Finite Group Databasebeta

← #29653864

Subgroups — #29653864

Order 15616.

1142 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1142 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×#16840388C211yesmaximal, characteristic, complemented
2#21456577C212yesmaximal, complemented
3#21451553C211yesmaximal, characteristic
4#21444812C214yesmaximal, complemented
5#21444744C214yesmaximal, complemented
6#21444392C212yesmaximal, complemented
7C2×C61×#323C211yesmaximal, characteristic, complemented
8#16840388—24no—
9#16840388—22no—
10#16840388C2214yes—
11C2×#14519756—21no—
12C2×#14519756C2212yes—
13C2×#14519754—22no—
14C2×#14519754—22no—
15C2×#14519751C2211yescharacteristic
16C2×#14519746—21no—
17C22×C22⋊Dic61—21no—
18#16840093C2214yescomplemented
19#16840081—22no—
20#16839991—24no—
21#16839915C2214yescomplemented
22#16839903—22no—
23#16839687C2212yes—
24#16839678—22no—
25#16839671C2212yes—
26#16839641—21no—
27#16839622C2211yescharacteristic
28#16839500C2218yescomplemented
29C22×C22⋊C4×C61C2211yescharacteristic
30C2×C22⋊C8×C61C2212yes—
31C61×#323C2214yes—
32C22×Dic122—21no—
33C22×Dic122D412yescomplemented
34C22×D244C2311yescharacteristic
35C22×C22⋊D61—21no—
36C22×C22⋊D61—22no—
37C23×Dic61—21no—
38C23×Dic61—21no—
39C2×#11758190—24no—
40#14519756—24no—
41#14519756—24no—
42#14519756—22no—
43#14519756—22no—
44#14519756D414yescomplemented
45#14519756C2314yes—
46#14519754—28no—
47#14519754—28no—
48#14519754—28no—
49#14519754—28no—
50#14519751—22no—
51#14519751—22no—
52#14519746—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61D412yescomplemented
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—24no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—24no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—21no—
65C2×#11758198—21no—
66C2×#11758198—22no—
67#14519709—24no—
68#14519702—24no—
69#14519699—24no—
70#14519684—22no—
71#14519684C2312yes—
72C25×C61D411yescharacteristic
73C23×C4×C61D411yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C22×C8×C61—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61D412yes—
79C2×C22⋊C4×C61C2312yes—
80C22⋊C8×C61—22no—
81C22⋊C8×C61C2314yes—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—22no—
85C24×C61—81no—
86C24×C61—81no—
87C24×C61SD1614yescomplemented
88C24×C61C2×D412yes—
89C24×C61C2×D411yescharacteristic
90C23×D61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—81no—
94C22×C4×C61—81no—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D412yes—
99C22×C4×C61C2411yescharacteristic
100C2×C22⋊D61—41no—
101C2×C22⋊D61—41no—
102C2×C22⋊D61—44no—
103C2×C22⋊D61—42no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—82no—
106C2×C22⋊D61—82no—
107C22×Dic61—22no—
108C22×Dic61—22no—
109C22×Dic61—24no—
110C22×Dic61—24no—
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61—82no—
114C22×Dic61—82no—
115C2×D244—22no—
116C2×D244—81no—
117C2×Dic122—24no—
118C2×Dic122—22no—
119C2×Dic122—82no—
120C2×Dic122—82no—
121C2×Dic122C2×D414yes—
122C2×C8×C61—24no—
123C2×C8×C61—82no—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C22⋊Dic61—28no—
127C22⋊Dic61—24no—
128C22⋊Dic61—24no—
129C22⋊Dic61—44no—
130C22⋊Dic61—42no—
131#11758198—42no—
132#11758198—82no—
133C4⋊Dic61—24no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61—81no—
137C4⋊Dic61C2×D414yes—
