Finite Group Databasebeta

← #29654132

Subgroups — #29654132

Order 15616.

1172 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 167 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#21456625C212yesmaximal, complemented
3#21444843C214yesmaximal, complemented
4#21444741C214yesmaximal, complemented
5#21444338C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C61×#2539C211yesmaximal, characteristic
8#16840388—24no—
9#16840388—22no—
10#16840388C2214yes—
11C2×#14519756—21no—
12C2×#14519756C2212yes—
13C2×#14519754—22no—
14C2×#14519754—22no—
15C22×C22⋊Dic61—21no—
16#16840097C2214yescomplemented
17#16840073—22no—
18#16840006—24no—
19#16839913C2214yescomplemented
20#16839900—22no—
21#16839726C2212yes—
22#16839498C2218yescomplemented
23#16839486C2214yes—
24C2×#123×C61—21no—
25C2×#121×C61C2211yescharacteristic
26C2×#120×C61—21no—
27C22×C22⋊C4×C61C2211yescharacteristic
28C61×#399C2212yes—
29C61×#392C2212yes—
30C61×#390—22no—
31C4×C22⋊C4×C61C2211yescharacteristic
32C22×Dic122—21no—
33C22×Dic122D412yescomplemented
34C23×Dic61—21no—
35C23×Dic61—21no—
36C2×#11758200—24no—
37C2×#11758194—22no—
38#14519756—24no—
39#14519756—24no—
40#14519756—22no—
41#14519756—22no—
42#14519756D414yescomplemented
43#14519756C2314yes—
44#14519754—28no—
45#14519754—28no—
46#14519754—28no—
47#14519754—28no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61D412yescomplemented
50C2×#11758196—22no—
51C2×#11758196—22no—
52C2×C22⋊Dic61—22no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—24no—
56#14519710—24no—
57#14519700—24no—
58#14519698—24no—
59C22⋊#92418—22no—
60C22⋊#92418C2314yes—
61C25×C61D411yescharacteristic
62C22×D4×C61—21no—
63C22×D4×C61—22no—
64C22×D4×C61C2311yescharacteristic
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67#123×C61—44no—
68#121×C61—22no—
69#121×C61—22no—
70#120×C61—44no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61—21no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61D412yes—
80C2×C22⋊C4×C61C2312yes—
81C2×C42×C61—21no—
82#95×C61C2312yes—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—24no—
86C24×C61—21no—
87C24×C61—42no—
88C24×C61—81no—
89C24×C61C2×D411yescharacteristic
90C24×C61C2×D412yes—
91C2×D4×C61—22no—
92C2×D4×C61—44no—
93C2×D4×C61—42no—
94C2×D4×C61—42no—
95C2×D4×C61—41no—
96C2×D4×C61—41no—
97C2×D4×C61—44no—
98C2×D4×C61—81no—
99C2×D4×C61—82no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—42no—
105C22×C4×C61—81no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—22no—
112C22×Dic61—22no—
113C22×Dic61—24no—
114C22×Dic61—24no—
115C22×Dic61—82no—
116C22×Dic61—82no—
117C2×Dic122—24no—
118C2×Dic122—22no—
119C2×Dic122—82no—
120C2×Dic122—82no—
121C2×Dic122C2×D414yes—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—44no—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C22⋊C4×C61—24no—
127C22⋊C4×C61—44no—
128C22⋊C4×C61—44no—
129C22⋊C4×C61—42no—
130C42×C61—81no—
131C22⋊Dic61—28no—
132C22⋊Dic61—24no—
133C22⋊Dic61—24no—
134#11758200—84no—
135C4⋊Dic61—24no—
136C4⋊Dic61—22no—
137C4⋊Dic61C2×D414yes—
138#11758196—28no—
139#11758196—24no—
140#11758196—24no—
141#11758194—82no—
142C2×#92418—24no—
143C23×C61—24no—
144C23×C61—24no—
145C23×C61—22no—
146C23×C61—22no—
147C23×C61—22no—
148C23×C61—22no—
149C23×C61—41no—
150C23×C61—42no—
151C23×C61—42no—
152C23×C61—42no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—81no—
160C23×C61—81no—
161C23×C61—88no—
162C23×C61#12011yescharacteristic
163C23×C61#12011yescharacteristic
164C23×C61#13714yes—
165C23×C61C22×D411yescharacteristic
166D4×C61—44no—
167D4×C61—88no—
168D4×C61—82no—
169D4×C61—82no—
170C2×C4×C61—24no—
171C2×C4×C61—22no—
172C2×C4×C61—42no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—81no—
184C2×C4×C61#12012yes—
185C2×C4×C61C22×D412yes—
186C2×Dic61—24no—
187C2×Dic61—24no—
188C2×Dic61—44no—
189C2×Dic61—44no—
190C2×Dic61—84no—
191C2×Dic61—84no—
192Dic122—44no—
193Dic122—82no—
194Dic122—84no—
195#92418—44no—
196#18318—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—24no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—44no—
206C22×C61—44no—
207C22×C61—48no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—82no—
211C22×C61—82no—
212C22×C61—82no—
213C22×C61—84no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61#45312yes—
218C22×C61C2×#12011yescharacteristic
219C22×C61C2×#13712yes—
220C4×C61—22no—
221C4×C61—22no—
222C4×C61—44no—
223C4×C61—42no—
224C4×C61—81no—
