Finite Group Databasebeta

← #29833884

Subgroups — #29833884

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×#16840576C211yesmaximal, characteristic, complemented
2C2×#16840202C211yesmaximal, characteristic, complemented
3#21455485C211yesmaximal, characteristic, complemented
4#21447756C212yesmaximal, complemented
5#21447751C212yesmaximal, complemented
6#21447739C212yesmaximal, complemented
7#21447734C212yesmaximal, complemented
8#21447531C211yesmaximal, characteristic, complemented
9C2×#16839567C211yesmaximal, characteristic, complemented
10#21447432C211yesmaximal, characteristic, complemented
11C2×C61×#508C211yesmaximal, characteristic, complemented
12C2×#14519824—21no—
13#16840576—24no—
14#16840576—24no—
15C2×#14519807—21no—
16C2×#14519806—22no—
17C2×#14519796C2212yes—
18C22×Dic244C2211yescharacteristic
19C2×Q16×D61C2212yes—
20C2×#14519783C2211yescharacteristic
21#16840340—21no—
22#16840202—24no—
23C2×#14519723C2212yes—
24C2×#14519721C2212yes—
25C2×#14519717C2211yescharacteristic
26#16840005—22no—
27#16840000C2212yes—
28#16839995C2212yes—
29C2×#14519711C2211yescharacteristic
30#16839763—21no—
31#16839567C2212yes—
32#16839567C2212yes—
33#16839566C2212yes—
34#16839563C2212yes—
35#16839541C2212yes—
36#16839541C2212yes—
37C2×#135×C61C2211yescharacteristic
38C2×#135×C61C2211yescharacteristic
39C61×#508—24no—
40C2×Q32×C61C2211yescharacteristic
41C2×SD32×C61C2211yescharacteristic
42C2×SD32×C61C2211yescharacteristic
43C2×D16×C61C2211yescharacteristic
44C22×C16×C61C2211yescharacteristic
45#14519824—28no—
46#14519824—44no—
47C2×#11758222—41no—
48C2×#11758222D411yescharacteristic
49C2×#11758224—22no—
50C2×#11758224—42no—
51C22×Dic122—21no—
52C22×Dic122—41no—
53C2×Q8×D61—22no—
54#14519807—28no—
55#14519806—216no—
56#14519796—24no—
57#14519796—24no—
58C2×Dic244—21no—
59C2×Dic244—22no—
60C2×Dic244D414yescomplemented
61C2×Dic244C2312yes—
62C2×#11758202—22no—
63Q16×D61—24no—
64Q16×D61—24no—
65C2×#11758200—22no—
66#14519783—22no—
67#14519783—22no—
68C2×#11758194D411yescharacteristic
69C2×#11758189D412yes—
70C2×C8×D61C2312yes—
71#14519756—41no—
72#14519753—22no—
73#14519751—41no—
74#14519749—42no—
75#14519748—22no—
76C2×C4⋊Dic61—21no—
77#14519723—24no—
78#14519721—24no—
79#14519717—22no—
80C2×#11758185C2312yes—
81#14519711—21no—
82#14519711C2312yes—
83#14519701—22no—
84#14519698—22no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87#135×C61—22no—
88#135×C61—22no—
89#135×C61—22no—
90#135×C61D414yescomplemented
91C2×Q16×C61C2311yescharacteristic
92C2×Q16×C61C2311yescharacteristic
93C2×SD16×C61—21no—
94C2×SD16×C61—21no—
95C2×D8×C61C2311yescharacteristic
96C2×D8×C61C2311yescharacteristic
97C22×C8×C61C2311yescharacteristic
98Q32×C61—22no—
99SD32×C61—22no—
100SD32×C61—22no—
101D16×C61—22no—
102C2×C16×C61—22no—
103C2×C16×C61C2311yescharacteristic
104C2×C16×C61C2311yescharacteristic
105#41×C61—24no—
106#41×C61—42no—
107#41×C61—42no—
108#41×C61—42no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61—41no—
117C22×C4×C61C2×D411yescharacteristic
118C2×C22⋊D61—42no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61D812yes—
121C22×Dic61—41no—
122C22×Dic61—42no—
123C22×Dic61—41no—
124Q8×D61—28no—
125Q8×D61—44no—
126#11758224—28no—
127#11758224—44no—
128#11758224—416no—
129#11758222—22no—
130#11758222—22no—
131#11758222—48no—
132C2×D244D811yescharacteristic
133C2×C4×D61—42no—
134C2×C4×D61C2×D412yes—
135C2×Dic122—24no—
136C2×Dic122—22no—
137C2×Dic122—48no—
138C2×Dic122—41no—
139C2×Dic122—42no—
140C2×Dic122—42no—
141C2×Dic122D811yescharacteristic
142Q16×C61—21no—
143Q16×C61—21no—
