Finite Group Databasebeta

← #29580253

Subgroups — #29580253

Order 15616.

856 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840081C211yesmaximal, characteristic, complemented
2#21451553C211yesmaximal, characteristic
3C2×#16839698C211yesmaximal, characteristic, complemented
4#21445910C211yesmaximal, characteristic, complemented
5C2×#16839497C211yesmaximal, characteristic, complemented
6#21443699C211yesmaximal, characteristic, complemented
7C2×C61×#323C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519746—21no—
10C22×C4⋊Dic61C2211yescharacteristic
11C22×#11758196—21no—
12C22×C22⋊Dic61—21no—
13#16840081—22no—
14#16840081—22no—
15C2×#14519709—21no—
16C2×#14519699—21no—
17#16839698—22no—
18#16839698—22no—
19#16839687—21no—
20#16839678—22no—
21#16839671—21no—
22#16839641—21no—
23#16839622C2211yescharacteristic
24C2×#14519684C2211yescharacteristic
25#16839525—22no—
26#16839497—22no—
27C22×C22⋊C4×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#323—22no—
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C22×C22⋊D61—22no—
34C23×Dic61—21no—
35C23×Dic61—21no—
36#14519751—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40#14519746—42no—
41#14519746—42no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61D411yescharacteristic
48C2×C4⋊Dic61D412yes—
49C2×C4⋊Dic61D412yescomplemented
50C2×#11758196—24no—
51C2×#11758196—44no—
52C2×C22⋊Dic61—22no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—44no—
56C2×#11758198—21no—
57C2×#11758198—22no—
58#14519709—24no—
59#14519699—24no—
60#14519684—22no—
61#14519684—21no—
62#14519684—21no—
63#14519684D412yescomplemented
64C25×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C22×C8×C61—21no—
68C22×C8×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61D412yescomplemented
79C24×C61—21no—
80C24×C61—21no—
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—42no—
86C24×C61—42no—
87C24×C61SD1612yes—
88C24×C61C2×D411yescharacteristic
89C23×D61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C2×C22⋊D61—44no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C2×C22⋊D61—41no—
105C2×C22⋊D61—41no—
106C2×C22⋊D61—41no—
107C2×C22⋊D61—41no—
108C2×C22⋊D61—82no—
109C2×C22⋊D61—82no—
110C22×Dic61—24no—
111C22×Dic61—22no—
112C22×Dic61—22no—
113C22×Dic61—21no—
114C22×Dic61—22no—
115C22×Dic61—42no—
116C22×Dic61—41no—
117C22×Dic61—41no—
118C22×Dic61—42no—
119C22×Dic61—42no—
120C22×Dic61—41no—
121C22×Dic61—41no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—42no—
125C2×C8×C61—22no—
126C2×C8×C61—42no—
127C2×C8×C61—42no—
128C2×C8×C61—41no—
129C22⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C22⋊Dic61—48no—
134C22⋊Dic61—44no—
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137#11758198—41no—
138#11758198—41no—
139#11758198—82no—
140C4⋊Dic61—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146C4⋊Dic61—42no—
147#11758196—48no—
148C23×C61—22no—
149C23×C61—22no—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61—22no—
153C23×C61—44no—
154C23×C61—42no—
155C23×C61—44no—
156C23×C61—42no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—42no—
161C23×C61—42no—
162C23×C61—41no—
163C23×C61—82no—
164C23×C61—82no—
165C23×C61—81no—
166C23×C61—81no—
167C23×C61#12011yescharacteristic
168C23×C61#12011yescharacteristic
169C23×C61#12111yescharacteristic
170C23×C61C2×SD1611yescharacteristic
171C23×C61#13511yescharacteristic
172C23×C61AGL(1,ℤ/8)11yescharacteristic
173C23×C61#13711yescharacteristic
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—42no—
178C22×D61—42no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—81no—
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12711yescharacteristic
193C22⋊D61—88no—
194C22⋊D61—81no—
195C22⋊D61—81no—
196C22⋊D61—81no—
197C22⋊D61—81no—
198C2×Dic61—24no—
199C2×Dic61—44no—
200C2×Dic61—44no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—82no—
213C2×Dic61—82no—
214D244—42no—
215D244—81no—
216C8×C61—42no—
217C8×C61—81no—
218#9920—611nomaximal, Sylow
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—44no—
228C22×C61—44no—
229C22×C61—42no—
230C22×C61—82no—
231C22×C61—82no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61#39211yescharacteristic
237C22×C61#44811yescharacteristic
238C22×C61#44911yescharacteristic
239C22×C61#45011yescharacteristic
240C22×C61#45211yescharacteristic
241C22×C61#45312yes—
242C22×C61#45812yes—
243C22×C61#46511yescharacteristic
244C22×C61#46811yescharacteristic
245C2×D61—82no—
246C2×D61—82no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—81no—
255Dic61—44no—
256Dic61—82no—
257Dic61—81no—
258Dic61—82no—
259C2×#481—611no—
260#2617—611no—
261C2×#399—611no—
262#1824—611no—
263C2×#327—611no—
264C2×#323D6111yescharacteristic, complemented
265#1294—611no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—82no—
271C2×C61#161312yes—
272C2×C61#161812yes—
273C2×C61#200011yescharacteristic
274C2×C61#200211yescharacteristic
