Finite Group Databasebeta

← #29653853

Subgroups — #29653853

Order 15616.

1408 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#16840324C211yesmaximal, characteristic
2#21456572C212yesmaximal, complemented
3#21444838C214yesmaximal, complemented
4#21444770C214yesmaximal, complemented
5#21444386C212yesmaximal, complemented
6C2×#16839486C211yesmaximal, characteristic, complemented
7C61×#2626C211yesmaximal, characteristic, complemented
8C23×C22⋊D61—21no—
9#16840324—24no—
10#16840324—22no—
11#16840324C2214yes—
12C2×#14519761—22no—
13C2×#14519751—21no—
14C2×#14519751C2212yes—
15C2×#14519747—22no—
16#16840090—22no—
17#16840080C2214yescomplemented
18#16839993—24no—
19#16839912—22no—
20#16839901C2214yescomplemented
21C2×C22⋊#92418C2212yescomplemented
22#16839495C2218yescomplemented
23#16839486C2214yescomplemented
24C2×C22⋊Q8×C61C2211yescharacteristic, complemented
25C22×C22⋊C4×C61C2211yescharacteristic
26C61×#399C2212yescomplemented
27C61×#396—22no—
28C61×#393C2212yescomplemented
29C61×#385—21no—
30C61×#380C2211yescharacteristic, complemented
31C61×#379—21no—
32C24×D61—21no—
33C22×D244—21no—
34C22×D244D412yes—
35C22×C22⋊D61—22no—
36C22×C22⋊D61—24no—
37C22×C22⋊D61—22no—
38C22×C22⋊D61—24no—
39C22×C22⋊D61—44no—
40C22×C22⋊D61—44no—
41C23×Dic61—21no—
42C2×#11758201—24no—
43C22×#92418—22no—
44#14519761—28no—
45#14519761—28no—
46#14519751—24no—
47#14519751—24no—
48#14519751—22no—
49#14519751—22no—
50#14519751D414yes—
51#14519751C2314yes—
52#14519747—28no—
53#14519747—28no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61D412yes—
56C2×#11758198—22no—
57C2×#11758198—22no—
58#14519708—24no—
59#14519701—24no—
60#14519697—24no—
61C22⋊#92418—22no—
62C22⋊#92418C2314yescomplemented
63C25×C61D411yescharacteristic
64C22×Q8×C61C2311yescharacteristic, complemented
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—22no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61C2311yescharacteristic, complemented
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61D412yes—
80C2×C22⋊C4×C61C2312yes—
81#95×C61—22no—
82#95×C61C2312yescomplemented
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—22no—
86C24×C61—81no—
87C24×C61—81no—
88C24×C61SD1614yescomplemented
89C24×C61C2×D412yes—
90C24×C61C2×D411yescharacteristic
91C23×D61—22no—
92C23×D61—24no—
93C23×D61—44no—
94C23×D61—82no—
95C23×D61—82no—
96C2×Q8×C61—22no—
97C2×Q8×C61—81no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—81no—
104C22×C4×C61—81no—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2411yescharacteristic
110C2×C22⋊D61—28no—
111C2×C22⋊D61—24no—
112C2×C22⋊D61—24no—
113C2×C22⋊D61—48no—
114C2×C22⋊D61—416no—
115C2×C22⋊D61—48no—
116C2×C22⋊D61—48no—
117C2×C22⋊D61—44no—
118C2×C22⋊D61—44no—
119C22×Dic61—24no—
120C22×Dic61—22no—
121C22×Dic61—44no—
122C2×D244—24no—
123C2×D244—22no—
124C2×D244—82no—
125C2×D244—82no—
126C2×D244C2×D414yes—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—81no—
130C4⋊C4×C61—82no—
131C22⋊C4×C61—24no—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C22⋊C4×C61—42no—
136C22⋊C4×C61—82no—
137C22⋊C4×C61—82no—
138#11758201—84no—
139#11758198—28no—
140#11758198—24no—
141#11758198—24no—
142C4⋊Dic61—24no—
143C4⋊Dic61—22no—
144C4⋊Dic61C2×D414yes—
145C2×#92418—24no—
146C2×#92418—82no—
147C23×C61—22no—
148C23×C61—24no—
149C23×C61—22no—
150C23×C61—24no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—42no—
