Finite Group Databasebeta

← #29577046

Subgroups — #29577046

Order 15616.

576 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 95 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#21456616C211yesmaximal, characteristic
2#21448976C211yesmaximal, characteristic, complemented
3#21446655C211yesmaximal, characteristic, complemented
4#21444349C211yesmaximal, characteristic, complemented
5#21443724C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#1507C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840080—21no—
10#16840078C412yescomplemented
11#16840013—22no—
12#16839896—21no—
13#16839895C412yescomplemented
14C2×#14519691—21no—
15C2×#14519691—21no—
16#16839726C2211yescharacteristic
17#16839665—21no—
18#16839655—21no—
19#16839648—21no—
20#16839642—21no—
21#16839642—21no—
22#16839637—21no—
23#16839625C2211yescharacteristic
24#16839622C2211yescharacteristic
25#16839497—21no—
26#16839497—21no—
27C2×#121×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C4×C22⋊C4×C61C2211yescharacteristic
30C61×#327—22no—
31C22×D244C2×C411yescharacteristic, complemented
32C22×C22⋊D61—21no—
33C23×Dic61—21no—
34C2×#11758199—22no—
35C2×#11758194—21no—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic, complemented
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61D411yescharacteristic
48C2×C22⋊Dic61D411yescharacteristic
49C2×#11758198—21no—
50C2×C4×Dic61—21no—
51#14519708—22no—
52#14519698—22no—
53#14519697—22no—
54#14519691—42no—
55#14519691—42no—
56C22⋊#92418—21no—
57C22⋊#92418C2×C412yescomplemented
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684D411yescharacteristic
63#14519684D411yescharacteristic
64C22×D4×C61—21no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C2×M16×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70#121×C61C2×C412yes—
71#121×C61D412yes—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C42×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78#95×C61—21no—
79C24×C61—21no—
80C24×C61C2×D411yescharacteristic
81C23×D61—21no—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—21no—
85C2×D4×C61—81no—
86C2×D4×C61—81no—
87C2×D4×C61—81no—
88C2×D4×C61C22⋊C412yes—
89C22×C4×C61—21no—
90C22×C4×C61—41no—
91C22×C4×C61—81no—
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—41no—
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—81no—
110C22×Dic61—81no—
111C22×Dic61—81no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—81no—
115M16×C61—81no—
116C2×C8×C61—41no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61C22⋊C412yes—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—22no—
121C22⋊C4×C61—22no—
122C22⋊C4×C61—41no—
123C42×C61—81no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—81no—
127C22⋊Dic61—81no—
128C22⋊Dic61—81no—
129C22⋊Dic61—81no—
130#11758199—82no—
131#11758198—41no—
132#11758198—41no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—81no—
136C4⋊Dic61—81no—
137C4×Dic61—81no—
138#11758194—81no—
139C2×#92418—22no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—22no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61C4×D411yescharacteristic
152C23×C61#12011yescharacteristic
153C23×C61#12111yescharacteristic
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—81no—
158C22×D61—81no—
159D4×C61—42no—
160D4×C61—81no—
161D4×C61—82no—
162D4×C61—82no—
163C2×C4×C61—22no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—81no—
174C2×C4×C61—81no—
175C2×C4×C61C2×C22⋊C411yescharacteristic
176C2×C4×C61C4×D411yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12311yescharacteristic
179C22⋊D61—81no—
180C22⋊D61—81no—
181C22⋊D61—81no—
182C22⋊D61—81no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—84no—
199D244—42no—
200D244—81no—
201C8×C61—81no—
202#92418—42no—
203#9343—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61#38611yescharacteristic
222C22×C61#45311yescharacteristic
223C22×C61#45411yescharacteristic
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—161no—
228C2×D61—161no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—82no—
235Dic61—81no—
236Dic61—81no—
237Dic61—81no—
238Dic61—81no—
239Dic61—81no—
240#3192—611no—
241#2377—611no—
242#1964—611no—
243#1510—611no—
244#1507D6111yescharacteristic, complemented
245#1305—611no—
246#1284—611no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—41no—
250C2×C61—41no—
251C2×C61—82no—