275C2×C61#202211yescharacteristic
276D61—161no—
277C2×#121—611no—
278C2×#120—1221no—
279C22×C4⋊C4—611no—
280C22×C4⋊C4—1221no—
281C22×C22⋊C4—1221no—
282C22×C22⋊C4C2×D6111yescharacteristic
283#481—1222no—
284#481—1222no—
285C2×#106—1221no—
286C2×#102—1221no—
287C2×C22⋊C8C2×D6111yescharacteristic
288C2×C22⋊C8C2×D6111yescharacteristic
289#399—1222no—
290#399—1222no—
291#397—1221no—
292#394—1222no—
293#392—1221no—
294#386—1221no—
295#380—611no—
296C2×#95—611no—
297#340—1222no—
298#327—1222no—
299#323—22no—
300C61#992011yescharacteristic, complemented, Sylow
301C25D24411yescharacteristic
302C22×D4—611no—
303C22×D4—1222no—
304C22×D4—1221no—
305C23×C4—1221no—
306C23×C4—1221no—
307C23×C4D24411yescharacteristic
308C23×C4C22×D6111yescharacteristic
309C22×C8—21no—
310C22×C8—21no—
311#121—1221no—
312#121—1221no—
313#121—1221no—
314#121—1221no—
315#120—2442no—
316#120—2442no—
317C2×C4⋊C4—612no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—612no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1224no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—2444no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1224no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—2444no—
339#106—1224no—
340#102—1224no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8C22⋊D6112yescomplemented
345#95—612no—
346#95—1222no—
347#95—1221no—
348#95—1221no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—42no—
356C24—42no—
357C24—1221no—
358C24#1175819012yes—
359C24C2×D24411yescharacteristic
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—4882no—
363C2×D4—4882no—
364C2×D4—2442no—
365C2×D4—2444no—
366C2×D4—2441no—
367C2×D4—2441no—
368C2×D4—2441no—
369C2×D4—2441no—
370C2×D4—2441no—
371C2×D4—2441no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—1224no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1222no—
381C22×C4—1221no—
382C22×C4—2442no—
383C22×C4—2442no—
384C22×C4—2441no—
385C22×C4—2441no—
386C22×C4—2442no—
387C22×C4—2441no—
388C22×C4—2441no—
389C22×C4#1175822211yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4D4×D6111yescharacteristic
393C22×C4D4×D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C2×C8—22no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—41no—
399C4⋊C4—1222no—
400C4⋊C4—1221no—
401C4⋊C4—1222no—
402C4⋊C4—1221no—
403C4⋊C4—2442no—
404C4⋊C4—2448no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—4882no—
412C22⋊C4—2448no—
413C22⋊C4—2444no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—44no—
424C23—42no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—82no—
434C23—82no—
435C23—81no—
436C23—81no—
437C23—2442no—
438C23—2441no—
439C23—2441no—
440C23—2441no—
441C23—2442no—
442C23#1451974711yescharacteristic
443C23#1451974711yescharacteristic
444C23#1451975111yescharacteristic
445C23C2×#1175819011yescharacteristic
446C23#1451979911yescharacteristic
447C23#1451980511yescharacteristic
448C23#1451981011yescharacteristic
449D4—4881no—
450D4—4888no—
451D4—4881no—
452D4—4881no—
453D4—4881no—
454D4—4881no—
455D4—2442no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—81no—
466C2×C4—1224no—
467C2×C4—4882no—
468C2×C4—4882no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2444no—
479C2×C4—2442no—
480C2×C4—2444no—
481C2×C4—2442no—
482C2×C4#1451974511yescharacteristic
483C2×C4#1451975011yescharacteristic
484C2×C4#1451975011yescharacteristic
485C2×C4#1451977311yescharacteristic
486C8—42no—
487C8—81no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—44no—
497C22—42no—
498C22—44no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—81no—
505C22—4882no—
506C22—4881no—
507C22—4881no—
508C22—4881no—
509C22—4882no—
510C22#1683967111yescharacteristic
511C22#1683990311yescharacteristic
512C22#1683990911yescharacteristic
513C22#1683991511yescharacteristic
514C22#1683992611yescharacteristic
515C22#1683993212yes—
516C22#1683995312yes—
517C22#1683999111yescharacteristic
518C22#1684000811yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—81no—
524C4—4881no—
525C4—4882no—
526C4—4882no—
527C4—2444no—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—82no—
533C2—9761no—
534C2#2144484112yes—
535C2#2144486912yes—
536C2#2144682411yescharacteristic
537C2#2144683611yescharacteristic
538C2#2144694911yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#323C2
socleC23×C61#120
prime radical#21451553C2
hypercenterC2×#323D61
hypocenterC61#9920
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
1. C2×#16840081C2semidirect
3. C2×#16839698C2semidirect
4. #21445910C2semidirect
5. C2×#16839497C2semidirect
6. #21443699C2semidirect
7. C2×C61×#323C2semidirect
49. C2×C4⋊Dic61D4semidirect
63. #14519684D4semidirect
78. C22⋊C8×C61D4semidirect
264. C2×#323D61semidirect
300. C61#9920semidirect
344. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400812
#214515532
C2×#168396982
#214459102
C2×#168394972
#214436992
C2×C61×#3232
#992061