Finite Group Databasebeta

← #29734274

Subgroups — #29734274

Order 15616.

1374 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic
2C2×#16840110C211yesmaximal, characteristic, complemented
3#21454166C211yesmaximal, characteristic, complemented
4#21447010C214yesmaximal, complemented
5#21446781C211yesmaximal, characteristic, complemented
6#21446625C211yesmaximal, characteristic, complemented
7#21446077C214yesmaximal, complemented
8C4×#14519689C211yesmaximal, characteristic, complemented
9C2×C61×#498C211yesmaximal, characteristic, complemented
10C22×#11758226—21no—
11#16840313—24no—
12#16840313—24no—
13C2×#14519774C2211yescharacteristic
14C2×#14519766—21no—
15C2×#14519753—21no—
16C2×#14519752—21no—
17C2×#14519752C2212yes—
18C2×#14519739—21no—
19C2×#14519739C2212yes—
20C2×C42×D61C2211yescharacteristic
21#16840110—24no—
22C2×#14519699C2214yes—
23C2×#14519697C2214yes—
24C2×#14519689C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839669—21no—
27#16839658—21no—
28#16839658C2212yes—
29#16839632C2211yescharacteristic
30#16839619—21no—
31#16839619C2212yes—
32#16839524C2214yes—
33#16839516C2212yes—
34C2×C4⋊Q8×C61C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#498—24no—
37C2×#106×C61C2214yes—
38C2×C4×C8×C61C2211yescharacteristic
39C2×#11758226—28no—
40C2×#11758226—44no—
41C2×#11758226—44no—
42C22×D244—22no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C22×C4×D61—22no—
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50C22×#92418C2312yes—
51#14519774—22no—
52#14519774D414yes—
53#14519766—28no—
54#14519753—44no—
55#14519752—28no—
56#14519752—24no—
57#14519752—24no—
58#14519739—28no—
59#14519739—24no—
60#14519739—24no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61C2311yescharacteristic
71C42×D61—22no—
72C42×D61—22no—
73#14519699—28no—
74#14519697—28no—
75#14519689—24no—
76C4×#92418—22no—
77#14519684—22no—
78#14519684—22no—
79C22×Q8×C61—21no—
80C22×Q8×C61—21no—
81C22×C8×C61C2312yes—
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85C4⋊Q8×C61D414yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91#106×C61—28no—
92C4×C8×C61—22no—
93C23×D61—22no—
94C23×D61C2×D411yescharacteristic
95C2×Q8×C61—24no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C2×Q8×C61—42no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—22no—
107C22×Dic61C2×D411yescharacteristic
108C22×Dic61C2×D411yescharacteristic
109C22×Dic61C2×D411yescharacteristic
110#11758226—432no—
111C2×D244—24no—
112C2×D244—24no—
113C2×D244—28no—
114C2×D244—24no—
115C2×D244—24no—
116C2×D244—22no—
117C2×D244—44no—
118C2×D244—44no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—42no—
123C2×D244C2×D414yes—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—28no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—44no—
135C2×C4×D61—44no—
136C2×C4×D61SD1614yes—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61#4114yescomplemented
140C4⋊C4×C61—24no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61C2×D414yes—
146C42×C61C2×D412yes—
147C42×C61C2×D412yes—
148#11758198—28no—
149#11758198—28no—
150#11758198—44no—
151C4⋊Dic61—24no—
152C4⋊Dic61—42no—
153C4⋊Dic61—42no—
154#11758196—44no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C4×Dic61—21no—
158C4×Dic61—21no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—22no—
162C23×C61C22×D411yescharacteristic