144Q16×C61—21no—
145Q16×C61C2×D412yes—
146SD16×C61—24no—
147SD16×C61—42no—
148D8×C61—21no—
149D8×C61—21no—
150D8×C61—21no—
151D8×C61C2×D412yes—
152C2×C8×C61C2×D411yescharacteristic
153C2×C8×C61C2×D412yes—
154C2×C8×C61C2×D412yes—
155C2×C8×C61C2411yescharacteristic
156C22⋊Dic61—42no—
157#11758202—28no—
158#11758200—28no—
159#11758198—42no—
160C4⋊Dic61—22no—
161C4⋊Dic61—41no—
162#11758196—42no—
163#11758194—22no—
164C2×#92418C2×D412yes—
165Dic244—24no—
166Dic244—22no—
167Dic244C2×D414yes—
168#11758189—24no—
169C8×D61—22no—
170C8×D61—22no—
171C16×C61—21no—
172C16×C61—21no—
173#11758185—22no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61C2×D811yescharacteristic
177C22×D61C2×D812yes—
178Q8×C61—22no—
179Q8×C61—41no—
180Q8×C61—41no—
181Q8×C61—41no—
182D4×C61—22no—
183D4×C61—44no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—82no—
188C2×C4×C61—41no—
189C2×C4×C61—44no—
190C2×C4×C61—41no—
191C2×C4×C61—82no—
192C2×C4×C61#12012yes—
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61AGL(1,ℤ/8)12yes—
195C2×C4×C61C22×D411yescharacteristic
196C22⋊D61—24no—
197C22⋊D61—48no—
198C22⋊D61—162no—
199C2×Dic61—48no—
200C2×Dic61—42no—
201C2×Dic61—44no—
202C2×Dic61—42no—
203C2×Dic61—161no—
204C2×Dic61C2×D812yes—
205D244—22no—
206C4×D61—22no—
207C4×D61—22no—
208C4×D61—48no—
209Dic122—22no—
210Dic122—24no—
211Dic122—48no—
212Dic122—42no—
213Dic122—44no—
214C8×C61#12012yes—
215C8×C61C22×D411yescharacteristic
216C8×C61C22×D411yescharacteristic
217#92418—22no—
218#33527—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—42no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61#44712yes—
228C22×C61C22×D811yescharacteristic
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—22no—
232C4×C61—42no—
233C4×C61—81no—
234C4×C61#44712yes—
235C4×C61C2×#12011yescharacteristic
236C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
237Dic61—22no—
238Dic61—44no—
239Dic61—82no—
240C2×#578—611no—
241C2×#510—611no—
242C2×#508D6111yescharacteristic, complemented
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—82no—
253C2×C61—161no—
254C2×C61C2×#44711yescharacteristic
255C2×C61#341611yescharacteristic
256C2×C61#341711yescharacteristic
257D61—42no—
258C2×2−1+4—1221no—
259#578—1224no—
260#578—1224no—
261C2×#137—1222no—
262C2×#137—1221no—
263C2×#135—612no—
264C2×#135C2×D6111yescharacteristic
265C2×#135C2×D6111yescharacteristic
266C22×Q16—612no—
267C22×Q16—611no—
268C2×#131—611no—
269#537—1221no—
270#510—1224no—
271#508—24no—
272C2×Q32—612no—
273C2×Q32C2×D6111yescharacteristic
274C2×SD32—612no—
275C2×SD32C2×D6111yescharacteristic
276C2×SD32C2×D6111yescharacteristic
277C2×D16C2×D6111yescharacteristic
278C2×M32—611no—
279C22×C16C2×D6111yescharacteristic
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—1228no—
2932−1+4—2444no—
294C2×#41—21no—
295C2×#41—21no—
296C2×#41—611no—
297C2×#41—1222no—
298C2×#41—2441no—
299C2×#41—2442no—
300C22×Q8—1222no—
301C22×Q8—1221no—
302C22×Q8—2441no—
303#137—12216no—
304#137—1228no—
305#135—22no—
306#135—22no—
307#135—22no—
308#135—1224no—
309#135—1224no—
310#135C22⋊D6114yescomplemented
311C2×Q16—614no—
312C2×Q16—612no—
313C2×Q16—1222no—
314C2×Q16—1224no—
315C2×Q16—1224no—
316C2×Q16—1221no—
317C2×Q16—1222no—
318C2×Q16C22×D6111yescharacteristic
319C2×Q16C22×D6111yescharacteristic
320C2×SD16—21no—
321C2×SD16—21no—
322C2×SD16—1222no—
323C2×D8C22×D6111yescharacteristic
324C2×D8C22×D6111yescharacteristic
325#131—1222no—
326#131—1222no—
327C2×M16—611no—
328C2×M16—612no—
329C22×C8—612no—
330C22×C8C22×D6111yescharacteristic
331C22⋊Q8—1222no—
332C22⋊Q8—2441no—
