Finite Group Databasebeta

← #29580754

Subgroups — #29580754

Order 15616.

602 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 99 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#21451177C211yesmaximal, characteristic, complemented
3#21444379C211yesmaximal, characteristic, complemented
4#21444248C211yesmaximal, characteristic, complemented
5#21444211C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#3103C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×C4×D244C2211yescharacteristic
10#16839974—22no—
11#16839964—21no—
12#16839954—21no—
13#16839815C2211yescharacteristic
14#16839786C2211yescharacteristic
15#16839763—21no—
16#16839697—21no—
17#16839673—21no—
18#16839664—21no—
19#16839660—21no—
20#16839633—22no—
21#16839628C2211yescharacteristic
22#16839496—22no—
23Q8⋊#92418—22no—
24#16839491—22no—
25C61×#532C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#481—21no—
28C61×#475—21no—
29C61×#460—21no—
30C61×#448—21no—
31C4×SD16×C61—22no—
32C61×#431C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C4×D61—21no—
35#14519802—42no—
36#14519802—42no—
37C2×#11758201—22no—
38C2×#11758199—22no—
39C2×#11758194—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—22no—
47C2×#11758198—22no—
48C2×#11758198—21no—
49C2×#11758198—22no—
50C4⋊#92418—21no—
51C4⋊#92418—21no—
52C4⋊#92418D412yes—
53#14519701—22no—
54#14519698—22no—
55C22⋊#92418—21no—
56C4×#92418—21no—
57#14519684—22no—
58#14519684—21no—
59#14519684D412yes—
60C2×#41×C61—21no—
61C22×Q8×C61D411yescharacteristic
62C2×SD16×C61—22no—
63C4⋊Q8×C61D411yescharacteristic
64#127×C61D411yescharacteristic
65#124×C61—21no—
66#121×C61—21no—
67#121×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71C4×Q8×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C42×C61—21no—
77C2×C42×C61C2311yescharacteristic
78#106×C61—22no—
79C4⋊C8×C61—21no—
80#103×C61—22no—
81#102×C61—22no—
82C22⋊C8×C61—21no—
83C4×C8×C61D412yes—
84C23×D61—21no—
85#41×C61—41no—
86#41×C61—41no—
87#41×C61—41no—
88#41×C61—41no—
89C2×Q8×C61—21no—
90C2×Q8×C61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—42no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—41no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×Dic61—21no—
104C22×Dic61—22no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244SD1612yescomplemented
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—82no—
114SD16×C61—44no—
115C2×C8×C61—22no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—41no—
124C42×C61—41no—
125C42×C61—41no—
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130#11758201—44no—
131#11758199—44no—
132#11758198—44no—
133#11758198—41no—
134#11758198—41no—
135#11758198—82no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—41no—
139C4⋊Dic61—41no—
140#11758196—82no—
141#11758194—41no—
142#11758194—41no—
143C2×#92418—22no—
144C2×#92418—41no—
145C23×C61—41no—
146C23×C61#12011yescharacteristic
147C22×D61—22no—
148C22×D61—42no—
149C22×D61—42no—
150C22×D61—41no—
151C22×D61—41no—
152Q8×C61—22no—
153Q8×C61—22no—
154Q8×C61—41no—
155Q8×C61—41no—
156Q8×C61—41no—
157Q8×C61—41no—
158D4×C61—42no—
159D4×C61—41no—
160D4×C61—41no—
161D4×C61—81no—
162D4×C61—81no—
163D4×C61—81no—
164C2×C4×C61—22no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—81no—
174C2×C4×C61—81no—
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12711yescharacteristic
181C2×C4×C61C2×SD1611yescharacteristic
182C2×C4×C61#13511yescharacteristic
183C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C61#13711yescharacteristic
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190D244—22no—
191D244—22no—
192D244—41no—
193D244—41no—
194C4×D61—82no—
195C4×D61—82no—
196C8×C61—42no—
197#92418—42no—
198#92418—81no—
199#9998—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61#39211yescharacteristic
206C22×C61#44811yescharacteristic
207C22×C61#44911yescharacteristic
208C2×D61—44no—
209C2×D61—42no—
210C2×D61—82no—
211C2×D61—81no—
212C2×D61—81no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—42no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61#45011yescharacteristic
221C4×C61#45211yescharacteristic
222C4×C61#46511yescharacteristic
223C4×C61#46811yescharacteristic
224Dic61—82no—
225Dic61—81no—
226#3103D6111yescharacteristic, complemented
227#3036—611no—
228#2580—611no—
229#1517—611no—
230#1487—611no—
231#1478—611no—
232#1286—611no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—161no—
236C2×C61#200011yescharacteristic
237C2×C61#200811yescharacteristic
