Finite Group Databasebeta

← #29575244

Subgroups — #29575244

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456374C211yesmaximal, characteristic
2#21454762C211yesmaximal, characteristic, complemented
3#21446739C211yesmaximal, characteristic
4#21444319C211yesmaximal, characteristic, complemented
5#21443658C211yesmaximal, characteristic, complemented
6#21443655C211yesmaximal, characteristic, complemented
7C61×#1480C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840158C411yescharacteristic, complemented
10#16840134C411yescharacteristic, complemented
11#16840114C411yescharacteristic, complemented
12#16840108C411yescharacteristic, complemented
13#16839989—22no—
14#16839891—22no—
15#16839889—21no—
16#16839886—21no—
17C4×#11758200—22no—
18#16839788—21no—
19C2×#14519704—21no—
20C2×#14519704—21no—
21#16839717—21no—
22C4×#11758194C2211yescharacteristic
23C2×C4×#92418C2211yescharacteristic
24C4×C4⋊Dic61C2211yescharacteristic
25#16839500—22no—
26C61×#532C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C2×C4⋊C8×C61C2211yescharacteristic
29C61×D4⋊C8—21no—
30C61×D4⋊C8—21no—
31C2×#11758200—22no—
32C2×#11758194D411yescharacteristic
33C2×#11758194D411yescharacteristic
34C22×#92418—21no—
35#14519777C2×C411yescharacteristic
36#14519777C2×C411yescharacteristic
37#14519769—21no—
38#14519756—21no—
39#14519756D412yes—
40C4×Dic122—21no—
41C4×Dic122D412yes—
42#14519736—21no—
43C2×C4⋊Dic61—21no—
44C2×C4×Dic61—21no—
45#14519708—22no—
46#14519706—22no—
47#14519706—22no—
48#14519704—42no—
49#14519704—42no—
50#14519700—22no—
51#14519698—22no—
52#14519688—21no—
53#14519687—22no—
54C4×#92418—21no—
55C4×#92418—21no—
56C4×#92418C2×C411yescharacteristic
57C4×#92418C2×C411yescharacteristic
58C8×Dic61—22no—
59#14519684—21no—
60C2×#41×C61—21no—
61C22×C8×C61—21no—
62C4⋊Q8×C61C2×C411yescharacteristic
63#127×C61C2×C411yescharacteristic
64#124×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C2×C42×C61C2311yescharacteristic
70C4⋊C8×C61—21no—
71C4⋊C8×C61—21no—
72C4⋊C8×C61D411yescharacteristic
73C4⋊C8×C61D411yescharacteristic
74C22⋊C8×C61—21no—
75C4×C8×C61—21no—
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79#41×C61—41no—
80C2×Q8×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×Dic61—41no—
90C2×Dic122—21no—
91C2×Dic122C22⋊C412yes—
92C2×C8×C61—21no—
93C2×C8×C61—21no—
94C2×C8×C61—41no—
95C2×C8×C61—41no—
96C2×C8×C61—41no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C22⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C42×C61C22×C411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61#4111yescharacteristic
105C22⋊Dic61—22no—
106#11758200—44no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—22no—
111C4⋊Dic61—41no—
112C4⋊Dic61C22⋊C412yes—
113#11758196—22no—
114#11758196—22no—
115C4×Dic61—22no—
116C4×Dic61—41no—
117C4×Dic61—41no—
118#11758194—21no—
119#11758194—21no—
120#11758194—21no—
121#11758194SD1612yescomplemented
122C2×#92418—21no—
123C2×#92418—21no—
124C2×#92418—21no—
125C2×#92418—21no—
126C2×#92418—42no—
127C23×C61—41no—
128C23×C61#12111yescharacteristic
129Q8×C61—41no—
130Q8×C61—41no—
131D4×C61—42no—
132D4×C61—41no—
133D4×C61—41no—
134D4×C61—41no—
135D4×C61—41no—
136D4×C61—81no—
137D4×C61—81no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—81no—
143C2×C4×C61—81no—
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61C4×D411yescharacteristic
147C2×C4×C61#12011yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12311yescharacteristic
150C2×C4×C61C2×SD1611yescharacteristic
151C2×C4×C61#13511yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C61#13711yescharacteristic
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—81no—
158C2×Dic61—81no—
159Dic122—41no—
160Dic122—41no—
161Dic122—42no—
162C8×C61—41no—
163C8×C61—41no—
164C8×C61—81no—
165#92418—22no—
166#92418—22no—
167#92418—41no—
168#92418—41no—
169#9054—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61#38611yescharacteristic
176C22×C61#46011yescharacteristic
177C22×C61#46411yescharacteristic
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—81no—
182C4×C61C4×SD1611yescharacteristic
183C4×C61#44011yescharacteristic
184C4×C61#44311yescharacteristic
185C4×C61#44411yescharacteristic
186C4×C61#44811yescharacteristic
187C4×C61#44911yescharacteristic
188Dic61—42no—
189Dic61—42no—
190Dic61—82no—
