Finite Group Databasebeta

← #29598301

Subgroups — #29598301

Order 15616.

840 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16839549C214yesmaximal, complemented
2#21445807C211yesmaximal, characteristic, complemented
3C4×#14519694C211yesmaximal, characteristic, complemented
4C61×#2408C211yesmaximal, characteristic
5C2×#14519694C2212yes—
6C2×#14519692C2212yes—
7C4×#11758194—21no—
8#16839622—21no—
9#16839549—28no—
10#16839531—21no—
11#16839529—21no—
12C23×D4×C61C411yescharacteristic, complemented
13C2×#121×C61—21no—
14C2×#120×C61C2212yes—
15C2×C4×D4×C61C2211yescharacteristic
16C61×#394—21no—
17C61×#392—21no—
18C61×#390—21no—
19C61×#380C411yescharacteristic, complemented
20C61×#379C412yescomplemented
21C2×#11758194—22no—
22C2×C4⋊Dic61—41no—
23C2×C22⋊Dic61—22no—
24#14519694—22no—
25#14519694D414yescomplemented
26#14519692—24no—
27#14519688—41no—
28C4×#92418—41no—
29#14519684—41no—
30C25×C61C2×C412yes—
31C22×D4×C61—22no—
32C22×D4×C61—21no—
33C22×D4×C61—21no—
34C22×D4×C61—41no—
35C22×D4×C61—41no—
36C22×D4×C61—44no—
37C22×D4×C61C2311yescharacteristic
38C23×C4×C61C2×C411yescharacteristic
39C23×C4×C61D411yescharacteristic
40#121×C61—44no—
41#120×C61—24no—
42#120×C61—24no—
43C4×D4×C61—41no—
44C4×D4×C61—41no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—22no—
50C2×C22⋊C4×C61C2×C412yes—
51C2×C22⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61D411yescharacteristic
53C2×C42×C61D411yescharacteristic
54#95×C61—21no—
55C24×C61—22no—
56C24×C61—24no—
57C24×C61—42no—
58C24×C61—44no—
59C24×C61—42no—
60C24×C61—82no—
61C24×C61C22×C411yescharacteristic
62C24×C61C2×D411yescharacteristic
63C2×D4×C61—24no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
66C2×D4×C61—42no—
67C2×D4×C61—42no—
68C2×D4×C61—44no—
69C2×D4×C61—48no—
70C2×D4×C61—44no—
71C2×D4×C61—44no—
72C2×D4×C61—44no—
73C2×D4×C61—42no—
74C2×D4×C61—42no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61—82no—
78C2×D4×C61C2×D412yes—
79C2×D4×C61#4112yes—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—42no—
86C22×C4×C61—42no—
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×Dic61—42no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—42no—
95C22⋊C4×C61—24no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊C4×C61—44no—
99C22⋊C4×C61—44no—
100C22⋊C4×C61—82no—
101C42×C61—21no—
102C42×C61—21no—
103C22⋊Dic61—42no—
104C22⋊Dic61—42no—
105C4⋊Dic61—82no—
106#11758194—24no—
107#11758194—42no—
108C2×#92418—42no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—24no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—42no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—44no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61—44no—
126C23×C61—84no—
127C23×C61—88no—
128C23×C61—88no—
129C23×C61—84no—
130C23×C61C2×C22⋊C411yescharacteristic
131C23×C61C4×D412yes—
132C23×C61#12411yescharacteristic
133C23×C61#12711yescharacteristic
134D4×C61—42no—
135D4×C61—42no—
136D4×C61—44no—
137D4×C61—48no—
138D4×C61—44no—
139D4×C61—84no—
140D4×C61—84no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—22no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61#9911yescharacteristic
158C2×C4×C61#9911yescharacteristic
159C2×C4×C61#12112yes—
160C2×Dic61—82no—
161C2×Dic61—82no—
162C2×Dic61—82no—
163#92418—44no—
164#13218—611nomaximal, Sylow
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—22no—
169C22×C61—21no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—42no—
173C22×C61—42no—
174C22×C61—44no—
175C22×C61—44no—
176C22×C61—41no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—44no—
181C22×C61—81no—
182C22×C61—82no—
183C22×C61—82no—
184C22×C61—88no—
185C22×C61—88no—
186C22×C61—82no—
187C22×C61—82no—
188C22×C61#39011yescharacteristic
189C22×C61C2×#9911yescharacteristic
190C22×C61#41011yescharacteristic
191C4×C61—21no—
192C4×C61—22no—
193C4×C61—41no—
194C4×C61—41no—
195C4×C61—41no—
196C4×C61—82no—
197C4×C61AΣL(1,𝔽16)12yescomplemented
198Dic61—162no—
199#2408D6111yescharacteristic
200C2×AΣL(1,𝔽16)—614no—
