Finite Group Databasebeta

← #29575682

Subgroups — #29575682

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839814C211yesmaximal, characteristic, complemented
2#21454165C211yesmaximal, characteristic
3#21445796C211yesmaximal, characteristic, complemented
4C2×#16839505C211yesmaximal, characteristic, complemented
5C2×#16839505C211yesmaximal, characteristic, complemented
6#21443702C211yesmaximal, characteristic, complemented
7C61×#1296C211yesmaximal, characteristic, complemented
8C2×#14519774C2211yescharacteristic
9C2×#14519752—21no—
10C2×#14519752—21no—
11C2×#14519752C412yes—
12C2×C42×D61C2211yescharacteristic
13#16839814—22no—
14#16839814—22no—
15C2×C4⋊#92418C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519689—21no—
18C2×#14519689—21no—
19C2×C8×Dic61—21no—
20#16839663—21no—
21#16839663—21no—
22#16839663C412yes—
23#16839516—21no—
24#16839505—22no—
25#16839505—22no—
26C2×C4⋊C8×C61C2211yescharacteristic
27C2×C4⋊C8×C61C2211yescharacteristic
28C61×#384C2211yescharacteristic
29C22×D244—22no—
30C22×D244—21no—
31C22×D244—21no—
32C22×D244C2×C412yes—
33C22×C4×D61C2×C411yescharacteristic
34C22×C4×D61D411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×#92418—21no—
37C22×#92418—21no—
38#14519774—21no—
39#14519774—21no—
40#14519774—21no—
41#14519774—21no—
42#14519752—42no—
43#14519752—42no—
44#14519752—42no—
45#14519752—42no—
46#14519752—44no—
47C2×#11758198—22no—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C42×D61—21no—
54C42×D61—21no—
55C42×D61—21no—
56C42×D61—21no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418D412yescomplemented
61#14519689—24no—
62#14519689—42no—
63C8×Dic61—24no—
64C22×C8×C61—21no—
65C22×C8×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61C2×C412yes—
69C2×C42×C61C2311yescharacteristic
70C4⋊C8×C61—21no—
71C4⋊C8×C61—21no—
72C4⋊C8×C61—21no—
73C4⋊C8×C61D412yescomplemented
74C23×D61—22no—
75C23×D61C22⋊C411yescharacteristic
76C22×C4×C61—22no—
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C2×D244—22no—
84C2×D244—22no—
85C2×D244—42no—
86C2×D244—42no—
87C2×D244—44no—
88C2×D244—44no—
89C2×D244—44no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—41no—
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—22no—
115C2×C4×D61—41no—
116C2×C4×D61D812yes—
117C2×C4×D61SD1612yes—
118C2×C8×C61—22no—
119C2×C8×C61—41no—
120C2×C8×C61—41no—
121C2×C8×C61—41no—
122C2×C8×C61—41no—
123C2×C8×C61—41no—
124C4⋊C4×C61—42no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—41no—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61#4111yescharacteristic
133#11758198—44no—
134#11758198—44no—
135#11758198—44no—
136C4×Dic61—21no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61—21no—
141C4×Dic61—21no—
142C2×#92418—22no—
143C2×#92418—41no—
144C2×#92418—41no—
145C2×#92418—41no—
146C2×#92418—41no—
147C2×#92418—41no—
148C23×C61C2×C22⋊C411yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—42no—
156C22×D61—42no—
157C22×D61—42no—
158C22×D61—42no—
159C22×D61—42no—
160C22×D61—42no—
161C22×D61—42no—
162C22×D61#10212yes—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61C4×D411yescharacteristic
169C2×C4×C61C4×D411yescharacteristic
170C2×C4×C61#12011yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12311yescharacteristic
174C2×C4×C61C2×D811yescharacteristic
175C2×C4×C61C2×SD1611yescharacteristic
176C2×C4×C61#13511yescharacteristic
177C2×C4×C61#13511yescharacteristic
178C2×C4×C61#13511yescharacteristic
179C2×C4×C61#13511yescharacteristic