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—24no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61C2×SD1612yes—
177Q8×C61—48no—
178Q8×C61—84no—
179C2×C4×C61—24no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61#12012yes—
186C2×C4×C61#12012yes—
187C2×C4×C61C2×SD1612yes—
188C2×C4×C61#13512yes—
189C2×C4×C61AGL(1,ℤ/8)12yes—
190C2×C4×C61AGL(1,ℤ/8)12yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4112yes—
194C2×C4×C612−1+412yes—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61C2×SD1612yes—
207D244—28no—
208D244—416no—
209D244—44no—
210D244—48no—
211D244—48no—
212C4×D61—28no—
213C4×D61—416no—
214C4×D61—42no—
215C4×D61—42no—
216C4×D61—44no—
217C8×C61—24no—
218#92418—42no—
219#25671—611nomaximal, Sylow
220C22×C61C2×#12011yescharacteristic
221C22×C61#53211yescharacteristic
222C22×C61#53611yescharacteristic
223C22×C61C22×SD1611yescharacteristic
224C22×C61C2×#13511yescharacteristic
225C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
226C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—41no—
231C2×D61—41no—
232C2×D61—41no—
233C2×D61—41no—
234C2×D61—44no—
235C2×D61—44no—
236C2×D61—44no—
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—82no—
241C4×C61—44no—
242C4×C61—82no—
243C4×C61#44914yes—
244C4×C61#45014yes—
245C4×C61#54712yes—
246C4×C61#54812yes—
247Dic61—22no—
248Dic61—41no—
249Dic61—41no—
250Dic61—82no—
251C2×#532—611no—
252C2×#498D6111yescharacteristic, complemented
253C2×#486—611no—
254#2857—611no—
255#2034—614no—
256#1990—611no—
257#1957—611no—
258#1864—614no—
259C4×C22⋊C8—611no—
260C2×C61#243211yescharacteristic
261C2×C61C2×#45011yescharacteristic
262C2×C61C2×#44911yescharacteristic
263C2×C61#316711yescharacteristic
264C2×C61#318411yescharacteristic
265C2×C61#322711yescharacteristic
266C2×C61#322811yescharacteristic
267D61—42no—
268D61—84no—
269C22×#41—1221no—
270#532—1224no—
271#532—1224no—
272C2×C4⋊Q8C2×D6111yescharacteristic
273C2×C4⋊Q8C2×D6111yescharacteristic
274C2×#127—611no—
275C2×#124—1221no—
276C2×C22⋊Q8—1221no—
277C2×#121—612no—
278C2×#121—1221no—
279C2×C4×D4—612no—
280C2×C4×D4—1221no—
281C22×C42—611no—
282#498—24no—
283#486—1224no—
284C2×#106C2×D6114yes—
285C2×#102—614no—
286C2×#102—614no—
287C2×C22⋊C8—612no—
288C2×C4×C8—611no—
289C2×C4×C8C2×D6111yescharacteristic
290#391—1221no—
291#389—612no—
292#389—1221no—
293#384—611no—
294#380—612no—
295#380—1221no—
296#340—614no—
297#336—612no—
298C61#2567111yescharacteristic, complemented, Sylow
299C2×#41—1228no—
300C2×#41—2444no—
301C2×#41—2444no—
302C22×Q8—21no—
303C22×Q8—21no—
304C22×D4—612no—
305C22×D4—1222no—
306C22×D4—1221no—
307C22×D4—1221no—
308C23×C4—611no—
309C23×C4—611no—
310C23×C4—611no—
311C23×C4—1222no—
312C22×C8—612no—
313C22×C8C22×D6112yes—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8C22⋊D6114yes—
318#127—614no—
319#127—1222no—
320#124—1228no—
321C22⋊Q8—2444no—
322#121—1224no—
323#121—1228no—
324#121—1224no—
325C4×D4—1224no—
326C4×D4—1228no—
327C4×D4—1224no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22×D6112yes—
335C2×C4⋊C4C22×D6112yes—
336C2×C22⋊C4—1222no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C42—611no—
340C2×C42—611no—
341C2×C42—611no—
342C2×C42—1222no—
343C2×C42—1222no—
344C2×C42C22×D6111yescharacteristic
345#106—28no—
346#102—1228no—
347#102—1228no—
348C22⋊C8—1224no—
349C4×C8—22no—
350C4×C8—1222no—
351#95—1222no—
352#95—1222no—
353C24—611no—
354C24—1222no—
355#41—24432no—
356C2×Q8—24no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×D4—614no—
