Finite Group Databasebeta

← #29652499

Subgroups — #29652499

Order 15616.

1210 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1210 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
1C2×#16840343C211yesmaximal, characteristic
2#21455759C212yesmaximal, complemented
3#21444855C214yesmaximal, complemented
4#21444821C214yesmaximal, complemented
5#21444347C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C61×#2484C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840343—24no—
10#16840343—22no—
11#16840343C2214yes—
12C2×#14519757—22no—
13C2×#14519752—22no—
14C2×#14519751—21no—
15C2×#14519751C2212yes—
16#16840086—24no—
17#16840080C2214yescomplemented
18#16840075C2214yescomplemented
19#16839805—22no—
20#16839766—22no—
21#16839726C2212yes—
22#16839495C2218yescomplemented
23#16839488C2214yescomplemented
24C2×#117×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#400C2212yescomplemented
27C61×#398C2212yescomplemented
28C61×#387—22no—
29C61×#385—21no—
30C61×#381—21no—
31C61×#381C2211yescharacteristic
32C2×#11758222—24no—
33C2×#11758222—24no—
34C2×#11758222—48no—
35C22×Dic122—21no—
36C22×D244—21no—
37C22×D244—21no—
38C22×D244D412yes—
39C22×C22⋊D61—22no—
40C22×C4×D61—22no—
41C2×#11758194—22no—
42#14519757—28no—
43#14519757—28no—
44#14519752—28no—
45#14519752—28no—
46#14519751—24no—
47#14519751—24no—
48#14519751—22no—
49#14519751—22no—
50#14519751D414yes—
51#14519751C2314yes—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61D412yes—
54C2×#11758198—22no—
55C2×#11758198—22no—
56#14519710—24no—
57#14519708—24no—
58#14519697—24no—
59#14519697—24no—
60C22⋊#92418—22no—
61C22⋊#92418C2314yes—
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65#117×C61—22no—
66#117×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61D412yes—
71C2×C4⋊C4×C61C2311yescharacteristic
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C4⋊C4×C61C2312yes—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C42×C61—21no—
77#95×C61—22no—
78#95×C61—22no—
79#95×C61—21no—
80#95×C61—22no—
81#95×C61C2312yes—
82C24×C61C2×D411yescharacteristic
83C23×D61—22no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—24no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—42no—
93C22×C4×C61—42no—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D412yes—
99C22×C4×C61C2411yescharacteristic
100C2×C22⋊D61—24no—
101C2×C22⋊D61—24no—
102C2×C22⋊D61—48no—
103C22×Dic61—22no—
104#11758222—416no—
105#11758222—48no—
106#11758222—48no—
107C2×D244—24no—
108C2×D244—24no—
109C2×D244—22no—
110C2×D244—44no—
111C2×D244—44no—
112C2×D244—82no—
113C2×D244C2×D414yes—
114C2×C4×D61—28no—
115C2×C4×D61—48no—
116C2×Dic122—24no—
117C2×Dic122—44no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—44no—
123C22⋊C4×C61—42no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C42×C61—42no—
127#11758198—28no—
128#11758198—24no—
129#11758198—24no—
130C4⋊Dic61—24no—
131C4⋊Dic61—22no—
132C4⋊Dic61C2×D414yes—
133#11758194—44no—
134C2×#92418—24no—
135C23×C61—22no—
136C23×C61—22no—
137C23×C61—41no—
138C23×C61#12011yescharacteristic
139C23×C61#12011yescharacteristic
140C23×C61C22×D411yescharacteristic
141C22×D61—24no—
142C22×D61—44no—
143C22×D61—44no—
144C22×D61—44no—
145C2×C4×C61—24no—
146C2×C4×C61—24no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—24no—
150C2×C4×C61—22no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—81no—
164C2×C4×C61#12012yes—
165C2×C4×C61AGL(1,ℤ/8)14yes—
166C2×C4×C61C22×D412yes—
167C22⋊D61—48no—
168C22⋊D61—48no—
169C2×Dic61—24no—
170C2×Dic61—44no—
171D244—44no—
172D244—48no—
173D244—82no—
174D244—84no—
175C4×D61—416no—
176Dic122—48no—
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#45812yes—
188C22×C61C2×#12011yescharacteristic
189C22×C61C2×AGL(1,ℤ/8)12yes—
190C2×D61—44no—
191C2×D61—44no—
192C2×D61—44no—
193C2×D61—88no—
194C4×C61—24no—
195C4×C61—22no—
196C4×C61—22no—
197C4×C61—42no—
198C4×C61—82no—
199C4×C61—81no—
200C4×C61—81no—
201Dic61—44no—
202C2×#537—611no—
203#3094—612no—
204#2484D6111yescharacteristic, complemented
205#1616—614no—
206#1610—614no—
207#1510—612no—
208#1463—611no—
209C2×C61—22no—