201#1799—611no—
202C4×#100—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—41no—
206C2×C61—42no—
207C2×C61—41no—
208C2×C61—84no—
209C2×C61#196411yescharacteristic
210C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
211C2×C61#212211yescharacteristic
212C23×D4Dic6111yescharacteristic, complemented
213C2×#121—21no—
214C2×#120C2×D6112yes—
215C2×C4×D4C2×D6111yescharacteristic
216C2×#100—612no—
217C2×#99—612no—
218C4×M16—1221no—
219#394—21no—
220#392—21no—
221#390—21no—
222#380—1221no—
223#380Dic6111yescharacteristic, complemented
224#379Dic6112yescomplemented
225AΣL(1,𝔽16)—1228no—
226#343—1221no—
227#342—1221no—
228C61#1321811yescharacteristic, complemented, Sylow
229C25C2×Dic6112yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—41no—
234C22×D4—41no—
235C22×D4—44no—
236C22×D4C22×D6111yescharacteristic
237C23×C4C2×Dic6111yescharacteristic
238C23×C4C22⋊D6111yescharacteristic
239C2×M16—1222no—
240#121—44no—
241#120—24no—
242#120—24no—
243C4×D4—41no—
244C4×D4—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—2441no—
247C2×C4⋊C4C22⋊D6111yescharacteristic
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4C2×Dic6112yes—
253C2×C22⋊C4C2×Dic6111yescharacteristic
254C2×C22⋊C4C22⋊D6111yescharacteristic
255C2×C42C22⋊D6111yescharacteristic
256#100—614no—
257#100—1222no—
258#99—1224no—
259#97—2441no—
260C4×C8—2441no—
261#95—21no—
262#95—2441no—
263C24—22no—
264C24—24no—
265C24—42no—
266C24—44no—
267C24—42no—
268C24—82no—
269C24C22×Dic6111yescharacteristic
270C24C2×C22⋊D6111yescharacteristic
271C2×D4—24no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—82no—
286C2×D4D4×D6112yes—
287C2×D4#1175822412yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—2442no—
296C22×C4C22⋊Dic6111yescharacteristic
297C22×C4C22⋊Dic6111yescharacteristic
298C22×C4C22⋊Dic6111yescharacteristic
299C22×C4C22⋊Dic6111yescharacteristic
300C22×C4C2×C22⋊D6111yescharacteristic
301M16—1224no—
302M16—2442no—
303C2×C8—2442no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—4882no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—2442no—
314C22⋊C4—2442no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—24no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—84no—
335C23—88no—
336C23—88no—
337C23—84no—
338C23C2×C22⋊Dic6111yescharacteristic
339C23D4×Dic6112yes—
340C23#1451976411yescharacteristic
341C23#1451977311yescharacteristic
342D4—42no—
343D4—42no—
344D4—44no—
345D4—48no—
346D4—44no—
347D4—84no—
348D4—84no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—4882no—
366C2×C4—4882no—
367C2×C4—4882no—
368C2×C4#1451969211yescharacteristic
369C2×C4#1451969211yescharacteristic
370C2×C4#1451974812yes—
371C8—2444no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—44no—
382C22—44no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—44no—
388C22—81no—
389C22—82no—
390C22—82no—
391C22—88no—
392C22—88no—
393C22—82no—
394C22—82no—
395C22#1683966511yescharacteristic
396C22C2×#1451969211yescharacteristic
397C22#1683973611yescharacteristic
398C4—22no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—9762no—
405C4#1683954912yescomplemented
406C2—21no—
407C2—21no—
408C2—41no—
409C2—42no—
410C2—41no—
411C2—84no—
412C2#2144665611yescharacteristic
413C2C2×#1683954911yescharacteristic
414C2#2144746911yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
Fitting, prime radicalC61×#2408C2
socleC22×C61C2×#99
cosocleC22×D4×C61C23
hypercenter#2408D61
hypocenterC61#13218
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. C2×#16839549C2semidirect
2. #21445807C2semidirect
3. C4×#14519694C2semidirect
12. C23×D4×C61C4semidirect
19. C61×#380C4semidirect
20. C61×#379C4semidirect
25. #14519694D4semidirect
197. C4×C61AΣL(1,𝔽16)semidirect
212. C23×D4Dic61semidirect
223. #380Dic61semidirect
224. #379Dic61semidirect
228. C61#13218semidirect
405. C4#16839549semidirect

Maximal subgroups

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

subgroupindex
C2×#168395492
#214458072
C4×#145196942
C61×#24082
#1321861