Finite Group Databasebeta

← #29598313

Subgroups — #29598313

Order 15616.

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

normal subgroups only

This group has 844 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#21449175C211yesmaximal, characteristic
2C2×#16839548C214yesmaximal, complemented
3#21445811C211yesmaximal, characteristic, complemented
4C4×#99×C61C211yesmaximal, characteristic, complemented
5C23×D244C411yescharacteristic, complemented
6C2×#14519751—21no—
7C2×#14519745C2212yes—
8#16839786—21no—
9C2×#14519694C2212yes—
10#16839679—21no—
11#16839671—21no—
12#16839664—21no—
13#16839622C411yescharacteristic, complemented
14#16839617C412yescomplemented
15#16839548—28no—
16#16839531—21no—
17C2×C4×D4×C61C2211yescharacteristic
18C2×#99×C61C2212yes—
19C4×C22⋊C4×C61—21no—
20C61×#342—21no—
21C24×D61C2×C412yes—
22C22×D244—21no—
23C22×D244—21no—
24C22×D244—44no—
25C22×D244—41no—
26C22×D244—41no—
27C22×C22⋊D61—22no—
28C2×#11758194—22no—
29#14519751—44no—
30#14519745—24no—
31#14519745—24no—
32C2×C4⋊Dic61—21no—
33C2×C22⋊Dic61C2×C412yes—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C2×#11758198—22no—
37C4⋊#92418—41no—
38C4⋊#92418—41no—
39#14519694—24no—
40#14519684—21no—
41C22×D4×C61C2311yescharacteristic
42C23×C4×C61C2×C411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C4×D4×C61—41no—
45C4×D4×C61—41no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61C2×C411yescharacteristic
49C2×C22⋊C4×C61D411yescharacteristic
50C2×C42×C61—41no—
51C2×C42×C61D411yescharacteristic
52#99×C61—22no—
53#99×C61D414yescomplemented
54#95×C61—41no—
55C24×C61C22×C411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C23×D61—22no—
58C23×D61—24no—
59C23×D61—44no—
60C23×D61—42no—
61C23×D61—42no—
62C23×D61—82no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61C2×D412yes—
66C2×D4×C61#4112yes—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—44no—
80C2×C22⋊D61—44no—
81C2×C22⋊D61—82no—
82C22×Dic61—22no—
83C2×D244—24no—
84C2×D244—42no—
85C2×D244—42no—
86C2×D244—44no—
87C2×D244—48no—
88C2×D244—44no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244—44no—
92C4⋊C4×C61—41no—
93C22⋊C4×C61—24no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—42no—
97C42×C61—21no—
98C42×C61—21no—
99C42×C61—82no—
100C22⋊Dic61—24no—
101#11758198—44no—
102#11758198—42no—
103#11758198—42no—
104#11758198—82no—
105C4⋊Dic61—42no—
106#11758194—42no—
107#11758194—42no—
108C2×#92418—42no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61C2×C22⋊C411yescharacteristic
118C23×C61C4×D412yes—
119C23×C61#12112yes—
120C22×D61—24no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—44no—
125C22×D61—42no—
126C22×D61—42no—
127C22×D61—42no—
128C22×D61—44no—
129C22×D61—88no—
130C22×D61—88no—
131C22×D61—84no—
132C22×D61—84no—
133D4×C61—42no—
134D4×C61—42no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—22no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—44no—
148C2×C4×C61—81no—
149C2×C4×C61—82no—
150C2×C4×C61—82no—
151C2×C4×C61#9911yescharacteristic
152C2×C4×C61#9911yescharacteristic
153C2×C4×C61#12411yescharacteristic
154C2×C4×C61#12711yescharacteristic
155C22⋊D61—88no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159D244—44no—
160D244—48no—
161D244—44no—
162#92418—82no—
163#13221—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—22no—
167C22×C61—22no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—81no—
176C22×C61#39011yescharacteristic
177C22×C61C2×#9911yescharacteristic
178C22×C61#41011yescharacteristic
179C2×D61—44no—
180C2×D61—44no—
181C2×D61—44no—
182C2×D61—82no—
183C2×D61—82no—
184C2×D61—88no—
185C2×D61—82no—
186C2×D61—82no—
187C2×D61—88no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—41no—
192C4×C61—41no—
193C4×C61—41no—
194C4×C61—84no—
195C4×C61AΣL(1,𝔽16)12yescomplemented
196Dic61—82no—
197#2408—611no—
198C2×AΣL(1,𝔽16)—614no—
199#1800—611no—
200C4×#99D6111yescharacteristic, complemented
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—42no—
204C2×C61—42no—
205C2×C61#196411yescharacteristic
