Finite Group Databasebeta

← #29580622

Subgroups — #29580622

Order 15616.

716 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456591C211yesmaximal, characteristic, complemented
2#21455079C211yesmaximal, characteristic
3#21445731C211yesmaximal, characteristic
4#21444184C211yesmaximal, characteristic, complemented
5#21444183C211yesmaximal, characteristic, complemented
6#21443695C211yesmaximal, characteristic, complemented
7C61×#3007C211yesmaximal, characteristic
8C2×#14519807—21no—
9C2×#14519756C2211yescharacteristic
10#16840097—21no—
11#16840095—21no—
12#16840001—22no—
13#16839924—22no—
14#16839918—21no—
15#16839918—21no—
16#16839917—21no—
17#16839913—21no—
18#16839763—21no—
19#16839726C2211yescharacteristic
20C2×C22⋊#92418—22no—
21C2×C22⋊#92418C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23#16839488—22no—
24#16839487—22no—
25C22×#41×C61C2211yescharacteristic
26C2×#137×C61—21no—
27C2×C22⋊Q8×C61C2211yescharacteristic
28C61×#452—22no—
29C61×#451—21no—
30C61×#450—21no—
31C61×#417—21no—
32C61×#407C2211yescharacteristic
33C22×Dic122D411yescharacteristic
34#14519807—44no—
35C2×#11758202—22no—
36C2×#11758202—22no—
37C2×#11758200—22no—
38C2×#11758194—21no—
39C22×#92418—21no—
40C22×#92418D412yes—
41#14519756—21no—
42#14519756—21no—
43#14519756—21no—
44#14519756—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48#14519710—22no—
49#14519701—22no—
50#14519701—22no—
51#14519700—22no—
52#14519698—22no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418—21no—
58C22⋊#92418—44no—
59C22⋊#92418D412yes—
60C2×#41×C61—21no—
61C2×#41×C61—21no—
62C2×#41×C61—21no—
63C2×#41×C61—21no—
64C2×#41×C61—44no—
65C22×Q8×C61D411yescharacteristic
66C22×Q8×C61D411yescharacteristic
67C22×D4×C61—21no—
68C22×D4×C61D411yescharacteristic
69C23×C4×C61—21no—
70C23×C4×C61C2311yescharacteristic
71#137×C61—44no—
72C2×Q16×C61—22no—
73C2×SD16×C61—22no—
74C2×M16×C61—21no—
75C22⋊Q8×C61—21no—
76C22⋊Q8×C61—21no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61D411yescharacteristic
81C2×C22⋊C4×C61—21no—
82#103×C61—22no—
83#102×C61—22no—
84C22⋊C8×C61—21no—
85C22⋊C8×C61D412yes—
86C24×C61—21no—
87C24×C61#4111yescharacteristic
88#41×C61—24no—
89#41×C61—24no—
90#41×C61—42no—
91#41×C61—42no—
92#41×C61—42no—
93#41×C61—42no—
94#41×C61—42no—
95#41×C61—42no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—42no—
101C2×Q8×C61—42no—
102C2×D4×C61—21no—
103C2×D4×C61—21no—
104C2×D4×C61—42no—
105C2×D4×C61—42no—
106C2×D4×C61—42no—
107C2×D4×C61—41no—
108C2×D4×C61—41no—
109C22×C4×C61—24no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—42no—
114C22×C4×C61—42no—
115C22×C4×C61—41no—
116C22×C4×C61—41no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—21no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—42no—
128Q16×C61—44no—
129SD16×C61—44no—
130M16×C61—42no—
131C2×C8×C61—22no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C22⋊Dic61—41no—
139C22⋊Dic61—41no—
140#11758202—44no—
141#11758202—82no—
142#11758200—44no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145#11758196—41no—
146#11758196—41no—
147#11758194—42no—
148C2×#92418—22no—
149C2×#92418—22no—
150C2×#92418—42no—
151C2×#92418—42no—
152C2×#92418—41no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—41no—
160C23×C61—42no—
161C23×C61#12111yescharacteristic
162C23×C61#12111yescharacteristic
163C23×C61#12111yescharacteristic
164Q8×C61—22no—
165Q8×C61—22no—
166Q8×C61—41no—
167Q8×C61—41no—
168Q8×C61—42no—
169Q8×C61—42no—
170D4×C61—22no—
171D4×C61—22no—
172D4×C61—41no—
173D4×C61—41no—
174D4×C61—44no—
175D4×C61—44no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—44no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—44no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—81no—
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12711yescharacteristic
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—81no—
199Dic122—42no—
200Dic122—42no—
201C8×C61—42no—
202#92418—42no—
203#92418—42no—
204#92418—81no—
205#9976—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—44no—
215C22×C61—42no—
216C22×C61—82no—
217C22×C61—82no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61#39211yescharacteristic
221C22×C61#47312yes—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—81no—
227C4×C61—81no—
228C4×C61B(4,𝔽2)12yes—
