Finite Group Databasebeta

← #29575948

Subgroups — #29575948

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455283C211yesmaximal, characteristic
2#21444378C211yesmaximal, characteristic, complemented
3#21444313C211yesmaximal, characteristic, complemented
4#21444210C211yesmaximal, characteristic, complemented
5#21443722C211yesmaximal, characteristic, complemented
6#21443677C211yesmaximal, characteristic, complemented
7C61×#2349C211yesmaximal, characteristic, complemented
8C2×#14519803—21no—
9#16840312C2211yescharacteristic
10#16839977—22no—
11#16839967C412yescomplemented
12#16839957C412yescomplemented
13#16839815C2211yescharacteristic
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839761—21no—
17#16839495—22no—
18#16839495—21no—
19#16839495—21no—
20D4⋊#92418—22no—
21C2×C4×D4×C61—21no—
22C2×C4×D4×C61C2211yescharacteristic
23C61×#389—21no—
24C61×#385—22no—
25C61×#382C2211yescharacteristic
26C61×#380—21no—
27C4×C4⋊C4×C61C2211yescharacteristic
28C2×#95×C61—21no—
29C2×#11758222—21no—
30#14519803—42no—
31#14519803—42no—
32C2×#11758200—22no—
33C2×#11758199—22no—
34C2×#11758194—21no—
35#14519777C2×C411yescharacteristic, complemented
36#14519774C2×C411yescharacteristic, complemented
37#14519765—21no—
38#14519751—21no—
39#14519751—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418—21no—
45C4⋊#92418C2×C412yescomplemented
46#14519700—22no—
47#14519697—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C4×#92418—21no—
51C4×#92418—21no—
52C22×D4×C61D411yescharacteristic
53C23×C4×C61—21no—
54C23×C4×C61—22no—
55C4×D4×C61—21no—
56C4×D4×C61—21no—
57C4×D4×C61—42no—
58C4×D4×C61—42no—
59C4×D4×C61C2×C412yes—
60C4×D4×C61D412yes—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61D411yescharacteristic
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—22no—
67C2×C42×C61—21no—
68C2×C42×C61C2311yescharacteristic
69#95×C61—22no—
70#95×C61—21no—
71#95×C61D411yescharacteristic
72#95×C61D411yescharacteristic
73#95×C61D412yes—
74C24×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—21no—
77C2×D4×C61—21no—
78C2×D4×C61C22⋊C412yes—
79C2×D4×C61D812yes—
80C2×D4×C61SD1612yes—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—44no—
88C22×C4×C61—44no—
89C22×C4×C61—82no—
90C22×C4×C61—82no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95#11758222—41no—
96#11758222—41no—
97#11758222—41no—
98#11758222—41no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×Dic122—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—42no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—82no—
109C4⋊C4×C61C22⋊C412yes—
110C22⋊C4×C61—22no—
111C22⋊C4×C61—22no—
112C22⋊C4×C61—44no—
113C42×C61—41no—
114C42×C61—41no—
115C42×C61C22×C411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61#4111yescharacteristic
118C42×C61#4111yescharacteristic
119#11758200—44no—
120#11758199—44no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125#11758194—41no—
126#11758194—41no—
127C2×#92418—22no—
128C2×#92418—41no—
129C2×#92418—41no—
130C23×C61—22no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61#12011yescharacteristic
136C22×D61—41no—
137D4×C61—22no—
138D4×C61—22no—
139D4×C61—22no—
140D4×C61#10214yes—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—82no—
155C2×C4×C61—82no—
156C2×C4×C61—88no—
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C2×C4×C61C4×D411yescharacteristic
159C2×C4×C61C4×D411yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12311yescharacteristic
163C2×C4×C61C2×D811yescharacteristic
164C2×C4×C61C2×SD1611yescharacteristic
165C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C61#13711yescharacteristic
167C22⋊D61—81no—
168C22⋊D61—81no—
169C2×Dic61—41no—
170D244—41no—
171D244—41no—
172D244—81no—
173D244—81no—
174C4×D61—81no—
175C4×D61—81no—
176Dic122—41no—
177Dic122—41no—
178#92418—42no—
179#92418—81no—
180#92418—81no—
181#9179—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—24no—
185C22×C61—44no—
186C22×C61—44no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61#38611yescharacteristic
190C22×C61#44711yescharacteristic
191C22×C61#44811yescharacteristic
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—42no—
200C4×C61—42no—
201C4×C61—81no—
202C4×C61—82no—
203C4×C61C2×#10211yescharacteristic
204C4×C61#41711yescharacteristic
205C4×C61#48011yescharacteristic
206C4×C61#48111yescharacteristic
207Dic61—81no—
208#3036—611no—
209#2349D6111yescharacteristic, complemented
210#1517—611no—
211#1502—611no—
212#1478—611no—
