Finite Group Databasebeta

← #29582385

Subgroups — #29582385

Order 15616.

658 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 103 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#21455279C211yesmaximal, characteristic
2#21450773C211yesmaximal, characteristic, complemented
3#21444500C211yesmaximal, characteristic, complemented
4#21444314C211yesmaximal, characteristic, complemented
5#21444209C211yesmaximal, characteristic, complemented
6#21443721C211yesmaximal, characteristic, complemented
7C61×#3206C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×#14519753—22no—
10C2×C4×D244C2211yescharacteristic
11#16839974—22no—
12#16839964—21no—
13#16839954—21no—
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839763—21no—
17#16839697—22no—
18#16839663—21no—
19#16839631—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21#16839511—22no—
22#16839496—22no—
23#16839489—22no—
24C61×#532C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#492—21no—
27C61×#487—21no—
28C61×#461—22no—
29C61×#448—22no—
30C61×#431C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C4×D61—21no—
33#14519802—42no—
34#14519802—42no—
35C2×#11758201—22no—
36C2×#11758199—22no—
37C2×#11758194—21no—
38#14519753—48no—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—24no—
46C2×#11758198—24no—
47C2×#11758198—21no—
48C2×C4×Dic61—21no—
49C4⋊#92418—21no—
50C4⋊#92418—21no—
51C4⋊#92418D412yes—
52#14519701—22no—
53#14519698—22no—
54C22⋊#92418—21no—
55C4×#92418—21no—
56#14519684D412yes—
57C2×#41×C61—21no—
58C22×Q8×C61D412yes—
59C2×SD16×C61—24no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61D411yescharacteristic
65#127×C61D411yescharacteristic
66#124×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—21no—
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61D412yes—
75#103×C61—24no—
76C22⋊C8×C61—21no—
77C4×C8×C61—21no—
78C23×D61—21no—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83C2×Q8×C61—22no—
84C2×Q8×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61—44no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—41no—
91C22×C4×C61—22no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×Dic61—22no—
97C22×Dic61—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244SD1612yescomplemented
104C2×C4×D61—44no—
105C2×C4×D61—41no—
106C2×C4×D61—41no—
107SD16×C61—84no—
108C2×C8×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121#11758201—44no—
122#11758199—44no—
123#11758198—48no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—24no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129#11758196—48no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132#11758194—41no—
133#11758194—41no—
134C2×#92418—22no—
135C2×#92418—41no—
136C23×C61—41no—
137C23×C61#12011yescharacteristic
138C22×D61—22no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61—41no—
142C22×D61—41no—
143Q8×C61—41no—
144Q8×C61—41no—
145Q8×C61—44no—
146Q8×C61—82no—
147D4×C61—41no—
148D4×C61—41no—
149D4×C61—42no—
150D4×C61—81no—
151D4×C61—81no—
152D4×C61—81no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—44no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12711yescharacteristic
166C2×C4×C61C2×SD1611yescharacteristic
167C2×C4×C61#13511yescharacteristic
168C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C61#13711yescharacteristic
170C2×Dic61—24no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—41no—
175D244—22no—
176D244—22no—
177D244—41no—
178D244—41no—
179C4×D61—82no—
180C4×D61—82no—
181C8×C61—42no—
182#92418—42no—
183#92418—81no—
184#10240—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#39211yescharacteristic
191C22×C61#44811yescharacteristic
192C22×C61#44911yescharacteristic
193C2×D61—44no—
194C2×D61—42no—
195C2×D61—82no—
196C2×D61—81no—
197C2×D61—81no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—81no—
202C4×C61—82no—
203C4×C61#45011yescharacteristic
204C4×C61#45211yescharacteristic
205C4×C61#45312yes—
206C4×C61#46511yescharacteristic
207C4×C61#46811yescharacteristic
208Dic61—44no—
209Dic61—81no—
210#3206D6111yescharacteristic, complemented
211#3036—611no—
212#2542—611no—
213#1548—611no—
214#1502—611no—
215#1478—611no—
216#1303—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—161no—
220C2×C61#200011yescharacteristic
221C2×C61#200211yescharacteristic
222C2×C61#200911yescharacteristic
223D61—82no—
224C2×AGL(1,ℤ/8)—1221no—
