Finite Group Databasebeta

← #29576283

Subgroups — #29576283

Order 15616.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455850C211yesmaximal, characteristic
2C4×Q8×Dic61C211yesmaximal, characteristic, complemented
3#21446646C211yesmaximal, characteristic
4#21444315C211yesmaximal, characteristic
5#21443720C211yesmaximal, characteristic, complemented
6#21443651C211yesmaximal, characteristic, complemented
7C61×#1487C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×Q8×Dic61—21no—
10#16840080—21no—
11#16840047C412yescomplemented
12#16839962C412yescomplemented
13#16839901—21no—
14C4×#11758201—22no—
15#16839815C2211yescharacteristic
16C2×#14519703—21no—
17C2×#14519703—21no—
18C4⋊C4×Dic61—22no—
19C4⋊C4×Dic61—21no—
20C4×C4⋊Dic61—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22C42×Dic61—21no—
23C42×Dic61C2211yescharacteristic
24#16839497—22no—
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C61×Q8⋊C8—22no—
29C2×#11758222—21no—
30C2×#11758201—22no—
31#14519777C2×C411yescharacteristic
32#14519774C2×C411yescharacteristic
33#14519765—21no—
34#14519751—21no—
35#14519751—21no—
36Q8×Dic61—28no—
37C4×D244—21no—
38C4×D244—21no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—21no—
41C2×C4×Dic61—22no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61D411yescharacteristic
47C2×C4×Dic61D411yescharacteristic
48#14519708—22no—
49C4⋊#92418C2×C412yes—
50#14519703—42no—
51#14519703—42no—
52#14519701—22no—
53#14519697—22no—
54C22⋊#92418—21no—
55C4×#92418—21no—
56#14519684—22no—
57#14519684D412yes—
58C22×Q8×C61D411yescharacteristic
59C2×M16×C61—21no—
60C4×Q8×C61—21no—
61C4×Q8×C61—21no—
62C4×Q8×C61C2×C412yes—
63C4×Q8×C61D412yes—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C42×C61—21no—
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61—21no—
69C4⋊C8×C61—21no—
70C4⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72C4×C8×C61—21no—
73C2×Q8×C61—24no—
74C2×Q8×C61—21no—
75C2×Q8×C61C22⋊C412yes—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C2×C22⋊D61—41no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—22no—
85#11758222—41no—
86#11758222—41no—
87#11758222—41no—
88#11758222—41no—
89C2×D244—21no—
90C2×D244—21no—
91C2×D244—21no—
92C2×C4×D61—41no—
93C2×Dic122—21no—
94M16×C61—41no—
95M16×C61—41no—
96C2×C8×C61—41no—
97C2×C8×C61—41no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61C22⋊C412yes—
102C42×C61—22no—
103C42×C61—22no—
104C42×C61C22×C411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61#4111yescharacteristic
107C42×C61#4111yescharacteristic
108#11758201—44no—
109#11758198—41no—
110#11758198—41no—
111C4⋊Dic61—28no—
112C4⋊Dic61—24no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4×Dic61—28no—
116C4×Dic61—24no—
117C4×Dic61—21no—
118C4×Dic61—21no—
119C4×Dic61—21no—
120C4×Dic61—42no—
121C4×Dic61—44no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124C4×Dic61SD1612yescomplemented
125C2×#92418—22no—
126C23×C61#12011yescharacteristic
127C22×D61—41no—
128Q8×C61—24no—
129Q8×C61—22no—
130Q8×C61#10414yescomplemented
131C2×C4×C61—24no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61#13511yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61#13711yescharacteristic
147C22⋊D61—81no—
148C22⋊D61—81no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—22no—
153C2×Dic61—24no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—41no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159D244—42no—
160D244—41no—
161D244—41no—
162D244—81no—
163C4×D61—81no—
164C4×D61—81no—
165Dic122—41no—
166Dic122—41no—
167C8×C61—81no—
168C8×C61—81no—
169#92418—42no—
170#9234—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61#38611yescharacteristic
174C22×C61#44811yescharacteristic
175C22×C61#44911yescharacteristic
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C4×C61—24no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—42no—
184C4×C61C2×#10412yes—
185C4×C61C4×SD1611yescharacteristic
186C4×C61#44011yescharacteristic
187C4×C61#46011yescharacteristic
188C4×C61#46411yescharacteristic
189Dic61—24no—
190Dic61—42no—
191Dic61—44no—
192Dic61—81no—
193#3103—611no—
194C42×Q8—611no—
195#1962—611no—
196#1502—611no—
197#1487D6111yescharacteristic, complemented
198#1303—611no—
199#1276—611no—
200C2×C61—21no—
201C2×C61—21no—
202C2×C61#188011yescharacteristic
203C2×C61#189611yescharacteristic
204C2×C61#190611yescharacteristic
205D61—161no—
