Finite Group Databasebeta

← #29580768

Subgroups — #29580768

Order 15616.

632 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455612C211yesmaximal, characteristic, complemented
2#21450720C211yesmaximal, characteristic
3#21444383C211yesmaximal, characteristic, complemented
4#21444244C211yesmaximal, characteristic, complemented
5#21444211C211yesmaximal, characteristic, complemented
6#21443678C211yesmaximal, characteristic, complemented
7C61×#3101C211yesmaximal, characteristic, complemented
8C2×#14519751—21no—
9C2×C4×D244C2211yescharacteristic
10#16840059—22no—
11#16840049—21no—
12#16840039—21no—
13#16839815C2211yescharacteristic
14#16839805—21no—
15#16839786C2211yescharacteristic
16#16839766—21no—
17#16839688—21no—
18#16839670—21no—
19#16839638—22no—
20#16839628C2211yescharacteristic
21#16839622—21no—
22#16839499—22no—
23#16839491—22no—
24D4⋊#92418—22no—
25C61×#533C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#483—21no—
28C61×#476—21no—
29C61×#461—21no—
30C61×#450—21no—
31C4×SD16×C61—22no—
32C61×#431C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C22⋊D61—22no—
35C22×C4×D61—21no—
36C2×#11758194—21no—
37#14519751—42no—
38#14519751—42no—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244—21no—
42C4×D244—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C22⋊Dic61—22no—
46C2×#11758198—22no—
47C2×#11758198—21no—
48#14519710—22no—
49#14519708—22no—
50C4⋊#92418—21no—
51C4⋊#92418—21no—
52C4⋊#92418D412yescomplemented
53#14519697—22no—
54#14519697—22no—
55C22⋊#92418—21no—
56C4×#92418—21no—
57#14519684—22no—
58#14519684—21no—
59#14519684D412yes—
60C22×D4×C61D411yescharacteristic
61C23×C4×C61—21no—
62C2×SD16×C61—22no—
63C4⋊Q8×C61D411yescharacteristic, complemented
64C4⋊Q8×C61D411yescharacteristic, complemented
65#125×C61—21no—
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74#117×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78#106×C61—22no—
79C4⋊C8×C61—21no—
80#103×C61—22no—
81#102×C61—22no—
82C22⋊C8×C61—21no—
83C4×C8×C61D412yescomplemented
84C24×C61—21no—
85C23×D61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—42no—
91C22×C4×C61—21no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—44no—
102C2×C22⋊D61—44no—
103C2×C22⋊D61—82no—
104C22×Dic61—21no—
105C22×Dic61—22no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244SD1612yescomplemented
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61—82no—
115SD16×C61—44no—
116C2×C8×C61—22no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C42×C61—41no—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61#4111yescharacteristic
133C22⋊Dic61—82no—
134#11758198—44no—
135#11758198—41no—
136#11758198—41no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—41no—
140C4⋊Dic61—41no—
141#11758194—41no—
142#11758194—41no—
143C2×#92418—22no—
144C2×#92418—41no—
145C23×C61—22no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61#12011yescharacteristic
153C22×D61—22no—
154C22×D61—42no—
155C22×D61—42no—
156C22×D61—41no—
157C22×D61—41no—
158Q8×C61—42no—
159Q8×C61—81no—
160D4×C61—22no—
161D4×C61—22no—
162D4×C61—41no—
163D4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—81no—
172C2×C4×C61—81no—
173C2×C4×C61—81no—
174C2×C4×C61—81no—
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12711yescharacteristic
181C2×C4×C61C2×SD1611yescharacteristic
182C2×C4×C61#13511yescharacteristic
183C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C61#13711yescharacteristic
185C22⋊D61—88no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191D244—22no—
192D244—22no—
193D244—41no—
194D244—41no—
195C4×D61—82no—
196C4×D61—82no—
197C8×C61—42no—
198#92418—42no—
199#92418—81no—
200#10000—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61#39211yescharacteristic
211C22×C61#45011yescharacteristic
212C22×C61#45211yescharacteristic
213C2×D61—44no—
214C2×D61—42no—
215C2×D61—82no—
216C2×D61—81no—
217C2×D61—81no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61—81no—
225C4×C61#44811yescharacteristic
226C4×C61#44911yescharacteristic
227C4×C61#46511yescharacteristic
228C4×C61#46811yescharacteristic
229Dic61—82no—
230Dic61—81no—
231#3101D6111yescharacteristic, complemented
232#3077—611no—
233#2535—611no—
234#1518—611no—
235#1486—611no—
236#1478—611no—
237#1286—611no—
238C2×C61—21no—
239C2×C61—21no—
240C2×C61—82no—
241C2×C61#200011yescharacteristic
242C2×C61#200811yescharacteristic
243C2×C61#201011yescharacteristic
244D61—82no—
