Finite Group Databasebeta

← #29580455

Subgroups — #29580455

Order 15616.

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

normal subgroups only

This group has 786 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#21456577C211yesmaximal, characteristic
2C2×#16839901C211yesmaximal, characteristic
3#21449459C211yesmaximal, characteristic, complemented
4#21446654C211yesmaximal, characteristic
5#21444390C211yesmaximal, characteristic, complemented
6#21443666C211yesmaximal, characteristic, complemented
7C61×#1787C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519775—21no—
10C2×#14519756C2211yescharacteristic
11C2×#14519751C2211yescharacteristic
12#16840093—21no—
13#16840081—21no—
14#16839991—22no—
15#16839915—21no—
16#16839903—21no—
17#16839901—22no—
18#16839901—22no—
19C2×#14519701—21no—
20C2×#14519691—22no—
21C2×C22⋊#92418C2211yescharacteristic
22#16839681—22no—
23#16839669—21no—
24#16839635—22no—
25C4×C22⋊Dic61C2211yescharacteristic
26#16839498—22no—
27C23×Q8×C61C2211yescharacteristic
28C2×#103×C61—22no—
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#380C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61—21no—
34C2×#11758190—22no—
35C2×#11758201—24no—
36C2×#11758201—44no—
37C22×#92418—21no—
38#14519775—44no—
39#14519756—21no—
40#14519756—21no—
41#14519756—21no—
42#14519756—21no—
43#14519751—21no—
44#14519751—21no—
45#14519751—21no—
46#14519751—21no—
47C2×C4⋊Dic61D411yescharacteristic
48C2×C4⋊Dic61D411yescharacteristic
49C2×#11758196—24no—
50C2×#11758196—21no—
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61D412yes—
53C2×#11758198—21no—
54C2×C4×Dic61—21no—
55#14519709—22no—
56#14519702—22no—
57#14519701—24no—
58#14519699—22no—
59#14519691—44no—
60C22⋊#92418—21no—
61C22⋊#92418D412yes—
62#14519684—24no—
63#14519684—21no—
64C22×Q8×C61—21no—
65C22×Q8×C61—21no—
66C22×Q8×C61—21no—
67C22×Q8×C61—21no—
68C22×Q8×C61—21no—
69C22×Q8×C61—44no—
70C22×Q8×C61D412yes—
71C23×C4×C61—21no—
72C23×C4×C61C2311yescharacteristic
73C22×C8×C61D412yes—
74C2×C4⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76#103×C61—44no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79#95×C61—21no—
80C24×C61C2×D411yescharacteristic
81C23×D61—21no—
82C2×Q8×C61—24no—
83C2×Q8×C61—24no—
84C2×Q8×C61—24no—
85C2×Q8×C61—22no—
86C2×Q8×C61—22no—
87C2×Q8×C61—44no—
88C2×Q8×C61—42no—
89C2×Q8×C61—42no—
90C2×Q8×C61—42no—
91C2×Q8×C61—44no—
92C2×Q8×C61—44no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C22×C4×C61—22no—
96C22×C4×C61—42no—
97C22×C4×C61—44no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×C22⋊D61—41no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—42no—
110C2×C22⋊D61—42no—
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—21no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—42no—
118C2×Dic122—21no—
119C2×Dic122—21no—
120C2×Dic122—42no—
121C2×C8×C61—22no—
122C2×C8×C61—42no—
123C4⋊C4×C61—42no—
124C22⋊C4×C61—42no—
125C22⋊Dic61—22no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128C22⋊Dic61SD1614yescomplemented
129#11758201—48no—
130#11758198—41no—
131#11758198—41no—
132C4⋊Dic61—21no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136#11758196—48no—
137#11758196—41no—
138#11758196—41no—
139C4×Dic61—42no—
140C2×#92418—22no—
141C2×#92418—42no—
142#11758190—82no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61#12011yescharacteristic
146C23×C61#12111yescharacteristic
147C23×C61#12711yescharacteristic
148C23×C61C2×SD1612yes—
149C23×C61#13712yes—
150C22×D61—41no—
151C22×D61—42no—
152C22×D61—81no—
153C22×D61—81no—
154Q8×C61—24no—
155Q8×C61—24no—
156Q8×C61—42no—
157Q8×C61—44no—
158Q8×C61—42no—
159Q8×C61—44no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—24no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—44no—
169C2×C4×C61—44no—
170C2×C4×C61—42no—
171C2×C4×C61—81no—
172C2×C4×C61—81no—
173C2×C4×C61—81no—
174C2×C4×C61#12011yescharacteristic
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C22⋊D61—82no—
179C22⋊D61—81no—
180C22⋊D61—81no—
181C2×Dic61—24no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190D244—42no—
191D244—42no—
192Dic122—42no—
193Dic122—81no—
194C8×C61—42no—
195#92418—42no—
196#9949—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61#39211yescharacteristic
206C22×C61#44812yes—
207C22×C61#46112yes—
208C22×C61#49211yescharacteristic
209C22×C61#49711yescharacteristic
210C2×D61—81no—
211C2×D61—81no—
212C2×D61—81no—
213C2×D61—162no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—44no—
218C4×C61—161no—
219C4×C61B(4,𝔽2)12yes—
220Dic61—44no—
221Dic61—81no—
222Dic61—81no—
223Dic61—81no—
