Finite Group Databasebeta

← #29833888

Subgroups — #29833888

Order 15616.

1098 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840203C211yesmaximal, characteristic, complemented
2C2×#16840192C211yesmaximal, characteristic, complemented
3#21455489C211yesmaximal, characteristic, complemented
4#21447759C212yesmaximal, complemented
5#21447753C212yesmaximal, complemented
6#21447742C212yesmaximal, complemented
7#21447736C212yesmaximal, complemented
8#21447531C211yesmaximal, characteristic, complemented
9C2×#16839566C211yesmaximal, characteristic, complemented
10#21447433C211yesmaximal, characteristic, complemented
11C2×C61×#578C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840340—21no—
14#16840203—24no—
15#16840192—24no—
16C2×#14519723C2212yes—
17C2×#14519723C2211yescharacteristic
18C2×#14519721C2211yescharacteristic
19C2×#14519720C2212yes—
20C2×#14519720C2211yescharacteristic
21C2×#14519718C2211yescharacteristic
22C2×#14519717C2211yescharacteristic
23C22×#11758185C2211yescharacteristic
24#16840008—22no—
25#16840003C2212yes—
26#16839998C2212yes—
27C2×#14519711C2211yescharacteristic
28#16839764—21no—
29#16839566C2212yes—
30#16839566C2212yes—
31#16839566C2212yes—
32#16839563C2212yes—
33C22⋊#11758185C2212yes—
34C22⋊#11758185C2212yes—
35C2×2−1+4×C61—21no—
36C61×#578—24no—
37C61×#578—24no—
38C2×#137×C61—22no—
39C2×#137×C61—21no—
40C2×#135×C61C2212yes—
41C2×#135×C61C2211yescharacteristic
42C22×Q16×C61C2212yes—
43C22×Q16×C61C2211yescharacteristic
44C2×#131×C61C2211yescharacteristic
45C2×#11758222—21no—
46#14519799—22no—
47#14519799—22no—
48C2×Dic244C2311yescharacteristic
49C2×#11758190—21no—
50C2×D488C2311yescharacteristic
51#14519756—22no—
52#14519753—41no—
53#14519751—22no—
54#14519749—42no—
55#14519748—41no—
56C2×C4⋊Dic61—21no—
57#14519723—22no—
58#14519723—24no—
59#14519721—22no—
60#14519720—22no—
61#14519720—24no—
62#14519718—22no—
63#14519717—22no—
64C2×#11758185—22no—
65C2×#11758185C2312yes—
66C2×#11758185C2311yescharacteristic
67C2×#11758185C2311yescharacteristic
68#14519711—21no—
69#14519711C2312yes—
70#14519702—22no—
71#14519699—22no—
722−1+4×C61—28no—
732−1+4×C61—44no—
74C2×#41×C61—22no—
75C2×#41×C61—21no—
76C2×#41×C61—41no—
77C2×#41×C61—42no—
78C2×#41×C61D411yescharacteristic
79C22×Q8×C61—22no—
80C22×Q8×C61—21no—
81C22×Q8×C61—41no—
82#137×C61—28no—
83#137×C61—216no—
84#135×C61—24no—
85#135×C61—24no—
86#135×C61—22no—
87#135×C61D414yescomplemented
88C2×Q16×C61—24no—
89C2×Q16×C61—24no—
90C2×Q16×C61—22no—
91C2×Q16×C61—21no—
92C2×Q16×C61—22no—
93C2×Q16×C61D414yescomplemented
94C2×Q16×C61C2311yescharacteristic
95C2×Q16×C61C2312yes—
96C2×SD16×C61—22no—
97C2×SD16×C61—21no—
98C2×D8×C61C2311yescharacteristic
99#131×C61—22no—
100#131×C61—22no—
101C2×M16×C61D412yes—
102C2×M16×C61D411yescharacteristic
103C22×C8×C61C2312yes—
104C22×C8×C61C2311yescharacteristic
105#41×C61—22no—
106#41×C61—22no—
107#41×C61—28no—
108#41×C61—24no—
109#41×C61—44no—
110#41×C61—416no—
111#41×C61—48no—
112#41×C61—42no—
113C2×Q8×C61—28no—
114C2×Q8×C61—24no—
115C2×Q8×C61—22no—
116C2×Q8×C61—21no—
117C2×Q8×C61—44no—
118C2×Q8×C61—48no—
119C2×Q8×C61—42no—
120C2×Q8×C61—41no—
121C2×Q8×C61—42no—
122C2×Q8×C61D811yescharacteristic
123C2×D4×C61—21no—
124C2×D4×C61—42no—
125C2×D4×C61—41no—
126C2×D4×C61D811yescharacteristic
127C2×D4×C61D812yes—
128C22×C4×C61—42no—
129C22×C4×C61—41no—
130C22×C4×C61—42no—
131C22×C4×C61—41no—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C2×C22⋊D61—41no—
135C2×C22⋊D61—42no—
136C22×Dic61—41no—
137#11758222—42no—
138#11758222—42no—
139C2×D244—21no—
140C2×C4×D61—41no—
141C2×Dic122—21no—
142Q16×C61—28no—
143Q16×C61—24no—
