Finite Group Databasebeta

← #29652500

Subgroups — #29652500

Order 15616.

1210 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455755C212yesmaximal, complemented
2#21450106C211yesmaximal, characteristic, complemented
3#21444856C214yesmaximal, complemented
4#21444822C214yesmaximal, complemented
5#21444348C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C2×C61×#537C211yesmaximal, characteristic
8C2×#14519736C2211yescharacteristic
9#16840083—24no—
10#16840078C2214yescomplemented
11#16840073C2214yescomplemented
12#16839802—22no—
13#16839767—22no—
14#16839726C2212yes—
15#16839701C2212yescomplemented
16#16839692C2212yescomplemented
17#16839643—22no—
18#16839635—21no—
19#16839625—21no—
20#16839625C2211yescharacteristic
21#16839496C2218yescomplemented
22#16839488C2214yescomplemented
23C22×#41×C61—21no—
24C61×#537—24no—
25C61×#537—22no—
26C61×#537C2214yes—
27C2×C22⋊Q8×C61—22no—
28C2×#121×C61—22no—
29C2×#121×C61—21no—
30C2×#121×C61C2212yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C2×#11758194—22no—
33#14519736—22no—
34#14519736—22no—
35C2×C4⋊Dic61C2311yescharacteristic
36C2×C4⋊Dic61C2311yescharacteristic
37C2×#11758196—22no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C4×Dic61—21no—
41#14519710—24no—
42#14519708—24no—
43#14519698—24no—
44#14519698—24no—
45C22⋊#92418—22no—
46C22⋊#92418C2314yes—
47#14519684—22no—
48#14519684—22no—
49#14519684—22no—
50#14519684—21no—
51#14519684C2312yes—
52C2×#41×C61—24no—
53C2×#41×C61—24no—
54C2×#41×C61—48no—
55C22×Q8×C61—21no—
56C22×D4×C61—21no—
57C22×D4×C61—22no—
58C22×D4×C61—21no—
59C22×D4×C61D412yes—
60C23×C4×C61—22no—
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64C22⋊Q8×C61—28no—
65C22⋊Q8×C61—28no—
66#121×C61—24no—
67#121×C61—28no—
68#121×C61—28no—
69#121×C61—22no—
70#121×C61—22no—
71#121×C61—24no—
72#121×C61D414yes—
73#121×C61C2314yes—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61D412yes—
78C2×C4⋊C4×C61D412yes—
79C2×C4⋊C4×C61C2312yes—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C24×C61—22no—
83C24×C61C2×D411yescharacteristic
84#41×C61—416no—
85#41×C61—48no—
86#41×C61—48no—
87C2×Q8×C61—24no—
88C2×Q8×C61—44no—
89C2×D4×C61—24no—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—24no—
93C2×D4×C61—22no—
94C2×D4×C61—44no—
95C2×D4×C61—44no—
96C2×D4×C61—48no—
97C2×D4×C61—82no—
98C2×D4×C61C2×D414yes—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—24no—
102C22×C4×C61—28no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61—42no—
107C22×C4×C61—48no—
108C22×C4×C61—42no—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—22no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61C2×D414yes—
123C22⋊C4×C61—28no—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C22⋊Dic61—42no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131#11758196—44no—
132C4×Dic61—42no—
133#11758194—44no—
134C2×#92418—24no—
135C23×C61—22no—
136C23×C61—22no—
137C23×C61—24no—
138C23×C61—41no—
139C23×C61—44no—
140C23×C61—44no—
141C23×C61—44no—
142C23×C61#12011yescharacteristic
143C23×C61#12011yescharacteristic
144C23×C61C22×D411yescharacteristic
145Q8×C61—48no—
146D4×C61—44no—
147D4×C61—48no—
148D4×C61—48no—
149D4×C61—48no—
150D4×C61—82no—
151D4×C61—84no—
152C2×C4×C61—24no—
153C2×C4×C61—24no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—24no—
157C2×C4×C61—22no—
158C2×C4×C61—24no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—416no—
163C2×C4×C61—44no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—81no—
167C2×C4×C61#12012yes—
168C2×C4×C61AGL(1,ℤ/8)14yes—
169C2×C4×C61C22×D412yes—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177#92418—44no—
178#18147—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—44no—
187C22×C61—44no—
188C22×C61—44no—
189C22×C61—44no—
190C22×C61—88no—
191C22×C61#45812yes—
192C22×C61C2×#12011yescharacteristic
193C22×C61C2×AGL(1,ℤ/8)12yes—
194C4×C61—24no—
195C4×C61—22no—
196C4×C61—22no—
197C4×C61—44no—
198C4×C61—42no—
199Dic61—82no—
200Dic61—81no—
