Finite Group Databasebeta

← #29580761

Subgroups — #29580761

Order 15616.

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

normal subgroups only

This group has 554 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#21455287C211yesmaximal, characteristic
2#21451185C211yesmaximal, characteristic, complemented
3#21444379C211yesmaximal, characteristic
4#21444248C211yesmaximal, characteristic, complemented
5#21444215C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#3103C211yesmaximal, characteristic
8C2×#14519807—21no—
9C2×C4×Dic122C2211yescharacteristic
10#16839981—22no—
11#16839969—21no—
12#16839959—21no—
13#16839815C2211yescharacteristic
14#16839786C2211yescharacteristic
15#16839763—21no—
16#16839700—21no—
17#16839681—21no—
18#16839668—21no—
19#16839660—21no—
20#16839643—22no—
21#16839628C2211yescharacteristic
22#16839496—22no—
23#16839494—22no—
24Q8⋊#92418—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×Dic122D411yescharacteristic
34#14519807—42no—
35#14519807—42no—
36C2×#11758202—22no—
37C2×#11758200—22no—
38C2×#11758194—21no—
39C4×Dic122—21no—
40C4×Dic122—21no—
41C4×Dic122—21no—
42C4×Dic122—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—22no—
46C2×#11758196—21no—
47C2×C4×Dic61—21no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50C4⋊#92418D412yes—
51#14519701—22no—
52#14519698—22no—
53C22⋊#92418—21no—
54C4×#92418—21no—
55#14519684—22no—
56#14519684—22no—
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—
84#41×C61—41no—
85#41×C61—41no—
86#41×C61—41no—
87#41×C61—41no—
88C2×Q8×C61—21no—
89C2×Q8×C61—21no—
90C2×Q8×C61—21no—
91C2×Q8×C61—42no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—41no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—22no—
104C22×Dic61—21no—
105C2×Dic122—21no—
106C2×Dic122—21no—
107C2×Dic122—21no—
108C2×Dic122—21no—
109C2×Dic122—21no—
110C2×Dic122SD1612yescomplemented
111SD16×C61—44no—
112C2×C8×C61—22no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C42×C61—41no—
122C42×C61—41no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127#11758202—44no—
128#11758200—44no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—41no—
132C4⋊Dic61—41no—
133#11758196—44no—
134#11758196—41no—
135#11758196—41no—
136C4×Dic61—41no—
137C4×Dic61—41no—
138#11758194—41no—
139#11758194—41no—
140C2×#92418—22no—
141C2×#92418—41no—
142C23×C61—41no—
143C23×C61#12011yescharacteristic
144Q8×C61—22no—
145Q8×C61—22no—
146Q8×C61—41no—
147Q8×C61—41no—
148Q8×C61—41no—
149Q8×C61—41no—
150D4×C61—42no—
151D4×C61—41no—
152D4×C61—41no—
153D4×C61—81no—
154D4×C61—81no—
155D4×C61—81no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12711yescharacteristic
172C2×C4×C61C2×SD1611yescharacteristic
173C2×C4×C61#13511yescharacteristic
174C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C61#13711yescharacteristic
176C2×Dic61—22no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—41no—
183C2×Dic61—42no—
184Dic122—22no—
185Dic122—22no—
186Dic122—41no—
187Dic122—41no—
188C8×C61—42no—
189#92418—42no—
190#92418—81no—
191#9999—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61#39211yescharacteristic
198C22×C61#44811yescharacteristic
199C22×C61#44911yescharacteristic
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—42no—
205C4×C61—81no—
206C4×C61—81no—
207C4×C61#45011yescharacteristic
208C4×C61#45211yescharacteristic
209C4×C61#46511yescharacteristic
210C4×C61#46811yescharacteristic
211Dic61—42no—
212Dic61—82no—
213Dic61—81no—
214#3103D6111yescharacteristic
215#3037—611no—
216#2581—611no—
217#1517—611no—
218#1487—611no—
219#1479—611no—
220#1286—611no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—161no—
224C2×C61#200011yescharacteristic
225C2×C61#200911yescharacteristic
226C2×C61#201011yescharacteristic
227C2×#137—1221no—
228#532C2×D6111yescharacteristic
229C2×C4×Q8—611no—
230C2×C4×Q8C2×D6111yescharacteristic
231#481—21no—
232#475—21no—
233#464—1222no—
234#462—1221no—
235#460—21no—
