Finite Group Databasebeta

← #29577795

Subgroups — #29577795

Order 15616.

912 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 115 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#21454848C211yesmaximal, characteristic, complemented
2#21449175C211yesmaximal, characteristic
3#21445749C212yesmaximal, complemented
4#21443686C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic, complemented
6C23×D244C2211yescharacteristic
7C2×#14519751—21no—
8C2×#14519745—21no—
9#16839834—24no—
10#16839819—21no—
11C2×#14519697—24no—
12#16839726—21no—
13C2×C22⋊#92418C2212yes—
14#16839679C412yescomplemented
15#16839671C412yescomplemented
16#16839664—21no—
17#16839622C2211yescharacteristic
18#16839617—21no—
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61—21no—
21C61×#380C2212yes—
22C61×#342—21no—
23C24×D61—21no—
24C22×D244—22no—
25C22×D244—21no—
26C22×D244—22no—
27C22×D244—44no—
28C22×D244C2×C412yes—
29C22×C22⋊D61—22no—
30C2×#11758194—22no—
31C22×#92418D414yescomplemented
32#14519751—44no—
33#14519745—44no—
34C2×C4⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×#11758198—22no—
37C2×#11758198—22no—
38C2×#11758198—22no—
39C4⋊#92418—22no—
40#14519697—48no—
41C22⋊#92418—22no—
42C22⋊#92418—22no—
43C22⋊#92418—22no—
44#14519688—22no—
45#14519684C2×C412yes—
46C22×D4×C61D411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C23×C4×C61C2311yescharacteristic
49C4×D4×C61—22no—
50C4×D4×C61—22no—
51C2×C4⋊C4×C61—22no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C42×C61D411yescharacteristic
57#99×C61—42no—
58#95×C61—22no—
59C24×C61C22⋊C411yescharacteristic
60C24×C61C2×D411yescharacteristic
61C23×D61—22no—
62C23×D61—44no—
63C23×D61—42no—
64C23×D61—41no—
65C23×D61—42no—
66C23×D61—41no—
67C23×D61—82no—
68C2×D4×C61—22no—
69C2×D4×C61—21no—
70C2×D4×C61—22no—
71C2×D4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61#4112yes—
84C2×C22⋊D61—44no—
85C2×C22⋊D61—44no—
86C2×C22⋊D61—82no—
87C22×Dic61—22no—
88C2×D244—28no—
89C2×D244—24no—
90C2×D244—24no—
91C2×D244—44no—
92C2×D244—48no—
93C2×D244—44no—
94C2×D244—42no—
95C2×D244—41no—
96C2×D244—41no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—44no—
99C22⋊C4×C61—22no—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—42no—
102C22⋊C4×C61—42no—
103C42×C61—21no—
104C42×C61—21no—
105C22⋊Dic61—42no—
106#11758198—44no—
107#11758198—44no—
108#11758198—82no—
109C4⋊Dic61—42no—
110#11758194—82no—
111C2×#92418—24no—
112C2×#92418—44no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61C2×C22⋊C411yescharacteristic
121C23×C61#12112yes—
122C23×C61AGL(1,ℤ/8)11yescharacteristic
123C23×C61AGL(1,ℤ/8)12yes—
124C23×C61#13711yescharacteristic
125C22×D61—24no—
126C22×D61—42no—
127C22×D61—48no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—42no—
132C22×D61—42no—
133C22×D61—88no—
134C22×D61—84no—
135C22×D61—84no—
136C22×D61—84no—
137C22×D61—82no—
138C22×D61—82no—
139D4×C61—42no—
140D4×C61—42no—
141C2×C4×C61—24no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C2×C4×C61—82no—
155C2×C4×C61—82no—
156C2×C4×C61—82no—
157C2×C4×C61C4×D412yes—
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61#12311yescharacteristic
160C22⋊D61—88no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164D244—28no—
165D244—44no—
166D244—44no—
167D244—44no—
168#92418—44no—
169#9472—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—22no—
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—42no—
175C22×C61—41no—
176C22×C61—41no—
177C22×C61—42no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—41no—
181C22×C61—81no—
182C22×C61#38611yescharacteristic
183C22×C61#41711yescharacteristic
184C22×C61#41811yescharacteristic
185C22×C61#46812yes—
186C22×C61#46912yes—
187C2×D61—44no—
188C2×D61—44no—
189C2×D61—44no—
190C2×D61—82no—
191C2×D61—82no—
192C2×D61—88no—
193C2×D61—88no—
194C2×D61—82no—
195C2×D61—82no—
196C4×C61—22no—
197C4×C61—21no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—162no—
202C4×C61B(4,𝔽2)12yes—
203Dic61—82no—
204#2973—611no—
205#2408—611no—
206#1799D6111yescharacteristic, complemented
207#1786—612no—
208#1289—612no—
209C2×C61—21no—
210C2×C61—22no—
211C2×C61—21no—
212C2×C61—22no—
213C2×C61—81no—