138#11758196—28no—
139#11758196—24no—
140#11758196—24no—
141#11758190—84no—
142C23×C61—22no—
143C23×C61—24no—
144C23×C61—22no—
145C23×C61—24no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—42no—
149C23×C61—42no—
150C23×C61—42no—
151C23×C61—42no—
152C23×C61—81no—
153C23×C61—81no—
154C23×C61—84no—
155C23×C61—84no—
156C23×C61#12011yescharacteristic
157C23×C61#12011yescharacteristic
158C23×C61C2×SD1614yes—
159C23×C61C2×SD1612yes—
160C23×C61#13712yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—41no—
165C22×D61—41no—
166C22×D61—41no—
167C2×C4×C61—22no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—81no—
175C2×C4×C61#12012yes—
176C2×C4×C61C22×D412yes—
177C22⋊D61—88no—
178C22⋊D61—82no—
179C22⋊D61—82no—
180C2×Dic61—24no—
181C2×Dic61—24no—
182C2×Dic61—44no—
183C2×Dic61—44no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—84no—
191C2×Dic61—84no—
192D244—44no—
193Dic122—44no—
194Dic122—82no—
195Dic122—84no—
196C8×C61—44no—
197#18286—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—42no—
206C22×C61—42no—
207C22×C61—44no—
208C22×C61—48no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—82no—
212C22×C61—84no—
213C22×C61#44812yes—
214C22×C61#44812yes—
215C22×C61C2×#12011yescharacteristic
216C22×C61C22×SD1611yescharacteristic
217C22×C61C2×#13711yescharacteristic
218C2×D61—82no—
219C2×D61—82no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—22no—
224C4×C61—22no—
225C4×C61—42no—
226Dic61—44no—
227Dic61—44no—
228Dic61—82no—
229Dic61—81no—
230C2×#543—611no—
231#3187—612no—
232#2617—611no—
233#1608—614no—
234#1596—614no—
235#1520—612no—
236C2×#323D6111yescharacteristic, complemented
237C2×C61—21no—
238C2×C61—21no—
239C2×C61—22no—
240C2×C61—22no—
241C2×C61—44no—
242C2×C61C2×#44811yescharacteristic
243C2×C61#301911yescharacteristic
244C2×C61#302611yescharacteristic
245D61—161no—
246#543—614no—
247#543—1224no—
248#543—1222no—
249C2×C22⋊Q8—612no—
250C2×C22⋊Q8—1222no—
251C2×C22⋊Q8—1222no—
252C2×C22⋊Q8—1221no—
253C2×#121—611no—
254C2×#120—1221no—
255C22×C22⋊C4—1221no—
256C22×C22⋊C4C2×D6111yescharacteristic
257#483—614no—
258#481—1222no—
259#465—1224no—
260#450—614no—
261#448—1222no—
262C2×C22⋊C8C2×D6112yes—
263#397—612no—
264#394—1222no—
265#392—612no—
266#386—1221no—
267#380—611no—
268#328—618no—
269#323C2×D6114yes—
270C61#1828611yescharacteristic, complemented, Sylow
271C25D24411yescharacteristic
272C22×Q8—612no—
273C22×Q8—1221no—
274C22×D4—611no—
275C22×D4—1222no—
276C22×D4—1221no—
277C23×C4—1221no—
278C23×C4—1221no—
279C23×C4D24411yescharacteristic
280C23×C4C22×D6111yescharacteristic
281C2×SD16—1224no—
282C22×C8—22no—
283C22⋊Q8—614no—
284C22⋊Q8—614no—
285C22⋊Q8—1228no—
286C22⋊Q8—1228no—
287C22⋊Q8—1228no—
288C22⋊Q8—1228no—
289C22⋊Q8—1224no—
290C22⋊Q8—1222no—
291C22⋊Q8—1224no—
292C22⋊Q8—1222no—
293#121—1222no—
294#121—1222no—
295#120—2444no—
296C2×C4⋊C4—612no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—1222no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4—1222no—
301C2×C4⋊C4—1221no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—1222no—
305C2×C22⋊C4—1224no—
306C2×C22⋊C4—1222no—
307C2×C22⋊C4—1224no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4D24412yes—
313C2×C22⋊C4C22×D6112yes—
314#106—1224no—
315#103—1224no—
316#102—1224no—
317C22⋊C8—22no—
