Finite Group Databasebeta

← #29576285

Subgroups — #29576285

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
2#21446647C211yesmaximal, characteristic
3#21444313C211yesmaximal, characteristic
4#21444248C211yesmaximal, characteristic, complemented
5#21443722C211yesmaximal, characteristic, complemented
6#21443652C211yesmaximal, characteristic, complemented
7C42×Q8×C61C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840080—21no—
10#16840047C412yescomplemented
11#16839962C412yescomplemented
12#16839901—21no—
13C4×#11758201—22no—
14#16839815C2211yescharacteristic
15#16839815C2211yescharacteristic
16#16839786C2211yescharacteristic
17C2×#14519705—21no—
18C2×#14519705—21no—
19#16839495—22no—
20Q8⋊#92418—22no—
21C2×C4×Q8×C61—21no—
22C2×C4×Q8×C61C2211yescharacteristic
23C4×C4⋊C4×C61—21no—
24C4×C4⋊C4×C61—21no—
25C4×C4⋊C4×C61—22no—
26C4×C4⋊C4×C61C2211yescharacteristic
27C43×C61—21no—
28C43×C61C2211yescharacteristic
29C2×#11758222—21no—
30C2×#11758201—22no—
31C2×#11758194—21no—
32#14519777C2×C411yescharacteristic
33#14519774C2×C411yescharacteristic
34#14519765—21no—
35#14519751—21no—
36#14519751—21no—
37C4×D244—21no—
38C4×D244—21no—
39#14519708—22no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418C2×C412yes—
44#14519705—42no—
45#14519705—42no—
46#14519701—22no—
47#14519697—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C4×#92418—21no—
51C4×#92418—21no—
52C22×Q8×C61D411yescharacteristic
53C4×Q8×C61—28no—
54C4×Q8×C61—21no—
55C4×Q8×C61—21no—
56C4×Q8×C61C2×C412yes—
57C4×Q8×C61D412yes—
58C2×C4⋊C4×C61—22no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C42×C61—22no—
63C2×C42×C61—21no—
64C2×C42×C61—21no—
65C2×C42×C61—22no—
66C2×C42×C61—21no—
67C2×C42×C61—21no—
68C2×C42×C61D411yescharacteristic
69C2×C42×C61D411yescharacteristic
70C2×C42×C61C2311yescharacteristic
71#95×C61—22no—
72#95×C61D412yes—
73C2×Q8×C61—24no—
74C2×Q8×C61—21no—
75C2×Q8×C61C22⋊C412yes—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—22no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C2×C22⋊D61—41no—
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—
94C4⋊C4×C61—28no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61C22⋊C412yes—
100C42×C61—28no—
101C42×C61—24no—
102C42×C61—22no—
103C42×C61—22no—
104C42×C61—21no—
105C42×C61—21no—
106C42×C61—21no—
107C42×C61—42no—
108C42×C61—44no—
109C42×C61—41no—
110C42×C61—41no—
111C42×C61SD1612yescomplemented
112C42×C61C22×C411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C42×C61#4111yescharacteristic
116#11758201—44no—
117#11758198—41no—
118#11758198—41no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121#11758194—41no—
122#11758194—41no—
123C2×#92418—22no—
124C2×#92418—41no—
125C2×#92418—41no—
126C23×C61#12011yescharacteristic
127C22×D61—41no—
128Q8×C61—24no—
129Q8×C61—22no—
130Q8×C61#10414yescomplemented
131C2×C4×C61—22no—
132C2×C4×C61—24no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61C2×C22⋊C411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12311yescharacteristic
152C2×C4×C61C2×SD1611yescharacteristic
153C2×C4×C61#13511yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C61#13711yescharacteristic
156C22⋊D61—81no—
157C22⋊D61—81no—
158C2×Dic61—41no—
159D244—42no—
160D244—41no—
161D244—41no—
162D244—81no—
163C4×D61—81no—
164C4×D61—81no—
165Dic122—41no—
166Dic122—41no—
167#92418—42no—
168#92418—81no—
169#92418—81no—
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—24no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—42no—
185C4×C61—44no—
186C4×C61—42no—
187C4×C61C2×#10412yes—
188C4×C61C4×SD1611yescharacteristic
189C4×C61#44011yescharacteristic
190C4×C61#46011yescharacteristic
191C4×C61#46411yescharacteristic
192Dic61—81no—
193#3103—611no—
194C42×Q8D6111yescharacteristic, complemented
195#1962—611no—
196#1502—611no—
197#1487—611no—
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—21no—
208C2×C4×Q8C2×D6111yescharacteristic
209#481—1221no—
210#475—612no—
