Finite Group Databasebeta

← #29652859

Subgroups — #29652859

Order 15616.

1166 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 167 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#21459590C211yesmaximal, characteristic
2#21455941C212yesmaximal, complemented
3#21444821C214yesmaximal, complemented
4#21444787C214yesmaximal, complemented
5#21444326C212yesmaximal, complemented
6#21444163C211yesmaximal, characteristic, complemented
7C61×#2399C211yesmaximal, characteristic, complemented
8C2×#14519825—21no—
9#16840434—22no—
10Q8×D244—22no—
11#16840424—24no—
12#16840343—21no—
13#16840343C2214yes—
14#16840312—22no—
15#16840312C2212yes—
16#16840075C2214yes—
17#16840057—22no—
18#16839962—22no—
19#16839901C2214yes—
20#16839860—24no—
21#16839819C2212yes—
22#16839495C2218yes—
23#16839488C2214yes—
24C2×C4×Q8×C61C2211yescharacteristic
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#398—21no—
27C61×#398C2212yes—
28C61×#395—21no—
29C61×#395C2212yes—
30C4×C4⋊C4×C61—22no—
31C61×#376C2211yescharacteristic
32#14519825—48no—
33C2×#11758226—24no—
34C2×#11758222—24no—
35C2×#11758222—22no—
36C2×Q8×D61—24no—
37C2×#11758201—24no—
38#14519777—22no—
39#14519777D412yes—
40#14519774—22no—
41#14519774D412yes—
42#14519765—24no—
43#14519765—22no—
44#14519757—24no—
45#14519757—28no—
46#14519752—24no—
47#14519752—28no—
48#14519751—22no—
49#14519751—24no—
50#14519751C2314yes—
51C4×D244—24no—
52C4×D244—24no—
53C4×D244—22no—
54C4×D244D414yes—
55#14519710—24no—
56C4⋊#92418—22no—
57#14519701—24no—
58#14519697—24no—
59C22⋊#92418C2314yes—
60#14519688—22no—
61C22×Q8×C61D411yescharacteristic
62C22×Q8×C61C2311yescharacteristic
63C4×Q8×C61—22no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×Q8×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C4⋊C4×C61C2311yescharacteristic
72C2×C4⋊C4×C61C2312yes—
73C2×C42×C61—21no—
74C2×C42×C61—21no—
75C2×C42×C61D412yes—
76C2×C42×C61C2311yescharacteristic
77#95×C61—21no—
78#95×C61—22no—
79#95×C61—22no—
80#95×C61C2312yes—
81C2×Q8×C61—22no—
82C2×Q8×C61—24no—
83C2×Q8×C61—22no—
84C2×Q8×C61—42no—
85C22×C4×C61—22no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D412yes—
92C22×C4×C61C2411yescharacteristic
93C2×C22⋊D61—24no—
94#11758226—416no—
95Q8×D61—416no—
96#11758222—48no—
97#11758222—48no—
98#11758222—48no—
99C2×D244—24no—
100C2×D244—22no—
101C2×D244—22no—
102C2×D244—24no—
103C2×D244C2×D414yes—
104C2×C4×D61—28no—
105C2×C4×D61—24no—
106C2×Dic122—24no—
107C2×Dic122—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—44no—
112C4⋊C4×C61—42no—
113C4⋊C4×C61—44no—
114C42×C61—24no—
115C42×C61—22no—
116C42×C61—22no—
117C42×C61—41no—
118C42×C61—41no—
119C42×C61C2×D412yes—
120C42×C61C2×D412yes—
121#11758201—48no—
122#11758198—28no—
123#11758198—24no—
124#11758198—24no—
125C4⋊Dic61—24no—
126C4⋊Dic61—22no—
127C4⋊Dic61C2×D414yes—
128C2×#92418—24no—
129C23×C61C22×D411yescharacteristic
130C22×D61—24no—
131Q8×C61—28no—
132Q8×C61—24no—
133Q8×C61—42no—
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—22no—
141C2×C4×C61—41no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61#12011yescharacteristic
150C2×C4×C61#12012yes—
151C2×C4×C61#12011yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)14yes—
153C2×C4×C61C22×D412yes—
154C22⋊D61—48no—
155C2×Dic61—24no—
156D244—44no—
157D244—44no—
158D244—48no—
159D244—44no—
160D244—84no—
161C4×D61—48no—
162C4×D61—416no—
163Dic122—44no—
164Dic122—48no—
165#92418—44no—
166#18187—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61C2×#12011yescharacteristic
170C22×C61C2×AGL(1,ℤ/8)12yes—
171C2×D61—44no—
172C2×D61—44no—
173C2×D61—44no—
174C4×C61—22no—
175C4×C61—24no—
176C4×C61—21no—
177C4×C61—22no—
178C4×C61—44no—
179C4×C61—42no—
180C4×C61—42no—
181C4×C61—81no—
182C4×C61#45612yes—
183Dic61—44no—
184#3497—611no—
185#3112—612no—
186#2399D6111yescharacteristic, complemented