213#1303—611no—
214#1286—611no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—44no—
218C2×C61#187411yescharacteristic
219C2×C61#190411yescharacteristic
220C2×C61#190611yescharacteristic
221D61—161no—
222C2×AGL(1,ℤ/8)—1221no—
223#532—611no—
224C2×C4×D4—21no—
225C2×C4×D4C2×D6111yescharacteristic
226#463—1222no—
227#461—612no—
228#459—612no—
229#431—611no—
230#431—611no—
231#423—611no—
232#417—1221no—
233#389—21no—
234#385—22no—
235#382C2×D6111yescharacteristic
236#380—21no—
237C4×C4⋊C4C2×D6111yescharacteristic
238C2×#95—21no—
239#327—1221no—
240#327—1221no—
241#327—1222no—
242D4⋊C8—1222no—
243C61#917911yescharacteristic, complemented, Sylow
244C2×#41—1221no—
245C22×D4C22⋊D6111yescharacteristic
246C23×C4—21no—
247C23×C4—22no—
248AGL(1,ℤ/8)—2442no—
249AGL(1,ℤ/8)—2442no—
250C2×SD16—1222no—
251C2×D8—1222no—
252C2×M16—1221no—
253C4⋊Q8—611no—
254#127—611no—
255#124—1221no—
256#121—1221no—
257#121—1221no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—1221no—
263C4×D4—1221no—
264C4×D4C4×D6112yes—
265C4×D4D24412yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4C22⋊D6111yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C42—21no—
273C2×C42C22×D6111yescharacteristic
274C4⋊C8—612no—
275C4⋊C8—1221no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278#103—1222no—
279#102—1222no—
280C22⋊C8—1221no—
281C22⋊C8—1221no—
282C4×C8—1221no—
283C4×C8—1221no—
284#95—22no—
285#95—21no—
286#95C22⋊D6111yescharacteristic
287#95C22⋊D6111yescharacteristic
288#95C22⋊D6112yes—
289C24—21no—
290#41—2441no—
291#41—2441no—
292#41—2441no—
293#41—2441no—
294C2×Q8—1221no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—1221no—
299C2×D4—1221no—
300C2×D4—1221no—
301C2×D4—2441no—
302C2×D4#1175819812yes—
303C2×D4#1175819912yes—
304C2×D4#1175820112yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—2441no—
316C22×C4C2×C22⋊D6111yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319SD16—2444no—
320D8—2444no—
321M16—2441no—
322M16—2441no—
323C2×C8—1222no—
324C2×C8—2441no—
325C2×C8—2441no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4#1175819812yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—44no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—41no—
340C42—41no—
341C42C2×C4×D6111yescharacteristic
342C42C2×D24411yescharacteristic
343C42#1175822211yescharacteristic
344C42#1175822211yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—2441no—
351C23#1451974611yescharacteristic
352Q8—2441no—
353Q8—2441no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—4881no—
358D4—4881no—
359D4—4881no—
360D4—4881no—
361D4—2441no—
362D4—2441no—
363D4#1451969714yes—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—88no—
380C2×C4—4881no—
381C2×C4—4881no—
382C2×C4—2441no—
383C2×C4C2×#1175819811yescharacteristic
384C2×C4C4×C22⋊D6111yescharacteristic
385C2×C4C4×C22⋊D6111yescharacteristic
386C2×C4#1451975111yescharacteristic
387C2×C4#1451975111yescharacteristic
388C2×C4#1451976011yescharacteristic
389C2×C4C2×#1175819911yescharacteristic
390C2×C4C2×#1175820111yescharacteristic
391C2×C4#1451980311yescharacteristic
392C2×C4#1451980811yescharacteristic
393C8—4881no—
394C8—4881no—
395C8—2442no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—4881no—
404C22—4881no—
405C22—4881no—
406C22#1683964111yescharacteristic
407C22#1683989611yescharacteristic
408C22#1683990111yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—42no—
415C4—81no—
416C4—82no—
417C4—4881no—
418C4C2×#1451969711yescharacteristic
419C4#1683976111yescharacteristic
420C4#1684007511yescharacteristic
421C4#1684008011yescharacteristic
422C2—21no—
423C2—21no—
424C2—44no—
425C2—9761no—
426C2#2144613611yescharacteristic
427C2#2144630611yescharacteristic
428C2#2144632011yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2349C2
socleC22×C61#386
prime radical#21455283C2
cosocleC2×C42×C61C23
hypercenter#2349D61
hypocenterC61#9179
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #21444378C2semidirect
3. #21444313C2semidirect
4. #21444210C2semidirect
5. #21443722C2semidirect
6. #21443677C2semidirect
7. C61×#2349C2semidirect
11. #16839967C4semidirect
12. #16839957C4semidirect
35. #14519777C2×C4semidirect
36. #14519774C2×C4semidirect
45. C4⋊#92418C2×C4semidirect
209. #2349D61semidirect
243. C61#9179semidirect

Maximal subgroups

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

subgroupindex
#214552832
#214443782
#214443132
#214442102
#214437222
#214436772
C61×#23492
#917961