210C2×C61—22no—
211C2×C61—41no—
212C2×C61#300311yescharacteristic
213C2×C61#302011yescharacteristic
214C2×C61C2×#45811yescharacteristic
215D61—84no—
216C22×#41—1221no—
217#537—614no—
218#537—1224no—
219#537—1222no—
220C2×C22⋊Q8—1222no—
221C2×#121—612no—
222C2×#121—1222no—
223C2×#121—1221no—
224C2×#117C2×D6111yescharacteristic
225C22×C4⋊C4C2×D6111yescharacteristic
226#482—1224no—
227#481—614no—
228#480—614no—
229#428—1222no—
230#418—1222no—
231#407—612no—
232#400C2×D6112yescomplemented
233#398C2×D6112yescomplemented
234#387—22no—
235#385—21no—
236#381—21no—
237#381C2×D6111yescharacteristic
238#327—618no—
239#324—614no—
240C61#1814711yescharacteristic, complemented, Sylow
241C2×#41—1224no—
242C2×#41—1224no—
243C2×#41—2448no—
244C22×Q8—1221no—
245C22×D4—612no—
246C22×D4—1221no—
247C22×D4—1221no—
248C22×D4—1222no—
249C23×C4—1222no—
250C23×C4C22⋊D6111yescharacteristic
251C23×C4C22⋊D6111yescharacteristic
252C23×C4C22×D6111yescharacteristic
253C2×M16—1222no—
254C22⋊Q8—1228no—
255C22⋊Q8—1228no—
256#121—614no—
257#121—614no—
258#121—1228no—
259#121—1228no—
260#121—1224no—
261#121—1222no—
262#121—1222no—
263#121—1224no—
264#117—22no—
265#117—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—612no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4C22⋊D6112yes—
272C2×C4⋊C4C22×D6111yescharacteristic
273C2×C4⋊C4C22×D6111yescharacteristic
274C2×C4⋊C4C22×D6112yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1222no—
279C2×C42—21no—
280#107—1224no—
281#106—1224no—
282#102—1224no—
283#102—1224no—
284C22⋊C8—614no—
285C22⋊C8—1222no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—22no—
290#95C22×D6112yes—
291C24—1222no—
292C24C2×C22⋊D6111yescharacteristic
293#41—24416no—
294#41—2448no—
295#41—2448no—
296C2×Q8—1224no—
297C2×Q8—2444no—
298C2×D4—614no—
299C2×D4—1224no—
300C2×D4—1224no—
301C2×D4—1224no—
302C2×D4—1224no—
303C2×D4—1222no—
304C2×D4—4882no—
305C2×D4—2444no—
306C2×D4—2444no—
307C2×D4—2448no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—1228no—
319C22×C4—1222no—
320C22×C4—2448no—
321C22×C4D4×D6112yes—
322C22×C4D4×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—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—614no—
335C4⋊C4—1224no—
336C4⋊C4—1222no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—1228no—
341C22⋊C4—1224no—
342C22⋊C4—1224no—
343C42—42no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—1224no—
348C23—2444no—
349C23—2444no—
350C23—2444no—
351C23#1451974511yescharacteristic
352C23#1451974511yescharacteristic
353C23C22×C22⋊D6111yescharacteristic
354Q8—2448no—
355D4—4882no—
356D4—4884no—
357D4—2444no—
358D4—2448no—
359D4—2448no—
360D4—2448no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
380C2×C4—1224no—
381C2×C4—24416no—
382C2×C4—2444no—
383C2×C4#1451974512yes—
384C2×C4#1451980214yes—
385C2×C4C2×D4×D6112yes—
386C8—2444no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—44no—
395C22—4888no—
396C22—2444no—
397C22—2444no—
398C22—2444no—
399C22#1683995112yes—
400C22C2×#1451974511yescharacteristic
401C22C2×#1451980212yes—
402C4—22no—
403C4—22no—
404C4—24no—
405C4—42no—
406C4—82no—
407C4—81no—
408C4—81no—
409C4—2444no—
410C2—22no—
411C2—22no—
412C2—41no—
413C2—4884no—
414C2#2145504511yescharacteristic
415C2#2145518211yescharacteristic
416C2C2×#1683995111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2484C2
socleC22×C61C2×#120
prime radicalC2×#16840343C2
hypercenter#2484D61
hypocenterC61#18147
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
2. #21455759C2semidirect
3. #21444855C2semidirect
4. #21444821C2semidirect
5. #21444347C2semidirect
6. #21444149C2semidirect
7. C61×#2484C2semidirect
17. #16840080C22semidirect
18. #16840075C22semidirect
22. #16839495C22semidirect
23. #16839488C22semidirect
26. C61×#400C22semidirect
27. C61×#398C22semidirect
204. #2484D61semidirect
232. #400C2×D61semidirect
233. #398C2×D61semidirect
240. C61#18147semidirect

Maximal subgroups

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

subgroupindex
C2×#168403432
#214557592
#214448552
#214448212
#214443472
#214441492
C61×#24842
#1814761