Finite Group Databasebeta

← #29577752

Subgroups — #29577752

Order 15616.

1034 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1034 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
1#21449452C211yesmaximal, characteristic
2#21445740C211yesmaximal, characteristic, complemented
3#21444347C212yesmaximal, complemented
4#21443671C212yesmaximal, complemented
5C61×#2308C211yesmaximal, characteristic, complemented
6C2×#14519773C412yescomplemented
7C2×#14519751C2211yescharacteristic
8C2×#14519746—21no—
9C2×#14519746—21no—
10C2×#14519694—22no—
11#16839726C2212yes—
12#16839670—22no—
13#16839665C412yescomplemented
14C4×C22⋊Dic61C2211yescharacteristic
15#16839495—24no—
16C23×D4×C61C2211yescharacteristic
17C22×C22⋊C4×C61—21no—
18C61×#386—24no—
19C61×#381C2212yes—
20C61×#379—21no—
21C22×D244C2×C411yescharacteristic, complemented
22C22×C22⋊D61—21no—
23C22×C22⋊D61—22no—
24C22×C22⋊D61—22no—
25C2×#11758194—22no—
26#14519773—44no—
27#14519751—22no—
28#14519751—22no—
29#14519746—44no—
30#14519746—44no—
31C2×C4⋊Dic61C2×C411yescharacteristic, complemented
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61—22no—
35C2×#11758198—21no—
36C2×C4×Dic61C2×C412yescomplemented
37#14519694—44no—
38C22⋊#92418—22no—
39C22⋊#92418—22no—
40#14519684—21no—
41C25×C61D411yescharacteristic
42C25×C61D411yescharacteristic
43C22×D4×C61—22no—
44C22×D4×C61—22no—
45C22×D4×C61—44no—
46C22×D4×C61C2×C412yes—
47C22×D4×C61D412yes—
48C23×C4×C61—22no—
49C23×C4×C61C2311yescharacteristic
50C2×C22⋊C4×C61—22no—
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61—44no—
55#95×C61—22no—
56#95×C61D414yescomplemented
57C24×C61—21no—
58C24×C61—24no—
59C24×C61—21no—
60C24×C61—21no—
61C24×C61—21no—
62C24×C61—42no—
63C24×C61—42no—
64C24×C61—42no—
65C24×C61—81no—
66C24×C61—81no—
67C24×C61C22⋊C412yes—
68C24×C61C22⋊C412yes—
69C24×C61C2×D411yescharacteristic
70C23×D61—21no—
71C2×D4×C61—24no—
72C2×D4×C61—24no—
73C2×D4×C61—24no—
74C2×D4×C61—24no—
75C2×D4×C61—42no—
76C2×D4×C61—42no—
77C2×D4×C61—44no—
78C2×D4×C61—44no—
79C2×D4×C61—48no—
80C22×C4×C61—24no—
81C22×C4×C61—42no—
82C22×C4×C61—44no—
83C22×C4×C61—82no—
84C22×C4×C61—82no—
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D412yes—
88C22×C4×C61#4112yes—
89C2×C22⋊D61—44no—
90C2×C22⋊D61—44no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61—44no—
94C2×C22⋊D61—44no—
95C2×C22⋊D61—41no—
96C2×C22⋊D61—41no—
97C2×C22⋊D61—82no—
98C2×C22⋊D61—82no—
99C22×Dic61—22no—
100C22×Dic61—21no—
101C2×D244—22no—
102C2×D244—42no—
103C22⋊C4×C61—44no—
104C22⋊C4×C61—44no—
105C22⋊C4×C61—82no—
106C22⋊C4×C61—82no—
107C22⋊Dic61—24no—
108C22⋊Dic61—44no—
109C22⋊Dic61—44no—
110#11758198—42no—
111C4⋊Dic61—22no—
112C4×Dic61—42no—
113#11758194—44no—
114C2×#92418—42no—
115C23×C61—22no—
116C23×C61—22no—
117C23×C61—24no—
118C23×C61—22no—
119C23×C61—22no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—22no—
123C23×C61—22no—
124C23×C61—22no—
125C23×C61—41no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—81no—
137C23×C61—81no—
138C23×C61—81no—
139C23×C61—81no—
140C23×C61—88no—
141C23×C61—88no—
142C23×C61#9914yes—
143C23×C61C2×C22⋊C411yescharacteristic
144C23×C61#12011yescharacteristic
145C23×C61#12311yescharacteristic
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—42no—
149C22×D61—41no—
150C22×D61—42no—
151D4×C61—48no—
152D4×C61—48no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—44no—
158C2×C4×C61—44no—
159C2×C4×C61—44no—
160C2×C4×C61—88no—
161C2×C4×C61C4×D412yes—
162C2×C4×C61#12112yes—
163C22⋊D61—88no—
164C22⋊D61—88no—
165C22⋊D61—82no—
166C22⋊D61—82no—
167C2×Dic61—24no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173D244—82no—
174#92418—82no—
175#9463—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—24no—
183C22×C61—21no—
184C22×C61—44no—
185C22×C61—44no—
186C22×C61—44no—
187C22×C61—44no—
188C22×C61—44no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—82no—
195C22×C61—82no—
196C22×C61—84no—
197C22×C61—84no—
198C22×C61—84no—
199C22×C61#38611yescharacteristic
200C22×C61C2×#9912yes—
201C22×C61#45312yes—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—81no—
205C2×D61—81no—
206C2×D61—81no—