318C22⋊C8C22×D6114yes—
319#95—612no—
320#95—1222no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—81no—
325C24—81no—
326C24—1221no—
327C24#1175819014yescomplemented
328C24C2×D24412yes—
329C24C2×D24411yescharacteristic
330C2×Q8—614no—
331C2×Q8—1224no—
332C2×Q8—1222no—
333C2×Q8—4882no—
334C2×Q8—4882no—
335C2×D4—1222no—
336C2×D4—4881no—
337C2×D4—4882no—
338C2×D4—4882no—
339C2×D4—2442no—
340C2×D4—2441no—
341C2×D4—2441no—
342C2×D4—2444no—
343C2×D4—2442no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—1224no—
349C22×C4—1224no—
350C22×C4—1222no—
351C22×C4—1222no—
352C22×C4—1222no—
353C22×C4—1221no—
354C22×C4—4882no—
355C22×C4—4882no—
356C22×C4C2×D24412yes—
357C22×C4C2×D24411yescharacteristic
358C22×C4D4×D6112yes—
359C22×C4D4×D6112yes—
360C22×C4C23×D6111yescharacteristic
361SD16—4884no—
362C2×C8—24no—
363C2×C8—82no—
364C4⋊C4—614no—
365C4⋊C4—1228no—
366C4⋊C4—1224no—
367C4⋊C4—1224no—
368C4⋊C4—1222no—
369C4⋊C4—1224no—
370C4⋊C4—1222no—
371C4⋊C4—4881no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—1228no—
375C22⋊C4—1224no—
376C22⋊C4—1224no—
377C22⋊C4—4882no—
378C22⋊C4—2444no—
379C22⋊C4—2442no—
380C22⋊C4—2442no—
381C23—22no—
382C23—24no—
383C23—22no—
384C23—24no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—81no—
392C23—81no—
393C23—84no—
394C23—84no—
395C23—2442no—
396C23—2442no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23#1451974711yescharacteristic
401C23#1451974711yescharacteristic
402C23C2×#1175819014yes—
403C23C2×#1175819012yes—
404C23#1451981012yes—
405C23C22×D24411yescharacteristic
406Q8—4882no—
407Q8—4884no—
408Q8—2444no—
409D4—4888no—
410D4—4882no—
411D4—4882no—
412D4—2444no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—81no—
421C2×C4—1224no—
422C2×C4—1224no—
423C2×C4—4884no—
424C2×C4—4884no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2441no—
428C2×C4—2442no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2444no—
432C2×C4—2444no—
433C2×C4#1451974712yes—
434C2×C4C2×D4×D6112yes—
435C8—44no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—84no—
451C22—4882no—
452C22—4882no—
453C22—4881no—
454C22—4881no—
455C22—4881no—
456C22#1683990312yes—
457C22#1683990312yes—
458C22C2×#1451974711yescharacteristic
459C22C22×#1175819011yescharacteristic
460C22C2×#1451981011yescharacteristic
461C4—22no—
462C4—22no—
463C4—42no—
464C4—4882no—
465C4—4881no—
466C4—2444no—
467C4—2444no—
468C2—21no—
469C2—22no—
470C2—21no—
471C2—22no—
472C2—44no—
473C2—9761no—
474C2C2×#1683990311yescharacteristic
475C2#2145517911yescharacteristic
476C2#2145521811yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#323C2
socleC22×C61C2×#120
prime radical#21451553C2
hypercenterC2×#323D61
hypocenterC61#18286
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
1. C2×#16840388C2semidirect
2. #21456577C2semidirect
4. #21444812C2semidirect
5. #21444744C2semidirect
6. #21444392C2semidirect
7. C2×C61×#323C2semidirect
18. #16840093C22semidirect
21. #16839915C22semidirect
28. #16839500C22semidirect
33. C22×Dic122D4semidirect
44. #14519756D4semidirect
55. C2×C4⋊Dic61D4semidirect
87. C24×C61SD16semidirect
236. C2×#323D61semidirect
270. C61#18286semidirect
327. C24#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214565772
#214515532
#214448122
#214447442
#214443922
C2×C61×#3232
#1828661