Finite Group Databasebeta

← #29576312

Subgroups — #29576312

Order 15616.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455851C211yesmaximal, characteristic
2#21448296C211yesmaximal, characteristic, complemented
3#21446647C211yesmaximal, characteristic
4#21444315C211yesmaximal, characteristic
5#21444249C211yesmaximal, characteristic, complemented
6#21443721C211yesmaximal, characteristic, complemented
7C61×#1276C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×C4×Dic122C2211yescharacteristic
10#16840081—21no—
11#16840048C412yescomplemented
12#16839963C412yescomplemented
13#16839903—21no—
14C4×#11758190—22no—
15#16839786C2211yescharacteristic
16C2×#14519705—21no—
17C2×#14519705—21no—
18#16839668—21no—
19#16839660—21no—
20#16839646—22no—
21C4×#11758196—21no—
22C4×C4⋊Dic61C2211yescharacteristic
23#16839608—21no—
24#16839497—22no—
25#16839494—22no—
26C61×#431C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C43×C61C2211yescharacteristic
29C2×#11758222—21no—
30C22×Dic122D411yescharacteristic
31C2×#11758190—22no—
32C2×#11758194—21no—
33#14519777C2×C411yescharacteristic
34#14519774C2×C411yescharacteristic
35#14519765—21no—
36#14519751—21no—
37#14519751—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×Dic122C2×C412yes—
41C4×Dic122D412yes—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—22no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—21no—
50#14519709—22no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53#14519705—42no—
54#14519705—42no—
55#14519702—22no—
56#14519699—22no—
57#14519684—22no—
58#14519684—22no—
59#14519684—22no—
60#14519684D412yes—
61C2×C42×C61—21no—
62C2×C42×C61—21no—
63C2×C42×C61D411yescharacteristic
64C2×C42×C61D411yescharacteristic
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61C2×C412yes—
68C22⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70C4×C8×C61—21no—
71C4×C8×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—41no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—22no—
81#11758222—41no—
82#11758222—41no—
83#11758222—41no—
84#11758222—41no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244—21no—
88C2×C4×D61—41no—
89C2×Dic122—21no—
90C2×Dic122—21no—
91C2×Dic122—42no—
92C2×Dic122C22⋊C412yes—
93C2×C8×C61—22no—
94C2×C8×C61—41no—
95C42×C61—21no—
96C42×C61—21no—
97C42×C61—21no—
98C42×C61—42no—
99C42×C61—42no—
100C42×C61SD1612yescomplemented
101C42×C61C22×C411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61#4111yescharacteristic
104C42×C61#4111yescharacteristic
105#11758198—41no—
106#11758198—41no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—42no—
111C4⋊Dic61C22⋊C412yes—
112#11758196—22no—
113#11758196—22no—
114#11758196—44no—
115C4×Dic61—22no—
116C4×Dic61—22no—
117C4×Dic61—41no—
118C4×Dic61—41no—
119#11758194—41no—
120#11758194—41no—
121C2×#92418—41no—
122#11758190—44no—
123C23×C61#12011yescharacteristic
124C22×D61—41no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61C2×C22⋊C411yescharacteristic
134C2×C4×C61C4×D411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12311yescharacteristic
139C2×C4×C61C2×SD1611yescharacteristic
140C2×C4×C61#13511yescharacteristic
141C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C61#13711yescharacteristic
143C22⋊D61—81no—
144C22⋊D61—81no—
145C2×Dic61—22no—
146C2×Dic61—22no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153D244—42no—
154D244—41no—
155D244—41no—
156D244—81no—
157C4×D61—81no—
158C4×D61—81no—
159Dic122—22no—
160Dic122—22no—
161Dic122—42no—
162Dic122—41no—
163Dic122—41no—
164C8×C61—42no—
165C8×C61—81no—
166#92418—81no—
167#9238—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61#38611yescharacteristic
171C22×C61#44811yescharacteristic
172C22×C61#44911yescharacteristic
173C2×D61—81no—
174C2×D61—81no—
175C2×D61—81no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—42no—
180C4×C61—42no—
181C4×C61#42112yes—
182C4×C61C4×SD1611yescharacteristic
183C4×C61#44011yescharacteristic
184C4×C61#46011yescharacteristic
185C4×C61#46411yescharacteristic
186Dic61—42no—
187Dic61—42no—
188Dic61—82no—
189Dic61—81no—
190#3103—611no—
191#2295—611no—
192#1962—611no—
193#1502—611no—
194#1487—611no—
195#1303—611no—
196#1276D6111yescharacteristic, complemented
197C2×C61—21no—
198C2×C61—21no—
199C2×C61#188011yescharacteristic
