Finite Group Databasebeta

← #29595984

Subgroups — #29595984

Order 15616.

600 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 101 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#21448480C211yesmaximal, characteristic
2C2×#16839549C211yesmaximal, characteristic
3#21445809C211yesmaximal, characteristic, complemented
4#21445805C211yesmaximal, characteristic, complemented
5#21443938C211yesmaximal, characteristic, complemented
6#21443936C211yesmaximal, characteristic, complemented
7C2×C61×AΣL(1,𝔽16)C211yesmaximal, characteristic
8C2×D4×Dic61C2211yescharacteristic
9C22×C22⋊Dic61C411yescharacteristic, complemented
10#16839785—21no—
11C2×#14519694C2211yescharacteristic
12C2×#14519692C2211yescharacteristic
13#16839665—21no—
14#16839655—21no—
15#16839655—21no—
16#16839642—21no—
17#16839642C2211yescharacteristic
18#16839637C411yescharacteristic, complemented
19#16839620—21no—
20#16839618C411yescharacteristic, complemented
21C22⋊C4×Dic61C411yescharacteristic, complemented
22#16839549—22no—
23#16839530—21no—
24#16839528—21no—
25C2×#120×C61C2211yescharacteristic
26C2×#100×C61C2211yescharacteristic
27C2×#99×C61C2211yescharacteristic
28C61×AΣL(1,𝔽16)—22no—
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C23×Dic61D411yescharacteristic
32C2×#11758194—21no—
33D4×Dic61—41no—
34D4×Dic61—41no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61D411yescharacteristic
37C2×#11758196—41no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—41no—
45C2×C22⋊Dic61—41no—
46C2×C22⋊Dic61C2×C411yescharacteristic
47C2×C22⋊Dic61C2×C411yescharacteristic
48C2×C22⋊Dic61D411yescharacteristic
49C2×C4×Dic61—21no—
50C2×C4×Dic61D411yescharacteristic
51#14519706—41no—
52#14519706—41no—
53#14519694—21no—
54#14519694—21no—
55#14519692—21no—
56#14519692—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—41no—
61C25×C61C2×C411yescharacteristic
62C22×D4×C61—21no—
63C22×D4×C61C2311yescharacteristic
64C2×M16×C61—21no—
65#120×C61—21no—
66#120×C61—21no—
67#120×C61D412yes—
68#120×C61Q812yes—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61C2×C411yescharacteristic
72#100×C61—21no—
73#100×C61—21no—
74#99×C61—21no—
75#99×C61—21no—
76C24×C61—21no—
77C24×C61—21no—
78C24×C61—41no—
79C24×C61—41no—
80C24×C61—81no—
81C24×C61—81no—
82C24×C61C4⋊C412yescomplemented
83C24×C61C22×C411yescharacteristic
84C24×C61C2×D411yescharacteristic
85C2×D4×C61—22no—
86C2×D4×C61—22no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61—81no—
90C2×D4×C61—81no—
91C2×D4×C61C2×D411yescharacteristic
92C2×D4×C61C2×Q811yescharacteristic
93C2×D4×C61#4111yescharacteristic
94C2×D4×C61#4111yescharacteristic
95C22×C4×C61—21no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—41no—
105C22×Dic61—41no—
106C22×Dic61—81no—
107C22×Dic61—81no—
108C22×Dic61—81no—
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61C22⋊C411yescharacteristic
111C22×Dic61C22⋊C411yescharacteristic
112C22×Dic61C22⋊C411yescharacteristic
113M16×C61—41no—
114M16×C61—41no—
115C2×C8×C61—41no—
116C22⋊C4×C61—21no—
117C22⋊C4×C61—22no—
118C22⋊C4×C61—22no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61C4⋊C412yescomplemented
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124C22⋊Dic61—42no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129C22⋊Dic61—41no—