206#532—611no—
207C2×C4×Q8—1221no—
208C2×C4×Q8C2×D6111yescharacteristic
209#481—1221no—
210#475—612no—
211#460—612no—
212#448—1221no—
213C4×SD16—1222no—
214#431—611no—
215#431C2×D6111yescharacteristic
216#423C2×D6111yescharacteristic
217C2×#104—1221no—
218C2×#104—1221no—
219C4×C4⋊C4—611no—
220C4×C4⋊C4—1222no—
221C4×C4⋊C4—1221no—
222C4×C4⋊C4—1221no—
223C43—611no—
224C43—1221no—
225#327—1222no—
226Q8⋊C8—22no—
227C61#923411yescharacteristic, complemented, Sylow
228C2×#41—1221no—
229C22×Q8D24411yescharacteristic
230C2×SD16—1222no—
231C2×M16—21no—
232C4⋊Q8—611no—
233#127—611no—
234#124—1221no—
235#121—1221no—
236#121—1221no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—1228no—
240C4×Q8C4×D6112yes—
241C4×Q8C22⋊D6112yes—
242C4×D4—1221no—
243C4×D4—1221no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4D24411yescharacteristic
248C2×C42—21no—
249C2×C42—611no—
250C2×C42—611no—
251C2×C42—1222no—
252C2×C42—1221no—
253C2×C42—1222no—
254C2×C42—1221no—
255C2×C42—1221no—
256C2×C42C22×D6111yescharacteristic
257#106—1222no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—612no—
262#104—2442no—
263#104—2442no—
264#103—1222no—
265#102—1222no—
266C22⋊C8—21no—
267C22⋊C8—1221no—
268C4×C8—21no—
269C4×C8—1221no—
270#95—612no—
271#95—1222no—
272#41—2441no—
273#41—2441no—
274#41—2441no—
275#41—2441no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—1221no—
279C2×Q8#1175819812yes—
280C2×D4—1221no—
281C2×D4—1221no—
282C2×D4—1221no—
283C2×D4—2441no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1221no—
289C22×C4—2441no—
290C22×C4C2×D24411yescharacteristic
291C22×C4D4×D6111yescharacteristic
292C22×C4D4×D6111yescharacteristic
293SD16—2444no—
294M16—41no—
295M16—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—1222no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—1228no—
303C4⋊C4—1224no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4#1175819812yes—
307C22⋊C4—2441no—
308C22⋊C4—2441no—
309C42—22no—
310C42—22no—
311C42—612no—
312C42—1228no—
313C42—1224no—
314C42—1221no—
315C42—1221no—
316C42—1221no—
317C42—2442no—
318C42—2444no—
319C42—2441no—
320C42—2441no—
321C42C2×C4×D6111yescharacteristic
322C42#1175822411yescharacteristic
323C42#1175822411yescharacteristic
324C42C2×C22⋊D6111yescharacteristic
325C23—2441no—
326C23#1451974711yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—2441no—
330Q8—2441no—
331Q8#1451970314yescomplemented
332D4—4881no—
333D4—4881no—
334D4—4881no—
335D4—2442no—
336D4—2441no—
337D4—2441no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—1224no—
348C2×C4—1222no—
349C2×C4—4881no—
350C2×C4—4881no—
351C2×C4—2441no—
352C2×C4—2442no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4C2×#1175819811yescharacteristic
357C2×C4#1451973711yescharacteristic
358C2×C4#1451973711yescharacteristic
359C2×C4#1451974911yescharacteristic
360C2×C4#1451974911yescharacteristic
361C2×C4#1451976111yescharacteristic
362C2×C4SD16×D6111yescharacteristic
363C2×C4#1451980011yescharacteristic
364C2×C4#1451980411yescharacteristic
365C2×C4#1451980911yescharacteristic
366C8—81no—
367C8—81no—
368C8—2442no—
369C22—21no—
370C22—21no—
371C22—4881no—
372C22—4881no—
373C22—4881no—
374C22#1683963911yescharacteristic
375C22#1683990211yescharacteristic
376C22#1683991011yescharacteristic
377C4—24no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—1224no—
383C4—4881no—
384C4—2442no—
385C4—2444no—
386C4C2×#1451970312yes—
387C4#1683984711yescharacteristic
388C4#1683986511yescharacteristic
389C4#1683996011yescharacteristic
390C4#1683998611yescharacteristic
391C2—21no—
392C2—21no—
393C2—9761no—
394C2#2144617211yescharacteristic
395C2#2144626011yescharacteristic
396C2#2144631811yescharacteristic
subgroupG/H
centerC2×C4#14519749
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1487C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1487D61
hypocenterC61#9234
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. C4×Q8×Dic61C2semidirect
5. #21443720C2semidirect
6. #21443651C2semidirect
7. C61×#1487C2semidirect
11. #16840047C4semidirect
12. #16839962C4semidirect
124. C4×Dic61SD16semidirect
130. Q8×C61#104semidirect
197. #1487D61semidirect
227. C61#9234semidirect
331. Q8#14519703semidirect

Maximal subgroups

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

subgroupindex
#214558502
C4×Q8×Dic612
#214466462
#214443152
#214437202
#214436512
C61×#14872
#923461