Finite Group Databasebeta

← #29686908

Subgroups — #29686908

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840032C211yesmaximal, characteristic
2#21454632C212yesmaximal, complemented
3C2×#16839873C211yesmaximal, characteristic, complemented
4#21454245C211yesmaximal, characteristic, complemented
5#21446727C211yesmaximal, characteristic, complemented
6#21446597C211yesmaximal, characteristic, complemented
7#21446479C212yesmaximal, complemented
8#21446325C212yesmaximal, complemented
9#21445727C211yesmaximal, characteristic, complemented
10#21445610C212yesmaximal, complemented
11C61×#1815C211yesmaximal, characteristic, complemented
12C2×#14519802C2212yes—
13C22×D488C2211yescharacteristic
14C2×#14519784C2211yescharacteristic
15#41×Dic61C2211yescharacteristic
16C2×#14519739C2212yes—
17#16840032C418yescomplemented
18#16839873—21no—
19#16839873—21no—
20#16839873—21no—
21#16839873—21no—
22#16839871—22no—
23#16839865—22no—
24#16839847—22no—
25#16839842—22no—
26C2×#14519713C2211yescharacteristic
27#16839802C2212yes—
28#16839768C2212yes—
29C2×#14519697C2212yes—
30C2×#14519693C2212yes—
31C2×#14519693C2212yes—
32#16839720—21no—
33#16839720C2211yescharacteristic
34#16839717—21no—
35#16839717C2211yescharacteristic
36M16×Dic61—21no—
37M16×Dic61C2212yes—
38M16×Dic61C2211yescharacteristic
39C2×#14519687—21no—
40C2×#14519687C2211yescharacteristic
41#16839531C2212yes—
42#16839526C2212yes—
43C2×#131×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C2×#106×C61C2211yescharacteristic
46C61×#417C2212yes—
47C61×#343C2212yes—
48C22×D244C2312yes—
49C22×C4×D61—22no—
50#14519802—24no—
51#14519802C2×C418yescomplemented
52C2×#11758201—22no—
53C2×D488—21no—
54C2×D488—21no—
55C2×D488—22no—
56C2×D488C2×C414yescomplemented
57C2×#11758199—22no—
58#14519784—22no—
59#14519784—22no—
60C2×#11758194—21no—
61C2×#11758194C2312yes—
62C2×#11758194C2311yescharacteristic
63C22×#92418—21no—
64C22×#92418C2311yescharacteristic
65Q8×Dic61D412yes—
66D4×Dic61—22no—
67D4×Dic61D412yes—
68#14519739—22no—
69#14519739—22no—
70#14519739—22no—
71#14519739—22no—
72#14519736—21no—
73#14519736C2311yescharacteristic
74C2×#11758198—22no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2311yescharacteristic
77#14519713C2×C414yescomplemented
78#14519710—22no—
79#14519708—22no—
80#14519699—22no—
81#14519699—22no—
82#14519697—22no—
83#14519697—22no—
84#14519693—24no—
85#14519693C2×C418yescomplemented
86#14519687—21no—
87#14519687—21no—
88#14519687—22no—
89#14519687—21no—
90#14519687D411yescharacteristic
91#14519687D411yescharacteristic
92#14519687D411yescharacteristic
93#14519687D411yescharacteristic
94C8×Dic61—21no—
95C8×Dic61—21no—
96C8×Dic61—22no—
97C8×Dic61—21no—
98C2×#41×C61C2311yescharacteristic
99#131×C61—22no—
100#131×C61C2×C414yescomplemented
101C2×M16×C61—21no—
102C2×M16×C61C2312yes—
103C2×M16×C61C2311yescharacteristic
104C22×C8×C61—21no—
105C22×C8×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2312yes—
107#108×C61—22no—
108#106×C61—21no—
109#106×C61—21no—
110#103×C61—22no—
111#102×C61—22no—
112C23×D61—22no—
113#41×C61—21no—
114#41×C61—21no—
115#41×C61C22×C412yes—
116#41×C61#4112yes—
117C2×Q8×C61C2×D411yescharacteristic
118C2×D4×C61—21no—
119C2×D4×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—22no—
124C22×Dic61#4112yes—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244C22×C414yes—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—84no—
134M16×C61—21no—
135M16×C61—24no—
136M16×C61—22no—
137M16×C61—22no—
138M16×C61C22×C412yes—
139C2×C8×C61—21no—
140C2×C8×C61—24no—
141C2×C8×C61—22no—
142C2×C8×C61C22×C412yes—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61#4112yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C22⋊Dic61—21no—
