Finite Group Databasebeta

← #29736146

Subgroups — #29736146

Order 15616.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840468C211yesmaximal, characteristic
2C2×#16840049C211yesmaximal, characteristic
3#21450808C211yesmaximal, characteristic, complemented
4#21447155C212yesmaximal, complemented
5#21447150C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446759C211yesmaximal, characteristic, complemented
8#21446077C212yesmaximal
9#21446063C212yesmaximal
10#21445645C211yesmaximal, characteristic
11C2×C61×#501C211yesmaximal, characteristic, complemented
12#16840468—24no—
13#16840468—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519775—21no—
16C2×#14519768C2211yescharacteristic
17C2×#14519757—21no—
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×#14519739C2211yescharacteristic
24C2×C4⋊C4×D61—21no—
25#16840049—24no—
26C2×#14519708C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30#16839693—21no—
31#16839693—21no—
32#16839669—21no—
33#16839658C2212yes—
34#16839658C2212yes—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×#125×C61C2211yescharacteristic
43C61×#501—24no—
44C2×#107×C61C2212yes—
45C2×#106×C61C2212yes—
46C2×#97×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122D411yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C22×#92418C2312yes—
54#14519776—22no—
55#14519776—22no—
56#14519775—44no—
57#14519768—22no—
58#14519768—22no—
59#14519757—28no—
60#14519755—28no—
61#14519755—28no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64C4×D244—22no—
65C4×D244D414yes—
66#14519739—22no—
67#14519739—22no—
68#14519739—22no—
69#14519739—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×#11758196D411yescharacteristic
79C2×#11758196D411yescharacteristic
80C4⋊C4×D61—28no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519708—24no—
88C4⋊#92418—22no—
89#14519699Q818yescomplemented
90#14519697—24no—
91#14519684—21no—
92#14519684—22no—
93#14519684—22no—
94C22×Q8×C61—21no—
95C22×C8×C61C2312yes—
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—22no—
98#125×C61—22no—
99#125×C61D414yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61C2312yes—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107#107×C61—24no—
108#106×C61—24no—
109#97×C61—22no—
110C23×D61—21no—
111C2×Q8×C61—42no—
112C2×Q8×C61—42no—
113C2×Q8×C61—42no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61C2×D411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126C2×D244—21no—
127C2×D244—22no—
128C2×D244C2×D412yes—
129C2×D244C2×Q814yes—
130C2×C4×D61—24no—
131C2×C4×D61—24no—
132C2×C4×D61—24no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61—82no—
136C2×C4×D61—82no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—24no—
140C2×Dic122—41no—
141C2×Dic122—82no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61C2×Q814yes—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—42no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C42×C61C2×D412yes—
155C42×C61C2×D412yes—
156#11758198—24no—
157#11758198—24no—
158#11758198—24no—
159C4⋊Dic61—24no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61C2×D412yes—
164C4⋊Dic61C2×Q814yes—
165#11758196—24no—
166#11758196—24no—
167#11758196—22no—
168#11758196—22no—
169#11758196—44no—
170#11758196—44no—
171C4×Dic61—21no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—21no—
175C4×Dic61—42no—
176C2×#92418—22no—
177C2×#92418—24no—
178C23×C61C22×D411yescharacteristic
179C22×D61—22no—
180C22×D61—24no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184Q8×C61—84no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61#12012yes—
197C2×C4×C61#12012yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×Q812yes—
201C2×C4×C612−1+412yes—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—22no—
208C2×Dic61—24no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61AGL(1,ℤ/8)12yes—
217D244—22no—
218D244—24no—
219D244C4⋊Q814yescomplemented
220C4×D61—48no—
221C4×D61—84no—
222C4×D61—84no—
223Dic122—44no—
224Dic122—84no—
225C8×C61—24no—
226#92418—42no—
227#25851—611nomaximal, Sylow
228C22×C61C2×#12011yescharacteristic
229C22×C61D4×Q812yes—
230C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C61#57511yescharacteristic
232C22×C61#57712yes—
233C2×D61—24no—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—42no—
237C2×D61—42no—
238C4×C61—22no—
239C4×C61—22no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—82no—
243C4×C61C2×C4⋊Q811yescharacteristic
244C4×C61#53611yescharacteristic
245C4×C61D4×Q812yes—
246Dic61—22no—
247Dic61—44no—
248Dic61—41no—
249Dic61—41no—
250Dic61—42no—
251Dic61—41no—
252C2×#554—611no—
253C2×#501D6111yescharacteristic, complemented
254C2×#476—611no—
255#2546—611no—
256#2059—612no—
257#2058—612no—
258#1989—611no—
259#1985—611no—
260#1864—612no—