225#532C2×D6111yescharacteristic
226C2×C4⋊Q8C2×D6111yescharacteristic
227C2×C22⋊Q8—1222no—
228C2×C4×D4—611no—
229#492—21no—
230#487—21no—
231#463—1222no—
232#461—22no—
233#461—1221no—
234#459—1221no—
235#448—22no—
236#431—611no—
237#431C2×D6111yescharacteristic
238#423—611no—
239#417—1221no—
240#399—1222no—
241#390—1221no—
242#384—1221no—
243C4×C4⋊C4—611no—
244#333—1222no—
245#327—1222no—
246D4⋊C8—1222no—
247C61#1024011yescharacteristic, complemented, Sylow
248C2×#41—21no—
249C22×Q8C22⋊D6112yes—
250C22×D4—611no—
251C23×C4—1221no—
252AGL(1,ℤ/8)—2442no—
253AGL(1,ℤ/8)—2442no—
254C2×SD16—24no—
255C2×SD16—1222no—
256C2×D8—1222no—
257C2×M16—1221no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8C22⋊D6111yescharacteristic
263#127C22⋊D6111yescharacteristic
264#124—21no—
265C22⋊Q8—2448no—
266#121—21no—
267#121—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—1221no—
271C4×D4—1221no—
272C4×D4—1221no—
273C4×D4—1221no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4—1224no—
279C2×C22⋊C4—1224no—
280C2×C22⋊C4—1221no—
281C2×C42—1221no—
282C2×C42C22×D6111yescharacteristic
283C4⋊C8—612no—
284C4⋊C8—1221no—
285C4⋊C8—1221no—
286C4⋊C8C22⋊D6112yes—
287#103—24no—
288#103—1222no—
289#102—1222no—
290C22⋊C8—21no—
291C22⋊C8—1221no—
292C4×C8—21no—
293C4×C8—1221no—
294#95—612no—
295C24—1221no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—44no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4—612no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—1221no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—1222no—
317C22×C4—1221no—
318C22×C4—2444no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4D4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324SD16—84no—
325SD16—2444no—
326D8—2444no—
327M16—2441no—
328M16—2441no—
329C2×C8—22no—
330C2×C8—1222no—
331C2×C8—2441no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—1224no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—2448no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—2448no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C42—2441no—
348C42—2441no—
349C42D4×D6111yescharacteristic
350C42#1175822411yescharacteristic
351C42C2×C22⋊D6111yescharacteristic
352C42C2×C22⋊D6111yescharacteristic
353C23—41no—
354C23—1222no—
355C23—2442no—
356C23—2442no—
357C23—2441no—
358C23—2441no—
359C23#1451974511yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—44no—
363Q8—82no—
364D4—41no—
365D4—41no—
366D4—42no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—1222no—
371D4—1222no—
372D4—2441no—
373D4—2441no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—1224no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—2442no—
385C2×C4—2441no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4#1451974511yescharacteristic
389C2×C4#1451974611yescharacteristic
390C2×C4#1451974811yescharacteristic
391C2×C4#1451974911yescharacteristic
392C2×C4#1451974911yescharacteristic
393C2×C4#1451977311yescharacteristic
394C2×C4SD16×D6111yescharacteristic
395C2×C4#1451979811yescharacteristic
396C2×C4#1451980411yescharacteristic
397C2×C4#1451980611yescharacteristic
398C8—42no—
399C8—4881no—
400C8—2442no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—4882no—
407C22—4881no—
408C22—4881no—
409C22—2444no—
410C22—2442no—
411C22#1683967011yescharacteristic
412C22#1683989911yescharacteristic
413C22#1683990611yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—82no—
419C4—4881no—
420C4—2444no—
421C4#1683991111yescharacteristic
422C4#1683992111yescharacteristic
423C4#1683993012yes—
424C4#1683998811yescharacteristic
425C4#1684000511yescharacteristic
426C2—21no—
427C2—21no—
428C2—161no—
429C2—4882no—
430C2#2144681911yescharacteristic
431C2#2144683211yescharacteristic
432C2#2144687711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3206C2
socleC22×C61#392
prime radical#21455279C2
hypercenter#3206D61
hypocenterC61#10240
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
2. #21450773C2semidirect
3. #21444500C2semidirect
4. #21444314C2semidirect
5. #21444209C2semidirect
6. #21443721C2semidirect
7. C61×#3206C2semidirect
103. C2×D244SD16semidirect
210. #3206D61semidirect
247. C61#10240semidirect

Maximal subgroups

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

subgroupindex
#214552792
#214507732
#214445002
#214443142
#214442092
#214437212
C61×#32062
#1024061