Finite Group Databasebeta

← #29577200

Subgroups — #29577200

Order 15616.

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

normal subgroups only

This group has 802 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#21455071C211yesmaximal, characteristic
2#21448649C211yesmaximal, characteristic, complemented
3#21446655C211yesmaximal, characteristic, complemented
4#21444336C211yesmaximal, characteristic, complemented
5#21443726C211yesmaximal, characteristic, complemented
6#21443705C211yesmaximal, characteristic, complemented
7C61×#1789C211yesmaximal, characteristic, complemented
8C2×#14519802C412yescomplemented
9C2×#14519751C2211yescharacteristic
10C2×D4×Dic61—21no—
11#16839907—22no—
12#16839901—21no—
13#16839896—21no—
14#16839763C412yescomplemented
15C2×#14519691—22no—
16#16839726C2211yescharacteristic
17#16839669—21no—
18#16839642—22no—
19#16839625—21no—
20#16839622C2211yescharacteristic
21C4×C22⋊Dic61C2211yescharacteristic
22C2×#14519684—21no—
23#16839497—21no—
24#16839497—21no—
25C22×#41×C61C2211yescharacteristic
26C2×#104×C61—22no—
27C61×#407C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C22×D244C2×C411yescharacteristic, complemented
30C22×C22⋊D61—21no—
31C23×Dic61—22no—
32#14519802—44no—
33C2×#11758201—22no—
34C2×#11758199—22no—
35C2×#11758194C2×C412yescomplemented
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40D4×Dic61—44no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic, complemented
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61D412yes—
46C2×#11758198—21no—
47C2×C4×Dic61—21no—
48#14519701—22no—
49#14519698—22no—
50#14519691—44no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53#14519684—22no—
54#14519684—22no—
55#14519684—21no—
56#14519684D411yescharacteristic
57#14519684D411yescharacteristic
58C2×#41×C61—21no—
59C2×#41×C61—21no—
60C2×#41×C61—44no—
61C2×#41×C61C2×C412yes—
62C2×#41×C61D412yes—
63C22×Q8×C61D411yescharacteristic
64C22×D4×C61—21no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61—21no—
67C23×C4×C61C2311yescharacteristic
68C2×M16×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C42×C61—21no—
71#104×C61—44no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74#95×C61—21no—
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—21no—
85C2×Q8×C61—42no—
86C2×Q8×C61C22⋊C412yes—
87C2×D4×C61—24no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—21no—
91C2×D4×C61—44no—
92C2×D4×C61C22⋊C412yes—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—42no—
97C22×C4×C61—44no—
98C22×C4×C61—41no—
99C22×C4×C61C22×C411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C2×C22⋊D61—42no—
106C2×C22⋊D61—42no—
107C2×C22⋊D61—41no—
108C2×C22⋊D61—41no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—44no—
114C22×Dic61—44no—
115C22×Dic61—82no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—42no—
119M16×C61—42no—
120C2×C8×C61—41no—
121C22⋊C4×C61—22no—
122C22⋊C4×C61—41no—
123C42×C61—42no—
124C22⋊Dic61—42no—
125C22⋊Dic61—44no—
126C22⋊Dic61—42no—
127#11758201—44no—
128#11758199—44no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—42no—
134C4×Dic61—42no—
135#11758194—42no—
136C2×#92418—22no—
137C23×C61—24no—
138C23×C61—22no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—41no—
142C23×C61—41no—
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—22no—
154Q8×C61—22no—
155Q8×C61—42no—
156D4×C61—24no—
157D4×C61—22no—
158D4×C61—44no—
159D4×C61—44no—
160D4×C61#9914yescomplemented
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—44no—
169C2×C4×C61—44no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—81no—
173C2×C4×C61—81no—
174C2×C4×C61C2×C22⋊C411yescharacteristic
175C2×C4×C61C4×D411yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12311yescharacteristic
178C22⋊D61—82no—
179C22⋊D61—81no—
180C22⋊D61—81no—
181C2×Dic61—22no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—88no—
193D244—42no—
194D244—81no—
195C8×C61—81no—
196#92418—42no—
197#9365—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—24no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—44no—
208C22×C61—44no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61#38611yescharacteristic
214C22×C61#45311yescharacteristic
215C22×C61#45411yescharacteristic
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—162no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—42no—
224C4×C61—82no—
225C4×C61C2×#9912yes—
226Dic61—42no—
227Dic61—81no—
228Dic61—82no—
229Dic61—81no—
230#3006—611no—
231#2339—611no—
232#1964—611no—
233#1789D6111yescharacteristic, complemented
234#1507—611no—
235#1305—611no—
236#1297—611no—
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—41no—
240C2×C61—41no—
241C2×C61—44no—
242C2×C61#188311yescharacteristic
243C2×C61#189811yescharacteristic
