411yescharacteristic
496C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#332C2
socleC22×C61C2×#121
prime radical#21455276C2
hypercenterC2×#332D61
hypocenterC61#19408
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. #21456744C2semidirect
2. #21455628C2semidirect
4. #21454928C2semidirect
5. #21445248C2semidirect
6. #21445241C2semidirect
7. #21445170C2semidirect
8. #21445163C2semidirect
9. #21444589C2semidirect
10. #21444494C2semidirect
11. C2×C61×#332C2semidirect
123. #11758222SD16semidirect
253. C2×#332D61semidirect
297. C61#19408semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214556282
#214552762
#214549282
#214452482
#214452412
#214451702
#214451632
#214445892
#214444942
C2×C61×#3322
#1940861