144Q16×C61—21no—
145Q16×C61—22no—
146Q16×C61C2×D412yes—
147Q16×C61C2×D414yes—
148SD16×C61—28no—
149SD16×C61—24no—
150D8×C61—21no—
151D8×C61C2×D412yes—
152M16×C61—22no—
153M16×C61—24no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61C2×D412yes—
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D412yes—
159C2×C8×C61C2×D412yes—
160C2×C8×C61C2411yescharacteristic
161C22⋊Dic61—42no—
162#11758198—42no—
163C4⋊Dic61—22no—
164C4⋊Dic61—41no—
165#11758196—42no—
166Dic244—21no—
167Dic244—21no—
168D488—21no—
169D488—21no—
170#11758190—42no—
171#11758185—21no—
172#11758185—21no—
173#11758185—22no—
174C23×C61—41no—
175C23×C61C2×D812yes—
176C23×C61C2×D811yescharacteristic
177C22×D61—41no—
178Q8×C61—22no—
179Q8×C61—24no—
180Q8×C61—22no—
181Q8×C61—48no—
182Q8×C61—41no—
183Q8×C61—42no—
184Q8×C61—44no—
185D4×C61—24no—
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—48no—
189D4×C61—44no—
190D4×C61—41no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—48no—
194C2×C4×C61—48no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—44no—
198C2×C4×C61—44no—
199C2×C4×C61#12012yes—
200C2×C4×C61C2×D812yes—
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—82no—
205C22⋊D61—162no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—161no—
209D244—41no—
210D244—41no—
211C4×D61—82no—
212Dic122—41no—
213Dic122—41no—
214C8×C61—22no—
215C8×C61#12012yes—
216C8×C61C22×D411yescharacteristic
217C8×C61C22×D411yescharacteristic
218#33527—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—42no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61#44712yes—
228C22×C61C22×D811yescharacteristic
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C4×C61—22no—
233C4×C61—44no—
234C4×C61—42no—
235C4×C61#44712yes—
236C4×C61C2×#12011yescharacteristic
237C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
238Dic61—82no—
239Dic61—81no—
240C2×#578D6111yescharacteristic, complemented
241C2×#510—611no—
242C2×#508—611no—
243#3061—611no—
244#2189—612no—
245#2188—612no—
246#2186—612no—
247#2185—612no—
248#2139—611no—
249C2×#358—611no—
250#2113—611no—
251C2×C61—22no—
252C2×C61—42no—
253C2×C61—82no—
254C2×C61C2×#44711yescharacteristic
255C2×C61#341611yescharacteristic
256C2×C61#341711yescharacteristic
257D61—161no—
258C2×2−1+4—21no—
259#578—24no—
260#578—24no—
261C2×#137—21no—
262C2×#137—22no—
263C2×#135—611no—
264C2×#135C2×D6112yes—
265C2×#135C2×D6111yescharacteristic
266C22×Q16C2×D6112yes—
267C22×Q16C2×D6111yescharacteristic
268C2×#131C2×D6111yescharacteristic
269#537—1221no—
270#510—1224no—
271#508—1224no—
272C2×Q32—611no—
273C2×Q32—612no—
274C2×SD32—611no—
275C2×SD32—612no—
276C2×SD32—611no—
277C2×D16—611no—
278C2×M32—611no—
279C22×C16—611no—
280#468—1222no—
281#467—612no—
282#466—612no—
283C2×#107—611no—
284#417—1221no—
285#358—612no—
286#358—612no—
287#358—612no—
288#357—612no—
289C22⋊C16—612no—
290C22⋊C16—612no—
291C61#3352711yescharacteristic, complemented, Sylow
2922−1+4—28no—
2932−1+4—44no—
294C2×#41—22no—
295C2×#41—21no—
296C2×#41—41no—
297C2×#41—42no—
298C2×#41—1221no—
299C2×#41C22⋊D6111yescharacteristic
300C22×Q8—22no—
301C22×Q8—21no—
302C22×Q8—41no—
303#137—28no—
304#137—216no—
305#135—24no—
306#135—24no—
307#135—22no—
308#135—1222no—
309#135—1222no—
310#135C22⋊D6114yescomplemented
311C2×Q16—24no—
312C2×Q16—24no—
313C2×Q16—22no—
314C2×Q16—21no—
315C2×Q16—22no—
316C2×Q16—611no—
317C2×Q16C22⋊D6114yescomplemented
318C2×Q16C22×D6111yescharacteristic
319C2×Q16C22×D6112yes—
320C2×SD16—22no—
321C2×SD16—21no—
322C2×SD16—1221no—
323C2×D8—611no—
324C2×D8C22×D6111yescharacteristic
325#131—22no—
326#131—22no—
327C2×M16C22⋊D6111yescharacteristic