180C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61#10012yes—
190C2×Dic61#10212yes—
191D244—82no—
192D244—82no—
193D244—88no—
194D244—81no—
195D244—81no—
196D244—81no—
197D244—81no—
198D244—81no—
199D244—81no—
200D244—81no—
201D244—81no—
202C4×D61—22no—
203C4×D61—22no—
204C4×D61—22no—
205C4×D61—22no—
206C4×D61—42no—
207C4×D61—42no—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—41no—
211C4×D61—41no—
212C8×C61—42no—
213C8×C61—81no—
214#92418—42no—
215#92418—81no—
216#9134—611nomaximal, Sylow
217C22×C61#38611yescharacteristic
218C22×C61C2×#10011yescharacteristic
219C22×C61#41311yescharacteristic
220C22×C61C2×#10211yescharacteristic
221C22×C61#41611yescharacteristic
222C22×C61#41611yescharacteristic
223C22×C61#41811yescharacteristic
224C2×D61—22no—
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—41no—
231C2×D61—82no—
232C2×D61—82no—
233C2×D61—82no—
234C2×D61—82no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—82no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—82no—
241C4×C61#44911yescharacteristic
242C4×C61#44911yescharacteristic
243C4×C61#46511yescharacteristic
244C4×C61#46511yescharacteristic
245C4×C61#46611yescharacteristic
246C4×C61#46611yescharacteristic
247C4×C61#48211yescharacteristic
248C4×C61#48211yescharacteristic
249Dic61—22no—
250Dic61—22no—
251Dic61—41no—
252C2×#431—611no—
253#2856—611no—
254#1797—611no—
255C2×#331—611no—
256C2×#331—611no—
257#1296—611no—
258#1296D6111yescharacteristic, complemented
259C2×C61#154011yescharacteristic
260C2×C61#154311yescharacteristic
261C2×C61#154511yescharacteristic
262C2×C61#155311yescharacteristic
263C2×C61#187511yescharacteristic
264C2×C61#189311yescharacteristic
265C2×C61#190011yescharacteristic
266D61—42no—
267D61—162no—
268C2×#127—611no—
269C2×#121—612no—
270C2×#121—1221no—
271C2×#121—1221no—
272C22×C42—611no—
273#431—1222no—
274#431—1222no—
275C2×C4⋊C8—611no—
276C2×C4⋊C8—611no—
277C2×C4⋊C8C2×D6111yescharacteristic
278C2×C4⋊C8C2×D6111yescharacteristic
279C2×C22⋊C8—1221no—
280C2×C22⋊C8—1221no—
281C2×C4×C8—1221no—
282#390—612no—
283#390—1221no—
284#390—1221no—
285#384C2×D6111yescharacteristic
286#336—1221no—
287#331—1222no—
288#331—1222no—
289C61#913411yescharacteristic, complemented, Sylow
290C22×D4—612no—
291C22×D4—1222no—
292C22×D4—1221no—
293C22×D4—1221no—
294C23×C4—611no—
295C23×C4—611no—
296C23×C4—611no—
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—1221no—
300C22×C8—1221no—
301#127—1221no—
302#127—1221no—
303#127—1221no—
304#127—1221no—
305#121—2442no—
306#121—2442no—
307#121—2442no—
308#121—2442no—
309#121—2444no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C4×D6112yes—
313C2×C22⋊C4—1222no—
314C2×C22⋊C4—1222no—
315C2×C22⋊C4—1222no—
316C2×C42—611no—
317C2×C42—611no—
318C2×C42—611no—
319C2×C42—1221no—
320C2×C42—1221no—
321C2×C42—1221no—
322C2×C42—1221no—
323C2×C42C22×D6111yescharacteristic
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—612no—
328C4⋊C8—1221no—
329C4⋊C8—1221no—
330C4⋊C8—1221no—
331C4⋊C8D24412yescomplemented
332C22⋊C8—1224no—
333C22⋊C8—2442no—
334C4×C8—1224no—
335C24—611no—
336C24—1222no—
337C2×D4—1222no—
338C2×D4—1222no—
339C2×D4—2442no—
340C2×D4—2442no—
341C2×D4—2444no—
342C2×D4—2444no—
343C2×D4—2444no—
344C2×D4—2441no—
345C2×D4—2441no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2441no—
350C2×D4—2441no—
351C2×D4—2441no—
352C2×D4—2441no—
353C2×D4—2441no—
354C2×D4—2441no—
355C2×D4—2441no—
356C22×C4—22no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—612no—