154C23×C61—42no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—81no—
158C23×C61—81no—
159C23×C61—84no—
160C23×C61—84no—
161C23×C61#12011yescharacteristic
162C23×C61#12011yescharacteristic
163C23×C61C2×SD1614yes—
164C23×C61C2×SD1612yes—
165C23×C61AGL(1,ℤ/8)12yes—
166C23×C61C22×D411yescharacteristic
167C22×D61—24no—
168C22×D61—44no—
169C22×D61—44no—
170C22×D61—48no—
171C22×D61—44no—
172C22×D61—84no—
173C22×D61—84no—
174C22×D61—816no—
175Q8×C61—44no—
176C2×C4×C61—22no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—81no—
191C2×C4×C61#12012yes—
192C2×C4×C61C22×D412yes—
193C22⋊D61—416no—
194C22⋊D61—44no—
195C22⋊D61—44no—
196C22⋊D61—48no—
197C2×Dic61—24no—
198C2×Dic61—48no—
199C2×Dic61—44no—
200D244—44no—
201D244—82no—
202D244—84no—
203#92418—44no—
204#18285—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—24no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—44no—
215C22×C61—48no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—82no—
219C22×C61—84no—
220C22×C61#45012yes—
221C22×C61#45012yes—
222C22×C61C2×#12011yescharacteristic
223C22×C61C22×SD1611yescharacteristic
224C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
225C2×D61—44no—
226C2×D61—44no—
227C2×D61—44no—
228C2×D61—88no—
229C2×D61—816no—
230C4×C61—22no—
231C4×C61—22no—
232C4×C61—42no—
233C4×C61—82no—
234C4×C61—81no—
235C4×C61—81no—
236Dic61—44no—
237C2×#534—611no—
238#3187—612no—
239#2626D6111yescharacteristic, complemented
240#1613—614no—
241#1601—614no—
242#1519—612no—
243C2×#323—611no—
244C2×C61—21no—
245C2×C61—22no—
246C2×C61—21no—
247C2×C61—22no—
248C2×C61—44no—
249C2×C61C2×#45011yescharacteristic
250C2×C61#301911yescharacteristic
251C2×C61#302611yescharacteristic
252D61—84no—
253C23×D4—1221no—
254#534—614no—
255#534—1224no—
256#534—1222no—
257C2×#123—1222no—
258C2×C22⋊Q8C2×D6111yescharacteristic, complemented
259C2×#121—612no—
260C2×#121—1221no—
261C2×#120—1222no—
262C22×C22⋊C4C2×D6111yescharacteristic
263#483—1222no—
264#481—614no—
265#465—1224no—
266#450—1222no—
267#448—614no—
268C2×C22⋊C8—612no—
269#399C2×D6112yescomplemented
270#396—22no—
271#393C2×D6112yescomplemented
272#385—21no—
273#380C2×D6111yescharacteristic, complemented
274#379—21no—
275#327—618no—
276#323—614no—
277C61#1828511yescharacteristic, complemented, Sylow
278C25—1221no—
279C25C22⋊D6111yescharacteristic
280C22×Q8C22×D6111yescharacteristic, complemented
281C22×D4—612no—
282C22×D4—1221no—
283C22×D4—1222no—
284C22×D4—1224no—
285C22×D4—1222no—
286C22×D4—1224no—
287C22×D4—2444no—
288C22×D4—2444no—
289C23×C4—1221no—
290C23×C4C22⋊D6111yescharacteristic
291C23×C4C22×D6111yescharacteristic
292C2×SD16—1224no—
293C22×C8—1222no—
294#123—1228no—
295#123—1228no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298#121—614no—
299#121—614no—
300#121—1224no—
301#121—1222no—
302#121—1222no—
303#121—1224no—
304#120—1228no—
305#120—1228no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—612no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4C22×D6111yescharacteristic, complemented
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—1222no—
319C2×C22⋊C4—1222no—
320C2×C22⋊C4C22⋊D6112yes—
321C2×C22⋊C4C22×D6112yes—
322#106—1224no—
323#103—1224no—
324#102—1224no—
325C22⋊C8—614no—
326C22⋊C8—1222no—
327#95—22no—
328#95C22×D6112yescomplemented
329C24—21no—
330C24—21no—
331C24—22no—
332C24—81no—
