Finite Group Databasebeta

← #29722589

Subgroups — #29722589

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840037C211yesmaximal, characteristic, complemented
2#21455452C212yesmaximal
3C2×#16840021C211yesmaximal, characteristic, complemented
4#21454520C211yesmaximal, characteristic
5C2×#16839809C211yesmaximal, characteristic, complemented
6#21447187C212yesmaximal, complemented
7#21447116C212yesmaximal, complemented
8#21446732C211yesmaximal, characteristic, complemented
9#21446597C211yesmaximal, characteristic, complemented
10#21446286C212yesmaximal, complemented
11C61×#1815C211yesmaximal, characteristic, complemented
12C2×#14519807C2212yes—
13C22×Dic244C2211yescharacteristic
14C22×#11758189C2211yescharacteristic
15C2×M16×D61C2212yes—
16#16840269C2211yescharacteristic
17C2×#14519757C2212yes—
18#16840037—24no—
19#16840021—22no—
20#16840021—22no—
21#16840019—22no—
22#16840007—22no—
23#16840002—22no—
24#16839990—22no—
25#16839838—22no—
26#16839833—22no—
27#16839809—24no—
28C2×#14519713C2211yescharacteristic
29C2×#14519712C2212yes—
30#16839802C2212yes—
31#16839790C2211yescharacteristic
32#16839772C2212yes—
33C2×#14519700C2212yes—
34C2×#14519695C2212yes—
35C2×#14519695C2212yes—
36#16839729C2211yescharacteristic
37#16839725C2211yescharacteristic
38#16839717C2211yescharacteristic
39#16839531C2212yes—
40C2×#131×C61C2211yescharacteristic
41C2×#108×C61C2211yescharacteristic
42C2×#106×C61C2211yescharacteristic
43C61×#417C2212yes—
44C61×#343C2212yes—
45C22×Dic122C2312yes—
46C22×C4×D61C2311yescharacteristic
47#14519807—24no—
48#14519807—42no—
49C2×Dic244—21no—
50C2×Dic244—21no—
51C2×Dic244—22no—
52C2×Dic244—22no—
53C2×#11758202—22no—
54C2×#11758200—22no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2312yes—
57C2×#11758189—22no—
58C2×#11758189—22no—
59C2×#11758189—22no—
60C2×#11758189D412yes—
61C2×#11758189D412yes—
62M16×D61—24no—
63M16×D61—42no—
64C22×#92418C2311yescharacteristic
65C2×C8×D61—22no—
66#14519758—21no—
67#14519757—22no—
68#14519757—22no—
69#14519757—22no—
70#14519757—22no—
71#14519753D412yes—
72#14519748D412yes—
73#14519745—21no—
74C4×C22⋊D61—22no—
75#14519736C2311yescharacteristic
76C2×C4⋊Dic61—22no—
77C2×#11758198—22no—
78#14519713D414yescomplemented
79#14519712—24no—
80#14519710—22no—
81#14519708—22no—
82#14519706—22no—
83#14519702—22no—
84#14519702—22no—
85#14519700—24no—
86#14519695—24no—
87#14519695—42no—
88C22⋊#92418—22no—
89#14519689—22no—
90#14519687—21no—
91C8×Dic61—21no—
92C2×#41×C61C2311yescharacteristic
93#131×C61—22no—
94#131×C61—22no—
95C2×M16×C61—21no—
96C2×M16×C61C2311yescharacteristic
97C2×M16×C61C2312yes—
98C22×C8×C61—21no—
99C22×C8×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101#108×C61—22no—
102#106×C61—21no—
103#106×C61—21no—
104#103×C61—22no—
105#102×C61—22no—
106C23×D61#4111yescharacteristic
107#41×C61—21no—
108#41×C61—21no—
109#41×C61—21no—
110#41×C61—21no—
111C2×Q8×C61C2×D411yescharacteristic
112C2×D4×C61—21no—
113C2×D4×C61C2×D411yescharacteristic
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—22no—
118C22×Dic61—22no—
119C22×Dic61#4111yescharacteristic
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61C2×D412yes—
125C2×C4×D61C2×D412yes—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132M16×C61—22no—
133M16×C61—21no—
134M16×C61—21no—
135M16×C61—22no—
136M16×C61—42no—
137C2×C8×C61—22no—
138C2×C8×C61—21no—
139C2×C8×C61—42no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D412yes—
143C2×C8×C61#4112yes—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C22⋊Dic61—21no—
147C22⋊Dic61—22no—
148#11758202—44no—