200C2×C61#189711yescharacteristic
201C2×C61#190611yescharacteristic
202D61—161no—
203#532—611no—
204C2×C4×Q8—611no—
205#481—1221no—
206#475—612no—
207#460—612no—
208#448—1221no—
209C4×SD16—1222no—
210#431C2×D6111yescharacteristic
211#431C2×D6111yescharacteristic
212#423—611no—
213C2×#104—1221no—
214C2×#104—1221no—
215#391—1221no—
216#389—1221no—
217#387—1222no—
218C4×C4⋊C4—611no—
219C4×C4⋊C4—1221no—
220#376—1221no—
221C43C2×D6111yescharacteristic
222#327—1222no—
223Q8⋊C8—1222no—
224C61#923811yescharacteristic, complemented, Sylow
225C2×#41—1221no—
226C22×Q8—611no—
227C2×SD16—1222no—
228C2×M16—1221no—
229C4⋊Q8—611no—
230#127—611no—
231#124—1221no—
232#121—1221no—
233#121—1221no—
234C4×Q8—612no—
235C4×Q8—612no—
236C4×Q8—1221no—
237C4×Q8—1221no—
238C4×D4—1221no—
239C4×D4—1221no—
240C2×C4⋊C4—611no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4—1222no—
243C2×C4⋊C4—1221no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—1221no—
247C2×C42—1221no—
248C2×C42D24411yescharacteristic
249C2×C42D24411yescharacteristic
250C2×C42C22×D6111yescharacteristic
251#106—1222no—
252C4⋊C8—21no—
253C4⋊C8—1221no—
254C4⋊C8—1221no—
255C4⋊C8C4×D6112yes—
256#104—2442no—
257#104—2442no—
258#103—1222no—
259#102—1222no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264#95—612no—
265#95—1222no—
266#95—1222no—
267#95—1222no—
268#41—2441no—
269#41—2441no—
270#41—2441no—
271#41—2441no—
272C2×Q8—612no—
273C2×Q8—1221no—
274C2×Q8—1221no—
275C2×Q8—2442no—
276C2×D4—1221no—
277C2×D4—1221no—
278C2×D4—1221no—
279C2×D4—2441no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—1222no—
285C22×C4—2441no—
286C22×C4C2×D24411yescharacteristic
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289SD16—2444no—
290M16—2441no—
291M16—2441no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—2441no—
295C4⋊C4—612no—
296C4⋊C4—1222no—
297C4⋊C4—1222no—
298C4⋊C4—1221no—
299C4⋊C4—1221no—
300C4⋊C4—1221no—
301C4⋊C4—2444no—
302C4⋊C4—2442no—
303C22⋊C4—2441no—
304C22⋊C4—2441no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—42no—
309C42—42no—
310C42—1222no—
311C42—1222no—
312C42—2441no—
313C42—2441no—
314C42#1175819012yescomplemented
315C42C2×C4×D6111yescharacteristic
316C42#1175822211yescharacteristic
317C42D4×D6111yescharacteristic
318C42#1175822611yescharacteristic
319C23—2441no—
320C23#1451974711yescharacteristic
321Q8—1222no—
322Q8—1222no—
323Q8—2442no—
324Q8—2441no—
325Q8—2441no—
326D4—4881no—
327D4—4881no—
328D4—4881no—
329D4—2442no—
330D4—2441no—
331D4—2441no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—4881no—
343C2×C4—4881no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2441no—
347C2×C4—2442no—
348C2×C4—2442no—
349C2×C4—2442no—
350C2×C4C22⋊C4×D6111yescharacteristic
351C2×C4#1451973911yescharacteristic
352C2×C4C4×D24411yescharacteristic
353C2×C4#1451975011yescharacteristic
354C2×C4#1451975211yescharacteristic
355C2×C4#1451976211yescharacteristic
356C2×C4C2×#1175819011yescharacteristic
357C2×C4#1451979911yescharacteristic
358C2×C4#1451980511yescharacteristic
359C2×C4#1451981011yescharacteristic
360C8—42no—
361C8—81no—
362C8—4881no—
363C22—21no—
364C22—21no—
365C22—4881no—
366C22—4881no—
367C22—4881no—
368C22#1683964011yescharacteristic
369C22#1683990311yescharacteristic
370C22#1683990911yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—4882no—
377C4—4881no—
378C4—2442no—
379C4—2442no—
380C4#1683977712yes—
381C4C4×#1175819011yescharacteristic
382C4#1683986411yescharacteristic
383C4#1683996311yescharacteristic
384C4#1683998511yescharacteristic
385C2—21no—
386C2—21no—
387C2—9761no—
388C2#2144617311yescharacteristic
389C2#2144626611yescharacteristic
390C2#2144631911yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1276C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1276D61
hypocenterC61#9238
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #21448296C2semidirect
5. #21444249C2semidirect
6. #21443721C2semidirect
7. C61×#1276C2semidirect
11. #16840048C4semidirect
12. #16839963C4semidirect
100. C42×C61SD16semidirect
196. #1276D61semidirect
224. C61#9238semidirect
314. C42#11758190semidirect

Maximal subgroups

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

subgroupindex
#214558512
#214482962
#214466472
#214443152
#214442492
#214437212
C61×#12762
#923861