130C22⋊Dic61—81no—
131C22⋊Dic61—81no—
132C22⋊Dic61—81no—
133C4⋊Dic61—41no—
134C4⋊Dic61—81no—
135#11758196—82no—
136C4×Dic61—21no—
137C4×Dic61—21no—
138C4×Dic61—81no—
139#11758194—41no—
140#11758194—41no—
141C2×#92418—41no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—22no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—82no—
160C23×C61—82no—
161C23×C61—84no—
162C23×C61—82no—
163C23×C61—82no—
164C23×C61C2×C22⋊C411yescharacteristic
165C23×C61C2×C4⋊C411yescharacteristic
166C23×C61#11711yescharacteristic
167C23×C61#12111yescharacteristic
168C23×C61C22⋊Q811yescharacteristic
169D4×C61—41no—
170D4×C61—41no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—82no—
174D4×C61—82no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—81no—
181C2×C4×C61—81no—
182C2×C4×C61—81no—
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61C22⋊Q811yescharacteristic
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—81no—
200C2×Dic61—81no—
201C2×Dic61—81no—
202C2×Dic61—82no—
203C2×Dic61—81no—
204C2×Dic61—81no—
205C2×Dic61—81no—
206C2×Dic61#9911yescharacteristic
207C2×Dic61#9911yescharacteristic
208C8×C61—81no—
209#92418—81no—
210#12795—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—41no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—42no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—84no—
233C22×C61—84no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61#38011yescharacteristic
237C22×C61C2×#9911yescharacteristic
238C22×C61#41011yescharacteristic
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—42no—
242C4×C61—161no—
243Dic61—21no—
244Dic61—21no—
245Dic61—41no—
246Dic61—41no—
247Dic61—41no—
248Dic61—81no—
249Dic61—81no—
250Dic61—161no—
251#2321—611no—
252C2×AΣL(1,𝔽16)—611no—
253C2×AΣL(1,𝔽16)D6111yescharacteristic
254#1800—611no—
255#1799—611no—
256#1390—611no—
257#1390—611no—
258C2×C61—21no—
259C2×C61—21no—
260C2×C61—41no—
261C2×C61—41no—
262C2×C61—41no—
263C2×C61—41no—
264C2×C61—82no—
265C2×C61#179911yescharacteristic
266C2×C61#212111yescharacteristic
267C2×C61#212211yescharacteristic
268C2×#120C2×D6111yescharacteristic
269C2×C4×D4—611no—
270C22×C22⋊C4—611no—
271#423—1221no—
272C2×#100—611no—
273C2×#100C2×D6111yescharacteristic
274C2×#99—611no—
275C2×#99C2×D6111yescharacteristic
276#390—1221no—
277#388—1221no—
278#388—1221no—
279#386—611no—
280#386—1221no—
281#385—611no—
282#380—1221no—
283#379—611no—
284C4×C22⋊C4—611no—
285AΣL(1,𝔽16)—22no—
286AΣL(1,𝔽16)—1222no—
287#343—1221no—
288#342—1221no—
289C61#1279511yescharacteristic, complemented, Sylow
290C25C4×D6111yescharacteristic
291C22×D4—21no—
292C22×D4C22×D6111yescharacteristic
293C23×C4—611no—
294C23×C4—611no—
295C23×C4—611no—
296C2×M16—21no—
297C2×M16—1221no—
298#120—21no—
299#120—21no—
300#120Dic12212yes—
301#120C22⋊D6112yes—
302C4×D4—2441no—
303C4×D4—2441no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—2441no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—611no—
310C2×C22⋊C4—611no—
311C2×C22⋊C4—611no—
312C2×C22⋊C4—1221no—
313C2×C22⋊C4—1221no—
314C2×C22⋊C4—1221no—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—2441no—
319C2×C22⋊C4—2441no—
320C2×C22⋊C4C4×D6111yescharacteristic
321C2×C42—611no—
322C2×C42—1221no—
323C4⋊C8—2441no—
324C4⋊C8—2441no—
325#100—21no—
