Finite Group Databasebeta

← #29575949

Subgroups — #29575949

Order 15616.

620 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 113 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#21455285C211yesmaximal, characteristic
2#21448727C211yesmaximal, characteristic, complemented
3#21444378C211yesmaximal, characteristic, complemented
4#21444315C211yesmaximal, characteristic, complemented
5#21444209C211yesmaximal, characteristic, complemented
6#21443720C211yesmaximal, characteristic, complemented
7C61×#1286C211yesmaximal, characteristic, complemented
8C2×#14519805—21no—
9#16840312C2211yescharacteristic
10C2×C4×D244C2211yescharacteristic
11C2×#14519739—21no—
12#16839978—22no—
13#16839968C412yescomplemented
14#16839958C412yescomplemented
15#16839815C2211yescharacteristic
16#16839764—21no—
17#16839658—21no—
18#16839636—22no—
19#16839621—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21#95×D61—21no—
22#16839497—21no—
23#16839497—21no—
24#16839495—22no—
25#16839489—22no—
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C61×#382C2211yescharacteristic
29C2×#11758222—21no—
30C22×D244D411yescharacteristic
31C22×C4×D61—21no—
32C22×C4×D61—22no—
33#14519805—42no—
34#14519805—42no—
35C2×#11758190—22no—
36C2×D488—22no—
37#14519777C2×C411yescharacteristic, complemented
38#14519774C2×C411yescharacteristic, complemented
39#14519765—21no—
40#14519751—21no—
41#14519751—21no—
42C4×D244—21no—
43C4×D244—21no—
44C4×D244—21no—
45C4×D244—21no—
46C4×D244C2×C412yes—
47C4×D244D412yes—
48#14519739—42no—
49#14519739—42no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61D411yescharacteristic
52C2×#11758196—22no—
53C2×#11758198—21no—
54C2×#11758198—22no—
55C2×C4×Dic61—21no—
56C4⋊#92418—21no—
57#14519702—22no—
58#14519699—22no—
59C22⋊#92418—21no—
60C4×#92418—21no—
61#14519684—22no—
62#14519684D411yescharacteristic
63#14519684D411yescharacteristic
64C2×M16×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C42×C61C2311yescharacteristic
67C4⋊C8×C61—21no—
68C4⋊C8×C61—21no—
69C4⋊C8×C61C2×C412yescomplemented
70C22⋊C8×C61—21no—
71C4×C8×C61—21no—
72#95×C61—21no—
73#95×C61D412yes—
74C23×D61—21no—
75C22×C4×C61—22no—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—41no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83#11758222—41no—
84#11758222—41no—
85#11758222—41no—
86#11758222—41no—
87C2×D244—21no—
88C2×D244—21no—
89C2×D244—21no—
90C2×D244—21no—
91C2×D244—21no—
92C2×D244—21no—
93C2×D244C22⋊C412yes—
94C2×D244D812yes—
95C2×D244SD1612yes—
96C2×C4×D61—22no—
97C2×C4×D61—22no—
98C2×C4×D61—44no—
99C2×C4×D61—44no—
100C2×C4×D61—41no—
101C2×C4×D61—82no—
102C2×C4×D61—82no—
103C2×Dic122—21no—
104M16×C61—41no—
105M16×C61—41no—
106C2×C8×C61—22no—
107C2×C8×C61—41no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C42×C61C22×C411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113C42×C61#4111yescharacteristic
114#11758198—22no—
115#11758198—22no—
116#11758198—44no—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—42no—
123C4⋊Dic61C22⋊C412yes—
124#11758196—82no—
125C4×Dic61—41no—
126C4×Dic61—41no—
127C2×#92418—41no—
128D488—44no—
129#11758190—44no—
130C23×C61#12011yescharacteristic
131C22×D61—22no—
132C22×D61—22no—
133C22×D61—22no—
134C22×D61—42no—
135C22×D61—42no—
136C22×D61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×C4×C61C4×D411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12311yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C2×SD1611yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C22⋊D61—81no—
153C22⋊D61—81no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163D244—22no—
164D244—22no—
165D244—22no—
166D244—41no—
167D244—41no—
168D244—81no—
169D244—81no—
170D244#10214yes—
171C4×D61—82no—
172C4×D61—82no—
173C4×D61—88no—
174C4×D61—81no—
175C4×D61—81no—
176Dic122—41no—
177Dic122—41no—
178C8×C61—42no—
179C8×C61—81no—
180#92418—81no—
181#9179—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61#38611yescharacteristic
185C22×C61#44711yescharacteristic
186C22×C61#44811yescharacteristic
187C2×D61—24no—
188C2×D61—44no—
189C2×D61—44no—
190C2×D61—42no—
191C2×D61—42no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—82no—
200C4×C61C2×#10211yescharacteristic
201C4×C61#41711yescharacteristic
202C4×C61#48011yescharacteristic
203C4×C61#48111yescharacteristic
204Dic61—42no—
205Dic61—42no—
206Dic61—81no—
207Dic61—81no—
208#3036—611no—
209#2349—611no—
210#1517—611no—
211#1502—611no—
212#1478—611no—
213#1303—611no—
214#1286D6111yescharacteristic, complemented
215C2×C61—21no—
216C2×C61—21no—
217C2×C61#187411yescharacteristic
218C2×C61#190411yescharacteristic
219C2×C61#190611yescharacteristic
