Finite Group Databasebeta

← #29574247

Subgroups — #29574247

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#21448770C211yesmaximal, characteristic
2#21446580C211yesmaximal, characteristic, complemented
3#21445816C211yesmaximal, characteristic
4C2×#16839491C211yesmaximal, characteristic
5#21443702C211yesmaximal, characteristic, complemented
6#21443659C211yesmaximal, characteristic, complemented
7C61×#1834C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519735—21no—
10C2×#14519708—21no—
11C2×C4⋊#92418C2211yescharacteristic
12C2×C4⋊#92418C2211yescharacteristic
13C2×#14519697—21no—
14#16839663—21no—
15#16839652—22no—
16#16839619—21no—
17C4⋊C4×Dic61—21no—
18C4×C4⋊Dic61C2211yescharacteristic
19#16839524—21no—
20#16839524—21no—
21#16839491C414yescomplemented
22C2×C4×C8×C61C2211yescharacteristic
23C61×#384C2211yescharacteristic
24C61×#384C2211yescharacteristic
25C61×#340—22no—
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⋊Dic61D411yescharacteristic
37C2×C4⋊Dic61D411yescharacteristic
38C2×C4⋊Dic61D411yescharacteristic
39C2×#11758198—21no—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×C4×Dic61—22no—
43C2×C4×Dic61—21no—
44#14519708—42no—
45C4⋊#92418—21no—
46C4⋊#92418—21no—
47C4⋊#92418—21no—
48C4⋊#92418C2×C412yes—
49#14519697—42no—
50#14519684—22no—
51#14519684—21no—
52C22×C8×C61—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61D412yes—
59C2×C42×C61C2311yescharacteristic
60C4×C8×C61—21no—
61C4×C8×C61C2×C412yes—
62C23×D61—21no—
63C22×C4×C61—22no—
64C22×C4×C61—41no—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
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—42no—
87C4⋊C4×C61—42no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—41no—
95C4⋊C4×C61—41no—
96C42×C61C22×C411yescharacteristic
97C42×C61C2×Q811yescharacteristic
98C42×C61#4111yescharacteristic
99C42×C61#4111yescharacteristic
100#11758198—22no—
101#11758198—22no—
102#11758198—44no—
103#11758198—82no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61C4⋊C412yes—
109C4⋊Dic61SD1612yes—
110C4×Dic61—44no—
111C4×Dic61—41no—
112C4×Dic61—41no—
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—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—81no—
133C2×C4×C61—81no—
134C2×C4×C61—81no—
135C2×C4×C61C2×C4⋊C411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61C22⋊Q811yescharacteristic
139C2×C4×C61C22⋊Q811yescharacteristic
140C2×C4×C61#12311yescharacteristic
141C2×C4×C61C2×SD1611yescharacteristic
142C2×C4×C61#13511yescharacteristic
143C2×C4×C61#13511yescharacteristic
144C2×C4×C61#13511yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C61#13711yescharacteristic
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
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—82no—
187C4×C61—161no—
188C4×C61#42711yescharacteristic
189C4×C61#42811yescharacteristic
190C4×C61C4×SD1611yescharacteristic
191C4×C61#44211yescharacteristic
192C4×C61#47611yescharacteristic
193C4×C61#47811yescharacteristic
194C4×C61#48211yescharacteristic
195C4×C61#48511yescharacteristic
196Dic61—42no—
197Dic61—42no—
198Dic61—81no—
199#2354—611no—
200#1948—611no—
201#1834D6111yescharacteristic, complemented
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—1221no—
217C2×C4⋊C8—611no—
218C2×C4⋊C8—611no—
219C2×#102—1221no—
220C2×C4×C8C2×D6111yescharacteristic
221#390—1221no—
222#388—1222no—
223#384C2×D6111yescharacteristic
224#384C2×D6111yescharacteristic
225#380—1221no—
226C4×C4⋊C4—611no—
227C4×C4⋊C4—1221no—
228#340—22no—
229#340—1221no—
230#340—1221no—
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—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—611no—
252C2×C4⋊C4C22⋊D6112yes—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1222no—
255C2×C22⋊C4—1222no—
256C2×C42—1222no—
257C2×C42—1221no—
258C2×C42C22×D6111yescharacteristic
259#106—2442no—
260C4⋊C8—612no—
261C4⋊C8—1221no—
262C4⋊C8—1221no—
263C4⋊C8—1221no—
264#102—2442no—
265C4×C8—21no—
266C4×C8C4×D6112yes—
267#95—1222no—
268#95—1221no—
269C24—1221no—
270C2×D4—612no—
271C2×D4—1221no—
272C2×D4—1221no—
273C2×D4—1221no—
274C2×D4—1221no—
275C2×D4—1221no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4—1221no—
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—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—612no—
310C4⋊C4—612no—
311C4⋊C4—1221no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C22⋊C4—1222no—
316C22⋊C4—1222no—
317C22⋊C4—4882no—
318C22⋊C4—2444no—
319C42—2444no—
320C42—2441no—
321C42—2441no—
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—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—1222no—
347C2×C4—1222no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
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×C4SD16×D6111yescharacteristic
366C2×C4#1451979611yescharacteristic
367C2×C4#1451979811yescharacteristic
368C2×C4#1451980011yescharacteristic
369C2×C4#1451980111yescharacteristic
370C2×C4#1451980411yescharacteristic
371C2×C4#1451980411yescharacteristic
372C2×C4#1451980911yescharacteristic
373C8—42no—
374C8—4881no—
375C8—2442no—
376C22—4881no—
377C22—4881no—
378C22—4881no—
379C22—4881no—
380C22—2442no—
381C22—2442no—
382C22—2442no—
383C22#1683963311yescharacteristic
384C22#1683990411yescharacteristic
385C22#1683990611yescharacteristic
386C22#1683991111yescharacteristic
387C22#1683992311yescharacteristic
388C22#1683993411yescharacteristic
389C22#1683993911yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—161no—
394C4—4881no—
395C4—2442no—
396C4—2442no—
397C4#1683979911yescharacteristic
398C4#1683980311yescharacteristic
399C4#1683984911yescharacteristic
400C4#1683987311yescharacteristic
401C4#1684005111yescharacteristic
402C4#1684006111yescharacteristic
403C4#1684008911yescharacteristic
404C4#1684010511yescharacteristic
405C2—4882no—
406C2#2144492611yescharacteristic
407C2#2144496711yescharacteristic
408C2#2144501611yescharacteristic
409C2#2144504411yescharacteristic
410C2#2144601511yescharacteristic
411C2#2144605111yescharacteristic
412C2#2144607711yescharacteristic
subgroupG/H
centerC23#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1834C2
socleC23×C61#120
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1834D61
hypocenterC61#8893
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
2. #21446580C2semidirect
5. #21443702C2semidirect
6. #21443659C2semidirect
7. C61×#1834C2semidirect
21. #16839491C4semidirect
201. #1834D61semidirect
232. C61#8893semidirect

Maximal subgroups

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

subgroupindex
#214487702
#214465802
#214458162
C2×#168394912
#214437022
#214436592
C61×#18342
#889361