236#460—1221no—
237#448—21no—
238C4×SD16—22no—
239#431—611no—
240#431C2×D6111yescharacteristic
241#423—611no—
242#417—1221no—
243#400—1221no—
244#395—1221no—
245#391—1221no—
246#389—1221no—
247#387—1222no—
248#382—611no—
249#327—1222no—
250Q8⋊C8—1222no—
251Q8⋊C8—1222no—
252C61#999911yescharacteristic, complemented, Sylow
253C2×#41—21no—
254C22×Q8—611no—
255C22×Q8C22⋊D6111yescharacteristic
256#137—2442no—
257#137—2442no—
258C2×Q16—1222no—
259C2×SD16—22no—
260C2×SD16—1222no—
261C2×M16—1221no—
262C4⋊Q8C22⋊D6111yescharacteristic
263#127C22⋊D6111yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—1221no—
272C4×Q8—1221no—
273C4×Q8—1221no—
274C4×Q8—1221no—
275C4×D4—21no—
276C4×D4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—1222no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4C22⋊D6111yescharacteristic
283C2×C42—21no—
284C2×C42—1221no—
285C2×C42C22×D6111yescharacteristic
286#106—22no—
287C4⋊C8—21no—
288C4⋊C8—612no—
289C4⋊C8—1221no—
290C4⋊C8—1221no—
291#103—22no—
292#103—1222no—
293#102—22no—
294#102—1222no—
295C22⋊C8—21no—
296C22⋊C8—1221no—
297C4×C8—1221no—
298C4×C8C22⋊D6112yes—
299#95—612no—
300#95—1222no—
301#95—1222no—
302#95—1222no—
303#95—1221no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—42no—
312C2×Q8—612no—
313C2×Q8—1221no—
314C2×Q8—1221no—
315C2×Q8—1221no—
316C2×Q8—1221no—
317C2×Q8—1221no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—41no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—1221no—
328C22×C4D4×D6111yescharacteristic
329C22×C4D4×D6111yescharacteristic
330C22×C4C2×C22⋊D6111yescharacteristic
331Q16—2444no—
332SD16—44no—
333SD16—2444no—
334M16—2441no—
335M16—2441no—
336C2×C8—22no—
337C2×C8—1222no—
338C2×C8—2441no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—1221no—
346C4⋊C4—1221no—
347C4⋊C4—2444no—
348C4⋊C4—2441no—
349C4⋊C4—2441no—
350C4⋊C4—2441no—
351C4⋊C4—2441no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C42—41no—
355C42—41no—
356C42—2441no—
357C42—2441no—
358C42D4×D6111yescharacteristic
359C42#1175822411yescharacteristic
360C42C2×C22⋊D6111yescharacteristic
361C42C2×C22⋊D6111yescharacteristic
362C23—41no—
363C23#1451974511yescharacteristic
364Q8—22no—
365Q8—22no—
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370Q8—1222no—
371Q8—1222no—
372Q8—2441no—
373Q8—2441no—
374D4—42no—
375D4—41no—
376D4—41no—
377D4—81no—
378D4—81no—
379D4—81no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—1222no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4#1451974511yescharacteristic
399C2×C4#1451974611yescharacteristic
400C2×C4#1451974811yescharacteristic
401C2×C4#1451974911yescharacteristic
402C2×C4#1451974911yescharacteristic
403C2×C4#1451977311yescharacteristic
404C2×C4SD16×D6111yescharacteristic
405C2×C4#1451979811yescharacteristic
406C2×C4#1451980411yescharacteristic
407C2×C4#1451980611yescharacteristic
408C8—42no—
409C8—4881no—
410C8—2442no—
411C22—21no—
412C22—21no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22#1683967011yescharacteristic
417C22#1683989911yescharacteristic
418C22#1683990611yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—81no—
425C4—81no—
426C4—4882no—
427C4—4881no—
428C4—2442no—
429C4#1683991111yescharacteristic
430C4#1683992111yescharacteristic
431C4#1683998811yescharacteristic
432C4#1684000511yescharacteristic
433C2—21no—
434C2—21no—
435C2—161no—
436C2#2144681911yescharacteristic
437C2#2144687611yescharacteristic
438C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3103C2
socleC22×C61#392
prime radicalC61×#532C22
hypercenter#3103D61
hypocenterC61#9999
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. #21451185C2semidirect
4. #21444248C2semidirect
5. #21444215C2semidirect
6. #21443678C2semidirect
110. C2×Dic122SD16semidirect
252. C61#9999semidirect

Maximal subgroups

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

subgroupindex
#214552872
#214511852
#214443792
#214442482
#214442152
#214436782
C61×#31032
#999961