224#3187—611no—
225C2×#448—611no—
226#2434—611no—
227#1964—611no—
228#1787D6111yescharacteristic, complemented
229#1520—611no—
230#1282—611no—
231C2×C61—22no—
232C2×C61—21no—
233C2×C61—22no—
234C2×C61—21no—
235C2×C61#200911yescharacteristic
236C2×C61#201311yescharacteristic
237C2×C61#202111yescharacteristic
238D61—161no—
239C23×Q8C2×D6111yescharacteristic
240C22×SD16—1221no—
241C2×C4⋊Q8—1221no—
242C2×C22⋊Q8—611no—
243C2×#121—611no—
244#483—1221no—
245#481—1221no—
246#465—1222no—
247#450—1221no—
248#448—1221no—
249#448—1222no—
250#448—1222no—
251C2×#103—22no—
252C2×#103—1221no—
253C2×#99—1222no—
254C2×C22⋊C8—611no—
255C2×C22⋊C8C2×D6111yescharacteristic
256#395—1222no—
257#391—1221no—
258#385—1222no—
259#380C2×D6111yescharacteristic
260C4×C22⋊C4—611no—
261#328—1222no—
262C61#994911yescharacteristic, complemented, Sylow
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8—44no—
269C22×Q8—611no—
270C22×Q8D24412yes—
271C22×D4—611no—
272C22×D4—1221no—
273C23×C4—21no—
274C23×C4C22×D6111yescharacteristic
275C2×SD16—1222no—
276C2×SD16—1224no—
277C2×SD16—2444no—
278C22×C8—1221no—
279C22×C8C22⋊D6112yes—
280C4⋊Q8—2444no—
281C22⋊Q8—1221no—
282C22⋊Q8—1221no—
283C22⋊Q8—1221no—
284C22⋊Q8—1221no—
285#121—1221no—
286#121—1221no—
287#121—1221no—
288#121—1221no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—611no—
291C2×C4⋊C4—611no—
292C2×C4⋊C4—1224no—
293C2×C4⋊C4—1221no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—612no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4—1221no—
298C2×C42—1221no—
299#106—1222no—
300#103—44no—
301#103—1222no—
302#103—1224no—
303#102—1222no—
304#99—2444no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—612no—
308C22⋊C8—1221no—
309#95—21no—
310#95—1224no—
311#95—1221no—
312C24—1221no—
313C24D4×D6111yescharacteristic
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—44no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—1221no—
328C2×Q8—1221no—
329C2×Q8—2442no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—2442no—
333C2×D4—2442no—
334C2×D4—2441no—
335C2×D4—2441no—
336C2×D4—2442no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—1222no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4C2×D24411yescharacteristic
348C22×C4#1175822211yescharacteristic
349C22×C4D4×D6111yescharacteristic
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353SD16—4882no—
354SD16—2448no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—1222no—
358C2×C8—2442no—
359C4⋊C4—42no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C4⋊C4—2448no—
365C4⋊C4—2441no—
366C4⋊C4—2441no—
367C22⋊C4—42no—
368C22⋊C4—614no—
369C22⋊C4—1222no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C22⋊C4—2441no—
373C22⋊C4—2441no—
374C42—2442no—
375C23—41no—
376C23—41no—
377C23—4881no—
378C23—4881no—
379C23—2442no—
380C23—2441no—
381C23#1451974711yescharacteristic
382C23#1451975011yescharacteristic
383C23#1451977311yescharacteristic
384C23SD16×D6112yes—
385C23#1451980912yes—
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—44no—
390Q8—42no—
391Q8—44no—
392Q8—4881no—
393Q8—2442no—
394D4—4881no—
395D4—4881no—
396D4—4882no—
397D4—2442no—
398D4—2442no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—1224no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2442no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4#1451974511yescharacteristic
423C2×C4#1451974711yescharacteristic
424C2×C4#1451975011yescharacteristic
425C2×C4#1451975111yescharacteristic
426C8—42no—
427C8—2442no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—9762no—
437C22—4881no—
438C22—4881no—
439C22—4881no—
440C22#1683967111yescharacteristic
441C22#1683990212yes—
442C22#1683996612yes—
443C22#1684013311yescharacteristic
444C22#1684015511yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—44no—
449C4—161no—
450C4—4881no—
451C4—4881no—
452C4—4881no—
453C4—2444no—
454C4#1683994812yes—
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—9761no—
460C2#2144688111yescharacteristic
461C2#2144690311yescharacteristic
462C2#2144694511yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1787C2
socleC22×C61#392
prime radicalC2×#14519751C22
hypercenter#1787D61
hypocenterC61#9949
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
3. #21449459C2semidirect
5. #21444390C2semidirect
6. #21443666C2semidirect
7. C61×#1787C2semidirect
128. C22⋊Dic61SD16semidirect
228. #1787D61semidirect
262. C61#9949semidirect

Maximal subgroups

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

subgroupindex
#214565772
C2×#168399012
#214494592
#214466542
#214443902
#214436662
C61×#17872
#994961