244C2×C61#190611yescharacteristic
245D61—161no—
246C22×#41C2×D6111yescharacteristic
247C2×AGL(1,ℤ/8)—612no—
248C2×#121—611no—
249C2×C4×D4—1221no—
250#449—1222no—
251#448—1221no—
252#447—1221no—
253C2×#104—22no—
254#417—612no—
255C2×#99—1222no—
256#407—611no—
257#407C2×D6111yescharacteristic
258#391—1221no—
259#386—1222no—
260#381—1221no—
261#380—611no—
262C4×C22⋊C4—611no—
263C4×C22⋊C4C2×D6111yescharacteristic
264C2×#95—1221no—
265#327—1221no—
266#327—1221no—
267C61#936511yescharacteristic, complemented, Sylow
268C2×#41—21no—
269C2×#41—21no—
270C2×#41—44no—
271C2×#41C4×D6112yes—
272C2×#41C22⋊D6112yes—
273C22×Q8D24411yescharacteristic
274C22×D4—21no—
275C22×D4—611no—
276C22×D4—1221no—
277C22×D4D24411yescharacteristic
278C23×C4—21no—
279C23×C4—1222no—
280C23×C4C22×D6111yescharacteristic
281AGL(1,ℤ/8)—2444no—
282C2×SD16—1222no—
283C2×D8—1222no—
284C2×M16—21no—
285C2×M16—612no—
286#121—1221no—
287#121—1221no—
288#121—1221no—
289#121—1221no—
290C4×D4—2444no—
291C2×C4⋊C4—611no—
292C2×C4⋊C4—1221no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—612no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4—1221no—
298C2×C42—21no—
299C2×C42—1221no—
300#104—44no—
301#103—1222no—
302#102—1222no—
303#99—2444no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308#95—21no—
309#95—611no—
310#95—611no—
311#95—1222no—
312#95—1222no—
313#95—1221no—
314C24—21no—
315C24—1221no—
316C24D4×D6111yescharacteristic
317#41—24no—
318#41—24no—
319#41—42no—
320#41—42no—
321#41—44no—
322#41—44no—
323C2×Q8—21no—
324C2×Q8—42no—
325C2×Q8#1175819812yes—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—44no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—2442no—
334C2×D4—2442no—
335C2×D4—2442no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4#1175819812yes—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—41no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—1222no—
349C22×C4—4882no—
350C22×C4—2444no—
351C22×C4—2444no—
352C22×C4C2×C4×D6111yescharacteristic
353C22×C4C2×D24411yescharacteristic
354C22×C4D4×D6111yescharacteristic
355C22×C4#1175822411yescharacteristic
356C22×C4#1175822411yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358SD16—2444no—
359D8—2444no—
360M16—42no—
361M16—2442no—
362C2×C8—41no—
363C2×C8—1222no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—2442no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—2442no—
370C22⋊C4—2444no—
371C22⋊C4—2441no—
372C22⋊C4—2441no—
373C22⋊C4—2442no—
374C42—42no—
375C42—2442no—
376C23—24no—
377C23—22no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—4881no—
386C23—4881no—
387C23—2442no—
388C23—2441no—
389C23#1451973711yescharacteristic
390C23#1451974711yescharacteristic
391C23#1451974911yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—42no—
395D4—24no—
396D4—22no—
397D4—44no—
398D4—44no—
399D4—4881no—
400D4—4882no—
401D4—4881no—
402D4—4881no—
403D4—2442no—
404D4#1451969114yescomplemented
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—1222no—
419C2×C4—4888no—
420C2×C4—2441no—
421C2×C4—2442no—
422C2×C4—2441no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2442no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4C2×#1175819811yescharacteristic
431C2×C4#1451973711yescharacteristic
432C2×C4#1451974911yescharacteristic
433C2×C4#1451976111yescharacteristic
434C8—81no—
435C8—2442no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—21no—
444C22—21no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—9762no—
452C22—4881no—
453C22—4881no—
454C22—4881no—
455C22#1683963911yescharacteristic
456C22#1683993111yescharacteristic
457C22#1683993611yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—42no—
462C4—82no—
463C4—4882no—
464C4—4881no—
465C4—4881no—
466C4—2442no—
467C4C2×#1451969112yes—
468C2—21no—
469C2—21no—
470C2—41no—
471C2—41no—
472C2—44no—
473C2—9761no—
474C2#2144619311yescharacteristic
475C2#2144627011yescharacteristic
476C2#2144631811yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1789C2
socleC22×C61#386
prime radical#21455071C2
cosocleC23×C4×C61C23
hypercenter#1789D61
hypocenterC61#9365
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #21448649C2semidirect
3. #21446655C2semidirect
4. #21444336C2semidirect
5. #21443726C2semidirect
6. #21443705C2semidirect
7. C61×#1789C2semidirect
8. C2×#14519802C4semidirect
14. #16839763C4semidirect
29. C22×D244C2×C4semidirect
35. C2×#11758194C2×C4semidirect
42. C2×C4⋊Dic61C2×C4semidirect
160. D4×C61#99semidirect
233. #1789D61semidirect
267. C61#9365semidirect
404. D4#14519691semidirect

Maximal subgroups

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

subgroupindex
#214550712
#214486492
#214466552
#214443362
#214437262
#214437052
C61×#17892
#936561