333#121—1222no—
334#121—2441no—
335#121—2442no—
336C2×C4⋊C4—1221no—
337Q32—22no—
338Q32—1224no—
339SD32—22no—
340SD32—22no—
341SD32—1224no—
342D16—22no—
343M32—1222no—
344C2×C16—22no—
345C2×C16—612no—
346C2×C16C22×D6111yescharacteristic
347C2×C16C22×D6111yescharacteristic
348#107—612no—
349#107—1221no—
350#103—1222no—
351#102—1222no—
352#41—24no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—1222no—
357#41—1222no—
358#41—1228no—
359#41—2448no—
360#41—2444no—
361#41—24416no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—611no—
365C2×Q8—1228no—
366C2×Q8—1222no—
367C2×Q8—1224no—
368C2×Q8—2442no—
369C2×Q8—2448no—
370C2×Q8—2444no—
371C2×Q8—2441no—
372C2×Q8—2442no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—612no—
378C2×D4—611no—
379C2×D4—2442no—
380C2×D4—2442no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—612no—
384C22×C4—2442no—
385C22×C4—2442no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4D4×D6111yescharacteristic
389Q16—21no—
390Q16—21no—
391Q16—21no—
392Q16—614no—
393Q16—1228no—
394Q16—1222no—
395Q16—1224no—
396Q16C2×C22⋊D6112yes—
397SD16—24no—
398SD16—42no—
399SD16—1228no—
400D8—21no—
401D8—21no—
402D8—21no—
403D8C2×C22⋊D6112yes—
404M16—1222no—
405M16—1224no—
406C2×C8—612no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8D4×D6112yes—
410C2×C8D4×D6111yescharacteristic
411C2×C8C2×C22⋊D6112yes—
412C2×C8C23×D6111yescharacteristic
413C4⋊C4—1222no—
414C4⋊C4—2442no—
415C4⋊C4—2441no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C16—21no—
419C16—21no—
420C16—1222no—
421C23—41no—
422C23—41no—
423C23—612no—
424C23D8×D6111yescharacteristic
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—41no—
429Q8—1222no—
430Q8—1224no—
431Q8—2448no—
432Q8—2442no—
433Q8—2444no—
434D4—22no—
435D4—44no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—82no—
440D4—1224no—
441D4—1222no—
442D4—9762no—
443D4—2448no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—612no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—9761no—
452C2×C4—2448no—
453C2×C4—2448no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2444no—
457C2×C4#1451974512yes—
458C2×C4D8×D6111yescharacteristic
459C2×C4#1451980112yes—
460C2×C4C2×D4×D6111yescharacteristic
461C8—1222no—
462C8#1451974512yes—
463C8C2×D4×D6111yescharacteristic
464C8C22×C22⋊D6111yescharacteristic
465C22—21no—
466C22—21no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—1222no—
474C22—1222no—
475C22—1222no—
476C22#1683989412yes—
477C22C2×D8×D6111yescharacteristic
478C4—42no—
479C4—81no—
480C4—1222no—
481C4—4882no—
482C4—2444no—
483C4#1683989412yes—
484C4C2×#1451974511yescharacteristic
485C4C2×#1451980111yescharacteristic
486C2—22no—
487C2—82no—
488C2—161no—
489C2—2442no—
490C2C2×#1683989411yescharacteristic
491C2#2145896211yescharacteristic
492C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#508C2
socleC22×C61C22×D8
hypercenterC2×#508D61
hypocenterC61#33527
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. C2×#16840576C2semidirect
2. C2×#16840202C2semidirect
3. #21455485C2semidirect
4. #21447756C2semidirect
5. #21447751C2semidirect
6. #21447739C2semidirect
7. #21447734C2semidirect
8. #21447531C2semidirect
9. C2×#16839567C2semidirect
10. #21447432C2semidirect
11. C2×C61×#508C2semidirect
60. C2×Dic244D4semidirect
90. #135×C61D4semidirect
242. C2×#508D61semidirect
291. C61#33527semidirect
310. #135C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405762
C2×#168402022
#214554852
#214477562
#214477512
#214477392
#214477342
#214475312
C2×#168395672
#214474322
C2×C61×#5082
#3352761