363C2×D4—1228no—
364C2×D4—1222no—
365C2×D4—1224no—
366C2×D4—1224no—
367C2×D4—1224no—
368C2×D4—1224no—
369C2×D4—2444no—
370C2×D4—2444no—
371C2×D4—2442no—
372C2×D4—2442no—
373C2×D4—2442no—
374C2×D4—2442no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—611no—
378C22×C4—614no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—1228no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—2444no—
393C22×C4—2444no—
394C22×C4D4×D6111yescharacteristic
395C22×C4D4×D6111yescharacteristic
396C22×C4C23×D6111yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8#1175822614yescomplemented
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—1224no—
409C4⋊C4—2444no—
410C4⋊C4—2442no—
411C4⋊C4—2442no—
412C4⋊C4C2×C22⋊D6114yes—
413C22⋊C4—1228no—
414C22⋊C4—1228no—
415C22⋊C4—2444no—
416C42—1221no—
417C42—1221no—
418C42—1222no—
419C42—1222no—
420C42D4×D6112yes—
421C42C2×C22⋊D6112yes—
422C23—612no—
423C23—1221no—
424C23—1221no—
425C23—1221no—
426C23—1221no—
427C23—1221no—
428C23—1221no—
429C23—1224no—
430C23—2442no—
431C23—2442no—
432C23—2442no—
433C23—2442no—
434C23—2442no—
435C23—2442no—
436C23C2×D4×D6111yescharacteristic
437Q8—48no—
438Q8—84no—
439D4—1228no—
440D4—24416no—
441D4—2448no—
442D4—2448no—
443D4—2444no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—612no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—1228no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—24416no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2444no—
466C2×C4—2442no—
467C2×C4#1451974512yes—
468C2×C4#1451974512yes—
469C2×C4SD16×D6112yes—
470C2×C4#1451979812yes—
471C2×C4#1451980412yes—
472C2×C4#1451980412yes—
473C2×C4C2×D4×D6111yescharacteristic
474C2×C4C22×C22⋊D6111yescharacteristic
475C2×C4C2×#1175822612yes—
476C2×C4#1451982512yes—
477C8—24no—
478C8—2442no—
479C22—1222no—
480C22—1222no—
481C22—1222no—
482C22—4882no—
483C22—4882no—
484C22—4882no—
485C22—4882no—
486C22—2444no—
487C22—2444no—
488C22—2444no—
489C22—2441no—
490C22—2441no—
491C22—2441no—
492C22—2441no—
493C22C2×#1451974511yescharacteristic
494C22#1684031311yescharacteristic
495C22#1684033711yescharacteristic
496C22C2×SD16×D6111yescharacteristic
497C22C2×#1451979811yescharacteristic
498C22C2×#1451980411yescharacteristic
499C22C2×#1451980411yescharacteristic
500C4—44no—
501C4—82no—
502C4—1222no—
503C4—4882no—
504C4—2441no—
505C4—2441no—
506C4#1683990614yes—
507C4#1683991114yes—
508C4#1684041512yes—
509C4#1684042312yes—
510C2—4884no—
511C2—2442no—
512C2#2144944411yescharacteristic
513C2C2×#1683991111yescharacteristic
514C2C2×#1683990611yescharacteristic
515C2#2145638011yescharacteristic
516C2#2145655811yescharacteristic
517C2#2145694811yescharacteristic
518C2#2145695611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#498C2
socleC23×C61C22×D4
prime radicalC2×#16840313C2
hypercenterC2×#498D61
hypocenterC61#25671
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
2. C2×#16840110C2semidirect
3. #21454166C2semidirect
4. #21447010C2semidirect
5. #21446781C2semidirect
6. #21446625C2semidirect
7. #21446077C2semidirect
8. C4×#14519689C2semidirect
9. C2×C61×#498C2semidirect
139. C2×C8×C61#41semidirect
252. C2×#498D61semidirect
298. C61#25671semidirect
402. C2×C8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168403132
C2×#168401102
#214541662
#214470102
#214467812
#214466252
#214460772
C4×#145196892
C2×C61×#4982
#2567161