Finite Group Databasebeta

← #29574249

Subgroups — #29574249

Order 15616.

530 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448766C211yesmaximal, characteristic
2#21445946C211yesmaximal, characteristic, complemented
3#21445816C211yesmaximal, characteristic
4C2×#16839489C211yesmaximal, characteristic
5#21443703C211yesmaximal, characteristic, complemented
6#21443660C211yesmaximal, characteristic, complemented
7C61×#1948C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519735—21no—
10C2×C4⋊#92418C2211yescharacteristic
11C2×#14519697—21no—
12C2×C4×#92418C2211yescharacteristic
13#16839664—21no—
14#16839654—22no—
15#16839632C2211yescharacteristic
16#16839619—21no—
17C4×C4⋊Dic61—21no—
18#16839524—22no—
19#16839524—21no—
20#16839489C414yescomplemented
21C2×#106×C61—21no—
22C2×C4⋊C8×C61C2211yescharacteristic
23C61×#384C2211yescharacteristic
24C4×C4⋊C4×C61C2211yescharacteristic
25C61×#340—21no—
26C22×D244D411yescharacteristic
27C22×C4×D61—21no—
28C22×#92418—21no—
29C22×#92418—21no—
30C4×D244—21no—
31C4×D244—21no—
32C4×D244C2×C412yes—
33C4×D244Q812yes—
34#14519735—42no—
35#14519735—42no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61D411yescharacteristic
39C2×C4⋊Dic61D412yes—
40C2×#11758198—21no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×C4×Dic61—22no—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46#14519697—42no—
47C4×#92418—21no—
48C4×#92418C2×C412yes—
49#14519684—22no—
50C22×C8×C61—21no—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61D411yescharacteristic
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C42×C61—21no—
57C2×C42×C61C2311yescharacteristic
58#106×C61—42no—
59C4⋊C8×C61—21no—
60C4⋊C8×C61C2×C412yes—
61#95×C61—21no—
62C23×D61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—41no—
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—22no—
71C2×D244—21no—
72C2×D244—21no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244—21no—
76C2×D244C4⋊C412yes—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61—41no—
83C2×C8×C61—22no—
84C2×C8×C61—41no—
85C2×C8×C61—41no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61—42no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61SD1612yes—
95C42×C61—41no—
96C42×C61—41no—
97C42×C61C22×C411yescharacteristic
98C42×C61C2×Q811yescharacteristic
99C42×C61#4111yescharacteristic
100C42×C61#4111yescharacteristic
101#11758198—22no—
102#11758198—22no—
103#11758198—44no—
104#11758198—82no—
105C4⋊Dic61—21no—
106C4⋊Dic61—41no—
107C4⋊Dic61—41no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110C4⋊Dic61—42no—
111C4⋊Dic61C4⋊C412yes—
112C4×Dic61—44no—
113C2×#92418—22no—
114C2×#92418—41no—
115C2×#92418—41no—
116C2×#92418—41no—
117C2×#92418—41no—
118C2×#92418—41no—
119C23×C61#12011yescharacteristic
120C22×D61—22no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—22no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—81no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×C4×C61C2×C4⋊C411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61C22⋊Q811yescharacteristic
142C2×C4×C61C22⋊Q811yescharacteristic
143C2×C4×C61#12311yescharacteristic
144C2×C4×C61C2×SD1611yescharacteristic
145C2×C4×C61#13511yescharacteristic
146C2×C4×C61#13511yescharacteristic
147C2×C4×C61#13511yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158D244—22no—
159D244—22no—
160D244—41no—
161D244—41no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—42no—
166C8×C61—42no—
167#92418—42no—
168#92418—81no—
169#8893—611nomaximal, Sylow
170C22×C61#38511yescharacteristic
171C22×C61#44911yescharacteristic
172C22×C61#44911yescharacteristic
173C22×C61#45011yescharacteristic
174C22×C61#45211yescharacteristic
175C22×C61#45411yescharacteristic
176C22×C61#45511yescharacteristic
177C2×D61—42no—
178C2×D61—42no—
179C2×D61—42no—
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—42no—
187C4×C61—81no—
188C4×C61—161no—
189C4×C61#42711yescharacteristic
190C4×C61#42811yescharacteristic
191C4×C61C4×SD1611yescharacteristic
192C4×C61#44211yescharacteristic
193C4×C61#47611yescharacteristic
194C4×C61#47811yescharacteristic
195C4×C61#48211yescharacteristic
196C4×C61#48511yescharacteristic
197Dic61—42no—
198Dic61—82no—
199#2354—611no—
200#1948D6111yescharacteristic, complemented
201#1834—611no—
202#1802—611no—
203C2×D4⋊C8—611no—
204#1296—611no—
205#1279—611no—
206C2×C61#162811yescharacteristic
207C2×C61#163511yescharacteristic
208C2×C61#164211yescharacteristic
209C2×C61#164611yescharacteristic
210C2×C61#185411yescharacteristic
211C2×C61#186011yescharacteristic
