Finite Group Databasebeta

← #29577223

Subgroups — #29577223

Order 15616.

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

normal subgroups only

This group has 774 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#21455071C211yesmaximal, characteristic
2#21446654C211yesmaximal, characteristic, complemented
3#21445763C211yesmaximal, characteristic, complemented
4#21444334C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443705C211yesmaximal, characteristic, complemented
7C61×#2341C211yesmaximal, characteristic, complemented
8C2×#14519802C412yescomplemented
9C2×#14519751C2211yescharacteristic
10#16839907—22no—
11#16839901—21no—
12#16839896—21no—
13C2×#14519704—22no—
14#16839763C412yescomplemented
15C2×#14519691—22no—
16#16839726C2211yescharacteristic
17#16839726C2211yescharacteristic
18C4×C22⋊Dic61C2211yescharacteristic
19#16839495—21no—
20#16839495—21no—
21C22×#41×C61C2211yescharacteristic
22C2×C4×Q8×C61—21no—
23C61×#390—21no—
24C61×#390—22no—
25C61×#380—21no—
26C61×#380C2211yescharacteristic
27C4×C22⋊C4×C61—21no—
28C4×C22⋊C4×C61C2211yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31#14519802—44no—
32C2×#11758201—22no—
33C2×#11758199—22no—
34C2×#11758194—21no—
35C2×#11758194C2×C412yes—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×C22⋊Dic61—21no—
42C2×#11758198—21no—
43C2×C4×Dic61—21no—
44#14519704—44no—
45#14519701—22no—
46#14519698—22no—
47#14519691—44no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52#14519684—21no—
53C2×#41×C61—21no—
54C2×#41×C61—21no—
55C2×#41×C61—44no—
56C2×#41×C61C2×C412yes—
57C2×#41×C61D412yes—
58C22×Q8×C61D411yescharacteristic
59C22×D4×C61—21no—
60C22×D4×C61D411yescharacteristic
61C23×C4×C61—21no—
62C23×C4×C61C2311yescharacteristic
63C4×Q8×C61—44no—
64C2×C4⋊C4×C61—22no—
65C2×C4⋊C4×C61—21no—
66C2×C22⋊C4×C61—22no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61—22no—
69C2×C22⋊C4×C61D412yes—
70C2×C42×C61—22no—
71C2×C42×C61—21no—
72#95×C61—21no—
73#95×C61D411yescharacteristic
74#95×C61D411yescharacteristic
75C24×C61—21no—
76C24×C61C2×D411yescharacteristic
77C23×D61—21no—
78#41×C61—24no—
79#41×C61—24no—
80#41×C61—42no—
81#41×C61—42no—
82#41×C61—44no—
83#41×C61—44no—
84C2×Q8×C61—24no—
85C2×Q8×C61—21no—
86C2×Q8×C61C22⋊C412yes—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—21no—
90C2×D4×C61—42no—
91C2×D4×C61—44no—
92C2×D4×C61C22⋊C412yes—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—42no—
100C22×C4×C61—44no—
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×C22⋊D61—42no—
108C2×C22⋊D61—42no—
109C2×C22⋊D61—41no—
110C2×C22⋊D61—41no—
111C22×Dic61—21no—
112C22×Dic61—41no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244—42no—
116C4⋊C4×C61—44no—
117C4⋊C4×C61—42no—
118C22⋊C4×C61—42no—
119C22⋊C4×C61—44no—
120C22⋊C4×C61—42no—
121C22⋊C4×C61—82no—
122C42×C61—44no—
123C42×C61—42no—
124C22⋊Dic61—22no—
125C22⋊Dic61—41no—
126#11758201—44no—
127#11758199—44no—
128#11758198—41no—
129#11758198—41no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4×Dic61—42no—
133#11758194—42no—
134#11758194—42no—
135C2×#92418—22no—
136C2×#92418—41no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—22no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—42no—
143C23×C61—42no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61C4×D411yescharacteristic
147C23×C61#12011yescharacteristic
148C23×C61#12111yescharacteristic
149C22×D61—41no—
150C22×D61—42no—
151C22×D61—81no—
152C22×D61—81no—
153Q8×C61—24no—
154Q8×C61—22no—
155Q8×C61#9914yescomplemented
156D4×C61—22no—
157D4×C61—22no—
158D4×C61—44no—
159D4×C61—44no—
160D4×C61—42no—
161C2×C4×C61—24no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—44no—
171C2×C4×C61—44no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61C2×C22⋊C411yescharacteristic
181C2×C4×C61C4×D411yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12311yescharacteristic
184C22⋊D61—82no—
185C22⋊D61—81no—
186C22⋊D61—81no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—81no—
191C2×Dic61—81no—
192D244—42no—
193D244—81no—
194#92418—42no—
195#92418—81no—
196#9368—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—82no—
211C22×C61—82no—
212C22×C61#38611yescharacteristic
213C22×C61#45311yescharacteristic
214C22×C61#45411yescharacteristic
215C2×D61—81no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—162no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—24no—
222C4×C61—21no—
223C4×C61—42no—
224C4×C61—82no—
225C4×C61—81no—
226C4×C61C2×#9912yes—
227Dic61—82no—
228Dic61—81no—
229#3006—611no—
230#2341D6111yescharacteristic, complemented
