Finite Group Databasebeta

← #30607587

Subgroups — #30607587

Order 15616.

522 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462573C211yesmaximal, characteristic, complemented
2D61⋊#360C211yesmaximal, characteristic, complemented
3#21462128C211yesmaximal, characteristic, complemented
4#21461968C211yesmaximal, characteristic, complemented
5#21460891C211yesmaximal, characteristic, complemented
6#21460865C211yesmaximal, characteristic, complemented
7#509×D61C211yesmaximal, characteristic
8#41×#6923—21no—
9D61⋊M32C2211yescharacteristic
10#16840803—21no—
11#16840802—21no—
12#16840801—22no—
13#16840772C2211yescharacteristic
14C2×#14519845C2211yescharacteristic
15#16840752—21no—
16C2×#14519839—21no—
17C2×#14519831C2211yescharacteristic
18#16840714—21no—
19#16840679—21no—
20#16840678—21no—
21#16840674—22no—
22#135×D61C2211yescharacteristic
23C2×D8×D61C2211yescharacteristic
24#16840200C411yescharacteristic, complemented
25#16840199—21no—
26#16840198C411yescharacteristic, complemented
27#16840197C411yescharacteristic, complemented
28#16840196—21no—
29SD32×D61—21no—
30D16×D61—21no—
31M32×D61C2211yescharacteristic
32C61×#509C411yescharacteristic, complemented
33C2×D61⋊C8D411yescharacteristic
34Q8×#6923—22no—
35D4×#6923—22no—
36D4×#6923—42no—
37#14519856—21no—
38#14519856—41no—
39C2×C4⋊#6923—21no—
40C2×C22⋊#6923—42no—
41C2×C4×#6923—41no—
42C2×C4×#6923D411yescharacteristic
43#14519852—21no—
44#14519851—21no—
45D61⋊C16—21no—
46#14519846D412yes—
47#14519845—21no—
48#14519845—21no—
49#14519845—21no—
50#14519840—22no—
51#14519839—22no—
52#14519839—42no—
53#14519831—21no—
54#14519831—21no—
55#41×D61—21no—
56C2×D4×D61—21no—
57#14519800D411yescharacteristic
58#14519799C2×C411yescharacteristic
59#14519798—21no—
60#14519797—21no—
61#14519796D411yescharacteristic
62Q16×D61D411yescharacteristic
63SD16×D61—21no—
64C2×D488C2×C411yescharacteristic, complemented
65C2×#11758199—21no—
66D8×D61—21no—
67D8×D61—21no—
68D8×D61—21no—
69D8×D61—21no—
70D8×D61D411yescharacteristic
71C2×C8×D61C2311yescharacteristic
72#14519722—21no—
73#14519721—21no—
74#14519720—21no—
75D976—21no—
76#14519718—21no—
77#14519718—21no—
78#14519717C2×C411yescharacteristic, complemented
79#14519716—21no—
80C16×D61—21no—
81#135×C61C2×C411yescharacteristic
82C2×D8×C61C2×C411yescharacteristic, complemented
83SD32×C61—21no—
84D16×C61—21no—
85M32×C61C2×C411yescharacteristic, complemented
86C23×D61—41no—
87C22×#6923—42no—
88C22×#6923—41no—
89C22×#6923D812yes—
90#41×C61—21no—
91C2×D4×C61—21no—
92C2×C22⋊D61—41no—
93#11758226—21no—
94Q8×D61—21no—
95#11758224—21no—
96#11758224—41no—
97D4×D61—21no—
98D4×D61—41no—
99D4×D61—41no—
100D4×D61—41no—
101D4×D61—41no—
102D4×D61—41no—
103#11758222—21no—
104#11758222—41no—
105C2×D244—21no—
106C2×C4×D61—41no—
107C2×C4×D61C2×D411yescharacteristic
108C22⋊#6923—42no—
109C22⋊#6923—41no—
110C22⋊#6923—84no—
111C2×#92420—21no—
112C4⋊#6923—22no—
113C4⋊#6923—42no—
114C4⋊#6923—41no—
115C4⋊#6923—41no—
116C4×#6923—21no—
117C4×#6923—21no—
118C4×#6923—42no—
119C4×#6923—41no—
120D61⋊C8—21no—
121D61⋊C8—21no—
122Q16×C61C22⋊C411yescharacteristic, complemented
123SD16×C61—21no—
124D8×C61—21no—
125D8×C61—21no—
126D8×C61C22⋊C411yescharacteristic, complemented
127C2×C8×C61C22×C411yescharacteristic
128#11758202—21no—
129#11758201—21no—
130#11758200—21no—
131#11758199—21no—
132#11758199—41no—
133#11758199—41no—
134C2×#92418#4111yescharacteristic
135Dic244C22⋊C411yescharacteristic, complemented