207C4×C61—22no—
208C4×C61—22no—
209C4×C61—84no—
210Dic61—44no—
211Dic61—81no—
212#2433—611no—
213#2308D6111yescharacteristic, complemented
214#1783—611no—
215#1510—612no—
216#1284—612no—
217C2×C61—22no—
218C2×C61—22no—
219C2×C61—41no—
220C2×C61—44no—
221C2×C61—82no—
222C2×C61#188711yescharacteristic
223C2×C61#189811yescharacteristic
224C2×C61#191111yescharacteristic
225D61—161no—
226C23×D4C2×D6111yescharacteristic
227C2×#127—612no—
228C2×#121—611no—
229C2×#120—1221no—
230C2×#120—1221no—
231C22×C22⋊C4—21no—
232C2×#100—1222no—
233#407—612no—
234#392—1222no—
235#390—612no—
236#386—24no—
237#381C2×D6112yes—
238#379—21no—
239C4×C22⋊C4—611no—
240#327—1224no—
241C61#946311yescharacteristic, complemented, Sylow
242C25C22⋊D6111yescharacteristic
243C25C22⋊D6111yescharacteristic
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—44no—
247C22×D4—611no—
248C22×D4—1222no—
249C22×D4—1222no—
250C22×D4—1221no—
251C22×D4C4×D6112yes—
252C22×D4D24412yes—
253C23×C4—22no—
254C23×C4C22×D6111yescharacteristic
255C2×M16—1222no—
256#127—2444no—
257#121—1222no—
258#121—1222no—
259#120—2444no—
260#120—2444no—
261C2×C4⋊C4—611no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1222no—
270C2×C22⋊C4—1221no—
271C2×C42—612no—
272#100—2444no—
273C22⋊C8—1222no—
274C22⋊C8—1222no—
275#95—22no—
276#95—1221no—
277#95C22⋊D6114yescomplemented
278C24—21no—
279C24—24no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—81no—
287C24—81no—
288C24—1221no—
289C24#1175819812yes—
290C24#1175819812yes—
291C24C2×C22⋊D6111yescharacteristic
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—1222no—
302C2×D4—4882no—
303C2×D4—4882no—
304C2×D4—2442no—
305C2×D4—2444no—
306C2×D4—2444no—
307C2×D4—2442no—
308C2×D4—2442no—
309C2×D4—2444no—
310C2×D4—2444no—
311C2×D4—2441no—
312C2×D4—2441no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—1222no—
319C22×C4—1221no—
320C22×C4C2×C4×D6111yescharacteristic
321C22×C4C2×D24411yescharacteristic
322C22×C4#1175822212yes—
323C22×C4C2×C22⋊D6112yes—
324M16—2444no—
325C2×C8—2442no—
326C4⋊C4—1222no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—1224no—
332C22⋊C4—2444no—
333C22⋊C4—2444no—
334C22⋊C4—2442no—
335C42—2442no—
336C23—22no—
337C23—22no—
338C23—24no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—88no—
362C23—88no—
363C23—2442no—
364C23—2441no—
365C23—2441no—
366C23—2442no—
367C23—2441no—
368C23#1451969114yes—
369C23C2×#1175819811yescharacteristic
370C23#1451974611yescharacteristic
371C23#1451976011yescharacteristic
372D4—48no—
373D4—48no—
374D4—4882no—
375D4—4882no—
376D4—4882no—
377D4—4888no—
378D4—4888no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—88no—
387C2×C4—1224no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4C4×C22⋊D6112yes—
394C2×C4#1451975112yes—
395C8—4882no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—24no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—44no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—82no—
415C22—82no—
416C22—84no—
417C22—84no—
418C22—84no—
419C22—4882no—
420C22—4882no—
421C22—4881no—
422C22—4881no—
423C22—4881no—
424C22#1683964111yescharacteristic
425C22C2×#1451969112yes—
426C22#1683993012yes—
427C4—22no—
428C4—22no—
429C4—84no—
430C4—4881no—
431C4—2444no—
432C2—22no—
433C2—22no—
434C2—41no—
435C2—44no—
436C2—82no—
437C2—9761no—
438C2#2144621711yescharacteristic
439C2#2144627211yescharacteristic
440C2#2144635111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2308C2
socleC22×C61#386
prime radical#21449452C2
cosocleC23×C4×C61C23
hypercenter#2308D61
hypocenterC61#9463
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2308

Complements

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

NH ≅ G/Ntype
2. #21445740C2semidirect
3. #21444347C2semidirect
4. #21443671C2semidirect
5. C61×#2308C2semidirect
6. C2×#14519773C4semidirect
13. #16839665C4semidirect
21. C22×D244C2×C4semidirect
31. C2×C4⋊Dic61C2×C4semidirect
36. C2×C4×Dic61C2×C4semidirect
56. #95×C61D4semidirect
213. #2308D61semidirect
241. C61#9463semidirect
277. #95C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214494522
#214457402
#214443472
#214436712
C61×#23082
#946361