212C2×C61#186411yescharacteristic
213D61—82no—
214C2×C4×D4—611no—
215C2×#117—1221no—
216C2×#106—21no—
217C2×C4⋊C8—611no—
218C2×C4⋊C8C2×D6111yescharacteristic
219C2×#102—1221no—
220C2×C4×C8—611no—
221#390—1221no—
222#388—1222no—
223#384—611no—
224#384C2×D6111yescharacteristic
225#380—1221no—
226C4×C4⋊C4—1221no—
227C4×C4⋊C4C2×D6111yescharacteristic
228#340—21no—
229#340—1221no—
230#340—1222no—
231D4⋊C8—614no—
232C61#889311yescharacteristic, complemented, Sylow
233C22×D4—611no—
234C23×C4—1221no—
235C22×C8—21no—
236C22×C8—1221no—
237C22×C8—1221no—
238C4×D4—612no—
239C4×D4—612no—
240C4×D4—1221no—
241C4×D4—1221no—
242#117—2442no—
243#117—2442no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—611no—
248C2×C4⋊C4—612no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4C22⋊D6111yescharacteristic
252C2×C4⋊C4C22⋊D6111yescharacteristic
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1222no—
255C2×C22⋊C4—1222no—
256C2×C42—21no—
257C2×C42—1222no—
258C2×C42C22×D6111yescharacteristic
259#106—42no—
260C4⋊C8—21no—
261C4⋊C8—1221no—
262C4⋊C8—1221no—
263C4⋊C8C4×D6112yes—
264#102—2442no—
265C4×C8—612no—
266C4×C8—1221no—
267#95—21no—
268#95—1222no—
269C24—1221no—
270C2×D4—612no—
271C2×D4—1221no—
272C2×D4—1221no—
273C2×D4—1221no—
274C2×D4—1221no—
275C2×D4—1221no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—1221no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—2441no—
284C22×C4—2441no—
285C22×C4—2441no—
286C22×C4—2441no—
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289C22×C4C2×C22⋊D6111yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—1222no—
294C2×C8—2441no—
295C2×C8—2441no—
296C2×C8—2441no—
297C2×C8—2441no—
298C2×C8—2441no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—612no—
308C4⋊C4—1221no—
309C4⋊C4—2442no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4—2441no—
314C4⋊C4#1175820112yes—
315C22⋊C4—1222no—
316C22⋊C4—1222no—
317C22⋊C4—4882no—
318C22⋊C4—2444no—
319C42—41no—
320C42—41no—
321C42—2444no—
322C42C2×C4×D6111yescharacteristic
323C42#1175822211yescharacteristic
324C42Q8×D6111yescharacteristic
325C42#1175822611yescharacteristic
326C23—1222no—
327C23—2441no—
328C23—2441no—
329C23—2441no—
330C23—2441no—
331C23—2441no—
332C23—2441no—
333C23#1451974511yescharacteristic
334D4—1222no—
335D4—1222no—
336D4—2441no—
337D4—2441no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—1222no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4—2442no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4C4⋊C4×D6111yescharacteristic
360C2×C4C4×C22⋊D6111yescharacteristic
361C2×C4#1451973911yescharacteristic
362C2×C4#1451975311yescharacteristic
363C2×C4#1451975511yescharacteristic
364C2×C4#1451976211yescharacteristic
365C2×C4C2×#1175820111yescharacteristic
366C2×C4#1451979711yescharacteristic
367C2×C4#1451979811yescharacteristic
368C2×C4#1451979811yescharacteristic
369C2×C4#1451980211yescharacteristic
370C2×C4#1451980311yescharacteristic
371C2×C4#1451980411yescharacteristic
372C2×C4#1451980611yescharacteristic
373C8—42no—
374C8—4881no—
375C8—2442no—
376C22—4881no—
377C22—4881no—
378C22—4881no—
379C22—4881no—
380C22—2442no—
381C22—2442no—
382C22—2442no—
383C22#1683963311yescharacteristic
384C22#1683990511yescharacteristic
385C22#1683990611yescharacteristic
386C22#1683991211yescharacteristic
387C22#1683992111yescharacteristic
388C22#1683993311yescharacteristic
389C22#1683993811yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—81no—
394C4—161no—
395C4—4882no—
396C4—2442no—
397C4#1683979711yescharacteristic
398C4#1683980111yescharacteristic
399C4C4×#1175820111yescharacteristic
400C4#1683987211yescharacteristic
401C4#1684004911yescharacteristic
402C4#1684005911yescharacteristic
403C4#1684008811yescharacteristic
404C4#1684010611yescharacteristic
405C2—4882no—
406C2#2144492811yescharacteristic
407C2#2144496811yescharacteristic
408C2#2144501811yescharacteristic
409C2#2144504511yescharacteristic
410C2#2144601711yescharacteristic
411C2#2144604911yescharacteristic
412C2#2144607911yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1948C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1948D61
hypocenterC61#8893
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. #21445946C2semidirect
5. #21443703C2semidirect
6. #21443660C2semidirect
7. C61×#1948C2semidirect
20. #16839489C4semidirect
200. #1948D61semidirect
232. C61#8893semidirect

Maximal subgroups

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

subgroupindex
#214487662
#214459462
#214458162
C2×#168394892
#214437032
#214436602
C61×#19482
#889361