201Dic61—81no—
202C2×#537D6111yescharacteristic
203#3094—612no—
204#2484—611no—
205#1616—614no—
206#1610—614no—
207#1510—612no—
208#1463—611no—
209C2×C61—22no—
210C2×C61—22no—
211C2×C61—41no—
212C2×C61—84no—
213C2×C61#300311yescharacteristic
214C2×C61#302011yescharacteristic
215C2×C61C2×#45811yescharacteristic
216C22×#41—21no—
217#537—24no—
218#537—22no—
219#537C2×D6114yes—
220C2×C22⋊Q8—22no—
221C2×#121—22no—
222C2×#121—21no—
223C2×#121C2×D6112yes—
224C2×#117—611no—
225C22×C4⋊C4C2×D6111yescharacteristic
226#482—1224no—
227#481—614no—
228#480—614no—
229#428—1222no—
230#418—1222no—
231#407—612no—
232#400—612no—
233#398—612no—
234#387—1222no—
235#385—1221no—
236#381—611no—
237#381—1221no—
238#327—618no—
239#324—614no—
240C61#1814711yescharacteristic, complemented, Sylow
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—48no—
244C22×Q8—21no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4C22⋊D6112yes—
249C23×C4—22no—
250C23×C4C22⋊D6111yescharacteristic
251C23×C4C22⋊D6111yescharacteristic
252C23×C4C22×D6111yescharacteristic
253C2×M16—1222no—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256#121—24no—
257#121—28no—
258#121—28no—
259#121—24no—
260#121—22no—
261#121—22no—
262#121C22⋊D6114yes—
263#121C22×D6114yes—
264#117—1222no—
265#117—1222no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—1222no—
272C2×C4⋊C4C22⋊D6112yes—
273C2×C4⋊C4C22⋊D6112yes—
274C2×C4⋊C4C22×D6112yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C42—1221no—
280#107—1224no—
281#106—1224no—
282#102—1224no—
283#102—1224no—
284C22⋊C8—614no—
285C22⋊C8—1222no—
286#95—612no—
287#95—1221no—
288#95—1222no—
289#95—1222no—
290#95—1222no—
291C24—22no—
292C24C2×C22⋊D6111yescharacteristic
293#41—416no—
294#41—48no—
295#41—48no—
296C2×Q8—24no—
297C2×Q8—44no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—82no—
307C2×D4C2×C22⋊D6114yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—1222no—
319C22×C4—1221no—
320C22×C4—1221no—
321C22×C4C2×C22⋊D6112yes—
322C22×C4C2×C22⋊D6112yes—
323C22×C4C2×C22⋊D6112yes—
324C22×C4C2×C22⋊D6112yes—
325C22×C4C2×C22⋊D6111yescharacteristic
326C22×C4C23×D6111yescharacteristic
327M16—2444no—
328C2×C8—1224no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—1222no—
334C4⋊C4—1222no—
335C4⋊C4—2444no—
336C4⋊C4C2×C22⋊D6114yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—2442no—
341C22⋊C4—2441no—
342C22⋊C4—2441no—
343C42—2442no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—41no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23#1451974611yescharacteristic
352C23#1451974611yescharacteristic
353C23C22×C22⋊D6111yescharacteristic
354Q8—48no—
355D4—44no—
356D4—48no—
357D4—48no—
358D4—48no—
359D4—82no—
360D4—84no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—416no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4#1451974612yes—
384C2×C4#1451980314yes—
385C2×C4C22×C22⋊D6112yes—
386C8—2444no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—44no—
398C22—88no—
399C22#1683995212yes—
400C22C2×#1451974611yescharacteristic
401C22C2×#1451980312yes—
402C4—22no—
403C4—22no—
404C4—24no—
405C4—42no—
406C4—44no—
407C4—4882no—
408C4—4881no—
409C4—4881no—
410C2—22no—
411C2—22no—
412C2—41no—
413C2—84no—
414C2#2145504611yescharacteristic
415C2#2145518311yescharacteristic
416C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#537C2
socleC22×C61C2×#120
hypercenterC2×#537D61
hypocenterC61#18147
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #21455755C2semidirect
2. #21450106C2semidirect
3. #21444856C2semidirect
4. #21444822C2semidirect
5. #21444348C2semidirect
6. #21444149C2semidirect
10. #16840078C22semidirect
11. #16840073C22semidirect
15. #16839701C22semidirect
16. #16839692C22semidirect
21. #16839496C22semidirect
22. #16839488C22semidirect
240. C61#18147semidirect

Maximal subgroups

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

subgroupindex
#214557552
#214501062
#214448562
#214448222
#214443482
#214441492
C2×C61×#5372
#1814761