206C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
207C2×C61#212211yescharacteristic
208D61—84no—
209C23×D4—611no—
210C2×#121—1221no—
211C2×#120—612no—
212C2×C4×D4C2×D6111yescharacteristic
213#423—1221no—
214C2×#100—612no—
215C2×#99C2×D6112yes—
216#394—1221no—
217#392—1221no—
218#390—1221no—
219#380—611no—
220#379—612no—
221C4×C22⋊C4—21no—
222AΣL(1,𝔽16)—1228no—
223#343—1221no—
224#342—21no—
225C61#1322111yescharacteristic, complemented, Sylow
226C25—612no—
227C22×D4—1222no—
228C22×D4—1221no—
229C22×D4—1221no—
230C22×D4—2441no—
231C22×D4—2441no—
232C22×D4—2444no—
233C22×D4C22×D6111yescharacteristic
234C23×C4C4×D6111yescharacteristic
235C23×C4C22⋊D6111yescharacteristic
236C2×M16—1222no—
237#121—2444no—
238#120—1224no—
239#120—1224no—
240C4×D4—41no—
241C4×D4—41no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4D24411yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—612no—
246C2×C22⋊C4—1222no—
247C2×C22⋊C4—1222no—
248C2×C22⋊C4—1222no—
249C2×C22⋊C4C4×D6111yescharacteristic
250C2×C22⋊C4C22⋊D6111yescharacteristic
251C2×C42—41no—
252C2×C42D24411yescharacteristic
253C4⋊C8—2441no—
254C4⋊C8—2441no—
255#100—1224no—
256#99—22no—
257#99D24414yescomplemented
258#95—41no—
259#95—1221no—
260C24—1222no—
261C24—1224no—
262C24—4882no—
263C24—2442no—
264C24—2442no—
265C24—2444no—
266C24C2×C4×D6111yescharacteristic
267C24C2×C22⋊D6111yescharacteristic
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—1224no—
271C2×D4—4882no—
272C2×D4—2442no—
273C2×D4—2442no—
274C2×D4—2444no—
275C2×D4—2448no—
276C2×D4—2444no—
277C2×D4—2444no—
278C2×D4—2444no—
279C2×D4—2444no—
280C2×D4—2441no—
281C2×D4—2441no—
282C2×D4C2×D24412yes—
283C2×D4#1175822212yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—1222no—
292C22×C4#1175819811yescharacteristic
293C22×C4#1175819811yescharacteristic
294C22×C4#1175819811yescharacteristic
295C22×C4#1175819811yescharacteristic
296C22×C4C2×D24411yescharacteristic
297M16—2442no—
298M16—2442no—
299C2×C8—2442no—
300C4⋊C4—41no—
301C4⋊C4—2442no—
302C22⋊C4—24no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—1224no—
307C22⋊C4—4882no—
308C22⋊C4—2444no—
309C22⋊C4—2442no—
310C22⋊C4—2442no—
311C42—21no—
312C42—21no—
313C42—82no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—1224no—
323C23—4884no—
324C23—4884no—
325C23—4888no—
326C23—4888no—
327C23—2442no—
328C23—2442no—
329C23—2442no—
330C23—2444no—
331C23—2444no—
332C23—2442no—
333C23—2442no—
334C23—2442no—
335C23C2×#1175819811yescharacteristic
336C23C4×D24412yes—
337C23#1451975112yes—
338D4—42no—
339D4—42no—
340D4—4888no—
341D4—2444no—
342D4—2448no—
343D4—2444no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4#1451969111yescharacteristic
364C2×C4#1451969111yescharacteristic
365C2×C4#1451976511yescharacteristic
366C2×C4#1451977411yescharacteristic
367C8—4882no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—81no—
380C22—4882no—
381C22—4882no—
382C22—4888no—
383C22—4888no—
384C22—4882no—
385C22—4882no—
386C22—2444no—
387C22—2444no—
388C22—2444no—
389C22#1683966411yescharacteristic
390C22C2×#1451969111yescharacteristic
391C22#1683973511yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—84no—
399C4—4882no—
400C4#1683954812yescomplemented
401C2—21no—
402C2—21no—
403C2—42no—
404C2—42no—
405C2—4884no—
406C2#2144665511yescharacteristic
407C2C2×#1683954811yescharacteristic
408C2#2144746811yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC4×#99×C61C2
socleC22×C61C2×#99
prime radical#21449175C2
cosocleC22×D4×C61C23
hypercenterC4×#99D61
hypocenterC61#13221
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16839548C2semidirect
3. #21445811C2semidirect
4. C4×#99×C61C2semidirect
5. C23×D244C4semidirect
13. #16839622C4semidirect
14. #16839617C4semidirect
53. #99×C61D4semidirect
195. C4×C61AΣL(1,𝔽16)semidirect
200. C4×#99D61semidirect
225. C61#13221semidirect
257. #99D244semidirect
400. C4#16839548semidirect

Maximal subgroups

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

subgroupindex
#214491752
C2×#168395482
#214458112
C4×#99×C612
#1322161