187#1610—614no—
188#1604—614no—
189#1505—612no—
190#1467—611no—
191C2×C61—22no—
192C2×C61#300311yescharacteristic
193C2×C61C2×#45611yescharacteristic
194C2×C61#302511yescharacteristic
195D61—84no—
196C2×2−1+4—1221no—
197#550—1222no—
198D4×Q8—1222no—
199#548—1224no—
200#537—614no—
201#537—1221no—
202#532—612no—
203#532—1222no—
204C2×C4×Q8C2×D6111yescharacteristic
205C2×C4×Q8C2×D6111yescharacteristic
206#480—614no—
207#477—1222no—
208#460—1222no—
209#448—614no—
210#440—1224no—
211#432—612no—
212#398—21no—
213#398C2×D6112yes—
214#395—21no—
215#395C2×D6112yes—
216C4×C4⋊C4—22no—
217#376C2×D6111yescharacteristic
218#327—618no—
219#324—614no—
220C61#1818711yescharacteristic, complemented, Sylow
2212−1+4—2448no—
222C2×#41—1224no—
223C2×#41—1222no—
224C2×#41—1224no—
225C22×Q8—1224no—
226C22×Q8C22⋊D6111yescharacteristic
227C22×Q8C22×D6111yescharacteristic
228C2×SD16—1224no—
229C4⋊Q8—612no—
230C4⋊Q8—1222no—
231#127—612no—
232#127—1222no—
233#124—1224no—
234#124—1222no—
235C22⋊Q8—1228no—
236C22⋊Q8—1224no—
237#121—614no—
238#121—1228no—
239#121—1224no—
240#121—1222no—
241#121—1224no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×D4—614no—
247C4×D4—1224no—
248C4×D4—1224no—
249C4×D4—1222no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C22⋊D6111yescharacteristic
254C2×C4⋊C4C22×D6111yescharacteristic
255C2×C4⋊C4C22×D6112yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C22⋊D6112yes—
259C2×C42C22×D6111yescharacteristic
260#107—1224no—
261C4⋊C8—1222no—
262#103—1224no—
263#102—1224no—
264C22⋊C8—614no—
265#97—1222no—
266#95—21no—
267#95—22no—
268#95—22no—
269#95C22×D6112yes—
270#41—2448no—
271#41—2448no—
272#41—24416no—
273#41—2448no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—42no—
278C2×Q8—1224no—
279C2×Q8—1222no—
280C2×Q8—24416no—
281C2×D4—614no—
282C2×D4—1224no—
283C2×D4—1224no—
284C2×D4—1224no—
285C2×D4—1222no—
286C2×D4—1222no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—1228no—
291C22×C4—1224no—
292C22×C4C2×C22⋊D6111yescharacteristic
293C22×C4C2×C22⋊D6111yescharacteristic
294C22×C4C2×C22⋊D6112yes—
295C22×C4C2×C22⋊D6112yes—
296C22×C4C23×D6111yescharacteristic
297SD16—2448no—
298C2×C8—1224no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—614no—
306C4⋊C4—1224no—
307C4⋊C4—1222no—
308C22⋊C4—1228no—
309C22⋊C4—1224no—
310C22⋊C4—1224no—
311C42—24no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42D4×D6112yes—
317C42D4×D6112yes—
318C23—1224no—
319C23C22×C22⋊D6111yescharacteristic
320Q8—28no—
321Q8—24no—
322Q8—42no—
323Q8—2444no—
324Q8—2448no—
325D4—4884no—
326D4—2444no—
327D4—2444no—
328D4—2448no—
329D4—2448no—
330D4—2444no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—1224no—
347C2×C4—2448no—
348C2×C4—24416no—
349C2×C4#1451974511yescharacteristic
350C2×C4#1451974512yes—
351C2×C4#1451974511yescharacteristic
352C2×C4#1451980214yes—
353C2×C4C2×D4×D6112yes—
354C8—2444no—
355C22—21no—
356C22—21no—
357C22—2444no—
358C22—2444no—
359C22—2444no—
360C22C2×#1451974511yescharacteristic
361C22C2×#1451980212yes—
362C4—21no—
363C4—22no—
364C4—22no—
365C4—24no—
366C4—44no—
367C4—42no—
368C4—42no—
369C4—81no—
370C4—2444no—
371C4#1683994412yes—
372C2—22no—
373C2—4884no—
374C2#2145504511yescharacteristic
375C2C2×#1683994411yescharacteristic
376C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2399C2
socleC22×C61C2×#120
prime radical#21459590C2
hypercenter#2399D61
hypocenterC61#18187
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
2. #21455941C2semidirect
3. #21444821C2semidirect
4. #21444787C2semidirect
5. #21444326C2semidirect
6. #21444163C2semidirect
7. C61×#2399C2semidirect
186. #2399D61semidirect
220. C61#18187semidirect

Maximal subgroups

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

subgroupindex
#214595902
#214559412
#214448212
#214447872
#214443262
#214441632
C61×#23992
#1818761