238C2×C61#201011yescharacteristic
239D61—82no—
240C2×AGL(1,ℤ/8)—1221no—
241#532C2×D6111yescharacteristic
242C2×C4×Q8C2×D6111yescharacteristic
243C2×C4×D4—611no—
244#481—21no—
245#475—21no—
246#463—1222no—
247#461—1221no—
248#460—21no—
249#459—1221no—
250#448—21no—
251C4×SD16—22no—
252#431—611no—
253#431C2×D6111yescharacteristic
254#423—611no—
255#417—1221no—
256#399—1221no—
257#393—1221no—
258#390—1221no—
259#389—1221no—
260#385—1222no—
261#382—611no—
262#327—1222no—
263Q8⋊C8—1222no—
264D4⋊C8—1222no—
265C61#999811yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C22×Q8C22⋊D6111yescharacteristic
268C22×D4—611no—
269C23×C4—1221no—
270AGL(1,ℤ/8)—2442no—
271AGL(1,ℤ/8)—2442no—
272C2×SD16—22no—
273C2×SD16—1222no—
274C2×D8—1222no—
275C2×M16—1221no—
276C4⋊Q8C22⋊D6111yescharacteristic
277#127C22⋊D6111yescharacteristic
278#124—21no—
279#121—21no—
280#121—21no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—1221no—
288C4×D4—1221no—
289C4×D4—1221no—
290C4×D4—1221no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1222no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4C22⋊D6111yescharacteristic
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1222no—
298C2×C22⋊C4—1221no—
299C2×C42—21no—
300C2×C42C22×D6111yescharacteristic
301#106—22no—
302C4⋊C8—21no—
303C4⋊C8—612no—
304C4⋊C8—1221no—
305C4⋊C8—1221no—
306#103—22no—
307#103—1222no—
308#102—22no—
309#102—1222no—
310C22⋊C8—21no—
311C22⋊C8—1221no—
312C4×C8—1221no—
313C4×C8C22⋊D6112yes—
314#95—612no—
315#95—1222no—
316#95—1221no—
317C24—1221no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—42no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4—612no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—1222no—
340C22×C4—1221no—
341C22×C4—4882no—
342C22×C4—2441no—
343C22×C4—2441no—
344C22×C4D4×D6111yescharacteristic
345C22×C4D4×D6111yescharacteristic
346C22×C4C2×C22⋊D6111yescharacteristic
347SD16—44no—
348SD16—2444no—
349D8—2444no—
350M16—2441no—
351M16—2441no—
352C2×C8—22no—
353C2×C8—1222no—
354C2×C8—2441no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—4882no—
364C4⋊C4—2441no—
365C4⋊C4—2441no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—4882no—
369C22⋊C4—2444no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C42—41no—
373C42—41no—
374C42D4×D6111yescharacteristic
375C42#1175822411yescharacteristic
376C42C2×C22⋊D6111yescharacteristic
377C42C2×C22⋊D6111yescharacteristic
378C23—41no—
379C23—1222no—
380C23—2442no—
381C23—2442no—
382C23—2441no—
383C23—2441no—
384C23#1451974511yescharacteristic
385Q8—22no—
386Q8—22no—
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—41no—
391D4—42no—
392D4—41no—
393D4—41no—
394D4—81no—
395D4—81no—
396D4—81no—
397D4—1222no—
398D4—1222no—
399D4—2441no—
400D4—2441no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—4882no—
413C2×C4—4882no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4—2441no—
419C2×C4#1451974511yescharacteristic
420C2×C4#1451974611yescharacteristic
421C2×C4#1451974811yescharacteristic
422C2×C4#1451974911yescharacteristic
423C2×C4#1451974911yescharacteristic
424C2×C4#1451977311yescharacteristic
425C2×C4SD16×D6111yescharacteristic
426C2×C4#1451979811yescharacteristic
427C2×C4#1451980411yescharacteristic
428C2×C4#1451980611yescharacteristic
429C8—42no—
430C8—4881no—
431C8—2442no—
432C22—21no—
433C22—21no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—4882no—
438C22—4881no—
439C22—4881no—
440C22—2444no—
441C22—2442no—
442C22#1683967011yescharacteristic
443C22#1683989911yescharacteristic
444C22#1683990611yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—81no—
451C4—81no—
452C4—4882no—
453C4—4881no—
454C4#1683991111yescharacteristic
455C4#1683992111yescharacteristic
456C4#1683998811yescharacteristic
457C4#1684000511yescharacteristic
458C2—21no—
459C2—21no—
460C2—161no—
461C2—4882no—
462C2#2144681911yescharacteristic
463C2#2144686911yescharacteristic
464C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3103C2
socleC22×C61#392
prime radical#21455279C2
hypercenter#3103D61
hypocenterC61#9998
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. #21451177C2semidirect
3. #21444379C2semidirect
4. #21444248C2semidirect
5. #21444211C2semidirect
6. #21443678C2semidirect
7. C61×#3103C2semidirect
110. C2×D244SD16semidirect
226. #3103D61semidirect
265. C61#9998semidirect

Maximal subgroups

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

subgroupindex
#214552792
#214511772
#214443792
#214442482
#214442112
#214436782
C61×#31032
#999861