Finite Group Databasebeta

← #29577660

Subgroups — #29577660

Order 15616.

770 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 103 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#21454264C211yesmaximal, characteristic, complemented
2#21449434C211yesmaximal, characteristic
3#21444347C212yesmaximal, complemented
4#21443671C212yesmaximal, complemented
5C61×#1782C211yesmaximal, characteristic, complemented
6C22×#11758194—21no—
7C2×#14519773C411yescharacteristic, complemented
8C2×#14519766C411yescharacteristic, complemented
9C2×#14519751C2211yescharacteristic
10#16839729—24no—
11#16839726—21no—
12#16839726C2212yes—
13#16839670—21no—
14#16839670—21no—
15#16839669C411yescharacteristic, complemented
16#16839665C411yescharacteristic, complemented
17#16839641—21no—
18#16839641—21no—
19C4×C22⋊Dic61C2211yescharacteristic
20#16839495—24no—
21C22×#41×C61C2211yescharacteristic
22C2×#99×C61—21no—
23C2×#99×C61—21no—
24C61×#381C2212yes—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C22×C22⋊D61—21no—
29C2×#11758194—22no—
30C2×#11758194—22no—
31C2×#11758194—42no—
32C2×#11758194—42no—
33C22×#92418—22no—
34#14519773—42no—
35#14519766—42no—
36#14519751—22no—
37#14519751—22no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61C2×C411yescharacteristic
47C22⋊#92418—24no—
48C22⋊#92418—22no—
49C22⋊#92418D414yescomplemented
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C2×#41×C61—22no—
54C2×#41×C61—22no—
55C2×#41×C61—42no—
56C2×#41×C61—42no—
57C22×Q8×C61C2×C411yescharacteristic
58C22×D4×C61C2×C411yescharacteristic
59C22×D4×C61D411yescharacteristic
60C22×D4×C61D411yescharacteristic
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64C2×C22⋊C4×C61—22no—
65#99×C61—42no—
66#99×C61—42no—
67#95×C61—22no—
68#95×C61—22no—
69C24×C61C22⋊C411yescharacteristic
70C24×C61C22⋊C411yescharacteristic
71C24×C61C2×D411yescharacteristic
72C23×D61—21no—
73#41×C61—44no—
74#41×C61—44no—
75#41×C61—48no—
76C2×Q8×C61—22no—
77C2×Q8×C61—22no—
78C2×Q8×C61—41no—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—22no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—42no—
99C22×C4×C61C22⋊C411yescharacteristic
100C22×C4×C61C22⋊C411yescharacteristic
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D412yes—
104C22×C4×C61#4112yes—
105C2×C22⋊D61—42no—
106C2×C22⋊D61—42no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—42no—
109C2×C22⋊D61—42no—
110C2×C22⋊D61—81no—
111C2×C22⋊D61—81no—
112C2×C22⋊D61—81no—
113C2×C22⋊D61—81no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C2×D244—22no—
118C2×D244—41no—
119C2×D244—41no—
120C22⋊C4×C61—42no—
121C22⋊C4×C61—42no—
122C22⋊Dic61—42no—
123C22⋊Dic61—81no—
124C22⋊Dic61—81no—
125#11758198—42no—
126#11758198—42no—
127#11758198—41no—
128#11758198—41no—
129C4⋊Dic61—22no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132#11758194—44no—
133#11758194—42no—
134#11758194—42no—
135C2×#92418—24no—
136C2×#92418—44no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61#9912yes—
151C23×C61C2×C22⋊C411yescharacteristic
152C23×C61#12112yes—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160Q8×C61—44no—
161D4×C61—44no—
162D4×C61—44no—
163D4×C61—44no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—24no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—82no—
181C2×C4×C61—82no—
182C2×C4×C61—82no—
183C2×C4×C61#9912yes—
184C2×C4×C61C4×D412yes—
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12311yescharacteristic
187C22⋊D61—84no—
188C22⋊D61—84no—
189C22⋊D61—84no—
190C2×Dic61—22no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199D244—82no—
200#92418—44no—
201#9445—611nomaximal, Sylow
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—22no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61#38611yescharacteristic
220C22×C61C2×#9911yescharacteristic
221C22×C61C2×#9911yescharacteristic
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C4×C61—21no—
230C4×C61—22no—
231C4×C61—21no—
232C4×C61—22no—
233C4×C61—41no—
234C4×C61—162no—
235Dic61—42no—
236Dic61—81no—
237Dic61—81no—
238#2879—611no—
239#2431—611no—
240#1782D6111yescharacteristic, complemented