229Dic61—81no—
230Dic61—81no—
231#3189—611no—
232#3007—611no—
233#3007D6111yescharacteristic
234#1780—611no—
235#1472—611no—
236#1472—611no—
237#1293—611no—
238C2×C61—22no—
239C2×C61—41no—
240C2×C61—41no—
241C2×C61—82no—
242C2×C61#201711yescharacteristic
243C2×C61#202011yescharacteristic
244C2×C61#202111yescharacteristic
245C22×#41C2×D6111yescharacteristic
246C2×#137—21no—
247C2×#137—1221no—
248C2×C22⋊Q8—611no—
249C2×C22⋊Q8C2×D6111yescharacteristic
250#484—1221no—
251#484—1221no—
252#467—1222no—
253#452—22no—
254#452—1222no—
255#451—21no—
256#451—1221no—
257#451—1221no—
258#451—1221no—
259#450—21no—
260#450—1221no—
261#417—21no—
262#417—1221no—
263#407—611no—
264#407C2×D6111yescharacteristic
265C2×C22⋊C8—611no—
266C2×C22⋊C8—611no—
267C2×C22⋊C8—1222no—
268#324—1222no—
269#324—1222no—
270C61#997611yescharacteristic, complemented, Sylow
271C2×#41—21no—
272C2×#41—21no—
273C2×#41—21no—
274C2×#41—21no—
275C2×#41—44no—
276C22×Q8—611no—
277C22×Q8C22⋊D6111yescharacteristic
278C22×Q8C22⋊D6111yescharacteristic
279C22×D4—21no—
280C22×D4C22⋊D6111yescharacteristic
281C23×C4—21no—
282C23×C4C22×D6111yescharacteristic
283#137—44no—
284#137—2444no—
285C2×Q16—22no—
286C2×Q16—1222no—
287C2×Q16—1222no—
288C2×SD16—22no—
289C2×SD16—1222no—
290C2×M16—21no—
291C2×M16—1221no—
292C22×C8—612no—
293C22×C8—1221no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—1221no—
299C22⋊Q8—1221no—
300C22⋊Q8—1221no—
301C22⋊Q8—1221no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4C22⋊D6111yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—1221no—
308#107—1222no—
309#103—22no—
310#103—1222no—
311#103—1222no—
312#103—1222no—
313#102—22no—
314#102—1222no—
315C22⋊C8—21no—
316C22⋊C8—612no—
317C22⋊C8—1221no—
318C22⋊C8—1221no—
319C22⋊C8—1221no—
320C22⋊C8—1221no—
321C22⋊C8—1221no—
322C22⋊C8—2444no—
323C22⋊C8C22⋊D6112yes—
324C24—21no—
325C24#1175822411yescharacteristic
326#41—24no—
327#41—24no—
328#41—42no—
329#41—42no—
330#41—42no—
331#41—42no—
332#41—42no—
333#41—42no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—1221no—
341C2×Q8—1221no—
342C2×Q8—2442no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—41no—
349C2×D4—41no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4D4×D6111yescharacteristic
361C22×C4D4×D6111yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C2×C22⋊D6111yescharacteristic
366Q16—44no—
367Q16—4882no—
368Q16—2444no—
369SD16—44no—
370SD16—2444no—
371M16—42no—
372M16—2442no—
373C2×C8—22no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8—2442no—
377C2×C8—2441no—
378C2×C8—2442no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—2441no—
386C4⋊C4—2441no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C23—22no—
392C23—22no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23#1451974811yescharacteristic
400C23#1451974911yescharacteristic
401C23#1451974911yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406Q8—42no—
407Q8—42no—
408Q8—2442no—
409Q8—2442no—
410D4—22no—
411D4—22no—
412D4—41no—
413D4—41no—
414D4—44no—
415D4—44no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—4881no—
432C2×C4—2442no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4#1451974511yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4#1451974611yescharacteristic
438C2×C4#1451977311yescharacteristic
439C8—42no—
440C8—4881no—
441C8—2442no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—44no—
452C22—42no—
453C22—82no—
454C22—82no—
455C22—81no—
456C22—81no—
457C22#1683967011yescharacteristic
458C22#1684003512yes—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—42no—
463C4—81no—
464C4—81no—
465C4—4881no—
466C4—4881no—
467C4#1683994812yes—
468C2—22no—
469C2—41no—
470C2—41no—
471C2—82no—
472C2#2144692211yescharacteristic
473C2#2144693711yescharacteristic
474C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#3007C2
socleC22×C61#392
prime radicalC22×#41×C61C22
hypercenter#3007D61
hypocenterC61#9976
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
1. #21456591C2semidirect
4. #21444184C2semidirect
5. #21444183C2semidirect
6. #21443695C2semidirect
270. C61#9976semidirect

Maximal subgroups

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

subgroupindex
#214565912
#214550792
#214457312
#214441842
#214441832
#214436952
C61×#30072
#997661