149C22⋊Dic61—22no—
150#11758201—28no—
151#11758199—28no—
152#11758198—42no—
153#11758198—42no—
154C4⋊Dic61—22no—
155C4⋊Dic61C2×D412yes—
156C4×Dic61—21no—
157C4×Dic61—22no—
158C4×Dic61C2×D412yes—
159C4×Dic61C2×D411yescharacteristic
160C4×Dic61C2×D411yescharacteristic
161#11758194—24no—
162#11758194—21no—
163#11758194—22no—
164#11758194C22×C414yes—
165#11758194#4112yes—
166C2×#92418—24no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173D488—24no—
174D488—24no—
175C23×C61—21no—
176C23×C61C2×#4111yescharacteristic
177C22×D61—24no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—82no—
181C22×D61—82no—
182Q8×C61—21no—
183Q8×C61C4×D412yes—
184D4×C61—22no—
185D4×C61—22no—
186D4×C61—21no—
187D4×C61C4×D412yes—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—82no—
193C2×C4×C61C23×C411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×Dic61—22no—
200C2×Dic61—42no—
201C2×Dic61—82no—
202C2×Dic61#12112yes—
203C2×Dic61#12112yes—
204D244—24no—
205D244—24no—
206D244—42no—
207D244—42no—
208C4×D61—84no—
209C4×D61—84no—
210C8×C61—22no—
211C8×C61—42no—
212C8×C61C4×D412yes—
213C8×C61C4×D412yes—
214#92418—24no—
215#92418—21no—
216#92418—22no—
217#92418—21no—
218#21555—611nomaximal, Sylow
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—42no—
222C22×C61C4×#4111yescharacteristic
223C22×C61C2×#12111yescharacteristic
224C22×C61#52911yescharacteristic
225C2×D61—44no—
226C2×D61—84no—
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—84no—
230C4×C61—22no—
231C4×C61—82no—
232C4×C61C2×C4×D411yescharacteristic
233C4×C61C2×C4×D411yescharacteristic
234C4×C61#52511yescharacteristic
235C4×C61#53211yescharacteristic
236Dic61—22no—
237Dic61—22no—
238Dic61—42no—
239C2×#472—611no—
240C4×AGL(1,ℤ/8)—612no—
241C2×#442—611no—
242#2875—611no—
243#1978—611no—
244#1951—611no—
245#1931—612no—
246#1907—612no—
247#1815D6111yescharacteristic, complemented
248#1778—611no—
249C4×#100—612no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—42no—
253C2×C61C4×#12111yescharacteristic
254C2×C61#306812yes—
255D61—84no—
256C2×AGL(1,ℤ/8)—612no—
257C22×D8—611no—
258C2×#131—611no—
259C2×#131C2×D6111yescharacteristic
260C4×#41—611no—
261C2×C4×D4—612no—
262#472—618no—
263#442—1221no—
264#442—1221no—
265#442—1221no—
266#442—1221no—
267#442—1222no—
268#440—1222no—
269C4×SD16—1222no—
270C4×D8—1222no—
271C2×#108—611no—
272C2×#108C2×D6111yescharacteristic
273#428—612no—
274C2×#106C2×D6111yescharacteristic
275#418—612no—
276#417C2×D6112yes—
277C2×#102—612no—
278C2×#100—612no—
279C2×#100—612no—
280#405—611no—
281#405—611no—
282#405—1221no—
283#405—1221no—
284C4×M16—611no—
285C4×M16—612no—
286C4×M16—1221no—
287C2×#97—611no—
288C2×#97—1221no—
289#343—612no—
290#343C2×D6112yes—
291#341—612no—
292C61#2155511yescharacteristic, complemented, Sylow
293C2×#41C22×D6111yescharacteristic
294C22×D4—612no—
295C23×C4—1222no—
296AGL(1,ℤ/8)—618no—
297AGL(1,ℤ/8)—1224no—
298C2×SD16—1222no—
299C2×D8—614no—
300C2×D8—1222no—
301C2×D8—1221no—
302C2×D8—1221no—
303C2×D8—1222no—
304#131—22no—
305#131—1222no—
306#131—1222no—
307#131C4×D6114yescomplemented
308C2×M16—21no—
309C2×M16—611no—
310C2×M16—612no—
311C2×M16—1221no—
312C2×M16C22×D6112yes—
313C2×M16C22×D6111yescharacteristic
314C22×C8—21no—
315C22×C8—611no—
316C22×C8—1221no—
317C22×C8C22×D6111yescharacteristic
318C4×Q8—612no—
319C4×D4—612no—
320C4×D4—1222no—
321C4×D4—1222no—
322C4×D4—1222no—
323C4×D4—1222no—
324C4×D4—1222no—
325#117—611no—
326#117—1221no—
327C2×C4⋊C4C22×D6112yes—
328C2×C22⋊C4—1222no—
329C2×C42—611no—
330C2×C42—1221no—
331#108—22no—
332#108—614no—
333#107—1222no—
334#106—21no—
335#106—21no—
336#106—1222no—
337#103—22no—
338#102—22no—
339#102—1222no—
