Finite Group Databasebeta

← #29576713

Subgroups — #29576713

Order 15616.

594 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456762C211yesmaximal, characteristic
2#21448542C211yesmaximal, characteristic
3#21446799C211yesmaximal, characteristic, complemented
4#21444488C211yesmaximal, characteristic
5#21444382C211yesmaximal, characteristic, complemented
6#21443647C211yesmaximal, characteristic, complemented
7C61×#1286C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×#14519774C2211yescharacteristic
10C2×#14519739—21no—
11#16840135C412yescomplemented
12#16840109C412yescomplemented
13#16839963—22no—
14#16839915—22no—
15#16839815C2211yescharacteristic
16#16839663—21no—
17#16839640—22no—
18#16839640—22no—
19#16839608C2211yescharacteristic
20#16839523—21no—
21#16839523—21no—
22#16839505—22no—
23#16839498—22no—
24C61×#431C2211yescharacteristic
25C61×#423C2211yescharacteristic
26C61×#382C2211yescharacteristic
27C22×D244—21no—
28C22×D244—21no—
29C22×D244D412yes—
30C22×C4×D61—22no—
31C2×#11758190—24no—
32#14519777C2×C411yescharacteristic, complemented
33#14519777C2×C411yescharacteristic, complemented
34#14519774—21no—
35#14519774—21no—
36#14519774C2×C412yes—
37#14519774D412yes—
38#14519769—21no—
39#14519756—21no—
40#14519756—21no—
41C4×Dic122—21no—
42C4×Dic122—21no—
43#14519739—42no—
44#14519739—42no—
45#14519736—21no—
46C2×#11758198—22no—
47C2×#11758198—22no—
48C2×#11758198—22no—
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C4⋊#92418—21no—
52#14519699—24no—
53C22⋊#92418—21no—
54C4×#92418—21no—
55#14519684—21no—
56#14519684—21no—
57C2×M16×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60C4⋊C8×C61—21no—
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64C4×C8×C61C2×C412yescomplemented
65#95×C61—21no—
66#95×C61D412yes—
67C23×D61—22no—
68C22×C4×C61—22no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C2×D244—22no—
75C2×D244—22no—
76C2×D244—22no—
77C2×D244—22no—
78C2×D244—22no—
79C2×D244—41no—
80C2×D244—41no—
81C2×D244—41no—
82C2×D244—41no—
83C2×D244—82no—
84C2×D244—82no—
85C2×D244C22⋊C414yes—
86C2×C4×D61—44no—
87C2×C4×D61—82no—
88C2×C4×D61—82no—
89C2×Dic122—21no—
90C2×Dic122—21no—
91M16×C61—41no—
92M16×C61—41no—
93C2×C8×C61—22no—
94C2×C8×C61—41no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C42×C61C22×C411yescharacteristic
98C42×C61C2×D411yescharacteristic
99C42×C61#4111yescharacteristic
100C42×C61#4111yescharacteristic
101C22⋊Dic61—41no—
102#11758198—44no—
103#11758198—82no—
104#11758198—82no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109#11758196—41no—
110#11758196—41no—
111C4×Dic61—21no—
112C4×Dic61—21no—
113C4×Dic61—21no—
114C4×Dic61—41no—
115C4×Dic61SD1612yescomplemented
116C2×#92418—41no—
117#11758190—84no—
118C23×C61#12011yescharacteristic
119C22×D61—24no—
120C22×D61—42no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—42no—
125C22×D61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61C2×C22⋊C411yescharacteristic
132C2×C4×C61C4×D411yescharacteristic
133C2×C4×C61C4×D411yescharacteristic
134C2×C4×C61#12111yescharacteristic
135C2×C4×C61#12111yescharacteristic
136C2×C4×C61#12311yescharacteristic
137C2×C4×C61C2×SD1611yescharacteristic
138C2×C4×C61#13511yescharacteristic
139C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C61#13711yescharacteristic
141C2×Dic61—22no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149D244—44no—
150D244—42no—
151D244—42no—
152D244—42no—
153D244—42no—
154D244—82no—
155D244—82no—
156D244—82no—
157C4×D61—88no—
158Dic122—42no—
159Dic122—81no—
160C8×C61—42no—
161C8×C61—81no—
162#92418—81no—
163#9295—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#38611yescharacteristic
167C22×C61#45011yescharacteristic
168C22×C61#45211yescharacteristic
169C2×D61—44no—
170C2×D61—44no—
171C2×D61—44no—
172C2×D61—82no—
173C2×D61—82no—
174C2×D61—82no—
175C2×D61—82no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—82no—
181C4×C61C4×SD1611yescharacteristic
182C4×C61#44011yescharacteristic
183C4×C61#46111yescharacteristic
184C4×C61#46311yescharacteristic
185Dic61—42no—
186Dic61—81no—
187Dic61—81no—
188Dic61—81no—
189#3208—611no—
190#2328—611no—
191#1995—611no—
192#1545—611no—
193#1518—611no—
194#1286D6111yescharacteristic, complemented