245#533C2×D6111yescharacteristic
246C2×#121—1221no—
247C2×C4×D4—611no—
248C2×C4×D4C2×D6111yescharacteristic
249#483—21no—
250#478—1222no—
251#476—21no—
252#476—1221no—
253#474—1221no—
254#461—21no—
255#450—21no—
256C4×SD16—22no—
257#431—611no—
258#431C2×D6111yescharacteristic
259#428—1221no—
260#423—611no—
261#418—1221no—
262#397—1221no—
263#392—1221no—
264#386—1222no—
265#382—611no—
266#380—1221no—
267#328—1222no—
268D4⋊C8—1222no—
269D4⋊C8—1222no—
270C61#1000011yescharacteristic, complemented, Sylow
271C22×D4—611no—
272C22×D4—1222no—
273C22×D4C22⋊D6111yescharacteristic
274C23×C4—21no—
275C23×C4—1221no—
276C2×SD16—22no—
277C2×M16—1221no—
278C4⋊Q8C22⋊D6111yescharacteristic, complemented
279C4⋊Q8C22⋊D6111yescharacteristic, complemented
280#125—21no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283#121—2442no—
284#121—2442no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—1221no—
292C4×D4—1221no—
293C4×D4—1221no—
294C4×D4—1221no—
295#117—21no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4C22⋊D6111yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—1222no—
301C2×C22⋊C4—1222no—
302C2×C22⋊C4—1221no—
303C2×C42C22×D6111yescharacteristic
304#107—1222no—
305#106—22no—
306#106—1222no—
307C4⋊C8—21no—
308C4⋊C8—612no—
309C4⋊C8—1221no—
310C4⋊C8—1221no—
311#103—22no—
312#102—22no—
313#102—1222no—
314#102—1222no—
315C22⋊C8—21no—
316C22⋊C8—1221no—
317C4×C8—1221no—
318C4×C8C22⋊D6112yescomplemented
319#95—612no—
320#95—1222no—
321#95—1221no—
322C24—21no—
323C24—1221no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4—612no—
330C2×D4—1221no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—4882no—
336C2×D4—2444no—
337C2×D4—2444no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—1222no—
346C22×C4—1221no—
347C22×C4—4882no—
348C22×C4—2441no—
349C22×C4—2441no—
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353SD16—44no—
354M16—2441no—
355M16—2441no—
356C2×C8—22no—
357C2×C8—1222no—
358C2×C8—2441no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—2441no—
370C4⋊C4—2441no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—4882no—
375C22⋊C4—2444no—
376C22⋊C4—2441no—
377C22⋊C4—2441no—
378C42—41no—
379C42D4×D6111yescharacteristic
380C42#1175822411yescharacteristic
381C42C2×C22⋊D6111yescharacteristic
382C42C2×C22⋊D6111yescharacteristic
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—1222no—
391C23—2442no—
392C23—2442no—
393C23—2441no—
394C23—2441no—
395C23#1451974511yescharacteristic
396Q8—42no—
397Q8—81no—
398D4—22no—
399D4—22no—
400D4—41no—
401D4—41no—
402D4—1222no—
403D4—1222no—
404D4—4888no—
405D4—2441no—
406D4—2441no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—4882no—
419C2×C4—4882no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2441no—
425C2×C4#1451974511yescharacteristic
426C2×C4#1451974611yescharacteristic
427C2×C4#1451974811yescharacteristic
428C2×C4#1451974911yescharacteristic
429C2×C4#1451974911yescharacteristic
430C2×C4#1451977311yescharacteristic
431C2×C4SD16×D6111yescharacteristic
432C2×C4#1451979811yescharacteristic
433C2×C4#1451980411yescharacteristic
434C2×C4#1451980611yescharacteristic
435C8—42no—
436C8—4881no—
437C8—2442no—
438C22—21no—
439C22—21no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—4882no—
448C22—4881no—
449C22—4881no—
450C22—2444no—
451C22—2442no—
452C22#1683967011yescharacteristic
453C22#1683991111yescharacteristic
454C22#1683992111yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—4882no—
463C4—4881no—
464C4#1683989911yescharacteristic
465C4#1683990611yescharacteristic
466C4#1683998811yescharacteristic
467C4#1684000511yescharacteristic
468C2—21no—
469C2—21no—
470C2—82no—
471C2—4882no—
472C2#2144681911yescharacteristic
473C2#2144686911yescharacteristic
474C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3101C2
socleC22×C61#392
prime radical#21450720C2
hypercenter#3101D61
hypocenterC61#10000
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
3. #21444383C2semidirect
4. #21444244C2semidirect
5. #21444211C2semidirect
6. #21443678C2semidirect
7. C61×#3101C2semidirect
52. C4⋊#92418D4semidirect
63. C4⋊Q8×C61D4semidirect
64. C4⋊Q8×C61D4semidirect
83. C4×C8×C61D4semidirect
111. C2×D244SD16semidirect
231. #3101D61semidirect
270. C61#10000semidirect
278. C4⋊Q8C22⋊D61semidirect
279. C4⋊Q8C22⋊D61semidirect
318. C4×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214556122
#214507202
#214443832
#214442442
#214442112
#214436782
C61×#31012
#1000061