Finite Group Databasebeta

← #29580451

Subgroups — #29580451

Order 15616.

786 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 105 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#21456576C211yesmaximal, characteristic
2#21449459C211yesmaximal, characteristic, complemented
3#21446656C211yesmaximal, characteristic
4#21445754C211yesmaximal, characteristic, complemented
5#21444392C211yesmaximal, characteristic, complemented
6#21443665C211yesmaximal, characteristic, complemented
7C2×C61×#448C211yesmaximal, characteristic
8C2×#14519775—21no—
9C2×#14519756C2211yescharacteristic
10#16840092—21no—
11#16840078—21no—
12#16839989—22no—
13#16839913—21no—
14#16839900—21no—
15C2×#14519701—22no—
16C2×#14519692—22no—
17C2×C22⋊#92418C2211yescharacteristic
18#16839681—22no—
19#16839669—21no—
20#16839635—22no—
21#16839622C2211yescharacteristic
22C4×C22⋊Dic61C2211yescharacteristic
23#16839500—22no—
24C23×Q8×C61C2211yescharacteristic
25C22×SD16×C61—21no—
26C2×#121×C61C2211yescharacteristic
27C61×#448—22no—
28C61×#448—22no—
29C2×#103×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C2×#11758200—22no—
33C22×#92418D412yes—
34#14519775—44no—
35#14519756—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—24no—
42C2×#11758196—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61D412yes—
46C2×C4×Dic61—21no—
47#14519708—22no—
48#14519701—44no—
49#14519700—22no—
50#14519698—22no—
51#14519692—44no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54#14519684—24no—
55#14519684—21no—
56#14519684—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—21no—
62C22×Q8×C61—44no—
63C22×Q8×C61D412yes—
64C22×D4×C61—21no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61—21no—
67C23×C4×C61C2311yescharacteristic
68C2×SD16×C61—24no—
69C2×SD16×C61—44no—
70C22×C8×C61—21no—
71#121×C61—21no—
72#121×C61—21no—
73#121×C61—21no—
74#121×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77#103×C61—24no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61D412yes—
80C24×C61—21no—
81C24×C61C2×D411yescharacteristic
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—
95C2×D4×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—42no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—42no—
101C2×D4×C61—42no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—42no—
105C22×C4×C61—44no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—41no—
118C2×Dic122—21no—
119C2×Dic122—21no—
120C2×Dic122—42no—
121SD16×C61—48no—
122C2×C8×C61—22no—
123C2×C8×C61—42no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C22⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C22⋊Dic61—22no—
129C22⋊Dic61—41no—
130C22⋊Dic61—41no—
131C22⋊Dic61—42no—
132C22⋊Dic61SD1614yescomplemented
133#11758200—82no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—42no—
137#11758196—48no—
138#11758196—41no—
139#11758196—41no—
140C4×Dic61—42no—
141C2×#92418—22no—
142C2×#92418—42no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—42no—
147C23×C61—81no—
148C23×C61—81no—
149C23×C61#12011yescharacteristic
150C23×C61#12111yescharacteristic
151C23×C61#12711yescharacteristic
152C23×C61C2×SD1612yes—
153C23×C61#13712yes—
154Q8×C61—24no—
155Q8×C61—24no—
156Q8×C61—42no—
157Q8×C61—44no—
158Q8×C61—42no—
159Q8×C61—44no—
160D4×C61—42no—
161D4×C61—42no—
162D4×C61—82no—
163D4×C61—81no—
164D4×C61—81no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—24no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—44no—
174C2×C4×C61—44no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12011yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×Dic61—24no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—81no—
190C2×Dic61—81no—
191C2×Dic61—81no—
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—81no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61—162no—
209C22×C61#39211yescharacteristic
210C22×C61#44812yes—
211C22×C61#46112yes—
212C22×C61#49211yescharacteristic
213C22×C61#49711yescharacteristic
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—44no—
218C4×C61—81no—
219C4×C61B(4,𝔽2)12yes—
220Dic61—44no—
221Dic61—81no—
222Dic61—81no—
223Dic61—161no—
224#3187—611no—
225C2×#448D6111yescharacteristic