195#1274—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61#187711yescharacteristic
199C2×C61#190411yescharacteristic
200C2×C61#191111yescharacteristic
201D61—84no—
202#533—611no—
203C2×#127—611no—
204C2×C4×D4—1221no—
205#492—612no—
206#486—612no—
207#460—1222no—
208#450—1222no—
209#431—611no—
210#431C2×D6111yescharacteristic
211#423C2×D6111yescharacteristic
212#390—1221no—
213#386—1222no—
214#386—1222no—
215#382C2×D6111yescharacteristic
216#376—611no—
217#339—1221no—
218#339—1221no—
219#331—1222no—
220#328—1222no—
221C61#929511yescharacteristic, complemented, Sylow
222C22×D4—612no—
223C22×D4—1221no—
224C22×D4—1221no—
225C23×C4—1222no—
226C2×SD16—1224no—
227C2×M16—21no—
228C4⋊Q8—611no—
229C4⋊Q8—611no—
230#127—612no—
231#127—612no—
232#127—1221no—
233#127—1221no—
234#125—1221no—
235C22⋊Q8—1221no—
236C22⋊Q8—1221no—
237C4×Q8—1221no—
238C4×Q8—1221no—
239C4×D4—2442no—
240C4×D4—2442no—
241#117—1221no—
242C2×C4⋊C4—21no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1222no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—1221no—
253#102—1224no—
254C22⋊C8—21no—
255C22⋊C8—1221no—
256C4×C8—1221no—
257C4×C8C4×D6112yescomplemented
258#95—21no—
259#95—1221no—
260#95—1221no—
261#95D24412yes—
262C24—1222no—
263C2×Q8—1221no—
264C2×Q8—1221no—
265C2×D4—614no—
266C2×D4—1222no—
267C2×D4—1222no—
268C2×D4—1222no—
269C2×D4—1222no—
270C2×D4—1222no—
271C2×D4—4882no—
272C2×D4—4882no—
273C2×D4—2441no—
274C2×D4—2441no—
275C2×D4—2441no—
276C2×D4—2441no—
277C22×C4—22no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—4882no—
281C22×C4—4882no—
282C22×C4—2444no—
283C22×C4C2×D24411yescharacteristic
284C22×C4D4×D6111yescharacteristic
285C22×C4D4×D6111yescharacteristic
286SD16—4884no—
287M16—41no—
288M16—41no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—2441no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—1221no—
295C4⋊C4—1221no—
296C4⋊C4—1221no—
297C4⋊C4—1221no—
298C4⋊C4—2441no—
299C4⋊C4—2441no—
300C22⋊C4—4882no—
301C22⋊C4—4882no—
302C22⋊C4—2444no—
303C22⋊C4—2441no—
304C42—612no—
305C42—1221no—
306C42—1221no—
307C42—1221no—
308C42—2441no—
309C42C2×C4×D6111yescharacteristic
310C42#1175822211yescharacteristic
311C42D4×D6111yescharacteristic
312C42#1175822611yescharacteristic
313C23—1224no—
314C23—2442no—
315C23—2442no—
316C23—2442no—
317C23—2442no—
318C23—2442no—
319C23—2442no—
320C23#1451974711yescharacteristic
321Q8—4881no—
322Q8—2442no—
323D4—4882no—
324D4—4882no—
325D4—4882no—
326D4—2444no—
327D4—2442no—
328D4—2442no—
329D4—2442no—
330D4—2442no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—1222no—
337C2×C4—4888no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4C22⋊C4×D6111yescharacteristic
346C2×C4#1451973911yescharacteristic
347C2×C4C4×D24411yescharacteristic
348C2×C4#1451975011yescharacteristic
349C2×C4#1451975211yescharacteristic
350C2×C4#1451976211yescharacteristic
351C2×C4SD16×D6111yescharacteristic
352C2×C4#1451979811yescharacteristic
353C2×C4#1451980111yescharacteristic
354C2×C4#1451980911yescharacteristic
355C8—42no—
356C8—81no—
357C8—4881no—
358C22—21no—
359C22—21no—
360C22—4882no—
361C22—4882no—
362C22—4882no—
363C22—4882no—
364C22—2444no—
365C22—2444no—
366C22—2444no—
367C22#1683964011yescharacteristic
368C22#1683991411yescharacteristic
369C22#1683992711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—4881no—
376C4—4881no—
377C4—4881no—
378C4—2442no—
379C4#1683984911yescharacteristic
380C4#1683986111yescharacteristic
381C4#1683996611yescharacteristic
382C4#1683997611yescharacteristic
383C2—21no—
384C2—21no—
385C2—4884no—
386C2#2144615411yescharacteristic
387C2#2144630511yescharacteristic
388C2#2144635011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1286C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#1286D61
hypocenterC61#9295
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
3. #21446799C2semidirect
5. #21444382C2semidirect
6. #21443647C2semidirect
7. C61×#1286C2semidirect
11. #16840135C4semidirect
12. #16840109C4semidirect
32. #14519777C2×C4semidirect
33. #14519777C2×C4semidirect
64. C4×C8×C61C2×C4semidirect
115. C4×Dic61SD16semidirect
194. #1286D61semidirect
221. C61#9295semidirect
257. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214567622
#214485422
#214467992
#214444882
#214443822
#214436472
C61×#12862
#929561