326#100—21no—
327#100—1221no—
328#100—1221no—
329#99—21no—
330#99—21no—
331#99—1221no—
332#99—1221no—
333#95—1221no—
334#95—1221no—
335#95—1221no—
336#95—2441no—
337C24—21no—
338C24—21no—
339C24—41no—
340C24—41no—
341C24—81no—
342C24—81no—
343C24#1175819612yescomplemented
344C24C2×C4×D6111yescharacteristic
345C24D4×D6111yescharacteristic
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—81no—
351C2×D4—81no—
352C2×D4C2×Dic12211yescharacteristic
353C2×D4#1175822411yescharacteristic
354C2×D4#1175822411yescharacteristic
355C2×D4C2×C22⋊D6111yescharacteristic
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—611no—
359C22×C4—611no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—4881no—
369C22×C4—4881no—
370C22×C4—4881no—
371C22×C4—2441no—
372C22×C4—2441no—
373C22×C4D4×D6111yescharacteristic
374M16—41no—
375M16—41no—
376M16—2441no—
377M16—2441no—
378C2×C8—41no—
379C2×C8—2441no—
380C4⋊C4—4881no—
381C4⋊C4—4882no—
382C4⋊C4—2441no—
383C22⋊C4—21no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—4881no—
389C22⋊C4—4881no—
390C22⋊C4—4881no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C22⋊C4—2442no—
396C22⋊C4—2441no—
397C22⋊C4—2441no—
398C22⋊C4—2441no—
399C22⋊C4#1175819612yescomplemented
400C42—1221no—
401C42—1221no—
402C42—4881no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—84no—
425C23C22⋊C4×D6111yescharacteristic
426C23C2×#1175819611yescharacteristic
427C23#1451973311yescharacteristic
428C23#1451974911yescharacteristic
429C23#1451975411yescharacteristic
430D4—41no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—82no—
435D4—82no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—4881no—
450C2×C4—4881no—
451C2×C4—4881no—
452C2×C4—4881no—
453C2×C4—4881no—
454C2×C4—4881no—
455C2×C4—4882no—
456C2×C4—2441no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4#1451974911yescharacteristic
468C2×C4#1451975411yescharacteristic
469C8—81no—
470C8—4881no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—84no—
493C22—84no—
494C22—81no—
495C22—81no—
496C22#1683962011yescharacteristic
497C22#99×D6111yescharacteristic
498C22#1683973411yescharacteristic
499C4—21no—
500C4—21no—
501C4—42no—
502C4—161no—
503C4—1221no—
504C4—1221no—
505C4—9761no—
506C4—4881no—
507C4—4881no—
508C4—2441no—
509C4—2441no—
510C4—2441no—
511C2—21no—
512C2—21no—
513C2—41no—
514C2—41no—
515C2—41no—
516C2—41no—
517C2—82no—
518C2#2144580511yescharacteristic
519C2#2144746211yescharacteristic
520C2#2144746711yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC2×C61×AΣL(1,𝔽16)C2
socleC22×C61C2×#99
prime radicalC2×#120×C61C22
cosocleC22×D4×C61C23
hypercenterC2×AΣL(1,𝔽16)D61
hypocenterC61#12795
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
3. #21445809C2semidirect
4. #21445805C2semidirect
5. #21443938C2semidirect
6. #21443936C2semidirect
9. C22×C22⋊Dic61C4semidirect
18. #16839637C4semidirect
20. #16839618C4semidirect
21. C22⋊C4×Dic61C4semidirect
82. C24×C61C4⋊C4semidirect
121. C22⋊C4×C61C4⋊C4semidirect
289. C61#12795semidirect
343. C24#11758196semidirect
399. C22⋊C4#11758196semidirect

Maximal subgroups

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

subgroupindex
#214484802
C2×#168395492
#214458092
#214458052
#214439382
#214439362
C2×C61×AΣL(1,𝔽16)2
#1279561