226#2434—611no—
227#1964—611no—
228#1787—611no—
229#1520—611no—
230#1282—611no—
231C2×C61—22no—
232C2×C61—21no—
233C2×C61—22no—
234C2×C61—21no—
235C2×C61—161no—
236C2×C61#200911yescharacteristic
237C2×C61#201311yescharacteristic
238C2×C61#202111yescharacteristic
239C23×Q8C2×D6111yescharacteristic
240C22×SD16—21no—
241C2×C4⋊Q8—1221no—
242C2×C22⋊Q8—611no—
243C2×#121C2×D6111yescharacteristic
244#483—1221no—
245#481—1221no—
246#465—1222no—
247#450—1221no—
248#448—22no—
249#448—22no—
250#448—1221no—
251C2×#103—21no—
252C2×#103—1222no—
253C2×#99—1222no—
254C2×C22⋊C8—611no—
255C2×C22⋊C8C2×D6111yescharacteristic
256#395—1222no—
257#391—1221no—
258#385—1222no—
259#380—611no—
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×Q8C22⋊D6112yes—
271C22×D4—21no—
272C22×D4C22⋊D6111yescharacteristic
273C23×C4—21no—
274C23×C4C22×D6111yescharacteristic
275C2×SD16—24no—
276C2×SD16—44no—
277C2×SD16—1222no—
278C22×C8—21no—
279C22×C8—612no—
280C4⋊Q8—2444no—
281C22⋊Q8—1221no—
282C22⋊Q8—1221no—
283C22⋊Q8—1221no—
284C22⋊Q8—1221no—
285#121—21no—
286#121—21no—
287#121—21no—
288#121—21no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—1224no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—1221no—
293C2×C4⋊C4C22⋊D6111yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—612no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4—1221no—
298C2×C42—1221no—
299#106—1222no—
300#103—24no—
301#103—1222no—
302#103—2444no—
303#102—1222no—
304#99—2444no—
305C22⋊C8—21no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8C22⋊D6112yes—
309#95—1224no—
310#95—1221no—
311#95—1221no—
312C24—21no—
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—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—1222no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—2441no—
347C22×C4D4×D6111yescharacteristic
348C22×C4D4×D6111yescharacteristic
349C22×C4#1175822411yescharacteristic
350C22×C4C2×C22⋊D6111yescharacteristic
351C22×C4C2×C22⋊D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353SD16—48no—
354SD16—4882no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—1222no—
358C2×C8—2442no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—2442no—
364C4⋊C4—2448no—
365C4⋊C4—2441no—
366C4⋊C4—2441no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—614no—
370C22⋊C4—1222no—
371C22⋊C4—2442no—
372C22⋊C4—2441no—
373C22⋊C4—2441no—
374C42—2442no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23#1451974511yescharacteristic
382C23#1451974911yescharacteristic
383C23#1451977311yescharacteristic
384C23SD16×D6112yes—
385C23#1451980612yes—
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—44no—
390Q8—42no—
391Q8—44no—
392Q8—4881no—
393Q8—2442no—
394D4—42no—
395D4—42no—
396D4—81no—
397D4—81no—
398D4—82no—
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—42no—
411C2×C4—41no—
412C2×C4—1224no—
413C2×C4—4881no—
414C2×C4—4881no—
415C2×C4—4881no—
416C2×C4—2442no—
417C2×C4—2442no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4#1451974511yescharacteristic
423C2×C4#1451974611yescharacteristic
424C2×C4#1451974811yescharacteristic
425C2×C4#1451974911yescharacteristic
426C8—42no—
427C8—2442no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—162no—
440C22#1683967011yescharacteristic
441C22#1683989912yes—
442C22#1683996512yes—
443C22#1684013311yescharacteristic
444C22#1684015311yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—44no—
449C4—81no—
450C4—9761no—
451C4—4881no—
452C4—4881no—
453C4—2444no—
454C4#1683994812yes—
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2#2144687711yescharacteristic
461C2#2144690011yescharacteristic
462C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#448C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenterC2×#448D61
hypocenterC61#9949
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. #21449459C2semidirect
4. #21445754C2semidirect
5. #21444392C2semidirect
6. #21443665C2semidirect
132. C22⋊Dic61SD16semidirect
262. C61#9949semidirect

Maximal subgroups

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

subgroupindex
#214565762
#214494592
#214466562
#214457542
#214443922
#214436652
C2×C61×#4482
#994961