225Dic61—44no—
226Dic61—44no—
227C2×#543—611no—
228#3193—612no—
229#2539D6111yescharacteristic
230#1614—614no—
231#1596—614no—
232#1508—612no—
233#1462—611no—
234C2×C61—22no—
235C2×C61—22no—
236C2×C61—41no—
237C2×C61—44no—
238C2×C61—161no—
239C2×C61#300411yescharacteristic
240C2×C61C2×#45311yescharacteristic
241C2×C61#302611yescharacteristic
242#543—614no—
243#543—1224no—
244#543—1222no—
245C2×#123—21no—
246C2×C22⋊Q8—612no—
247C2×C22⋊Q8—1222no—
248C2×C22⋊Q8—1222no—
249C2×C22⋊Q8—1221no—
250C2×#121C2×D6111yescharacteristic
251C2×#120—21no—
252C22×C22⋊C4—1221no—
253C22×C22⋊C4C2×D6111yescharacteristic
254#484—614no—
255#480—1222no—
256#468—1224no—
257#450—614no—
258#448—1222no—
259#407—612no—
260#399C2×D6112yes—
261#392C2×D6112yes—
262#390—22no—
263C4×C22⋊C4C2×D6111yescharacteristic
264#328—618no—
265#323—614no—
266C61#1831811yescharacteristic, complemented, Sylow
267C25C22⋊D6111yescharacteristic
268C22×Q8—612no—
269C22×Q8—1221no—
270C22×D4—21no—
271C22×D4—22no—
272C22×D4C22×D6111yescharacteristic
273C23×C4—1221no—
274C23×C4—1221no—
275C23×C4C22⋊D6111yescharacteristic
276C23×C4C22×D6111yescharacteristic
277C2×SD16—1224no—
278C2×M16—1222no—
279#123—44no—
280C22⋊Q8—614no—
281C22⋊Q8—614no—
282C22⋊Q8—1228no—
283C22⋊Q8—1228no—
284C22⋊Q8—1228no—
285C22⋊Q8—1228no—
286C22⋊Q8—1224no—
287C22⋊Q8—1222no—
288C22⋊Q8—1222no—
289C22⋊Q8—1224no—
290#121—22no—
291#121—22no—
292#120—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—612no—
295C2×C4⋊C4—1222no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1222no—
298C2×C4⋊C4C22×D6111yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—1222no—
306C2×C22⋊C4—1224no—
307C2×C22⋊C4—1222no—
308C2×C22⋊C4—1224no—
309C2×C22⋊C4C22⋊D6112yes—
310C2×C22⋊C4C22×D6112yes—
311C2×C42—21no—
312#107—1224no—
313#103—1224no—
314#102—1224no—
315C22⋊C8—614no—
316C22⋊C8—1222no—
317#95C22×D6112yes—
318C24—21no—
319C24—21no—
320C24—24no—
321C24—21no—
322C24—42no—
323C24—81no—
324C24C2×C22⋊D6111yescharacteristic
325C24C2×C22⋊D6112yes—
326C2×Q8—614no—
327C2×Q8—1224no—
328C2×Q8—1222no—
329C2×Q8—4882no—
330C2×Q8—4882no—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—44no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—81no—
346C22×C4—1224no—
347C22×C4—1224no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—4882no—
351C22×C4—4882no—
352C22×C4D4×D6112yes—
353C22×C4D4×D6112yes—
354C22×C4C2×C22⋊D6112yes—
355C22×C4C2×C22⋊D6111yescharacteristic
356C22×C4C23×D6111yescharacteristic
357SD16—4884no—
358M16—4882no—
359C2×C8—1224no—
360C4⋊C4—22no—
361C4⋊C4—44no—
362C4⋊C4—614no—
363C4⋊C4—1228no—
364C4⋊C4—1224no—
365C4⋊C4—1224no—
366C4⋊C4—1222no—
367C4⋊C4—1224no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—42no—
374C22⋊C4—1228no—
375C22⋊C4—1224no—
376C22⋊C4—1224no—
377C42—81no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23—81no—
395C23—81no—
396C23—88no—
397C23#1451974511yescharacteristic
398C23#1451974511yescharacteristic
399C23#1451980714yes—
400C23C22×C22⋊D6111yescharacteristic
401Q8—4882no—
402Q8—4884no—
403Q8—2444no—
404D4—44no—
405D4—88no—
406D4—82no—
407D4—82no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—1224no—
423C2×C4—1224no—
424C2×C4—4884no—
425C2×C4—4884no—
426C2×C4—2444no—
427C2×C4—2444no—
428C2×C4#1451974512yes—
429C2×C4C2×D4×D6112yes—
430C8—2444no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—44no—
440C22—44no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—84no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22#1683992912yes—
452C22C2×#1451974511yescharacteristic
453C22C2×#1451980712yes—
454C4—22no—
455C4—22no—
456C4—44no—
457C4—42no—
458C4—81no—
459C4—2444no—
460C4—2444no—
461C2—22no—
462C2—22no—
463C2—41no—
464C2—44no—
465C2—161no—
466C2#2145505211yescharacteristic
467C2C2×#1683992911yescharacteristic
468C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2539C2
socleC22×C61C2×#120
hypercenter#2539D61
hypocenterC61#18318
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. C2×#16840388C2semidirect
2. #21456625C2semidirect
3. #21444843C2semidirect
4. #21444741C2semidirect
5. #21444338C2semidirect
6. #21444146C2semidirect
16. #16840097C22semidirect
19. #16839913C22semidirect
22. #16839498C22semidirect
33. C22×Dic122D4semidirect
42. #14519756D4semidirect
49. C2×C4⋊Dic61D4semidirect
266. C61#18318semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214566252
#214448432
#214447412
#214443382
#214441462
C61×#25392
#1831861