231#1964—611no—
232#1789—611no—
233#1507—611no—
234#1305—611no—
235#1297—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—41no—
239C2×C61—41no—
240C2×C61—82no—
241C2×C61#188311yescharacteristic
242C2×C61#189811yescharacteristic
243C2×C61#190611yescharacteristic
244D61—161no—
245C22×#41C2×D6111yescharacteristic
246C2×AGL(1,ℤ/8)—612no—
247C2×#121—611no—
248C2×C4×Q8—21no—
249#449—1222no—
250#448—1221no—
251#447—1221no—
252C2×#104—1222no—
253#417—612no—
254C2×#99—1222no—
255#407—611no—
256#407—611no—
257#390—21no—
258#390—22no—
259#380—21no—
260#380C2×D6111yescharacteristic
261C4×C22⋊C4—21no—
262C4×C22⋊C4—611no—
263C4×C22⋊C4C2×D6111yescharacteristic
264#327—1221no—
265#327—1221no—
266C61#936811yescharacteristic, complemented, Sylow
267C2×#41—21no—
268C2×#41—21no—
269C2×#41—44no—
270C2×#41C4×D6112yes—
271C2×#41D24412yes—
272C22×Q8C22⋊D6111yescharacteristic
273C22×D4—21no—
274C22×D4—611no—
275C22×D4—1221no—
276C22×D4C22⋊D6111yescharacteristic
277C23×C4—21no—
278C23×C4C22×D6111yescharacteristic
279AGL(1,ℤ/8)—2444no—
280C2×SD16—1222no—
281C2×D8—1222no—
282C2×M16—612no—
283C2×M16—1221no—
284#121—1221no—
285#121—1221no—
286#121—1221no—
287#121—1221no—
288C4×Q8—44no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—611no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4C22⋊D6112yes—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—1221no—
301#104—2444no—
302#103—1222no—
303#102—1222no—
304#99—2444no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309#95—21no—
310#95—1221no—
311#95C22⋊D6111yescharacteristic
312#95C22⋊D6111yescharacteristic
313C24—21no—
314C24—1221no—
315C24C2×C22⋊D6111yescharacteristic
316#41—24no—
317#41—24no—
318#41—42no—
319#41—42no—
320#41—44no—
321#41—44no—
322C2×Q8—24no—
323C2×Q8—21no—
324C2×Q8#1175819812yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4—44no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—2442no—
333C2×D4—2442no—
334C2×D4—2442no—
335C2×D4—2441no—
336C2×D4—2441no—
337C2×D4#1175819812yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—1221no—
347C22×C4—2441no—
348C22×C4C2×C4×D6111yescharacteristic
349C22×C4C2×D24411yescharacteristic
350C22×C4#1175822211yescharacteristic
351C22×C4#1175822211yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354SD16—2444no—
355D8—2444no—
356M16—2442no—
357M16—2442no—
358C2×C8—1222no—
359C2×C8—2441no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—42no—
367C22⋊C4—82no—
368C22⋊C4—1222no—
369C22⋊C4—2441no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C42—44no—
373C42—42no—
374C42—2442no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—4881no—
385C23—4881no—
386C23—2442no—
387C23—2441no—
388C23C4×C22⋊D6111yescharacteristic
389C23#1451974611yescharacteristic
390C23#1451975111yescharacteristic
391Q8—24no—
392Q8—22no—
393Q8#1451969114yescomplemented
394D4—22no—
395D4—22no—
396D4—44no—
397D4—44no—
398D4—42no—
399D4—4881no—
400D4—4882no—
401D4—4881no—
402D4—4881no—
403D4—2442no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—4881no—
424C2×C4—4881no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4C2×#1175819811yescharacteristic
429C2×C4C4×C22⋊D6111yescharacteristic
430C2×C4#1451975111yescharacteristic
431C2×C4#1451976011yescharacteristic
432C8—4881no—
433C8—2442no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—82no—
448C22—82no—
449C22—9762no—
450C22—4881no—
451C22—4881no—
452C22—4881no—
453C22#1683964111yescharacteristic
454C22#1683993011yescharacteristic
455C22#1683993511yescharacteristic
456C4—21no—
457C4—24no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—82no—
462C4—81no—
463C4—4882no—
464C4—4881no—
465C4C2×#1451969112yes—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—82no—
471C2—9761no—
472C2#2144619511yescharacteristic
473C2#2144627211yescharacteristic
474C2#2144632011yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2341C2
socleC22×C61#386
prime radical#21455071C2
cosocleC23×C4×C61C23
hypercenter#2341D61
hypocenterC61#9368
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
2. #21446654C2semidirect
3. #21445763C2semidirect
4. #21444334C2semidirect
5. #21443725C2semidirect
6. #21443705C2semidirect
7. C61×#2341C2semidirect
8. C2×#14519802C4semidirect
14. #16839763C4semidirect
155. Q8×C61#99semidirect
230. #2341D61semidirect
266. C61#9368semidirect
393. Q8#14519691semidirect

Maximal subgroups

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

subgroupindex
#214550712
#214466542
#214457632
#214443342
#214437252
#214437052
C61×#23412
#936861