214C2×C61#188911yescharacteristic
215C2×C61#189511yescharacteristic
216C2×C61#191011yescharacteristic
217D61—84no—
218C23×D4—611no—
219C2×#121—1221no—
220C2×#120—1221no—
221C2×C4×D4C2×D6111yescharacteristic
222#435—1224no—
223#432—1221no—
224C2×#102—1224no—
225C2×#99—21no—
226#407—1221no—
227C2×C22⋊C8—612no—
228#394—612no—
229#392—612no—
230#390—1221no—
231#380—611no—
232#380C2×D6112yes—
233#379—1221no—
234#342—21no—
235C61#947211yescharacteristic, complemented, Sylow
236C25—1221no—
237C22×D4—612no—
238C22×D4—1222no—
239C22×D4—1221no—
240C22×D4—1222no—
241C22×D4—1222no—
242C22×D4—2444no—
243C22×D4C22⋊D6111yescharacteristic
244C23×C4D24411yescharacteristic
245C23×C4C22×D6111yescharacteristic
246C2×M16—1222no—
247C22×C8—614no—
248#121—2444no—
249#120—2444no—
250C4×D4—22no—
251C4×D4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4C4×D6111yescharacteristic
255C2×C22⋊C4—22no—
256C2×C22⋊C4—1222no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1222no—
260C2×C22⋊C4C4×D6111yescharacteristic
261C2×C22⋊C4C22⋊D6111yescharacteristic
262C2×C42D24411yescharacteristic
263C4⋊C8—1222no—
264#102—2448no—
265#99—42no—
266C22⋊C8—1222no—
267C22⋊C8—1222no—
268C22⋊C8—1222no—
269#97—1222no—
270#95—22no—
271#95—612no—
272C24—1222no—
273C24—4882no—
274C24—2444no—
275C24—2441no—
276C24—2442no—
277C24—2441no—
278C24—2442no—
279C24#1175819811yescharacteristic
280C24C2×C22⋊D6111yescharacteristic
281C2×D4—21no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—1228no—
286C2×D4—1224no—
287C2×D4—1224no—
288C2×D4—4882no—
289C2×D4—2444no—
290C2×D4—2442no—
291C2×D4—2444no—
292C2×D4—2448no—
293C2×D4—2444no—
294C2×D4—2444no—
295C2×D4—2441no—
296C2×D4—2441no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—1222no—
303C22×C4#1175819811yescharacteristic
304C22×C4#1175819811yescharacteristic
305C22×C4#1175819811yescharacteristic
306C22×C4C2×C4×D6111yescharacteristic
307C22×C4C2×D24411yescharacteristic
308C22×C4D4×D6112yes—
309C22×C4#1175822412yes—
310M16—4882no—
311C2×C8—1224no—
312C2×C8—2444no—
313C4⋊C4—22no—
314C4⋊C4—44no—
315C4⋊C4—2442no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—4882no—
321C22⋊C4—2444no—
322C22⋊C4—2444no—
323C22⋊C4—2442no—
324C42—21no—
325C42—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—1224no—
334C23—4884no—
335C23—4884no—
336C23—4888no—
337C23—4884no—
338C23—4882no—
339C23—4882no—
340C23—2442no—
341C23—2448no—
342C23—2442no—
343C23—2442no—
344C23—2442no—
345C23—2442no—
346C23—2442no—
347C23C2×#1175819811yescharacteristic
348C23#1451974912yes—
349C23#1451980212yes—
350C23#1451980311yescharacteristic
351C23#1451980811yescharacteristic
352D4—42no—
353D4—42no—
354D4—1228no—
355D4—4888no—
356D4—2444no—
357D4—2444no—
358D4—2444no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4#1451973712yes—
379C2×C4#1451974711yescharacteristic
380C2×C4#1451976111yescharacteristic
381C8—2444no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—22no—
386C22—41no—
387C22—42no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—4888no—
395C22—4888no—
396C22—4882no—
397C22—4882no—
398C22—4882no—
399C22—4882no—
400C22—2444no—
401C22—2444no—
402C22—2444no—
403C22#1683963911yescharacteristic
404C22#1683976111yescharacteristic
405C22#1683976611yescharacteristic
406C22#1684001012yes—
407C22#1684001312yes—
408C4—21no—
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—162no—
414C4—4882no—
415C4#1683995012yes—
416C2—22no—
417C2—21no—
418C2—21no—
419C2—22no—
420C2—81no—
421C2—4884no—
422C2#2144622511yescharacteristic
423C2#2144625511yescharacteristic
424C2#2144634411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#386
prime radical#21449175C2
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#9472
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
1. #21454848C2semidirect
3. #21445749C2semidirect
4. #21443686C2semidirect
5. C61×#1799C2semidirect
14. #16839679C4semidirect
15. #16839671C4semidirect
31. C22×#92418D4semidirect
206. #1799D61semidirect
235. C61#9472semidirect

Maximal subgroups

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

subgroupindex
#214548482
#214491752
#214457492
#214436862
C61×#17992
#947261