360C22×C4—612no—
361C22×C4—611no—
362C22×C4—1222no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—2441no—
376C22×C4C2×C4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4D4×D6111yescharacteristic
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—1222no—
386C2×C8—2441no—
387C2×C8—2441no—
388C2×C8—2441no—
389C2×C8—2441no—
390C2×C8—2441no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C22⋊C4—2444no—
397C22⋊C4—2444no—
398C22⋊C4—2444no—
399C42—1221no—
400C42—1221no—
401C42—1221no—
402C42—1221no—
403C42—1221no—
404C42—1221no—
405C42C2×D24411yescharacteristic
406C42#1175822211yescharacteristic
407C42D4×D6111yescharacteristic
408C42#1175822611yescharacteristic
409C23—612no—
410C23—1221no—
411C23—1221no—
412C23—1221no—
413C23—1221no—
414C23—1221no—
415C23—1221no—
416C23—2442no—
417C23—2442no—
418C23—2442no—
419C23—2442no—
420C23—2442no—
421C23—2442no—
422C23—2442no—
423C23C22⋊C4×D6111yescharacteristic
424D4—4882no—
425D4—4882no—
426D4—4888no—
427D4—4881no—
428D4—4881no—
429D4—4881no—
430D4—4881no—
431D4—4881no—
432D4—4881no—
433D4—4881no—
434D4—4881no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—612no—
441C2×C4—612no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4#1451973911yescharacteristic
460C2×C4C4×D24411yescharacteristic
461C2×C4#1451974711yescharacteristic
462C2×C4#1451975011yescharacteristic
463C2×C4#1451975211yescharacteristic
464C2×C4#1451976211yescharacteristic
465C2×C4D8×D6111yescharacteristic
466C2×C4SD16×D6111yescharacteristic
467C2×C4#1451979811yescharacteristic
468C2×C4#1451979811yescharacteristic
469C2×C4#1451980011yescharacteristic
470C2×C4#1451980011yescharacteristic
471C2×C4#1451980111yescharacteristic
472C2×C4#1451980411yescharacteristic
473C8—42no—
474C8—81no—
475C8—4881no—
476C8—2442no—
477C22—1222no—
478C22—1222no—
479C22—1222no—
480C22—4882no—
481C22—4882no—
482C22—4882no—
483C22—4882no—
484C22—4882no—
485C22—4882no—
486C22—4882no—
487C22—2441no—
488C22—2441no—
489C22—2441no—
490C22—2441no—
491C22#1683964011yescharacteristic
492C22#100×D6111yescharacteristic
493C22#1683974511yescharacteristic
494C22#102×D6111yescharacteristic
495C22#1683975711yescharacteristic
496C22#1683975711yescharacteristic
497C22#1683976511yescharacteristic
498C4—21no—
499C4—21no—
500C4—82no—
501C4—1222no—
502C4—1222no—
503C4—2441no—
504C4#1683990811yescharacteristic
505C4#1683991011yescharacteristic
506C4#1683999211yescharacteristic
507C4#1683999411yescharacteristic
508C4#1683999711yescharacteristic
509C4#1683999911yescharacteristic
510C4#1684008811yescharacteristic
511C4#1684008911yescharacteristic
512C2—9762no—
513C2—2442no—
514C2#2144446711yescharacteristic
515C2#2144448011yescharacteristic
516C2#2144448511yescharacteristic
517C2#2144452311yescharacteristic
518C2#2144614011yescharacteristic
519C2#2144624611yescharacteristic
520C2#2144628111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1296C2
socleC23×C61C2×C22⋊C4
prime radical#21454165C2
cosocleC2×C42×C61C23
hypercenter#1296D61
hypocenterC61#9134
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
1. C2×#16839814C2semidirect
3. #21445796C2semidirect
4. C2×#16839505C2semidirect
5. C2×#16839505C2semidirect
6. #21443702C2semidirect
7. C61×#1296C2semidirect
60. C4⋊#92418D4semidirect
73. C4⋊C8×C61D4semidirect
258. #1296D61semidirect
289. C61#9134semidirect
331. C4⋊C8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168398142
#214541652
#214457962
C2×#168395052
#214437022
C61×#12962
#913461