328C2×M16C22⋊D6112yes—
329C22×C8C22×D6112yes—
330C22×C8C22×D6111yescharacteristic
331C22⋊Q8—1222no—
332C22⋊Q8—2441no—
333#121—1222no—
334#121—2441no—
335#121—2442no—
336C2×C4⋊C4—1221no—
337Q32—1222no—
338Q32—1224no—
339SD32—1222no—
340SD32—1224no—
341SD32—1222no—
342D16—1222no—
343M32—1222no—
344C2×C16—612no—
345C2×C16—611no—
346C2×C16—611no—
347C2×C16—1222no—
348#107—612no—
349#107—1221no—
350#103—1222no—
351#102—1222no—
352#41—22no—
353#41—22no—
354#41—28no—
355#41—24no—
356#41—44no—
357#41—416no—
358#41—48no—
359#41—42no—
360#41—2442no—
361#41—2442no—
362C2×Q8—28no—
363C2×Q8—22no—
364C2×Q8—21no—
365C2×Q8—24no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—42no—
369C2×Q8—41no—
370C2×Q8—42no—
371C2×Q8—1221no—
372C2×Q8#1175819911yescharacteristic
373C2×D4—21no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—1221no—
377C2×D4—2442no—
378C2×D4—2441no—
379C2×D4#1175819912yes—
380C2×D4#1175819911yescharacteristic
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—2441no—
386C22×C4—2441no—
387C22×C4C2×C22⋊D6112yes—
388C22×C4C2×C22⋊D6111yescharacteristic
389Q16—28no—
390Q16—21no—
391Q16—22no—
392Q16—24no—
393Q16—1221no—
394Q16—1221no—
395Q16C2×C22⋊D6112yes—
396Q16C2×C22⋊D6114yes—
397SD16—28no—
398SD16—24no—
399SD16—2442no—
400D8—21no—
401D8—1221no—
402D8—1221no—
403D8C2×C22⋊D6112yes—
404M16—22no—
405M16—24no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8C2×C22⋊D6112yes—
409C2×C8C2×C22⋊D6112yes—
410C2×C8C2×C22⋊D6112yes—
411C2×C8C2×C22⋊D6111yescharacteristic
412C2×C8C23×D6111yescharacteristic
413C4⋊C4—1222no—
414C4⋊C4—2441no—
415C4⋊C4—2442no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C16—1221no—
419C16—1221no—
420C16—1222no—
421C23—41no—
422C23—2441no—
423C23C2×#1175819912yes—
424C23C2×#1175819911yescharacteristic
425Q8—22no—
426Q8—24no—
427Q8—22no—
428Q8—48no—
429Q8—41no—
430Q8—42no—
431Q8—44no—
432Q8—2441no—
433Q8—2441no—
434D4—24no—
435D4—22no—
436D4—22no—
437D4—48no—
438D4—44no—
439D4—41no—
440D4—9762no—
441D4—4882no—
442D4—2441no—
443D4—2441no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—9761no—
453C2×C4—4882no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4#1451974612yes—
457C2×C4C2×#1175819912yes—
458C2×C4C2×#1175819911yescharacteristic
459C2×C4#1451980312yes—
460C2×C4C22×C22⋊D6111yescharacteristic
461C8—22no—
462C8#1451974612yes—
463C8C22×C22⋊D6111yescharacteristic
464C8C22×C22⋊D6111yescharacteristic
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—4881no—
474C22—4881no—
475C22—4881no—
476C22#1683989612yes—
477C22C22×#1175819911yescharacteristic
478C4—22no—
479C4—44no—
480C4—42no—
481C4—4882no—
482C4—4881no—
483C4#1683989612yes—
484C4C2×#1451974611yescharacteristic
485C4C2×#1451980311yescharacteristic
486C2—22no—
487C2—42no—
488C2—82no—
489C2—9761no—
490C2C2×#1683989611yescharacteristic
491C2#2145896311yescharacteristic
492C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#578C2
socleC22×C61C22×D8
hypercenterC2×#578D61
hypocenterC61#33527
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C2×#16840192C2semidirect
3. #21455489C2semidirect
4. #21447759C2semidirect
5. #21447753C2semidirect
6. #21447742C2semidirect
7. #21447736C2semidirect
8. #21447531C2semidirect
9. C2×#16839566C2semidirect
10. #21447433C2semidirect
11. C2×C61×#578C2semidirect
87. #135×C61D4semidirect
93. C2×Q16×C61D4semidirect
240. C2×#578D61semidirect
291. C61#33527semidirect
310. #135C22⋊D61semidirect
317. C2×Q16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C2×#168401922
#214554892
#214477592
#214477532
#214477422
#214477362
#214475312
C2×#168395662
#214474332
C2×C61×#5782
#3352761