220D61—44no—
221D61—161no—
222C2×AGL(1,ℤ/8)—1221no—
223#532—611no—
224C2×C4×D4—611no—
225C2×C4×D4—1221no—
226#463—1222no—
227#461—612no—
228#459—612no—
229#431—611no—
230#431C2×D6111yescharacteristic
231#423C2×D6111yescharacteristic
232#417—1221no—
233#389—1221no—
234#385—1222no—
235#382C2×D6111yescharacteristic
236#380—1221no—
237C4×C4⋊C4—611no—
238C2×#95—1221no—
239#327—1221no—
240#327—1221no—
241#327—1222no—
242D4⋊C8—1222no—
243C61#917911yescharacteristic, complemented, Sylow
244C2×#41—1221no—
245C22×D4—611no—
246C23×C4—1222no—
247C23×C4—1221no—
248AGL(1,ℤ/8)—2442no—
249AGL(1,ℤ/8)—2442no—
250C2×SD16—1222no—
251C2×D8—1222no—
252C2×M16—21no—
253C4⋊Q8—611no—
254#127—611no—
255#124—1221no—
256#121—1221no—
257#121—1221no—
258C4×D4—612no—
259C4×D4—612no—
260C4×D4—1221no—
261C4×D4—1221no—
262C4×D4—1221no—
263C4×D4—1221no—
264C4×D4—2442no—
265C4×D4—2442no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1222no—
272C2×C42—1221no—
273C2×C42C22×D6111yescharacteristic
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—1221no—
277C4⋊C8C4×D6112yescomplemented
278#103—1222no—
279#102—1222no—
280C22⋊C8—21no—
281C22⋊C8—1221no—
282C4×C8—21no—
283C4×C8—1221no—
284#95—21no—
285#95—611no—
286#95—611no—
287#95—1222no—
288#95D24412yes—
289C24—1221no—
290#41—2441no—
291#41—2441no—
292#41—2441no—
293#41—2441no—
294C2×Q8—1221no—
295C2×D4—612no—
296C2×D4—612no—
297C2×D4—612no—
298C2×D4—1221no—
299C2×D4—1221no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—1221no—
304C2×D4—2441no—
305C22×C4—22no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—1222no—
309C22×C4—1222no—
310C22×C4—1221no—
311C22×C4—4882no—
312C22×C4—4882no—
313C22×C4—2444no—
314C22×C4—2444no—
315C22×C4—2441no—
316C22×C4C2×D24411yescharacteristic
317C22×C4D4×D6111yescharacteristic
318C22×C4D4×D6111yescharacteristic
319SD16—2444no—
320D8—2444no—
321M16—41no—
322M16—41no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—2441no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—612no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—4882no—
333C4⋊C4—2442no—
334C22⋊C4—1222no—
335C22⋊C4—1222no—
336C22⋊C4—2444no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—2441no—
340C42—2441no—
341C42C2×C4×D6111yescharacteristic
342C42#1175822211yescharacteristic
343C42D4×D6111yescharacteristic
344C42#1175822611yescharacteristic
345C23—1222no—
346C23—1222no—
347C23—1222no—
348C23—2442no—
349C23—2442no—
350C23—2441no—
351C23#1451974711yescharacteristic
352Q8—2441no—
353Q8—2441no—
354D4—614no—
355D4—1222no—
356D4—1222no—
357D4—1222no—
358D4—4881no—
359D4—4881no—
360D4—4881no—
361D4—4881no—
362D4—2441no—
363D4—2441no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—1222no—
370C2×C4—1222no—
371C2×C4—4882no—
372C2×C4—4882no—
373C2×C4—4888no—
374C2×C4—4881no—
375C2×C4—4881no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2442no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4C22⋊C4×D6111yescharacteristic
384C2×C4#1451973911yescharacteristic
385C2×C4C4×D24411yescharacteristic
386C2×C4#1451975011yescharacteristic
387C2×C4#1451975211yescharacteristic
388C2×C4#1451976211yescharacteristic
389C2×C4D8×D6111yescharacteristic
390C2×C4SD16×D6111yescharacteristic
391C2×C4#1451980411yescharacteristic
392C2×C4#1451980911yescharacteristic
393C8—42no—
394C8—81no—
395C8—4881no—
396C22—21no—
397C22—21no—
398C22—1224no—
399C22—4881no—
400C22—4881no—
401C22—4881no—
402C22—2444no—
403C22—2444no—
404C22—2442no—
405C22—2442no—
406C22#1683964011yescharacteristic
407C22#1683989711yescharacteristic
408C22#1683990211yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—82no—
414C4—4881no—
415C4—4881no—
416C4—2442no—
417C4—2442no—
418C4#102×D6111yescharacteristic
419C4#1683976211yescharacteristic
420C4#1684007711yescharacteristic
421C4#1684008211yescharacteristic
422C2—21no—
423C2—21no—
424C2—9761no—
425C2—2444no—
426C2#2144613511yescharacteristic
427C2#2144630711yescharacteristic
428C2#2144632111yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1286C2
socleC22×C61#386
prime radical#21455285C2
cosocleC2×C42×C61C23
hypercenter#1286D61
hypocenterC61#9179
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
2. #21448727C2semidirect
3. #21444378C2semidirect
4. #21444315C2semidirect
5. #21444209C2semidirect
6. #21443720C2semidirect
7. C61×#1286C2semidirect
13. #16839968C4semidirect
14. #16839958C4semidirect
37. #14519777C2×C4semidirect
38. #14519774C2×C4semidirect
69. C4⋊C8×C61C2×C4semidirect
214. #1286D61semidirect
243. C61#9179semidirect
277. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214552852
#214487272
#214443782
#214443152
#214442092
#214437202
C61×#12862
#917961