252C2×C61#188311yescharacteristic
253C2×C61#190411yescharacteristic
254C2×C61#191311yescharacteristic
255D61—161no—
256C2×#121—611no—
257C2×#121C2×D6111yescharacteristic
258#481—612no—
259#481—1221no—
260#469—1222no—
261#447—612no—
262#447—1221no—
263C2×#99—1221no—
264C2×#99—1221no—
265#407—611no—
266#407C2×D6111yescharacteristic
267#390—1221no—
268#388—1221no—
269#387—1221no—
270#386—1221no—
271#386—1221no—
272#385—1221no—
273#381—611no—
274#380—611no—
275C4×C22⋊C4C2×D6111yescharacteristic
276#327—22no—
277#327—1221no—
278#327—1221no—
279C61#934311yescharacteristic, complemented, Sylow
280C22×D4—21no—
281C22×D4—611no—
282C22×D4—1221no—
283C22×D4D24411yescharacteristic
284C23×C4—1221no—
285C23×C4C22×D6111yescharacteristic
286C2×D8—1222no—
287C2×M16—21no—
288C2×M16—1221no—
289#121—21no—
290#121—21no—
291#121—1221no—
292#121—1221no—
293#121—1221no—
294#121—1221no—
295#121C4×D6112yes—
296#121C22⋊D6112yes—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4D24411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—611no—
304C2×C22⋊C4—611no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C42—21no—
311C2×C42—1221no—
312#106—1222no—
313#102—1222no—
314#102—1222no—
315#99—2442no—
316#99—2442no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—612no—
320C22⋊C8—1221no—
321#95—21no—
322#95—611no—
323#95—611no—
324#95—1221no—
325#95—1221no—
326#95—1221no—
327#95—1221no—
328C24—21no—
329C24—1221no—
330C24D4×D6111yescharacteristic
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—4881no—
340C2×D4—2441no—
341C2×D4—2441no—
342C2×D4—2441no—
343C2×D4—2441no—
344C2×D4—2441no—
345C2×D4—2441no—
346C2×D4#1175819812yes—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—81no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—4881no—
356C22×C4—4881no—
357C22×C4—4881no—
358C22×C4C2×C4×D6111yescharacteristic
359C22×C4C2×D24411yescharacteristic
360C22×C4D4×D6111yescharacteristic
361C22×C4#1175822411yescharacteristic
362C22×C4#1175822411yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364D8—4882no—
365M16—81no—
366M16—4881no—
367C2×C8—41no—
368C2×C8—1222no—
369C4⋊C4—21no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—4881no—
373C4⋊C4—4881no—
374C4⋊C4#1175819812yes—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—41no—
379C22⋊C4—4881no—
380C22⋊C4—4881no—
381C22⋊C4—4881no—
382C22⋊C4—4881no—
383C22⋊C4—2441no—
384C22⋊C4—2441no—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C42—81no—
388C42—4881no—
389C23—21no—
390C23—21no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—4881no—
401C23—4881no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23#1451973711yescharacteristic
406C23#1451974711yescharacteristic
407C23#1451974911yescharacteristic
408D4—42no—
409D4—81no—
410D4—82no—
411D4—82no—
412D4—4881no—
413D4—4881no—
414D4—4881no—
415D4—4881no—
416D4—4881no—
417D4—2442no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—4884no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4C2×#1175819811yescharacteristic
447C2×C4#1451973711yescharacteristic
448C2×C4#1451974911yescharacteristic
449C2×C4#1451976111yescharacteristic
450C8—81no—
451C8—2442no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—9761no—
470C22—9761no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22#1683963911yescharacteristic
475C22#1683993111yescharacteristic
476C22#1683993611yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—82no—
483C4—4881no—
484C4—4881no—
485C4—4881no—
486C4—4881no—
487C4—4881no—
488C2—21no—
489C2—21no—
490C2—41no—
491C2—41no—
492C2—82no—
493C2—9761no—
494C2#2144619311yescharacteristic
495C2#2144630411yescharacteristic
496C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1507C2
socleC22×C61#386
prime radical#21456616C2
cosocleC23×C4×C61C23
hypercenter#1507D61
hypocenterC61#9343
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #21448976C2semidirect
3. #21446655C2semidirect
4. #21444349C2semidirect
5. #21443724C2semidirect
6. #21443672C2semidirect
7. C61×#1507C2semidirect
10. #16840078C4semidirect
13. #16839895C4semidirect
31. C22×D244C2×C4semidirect
42. C2×C4⋊Dic61C2×C4semidirect
57. C22⋊#92418C2×C4semidirect
244. #1507D61semidirect
279. C61#9343semidirect

Maximal subgroups

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

subgroupindex
#214566162
#214489762
#214466552
#214443492
#214437242
#214436722
C61×#15072
#934361