191#3167—611no—
192#2962—611no—
193#1980—611no—
194#1503—611no—
195#1480D6111yescharacteristic
196#1279—611no—
197#1278—611no—
198C2×C61—21no—
199C2×C61—21no—
200C2×C61—161no—
201C2×C61#187611yescharacteristic
202C2×C61#188011yescharacteristic
203C2×C61#188411yescharacteristic
204#533—611no—
205#532C2×D6111yescharacteristic
206#498—611no—
207#492—611no—
208#487—611no—
209#486—611no—
210#465—1222no—
211#446—1222no—
212#446—1221no—
213C8×Q8—1221no—
214C4×SD16—1222no—
215#431C2×D6111yescharacteristic
216#424—1221no—
217C2×C4⋊C8C2×D6111yescharacteristic
218C2×#104—1221no—
219C2×#104—1221no—
220#405—1221no—
221C4×M16—611no—
222C2×C4×C8—611no—
223C4×C4⋊C4—611no—
224#328—1222no—
225D4⋊C8—21no—
226D4⋊C8—21no—
227C61#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—1222no—
230C2×M16—611no—
231C2×M16—611no—
232C22×C8—21no—
233C22×C8—1221no—
234C4⋊Q8—611no—
235C4⋊Q8—611no—
236C4⋊Q8C4×D6111yescharacteristic
237#127C4×D6111yescharacteristic
238#125—1221no—
239#124—21no—
240C22⋊Q8—612no—
241C22⋊Q8—1221no—
242#121—21no—
243#121—21no—
244C4×Q8—612no—
245C4×Q8—1221no—
246C4×D4—21no—
247C4×D4—21no—
248#117—1221no—
249C2×C4⋊C4—1221no—
250C2×C42—1221no—
251C2×C42C22×D6111yescharacteristic
252#106—1222no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—1222no—
256C4⋊C8—1222no—
257C4⋊C8C22⋊D6111yescharacteristic
258C4⋊C8C22⋊D6111yescharacteristic
259#104—2442no—
260#104—2442no—
261#103—1222no—
262#102—1222no—
263C22⋊C8—21no—
264#97—1222no—
265#97—1221no—
266C4×C8—21no—
267C4×C8—611no—
268C4×C8—611no—
269C4×C8—1222no—
270C4×C8—1221no—
271C4×C8—1221no—
272#95—1221no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—612no—
279C2×Q8—1221no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—2441no—
286C22×C4#1175822211yescharacteristic
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289SD16—2444no—
290M16—612no—
291M16—1221no—
292M16—1221no—
293M16—1221no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—1221no—
300C2×C8—1221no—
301C2×C8—1221no—
302C2×C8—1221no—
303C2×C8—2442no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—612no—
307C4⋊C4—1222no—
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C4⋊C4—1221no—
311C4⋊C4—1222no—
312C4⋊C4—1222no—
313C4⋊C4—2441no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—1222no—
317C42—1222no—
318C42—2441no—
319C42—2441no—
320C42C2×C4×D6111yescharacteristic
321C42D4×D6111yescharacteristic
322C42#1175822411yescharacteristic
323C42C2×C22⋊D6111yescharacteristic
324C23—41no—
325C23#1451975011yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—2442no—
329Q8—2441no—
330Q8—2441no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—2442no—
349C2×C4C22⋊C4×D6111yescharacteristic
350C2×C4#1451973711yescharacteristic
351C2×C4C4×C22⋊D6111yescharacteristic
352C2×C4#1451974511yescharacteristic
353C2×C4#1451974811yescharacteristic
354C2×C4#1451975911yescharacteristic
355C2×C4SD16×D6111yescharacteristic
356C2×C4#1451979611yescharacteristic
357C2×C4#1451980111yescharacteristic
358C2×C4#1451980611yescharacteristic
359C8—41no—
360C8—41no—
361C8—81no—
362C8—1222no—
363C8—1222no—
364C8—2441no—
365C8—2441no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1683963811yescharacteristic
372C22#1683996111yescharacteristic
373C22#1683998311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—4882no—
379C4—2442no—
380C4—2442no—
381C4#1683985311yescharacteristic
382C4#1683985911yescharacteristic
383C4#1683987711yescharacteristic
384C4#1683988211yescharacteristic
385C4#1683989911yescharacteristic
386C4#1683990411yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2144614911yescharacteristic
391C2#2144617111yescharacteristic
392C2#2144619911yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1480C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#1480D61
hypocenterC61#9054
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #21454762C2semidirect
4. #21444319C2semidirect
5. #21443658C2semidirect
6. #21443655C2semidirect
9. #16840158C4semidirect
10. #16840134C4semidirect
11. #16840114C4semidirect
12. #16840108C4semidirect
121. #11758194SD16semidirect
227. C61#9054semidirect

Maximal subgroups

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

subgroupindex
#214563742
#214547622
#214467392
#214443192
#214436582
#214436552
C61×#14802
#905461