261#1862—612no—
262#1762—611no—
263C2×C61#243411yescharacteristic
264C2×C61#300611yescharacteristic
265C2×C61#300911yescharacteristic
266C2×C61#338412yes—
267C2×C61#338812yes—
268D61—44no—
269#554—1224no—
270#554—1224no—
271C2×C4⋊Q8—611no—
272C2×C4⋊Q8—1221no—
273C2×C4⋊Q8C2×D6111yescharacteristic
274C2×#125—611no—
275C2×#125C2×D6111yescharacteristic
276C2×C22⋊Q8—1221no—
277C2×C22⋊Q8—1221no—
278C2×C22⋊Q8—1221no—
279C2×C4×Q8—611no—
280C2×C4×D4—611no—
281C2×C4×D4—611no—
282C2×C4×D4—611no—
283C22×C4⋊C4—1221no—
284#501—24no—
285#476—1224no—
286C2×#107C2×D6112yes—
287C2×#106—612no—
288C2×#106C2×D6112yes—
289C2×C4⋊C8—611no—
290C2×#102—612no—
291C2×#102—612no—
292C2×#97C2×D6111yescharacteristic
293#398—1221no—
294#398—1221no—
295#391—1221no—
296#389—612no—
297#389—612no—
298C4×C4⋊C4—611no—
299#340—612no—
300#340—612no—
301#340—612no—
302#336—612no—
303#336—612no—
304C61#2585111yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C22×Q8—611no—
307C22×Q8—1221no—
308C22×D4—611no—
309C23×C4—1221no—
310C23×C4—1221no—
311C23×C4—1221no—
312C22×C8—612no—
313C22×C8C22×D6112yes—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—1222no—
317C4⋊Q8—1222no—
318C4⋊Q8—2444no—
319#125—22no—
320#125—1222no—
321#125—1222no—
322#125C22⋊D6114yes—
323C22⋊Q8—1228no—
324C22⋊Q8—1228no—
325C22⋊Q8—1228no—
326C4×Q8—1222no—
327C4×Q8—1222no—
328C4×D4—614no—
329C4×D4—1222no—
330C4×D4—1222no—
331C4×D4—1222no—
332C4×D4—1222no—
333C4×D4—1222no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—1228no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4C22×D6112yes—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—1221no—
357C2×C42C22×D6111yescharacteristic
358#107—24no—
359#106—24no—
360#106—1224no—
361C4⋊C8—1222no—
362#102—618no—
363#102—1224no—
364#97—22no—
365#95—1222no—
366#95—1222no—
367#95—1221no—
368C24—1221no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—1222no—
373C2×Q8—1224no—
374C2×Q8—1222no—
375C2×Q8—4882no—
376C2×Q8—2441no—
377C2×D4—612no—
378C2×D4—614no—
379C2×D4—1221no—
380C2×D4—1222no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—1224no—
389C22×C4—1224no—
390C22×C4—1222no—
391C22×C4—1221no—
392C22×C4—1224no—
393C22×C4—4882no—
394C22×C4—4882no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4D4×D6111yescharacteristic
398C22×C4D4×D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—1222no—
403C2×C8—1224no—
404C2×C8Q8×D6114yes—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—612no—
413C4⋊C4—614no—
414C4⋊C4—1224no—
415C4⋊C4—1224no—
416C4⋊C4—1224no—
417C4⋊C4—1221no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—1222no—
421C4⋊C4—1222no—
422C4⋊C4—2444no—
423C4⋊C4—2444no—
424C4⋊C4C2×C22⋊D6112yes—
425C4⋊C4C2×C22⋊D6112yes—
426C22⋊C4—1224no—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C42—1221no—
430C42—1221no—
431C42—1221no—
432C42—1221no—
433C42—2442no—
434C42D4×D6112yes—
435C42C2×C22⋊D6112yes—
436C23—1222no—
437C23—1224no—
438C23—2441no—
439C23—2441no—
440C23—2442no—
441C23C2×D4×D6111yescharacteristic
442Q8—84no—
443Q8—4884no—
444Q8—2444no—
445D4—614no—
446D4—1222no—
447D4—1224no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—612no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1222no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1224no—
471C2×C4—4884no—
472C2×C4—4884no—
473C2×C4—2448no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4#1451974512yes—
478C2×C4#1451974512yes—
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4C2×Q8×D6112yes—
482C2×C4#1451982512yes—
483C8—24no—
484C8—2442no—
485C22—1224no—
486C22—2444no—
487C22—2444no—
488C22—2442no—
489C22—2442no—
490C22C2×#1451974511yescharacteristic
491C22#1684042712yes—
492C22AGL(1,ℤ/8)×D6111yescharacteristic
493C22#1684055811yescharacteristic
494C22#1684057112yes—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—1222no—
501C4—2444no—
502C4—2441no—
503C4—2441no—
504C4—2441no—
505C4—2442no—
506C4C4⋊Q8×D6111yescharacteristic
507C4#1684033711yescharacteristic
508C4Q8×C22⋊D6112yes—
509C2—2444no—
510C2#2144946011yescharacteristic
511C2#2145506711yescharacteristic
512C2#2145509511yescharacteristic
513C2#2145871512yes—
514C2#2145875912yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#501D61
hypocenterC61#25851
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #21450808C2semidirect
4. #21447155C2semidirect
5. #21447150C2semidirect
6. #21446777C2semidirect
7. #21446759C2semidirect
11. C2×C61×#501C2semidirect
89. #14519699Q8semidirect
219. D244C4⋊Q8semidirect
253. C2×#501D61semidirect
304. C61#25851semidirect

Maximal subgroups

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

subgroupindex
C2×#168404682
C2×#168400492
#214508082
#214471552
#214471502
#214467772
#214467592
#214460772
#214460632
#214456452
C2×C61×#5012
#2585161