340#102—1222no—
341#102—1222no—
342#102—1222no—
343#100—618no—
344#100—1224no—
345#97—611no—
346#97—611no—
347#97—611no—
348#97—611no—
349#97—1221no—
350#97—1222no—
351#97—1221no—
352#97—1221no—
353C4×C8—1221no—
354C4×C8—1222no—
355C4×C8—1221no—
356C4×C8—1221no—
357C24—1222no—
358#41—21no—
359#41—21no—
360#41C2×C4×D6112yes—
361#41#1175822612yes—
362C2×Q8D4×D6111yescharacteristic
363C2×D4—21no—
364C2×D4—614no—
365C2×D4—1222no—
366C2×D4—1222no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4D4×D6111yescharacteristic
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—612no—
374C22×C4—1222no—
375C22×C4—4884no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4C23×D6111yescharacteristic
379SD16—1228no—
380D8—1228no—
381D8—1224no—
382D8—1224no—
383M16—21no—
384M16—24no—
385M16—22no—
386M16—22no—
387M16—614no—
388M16—612no—
389M16—1224no—
390M16—1221no—
391M16—1222no—
392M16C2×C4×D6112yes—
393C2×C8—21no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—611no—
397C2×C8—611no—
398C2×C8—1224no—
399C2×C8—1221no—
400C2×C8—1222no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8C2×C4×D6112yes—
404C2×C8D4×D6111yescharacteristic
405C2×C8D4×D6111yescharacteristic
406C2×C8#1175822412yes—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—612no—
410C4⋊C4—1222no—
411C22⋊C4—1222no—
412C22⋊C4—1221no—
413C22⋊C4—2442no—
414C22⋊C4—2442no—
415C42—611no—
416C42—612no—
417C42—611no—
418C42—1222no—
419C42—1221no—
420C23—21no—
421C23—1224no—
422C23—4882no—
423C23—4882no—
424C23—2442no—
425C23—2442no—
426C23#41×D6111yescharacteristic
427Q8—21no—
428Q8#1451973912yes—
429D4—22no—
430D4—22no—
431D4—21no—
432D4—1224no—
433D4—1224no—
434D4—2442no—
435D4—2442no—
436D4#1451973912yes—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—1222no—
445C2×C4—4884no—
446C2×C4—4884no—
447C2×C4—4882no—
448C2×C4—2442no—
449C2×C4C22×C4×D6111yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C2×D4×D6111yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4C2×#1175822411yescharacteristic
454C2×C4C2×#1175822611yescharacteristic
455C8—22no—
456C8—42no—
457C8—1224no—
458C8—1221no—
459C8—1221no—
460C8—1222no—
461C8#1451973712yes—
462C8#1451973712yes—
463C22—22no—
464C22—22no—
465C22—42no—
466C22—4884no—
467C22—4884no—
468C22—4882no—
469C22—4882no—
470C22—2444no—
471C22#1684023011yescharacteristic
472C22#121×D6111yescharacteristic
473C22#1684029811yescharacteristic
474C4—22no—
475C4—82no—
476C4—1222no—
477C4—1222no—
478C4—2442no—
479C4C2×#1451973711yescharacteristic
480C4C2×#1451973911yescharacteristic
481C4#1684027511yescharacteristic
482C4#1684030711yescharacteristic
483C2—21no—
484C2—21no—
485C2—42no—
486C2—4884no—
487C2#2144832511yescharacteristic
488C2#2145554212yes—
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1815C2
socleC22×C61C2×#121
prime radicalC2×#16840032C2
cosocleC22×C4×C61C24
hypercenter#1815D61
hypocenterC61#21555
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1815

Complements

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

NH ≅ G/Ntype
2. #21454632C2semidirect
3. C2×#16839873C2semidirect
4. #21454245C2semidirect
5. #21446727C2semidirect
6. #21446597C2semidirect
7. #21446479C2semidirect
8. #21446325C2semidirect
9. #21445727C2semidirect
10. #21445610C2semidirect
11. C61×#1815C2semidirect
17. #16840032C4semidirect
51. #14519802C2×C4semidirect
56. C2×D488C2×C4semidirect
77. #14519713C2×C4semidirect
85. #14519693C2×C4semidirect
100. #131×C61C2×C4semidirect
247. #1815D61semidirect
292. C61#21555semidirect
307. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400322
#214546322
C2×#168398732
#214542452
#214467272
#214465972
#214464792
#214463252
#214457272
#214456102
C61×#18152
#2155561