241#1510—612no—
242#1284—612no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—42no—
246C2×C61—42no—
247C2×C61—81no—
248C2×C61#188411yescharacteristic
249C2×C61#189811yescharacteristic
250C2×C61#191011yescharacteristic
251D61—161no—
252C22×#41C2×D6111yescharacteristic
253C22×M16—1221no—
254C2×#127—611no—
255C2×#124—611no—
256C2×#121—611no—
257C2×#99—21no—
258C2×#99—21no—
259#408—1224no—
260#407—612no—
261#407—1221no—
262#392—1221no—
263#392—1221no—
264#391—611no—
265#390—611no—
266#386—1221no—
267#386—1221no—
268#381C2×D6112yes—
269C4×C22⋊C4—611no—
270#327—1224no—
271C61#944511yescharacteristic, complemented, Sylow
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—42no—
275C2×#41—42no—
276C22×Q8C4×D6111yescharacteristic
277C22×D4—611no—
278C22×D4—1221no—
279C22×D4—1221no—
280C22×D4—1221no—
281C22×D4C4×D6111yescharacteristic
282C22×D4C22⋊D6111yescharacteristic
283C22×D4C22⋊D6111yescharacteristic
284C23×C4D24411yescharacteristic
285C23×C4D24411yescharacteristic
286C23×C4C22×D6111yescharacteristic
287C2×M16—1222no—
288C2×M16—1222no—
289C2×M16—2442no—
290C2×M16—2442no—
291C22×C8—1222no—
292#127—2442no—
293#124—2442no—
294#121—1222no—
295#121—1222no—
296C2×C4⋊C4—611no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—1221no—
299C2×C22⋊C4—1221no—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4—1221no—
304C2×C42—611no—
305C2×C42—611no—
306#99—42no—
307#99—42no—
308C22⋊C8—614no—
309C22⋊C8—1224no—
310C22⋊C8—1222no—
311#95—22no—
312#95—22no—
313#95—1221no—
314#95—1221no—
315#95—1221no—
316C24—1221no—
317C24#1175819811yescharacteristic
318C24#1175819811yescharacteristic
319C24C2×C22⋊D6111yescharacteristic
320#41—44no—
321#41—44no—
322#41—48no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—1222no—
336C2×D4—4881no—
337C2×D4—4881no—
338C2×D4—4881no—
339C2×D4—4881no—
340C2×D4—2441no—
341C2×D4—2441no—
342C2×D4—2442no—
343C2×D4—2442no—
344C2×D4—2442no—
345C2×D4—2442no—
346C2×D4—2442no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4#1175819811yescharacteristic
362C22×C4#1175819811yescharacteristic
363C22×C4C2×C4×D6111yescharacteristic
364C22×C4C2×D24411yescharacteristic
365C22×C4D4×D6112yes—
366C22×C4#1175822412yes—
367M16—2444no—
368M16—2442no—
369M16—2442no—
370C2×C8—1224no—
371C2×C8—2444no—
372C4⋊C4—1222no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—4881no—
376C22⋊C4—4881no—
377C22⋊C4—2442no—
378C22⋊C4—2442no—
379C22⋊C4—2442no—
380C22⋊C4—2441no—
381C22⋊C4—2441no—
382C42—2441no—
383C42—2441no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23—2441no—
401C23—2441no—
402C23—2441no—
403C23—2441no—
404C23#1451969112yes—
405C23C2×#1175819811yescharacteristic
406C23#1451974912yes—
407Q8—44no—
408D4—44no—
409D4—44no—
410D4—44no—
411D4—4882no—
412D4—4884no—
413D4—4884no—
414D4—4884no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—1222no—
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#1451969112yes—
444C2×C4#1451973712yes—
445C2×C4#1451974711yescharacteristic
446C2×C4#1451976111yescharacteristic
447C8—2444no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22—4881no—
472C22#1683963911yescharacteristic
473C22C2×#1451969111yescharacteristic
474C22C2×#1451969111yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—41no—
480C4—162no—
481C4—4881no—
482C4—4881no—
483C4—2442no—
484C2—21no—
485C2—21no—
486C2—42no—
487C2—42no—
488C2—81no—
489C2—9761no—
490C2#2144620011yescharacteristic
491C2#2144627011yescharacteristic
492C2#2144634411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1782C2
socleC22×C61#386
prime radical#21449434C2
cosocleC23×C4×C61C23
hypercenter#1782D61
hypocenterC61#9445
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #21454264C2semidirect
3. #21444347C2semidirect
4. #21443671C2semidirect
5. C61×#1782C2semidirect
7. C2×#14519773C4semidirect
8. C2×#14519766C4semidirect
15. #16839669C4semidirect
16. #16839665C4semidirect
49. C22⋊#92418D4semidirect
240. #1782D61semidirect
271. C61#9445semidirect

Maximal subgroups

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

subgroupindex
#214542642
#214494342
#214443472
#214436712
C61×#17822
#944561