136D488—21no—
137D488—21no—
138D488C22⋊C411yescharacteristic, complemented
139#11758190—21no—
140C8×D61C2×D411yescharacteristic
141C8×D61C2×D411yescharacteristic
142C8×D61C2×D411yescharacteristic
143C8×D61#4111yescharacteristic
144#11758187—41no—
145C16×C61—21no—
146#11758185—21no—
147C23×C61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—82no—
151C22×D61—82no—
152C22×D61—81no—
153C22×D61—81no—
154C22×D61C2×D811yescharacteristic
155C2×#6923—22no—
156C2×#6923—42no—
157C2×#6923—42no—
158C2×#6923—84no—
159C2×#6923—82no—
160C2×#6923—81no—
161Q8×C61—21no—
162D4×C61—21no—
163D4×C61—41no—
164D4×C61—41no—
165D4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61C4×D411yescharacteristic
168C22⋊D61—41no—
169C22⋊D61—41no—
170C22⋊D61—41no—
171C22⋊D61—81no—
172C22⋊D61—81no—
173C2×Dic61—41no—
174C2×Dic61#13511yescharacteristic
175D244—21no—
176D244—41no—
177D244—41no—
178D244—41no—
179C4×D61—41no—
180C4×D61—41no—
181C4×D61—41no—
182C4×D61#12011yescharacteristic
183C4×D61#12111yescharacteristic
184C4×D61AGL(1,ℤ/8)11yescharacteristic
185C4×D61AGL(1,ℤ/8)11yescharacteristic
186Dic122—21no—
187Dic122—41no—
188#92420—41no—
189C8×C61C2×C22⋊C411yescharacteristic
190C8×C61C4×D411yescharacteristic
191#92418#12111yescharacteristic
192#92418#12311yescharacteristic
193#13025—611nomaximal, Sylow
194C22×C61—41no—
195C22×C61—82no—
196C22×C61—81no—
197C22×C61C4×D811yescharacteristic
198C2×D61—22no—
199C2×D61—41no—
200C2×D61—82no—
201C2×D61—82no—
202C2×D61—81no—
203C2×D61—161no—
204C2×D61—161no—
205C2×D61—162no—
206C2×D61#44711yescharacteristic
207C2×D61#45911yescharacteristic
208#6923—42no—
209#6923—82no—
210#6923—161no—
211C4×C61—41no—
212C4×C61#38611yescharacteristic
213C4×C61#44211yescharacteristic
214Dic61—41no—
215Dic61#44911yescharacteristic
216Dic61#46311yescharacteristic
217C2×#509—611no—
218#2943—611no—
219C2×#360—611no—
220#2140—611no—
221#1972—611no—
222#1386—611no—
223#1377—611no—
224C2×C61—21no—
225C2×C61—81no—
226C2×C61—161no—
227C2×C61#187911yescharacteristic
228D61—21no—
229D61—81no—
230D61—161no—
231D61#219012yes—
232C2×#135—611no—
233C22×D8—611no—
234C4×#41—1221no—
235#509—611no—
236#509—611no—
237#509—611no—
238#509—1221no—
239#509—1221no—
240#509#692311yescharacteristic, complemented
241C2×SD32—1221no—
242C2×D16—1221no—
243C2×M32—611no—
244C2×M32—611no—
245#442—1221no—
246#441—1221no—
247#440—1222no—
248#427—611no—
249C2×#106—611no—
250#416—1221no—
251C2×#102—1221no—
252C2×#97—611no—
253#390—1221no—
254#360—1221no—
255#360—1221no—
256#357—1222no—
257C61#1302511yescharacteristic, complemented, Sylow
258C2×#41—1221no—
259C22×D4—1221no—
260#135—611no—
261#135—611no—
262#135—611no—
263#135—1221no—
264#135—1221no—
265#135C2×#692311yescharacteristic
266C2×Q16—611no—
267C2×SD16—1221no—
268C2×D8—611no—
269C2×D8—611no—
270C2×D8—1221no—
271C2×D8—1221no—
272C2×D8—1221no—
273C2×D8—1221no—
274C2×D8—1221no—
275C2×D8C2×#692311yescharacteristic, complemented
276C22×C8—611no—
277C22×C8—611no—
278C4×Q8—1222no—
279C4×D4—1222no—
280C4×D4—2442no—
281#117—1221no—
282#117—2441no—
283C2×C4⋊C4—1221no—
284C2×C22⋊C4—2442no—
285C2×C42—611no—
286C2×C42—2441no—
287SD32—21no—
288SD32—1221no—
289SD32—1221no—
290SD32—1221no—
291D16—21no—
292D16—1221no—
293D16—1221no—
294D16—1221no—
295M32—611no—
296M32—1221no—
297M32—1221no—
298M32—1221no—
299M32C2×#692311yescharacteristic, complemented
300C2×C16—1221no—
301C2×C16—1221no—
302#107—612no—
303#106—1221no—
304#106—1221no—
305#106—1221no—