333C24—81no—
334C24—1224no—
335C24—1222no—
336C24—4882no—
337C24—4882no—
338C24—2444no—
339C24#1175820114yescomplemented
340C24C2×C22⋊D6112yes—
341C24C2×C22⋊D6111yescharacteristic
342C2×Q8—22no—
343C2×Q8—81no—
344C2×D4—614no—
345C2×D4—1224no—
346C2×D4—1228no—
347C2×D4—1224no—
348C2×D4—1224no—
349C2×D4—1222no—
350C2×D4—4882no—
351C2×D4—4882no—
352C2×D4—2448no—
353C2×D4—24416no—
354C2×D4—2448no—
355C2×D4—2448no—
356C2×D4—2444no—
357C2×D4—2444no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—1224no—
366C22×C4—1222no—
367C22×C4—2444no—
368C22×C4D4×D6112yes—
369C22×C4D4×D6112yes—
370C22×C4C2×C22⋊D6112yes—
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C23×D6111yescharacteristic
373SD16—4884no—
374C2×C8—1224no—
375C2×C8—4882no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—614no—
381C4⋊C4—1224no—
382C4⋊C4—1222no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—1228no—
391C22⋊C4—1224no—
392C22⋊C4—1224no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—24no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—81no—
404C23—81no—
405C23—84no—
406C23—84no—
407C23—1224no—
408C23—48816no—
409C23—4884no—
410C23—4884no—
411C23—2444no—
412C23—2444no—
413C23—2448no—
414C23—2444no—
415C23#1451974511yescharacteristic
416C23#1451974511yescharacteristic
417C23C2×#1175820114yes—
418C23C2×#1175820112yes—
419C23#1451980212yes—
420C23C22×C22⋊D6111yescharacteristic
421Q8—44no—
422D4—4882no—
423D4—4884no—
424D4—2444no—
425D4—24416no—
426D4—2444no—
427D4—2444no—
428D4—2448no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—1224no—
445C2×C4—2448no—
446C2×C4—2444no—
447C2×C4#1451974512yes—
448C2×C4C2×D4×D6112yes—
449C8—2444no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—82no—
464C22—84no—
465C22—4888no—
466C22—48816no—
467C22—2444no—
468C22—2444no—
469C22—2444no—
470C22#1683991212yes—
471C22#1683991212yes—
472C22C2×#1451974511yescharacteristic
473C22C22×#1175820111yescharacteristic
474C22C2×#1451980211yescharacteristic
475C4—22no—
476C4—22no—
477C4—42no—
478C4—82no—
479C4—81no—
480C4—81no—
481C4—2444no—
482C2—21no—
483C2—22no—
484C2—21no—
485C2—22no—
486C2—44no—
487C2—4884no—
488C2C2×#1683991211yescharacteristic
489C2#2145517511yescharacteristic
490C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2626C2
socleC22×C61C2×#120
prime radicalC2×#16840324C2
hypercenter#2626D61
hypocenterC61#18285
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #21456572C2semidirect
3. #21444838C2semidirect
4. #21444770C2semidirect
5. #21444386C2semidirect
6. C2×#16839486C2semidirect
7. C61×#2626C2semidirect
17. #16840080C22semidirect
20. #16839901C22semidirect
21. C2×C22⋊#92418C22semidirect
22. #16839495C22semidirect
23. #16839486C22semidirect
24. C2×C22⋊Q8×C61C22semidirect
26. C61×#399C22semidirect
28. C61×#393C22semidirect
30. C61×#380C22semidirect
62. C22⋊#92418C23semidirect
64. C22×Q8×C61C23semidirect
72. C2×C4⋊C4×C61C23semidirect
82. #95×C61C23semidirect
88. C24×C61SD16semidirect
239. #2626D61semidirect
258. C2×C22⋊Q8C2×D61semidirect
269. #399C2×D61semidirect
271. #393C2×D61semidirect
273. #380C2×D61semidirect
277. C61#18285semidirect
280. C22×Q8C22×D61semidirect
311. C2×C4⋊C4C22×D61semidirect
328. #95C22×D61semidirect
339. C24#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168403242
#214565722
#214448382
#214447702
#214443862
C2×#168394862
C61×#26262
#1828561