211#460—612no—
212#448—1221no—
213C4×SD16—1222no—
214#431—611no—
215#431—611no—
216#423—611no—
217C2×#104—1221no—
218C2×#104—1221no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2×D6111yescharacteristic
223C43—21no—
224C43C2×D6111yescharacteristic
225#327—1222no—
226Q8⋊C8—1222no—
227C61#923411yescharacteristic, complemented, Sylow
228C2×#41—1221no—
229C22×Q8C22⋊D6111yescharacteristic
230C2×SD16—1222no—
231C2×M16—1221no—
232C4⋊Q8—611no—
233#127—611no—
234#124—1221no—
235#121—1221no—
236#121—1221no—
237C4×Q8—28no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4×D6112yes—
241C4×Q8D24412yes—
242C4×D4—1221no—
243C4×D4—1221no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D6111yescharacteristic
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22⋊D6111yescharacteristic
255C2×C42C22⋊D6111yescharacteristic
256C2×C42C22×D6111yescharacteristic
257#106—1222no—
258C4⋊C8—612no—
259C4⋊C8—1221no—
260C4⋊C8—1221no—
261C4⋊C8—1221no—
262#104—2442no—
263#104—2442no—
264#103—1222no—
265#102—1222no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268C4×C8—1221no—
269C4×C8—1221no—
270#95—22no—
271#95C22⋊D6112yes—
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—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—2441no—
290C22×C4C2×C22⋊D6111yescharacteristic
291C22×C4C2×C22⋊D6111yescharacteristic
292C22×C4C2×C22⋊D6111yescharacteristic
293SD16—2444no—
294M16—2441no—
295M16—2441no—
296C2×C8—1222no—
297C2×C8—2441no—
298C2×C8—2441no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4#1175819812yes—
307C22⋊C4—2441no—
308C22⋊C4—2441no—
309C42—28no—
310C42—24no—
311C42—21no—
312C42—22no—
313C42—22no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—44no—
318C42—41no—
319C42—41no—
320C42#1175820112yescomplemented
321C42C2×C4×D6111yescharacteristic
322C42C2×D24411yescharacteristic
323C42#1175822211yescharacteristic
324C42#1175822211yescharacteristic
325C23—2441no—
326C23#1451974611yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—2441no—
330Q8—2441no—
331Q8#1451970514yescomplemented
332D4—4881no—
333D4—4881no—
334D4—4881no—
335D4—2442no—
336D4—2441no—
337D4—2441no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—4881no—
354C2×C4—4881no—
355C2×C4—2441no—
356C2×C4C2×#1175819811yescharacteristic
357C2×C4C4×C22⋊D6111yescharacteristic
358C2×C4C4×C22⋊D6111yescharacteristic
359C2×C4#1451975111yescharacteristic
360C2×C4#1451975111yescharacteristic
361C2×C4#1451976011yescharacteristic
362C2×C4C2×#1175820111yescharacteristic
363C2×C4#1451979711yescharacteristic
364C2×C4#1451980211yescharacteristic
365C2×C4#1451980811yescharacteristic
366C8—4881no—
367C8—4881no—
368C8—2442no—
369C22—21no—
370C22—21no—
371C22—4881no—
372C22—4881no—
373C22—4881no—
374C22#1683964111yescharacteristic
375C22#1683990111yescharacteristic
376C22#1683990711yescharacteristic
377C4—24no—
378C4—24no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—4881no—
386C4C2×#1451970512yes—
387C4C4×#1175820111yescharacteristic
388C4#1683986011yescharacteristic
389C4#1683996211yescharacteristic
390C4#1683998411yescharacteristic
391C2—21no—
392C2—21no—
393C2—9761no—
394C2#2144617411yescharacteristic
395C2#2144626211yescharacteristic
396C2#2144632011yescharacteristic
subgroupG/H
centerC2×C4#14519751
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC42×Q8×C61C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenterC42×Q8D61
hypocenterC61#9234
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
4. #21444248C2semidirect
5. #21443722C2semidirect
6. #21443652C2semidirect
7. C42×Q8×C61C2semidirect
10. #16840047C4semidirect
11. #16839962C4semidirect
111. C42×C61SD16semidirect
130. Q8×C61#104semidirect
194. C42×Q8D61semidirect
227. C61#9234semidirect
320. C42#11758201semidirect
331. Q8#14519705semidirect

Maximal subgroups

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

subgroupindex
#214558502
#214466472
#214443132
#214442482
#214437222
#214436522
C42×Q8×C612
#923461