306#103—1222no—
307#102—1222no—
308#102—2442no—
309#97—1221no—
310#97—1221no—
311C24—2441no—
312#41—21no—
313#41—1221no—
314#41—1221no—
315#41—1221no—
316#41—2441no—
317#41—2441no—
318C2×Q8—1221no—
319C2×D4—21no—
320C2×D4—1221no—
321C2×D4—1221no—
322C2×D4—2441no—
323C2×D4—2441no—
324C2×D4—2441no—
325C2×D4—2441no—
326C2×D4—2441no—
327C2×D4—2441no—
328C22×C4—611no—
329C22×C4—612no—
330C22×C4—2441no—
331C22×C4—2442no—
332C22×C4—2441no—
333Q16—611no—
334Q16—1221no—
335Q16C22⋊#692311yescharacteristic, complemented
336SD16—21no—
337SD16—1221no—
338SD16—1221no—
339SD16—1221no—
340D8—21no—
341D8—21no—
342D8—611no—
343D8—1221no—
344D8—1221no—
345D8—1221no—
346D8—2441no—
347D8—2441no—
348D8C22⋊#692311yescharacteristic, complemented
349C2×C8—611no—
350C2×C8—611no—
351C2×C8—611no—
352C2×C8—611no—
353C2×C8—611no—
354C2×C8—1221no—
355C2×C8—1221no—
356C2×C8—1221no—
357C2×C8C22×#692311yescharacteristic
358C4⋊C4—1222no—
359C4⋊C4—2441no—
360C4⋊C4—2441no—
361C4⋊C4—2442no—
362C22⋊C4—4884no—
363C22⋊C4—2442no—
364C22⋊C4—2441no—
365C42—1221no—
366C42—1221no—
367C42—2441no—
368C42—2442no—
369C16—21no—
370C16—1221no—
371C16—2441no—
372C23—41no—
373C23—611no—
374C23—4881no—
375C23—4881no—
376C23—4882no—
377C23—4882no—
378C23—2441no—
379C23—2441no—
380Q8—21no—
381Q8—1221no—
382Q8—2441no—
383D4—21no—
384D4—41no—
385D4—41no—
386D4—41no—
387D4—1221no—
388D4—4881no—
389D4—4881no—
390D4—2441no—
391D4—2441no—
392D4—2441no—
393D4—2441no—
394D4—2441no—
395D4—2441no—
396C2×C4—41no—
397C2×C4—611no—
398C2×C4—611no—
399C2×C4—611no—
400C2×C4—611no—
401C2×C4—611no—
402C2×C4—1222no—
403C2×C4—4881no—
404C2×C4—4882no—
405C2×C4—4884no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2442no—
410C2×C4—2441no—
411C2×C4—2442no—
412C2×C4D4×#692311yescharacteristic
413C8—611no—
414C8—611no—
415C8—2441no—
416C8C2×C22⋊#692311yescharacteristic
417C8D4×#692311yescharacteristic
418C22—41no—
419C22—82no—
420C22—81no—
421C22—611no—
422C22—611no—
423C22—1222no—
424C22—9762no—
425C22—9761no—
426C22—9761no—
427C22—4882no—
428C22—4882no—
429C22—4881no—
430C22—2441no—
431C22D8×#692311yescharacteristic
432C4—41no—
433C4—611no—
434C4—611no—
435C4—9761no—
436C4—4882no—
437C4—2442no—
438C4—2441no—
439C4#1684071011yescharacteristic
440C4#1684080311yescharacteristic
441C2—21no—
442C2—81no—
443C2—161no—
444C2—612no—
445C2—1221no—
446C2—9761no—
447C2—4881no—
448C2#2146181411yescharacteristic
subgroupG/H
centerC2#21461814
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#509C4
socleC2×C61#1879
prime radical#509×D61C2
cosocleC2×C8×D61C23
hypercenter#509#6923
hypocenterC61#13025
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #21462573C2semidirect
2. D61⋊#360C2semidirect
3. #21462128C2semidirect
4. #21461968C2semidirect
5. #21460891C2semidirect
6. #21460865C2semidirect
24. #16840200C4semidirect
26. #16840198C4semidirect
27. #16840197C4semidirect
32. C61×#509C4semidirect
64. C2×D488C2×C4semidirect
78. #14519717C2×C4semidirect
82. C2×D8×C61C2×C4semidirect
85. M32×C61C2×C4semidirect
122. Q16×C61C22⋊C4semidirect
126. D8×C61C22⋊C4semidirect
135. Dic244C22⋊C4semidirect
138. D488C22⋊C4semidirect
240. #509#6923semidirect
257. C61#13025semidirect
275. C2×D8C2×#6923semidirect
299. M32C2×#6923semidirect
335. Q16C22⋊#6923semidirect
348. D8C22⋊#6923semidirect

Maximal subgroups

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

subgroupindex
#214625732
D61⋊#3602
#214621282
#214619682
#214608912
#214608652
#509×D612
#1302561