149#11758200—44no—
150#11758198—22no—
151#11758198—42no—
152#11758198—42no—
153C4⋊Dic61—42no—
154C4⋊Dic61—42no—
155C4⋊Dic61C2×D412yes—
156#11758196—22no—
157C4×Dic61—21no—
158#11758194—22no—
159#11758194—21no—
160#11758194—21no—
161#11758194C2×D414yes—
162C2×#92418—22no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418C2×D412yes—
167Dic244—42no—
168Dic244—42no—
169#11758189—24no—
170#11758189—24no—
171#11758189—44no—
172C8×D61—44no—
173C23×C61—21no—
174C23×C61C2×#4111yescharacteristic
175C22×D61—21no—
176C22×D61—21no—
177C22×D61—42no—
178C22×D61#12112yes—
179Q8×C61—21no—
180Q8×C61—21no—
181D4×C61—21no—
182D4×C61—21no—
183D4×C61—41no—
184D4×C61—41no—
185C2×C4×C61—21no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—82no—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61H(5,𝔽2)11yescharacteristic
195C2×C4×C612−1+411yescharacteristic
196C22⋊D61—82no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—22no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61#12112yes—
204C4×D61—22no—
205C4×D61—22no—
206C4×D61—22no—
207C4×D61—22no—
208Dic122—42no—
209Dic122—42no—
210Dic122—42no—
211Dic122—42no—
212C8×C61—41no—
213C8×C61—42no—
214C8×C61#12112yes—
215C8×C61#12112yes—
216#92418—22no—
217#92418—24no—
218#92418—41no—
219#24807—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—41no—
222C22×C61—42no—
223C22×C61C2×#12111yescharacteristic
224C22×C61#54811yescharacteristic
225C22×C61#55311yescharacteristic
226C2×D61—22no—
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—41no—
232C4×C61—82no—
233C4×C61C2×#12111yescharacteristic
234C4×C61#53711yescharacteristic
235C4×C61D4211yescharacteristic
236C4×C61#54711yescharacteristic
237Dic61—22no—
238Dic61—42no—
239Dic61—82no—
240C2×#473—611no—
241#3056—612no—
242C2×#470—611no—
243#2930—611no—
244C2×#430—611no—
245#2064—612no—
246#2052—612no—
247#1979—611no—
248#1951—611no—
249#1901—612no—
250#1815D6111yescharacteristic, complemented
251C2×C61—21no—
252C2×C61—21no—
253C2×C61—81no—
254C2×C61#241611yescharacteristic
255C2×C61#306912yes—
256D61—42no—
257C2×#137—612no—
258C22×Q16—611no—
259C2×#131C2×D6111yescharacteristic
260C22×M16—611no—
261C22×M16—612no—
262#525—611no—
263C2×C22⋊Q8—612no—
264#473—1224no—
265#470—1222no—
266#470—1222no—
267#470—1222no—
268#468—1222no—
269#467—1222no—
270#465—1222no—
271#436—1222no—
272#435—1222no—
273#430—1224no—
274C2×#108—611no—
275C2×#108—612no—
276C2×#108C2×D6111yescharacteristic
277#428—612no—
278C2×#106C2×D6111yescharacteristic
279#424—611no—
280#419—612no—
281#417C2×D6112yes—
282C2×#103—612no—
283C2×#101—612no—
284C2×#101—612no—
285#408—611no—
286#407—611no—
287#405—611no—
288#343—612no—
289#343C2×D6112yes—
290C61#2480711yescharacteristic, complemented, Sylow
291C2×#41C22×D6111yescharacteristic
292C22×Q8—612no—
293C23×C4—611no—
294#137—1224no—
295#137—2442no—
296C2×Q16—1222no—
297C2×Q16—1222no—
298C2×Q16—1221no—
299C2×Q16—1221no—
300C2×Q16—1222no—
301C2×SD16—1222no—
302#131—22no—
303#131—22no—
304C2×M16—21no—
305C2×M16—612no—
306C2×M16—612no—
307C2×M16—612no—
308C2×M16—611no—
309C2×M16—1222no—
310C2×M16—1222no—
311C2×M16—1222no—
312C2×M16—1224no—
313C2×M16—2442no—
314C2×M16C22×D6112yes—
315C2×M16C22×D6111yescharacteristic
316C22×C8—21no—
317C22×C8—611no—
318C22×C8—1222no—
319C22×C8C22×D6111yescharacteristic
320#123—1221no—
321C22⋊Q8—612no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324C22⋊Q8—1222no—
325C22⋊Q8—1222no—
326#121—612no—
327#120—1221no—
328C4×D4—1222no—
329#117—611no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4C22×D6112yes—
332C2×C22⋊C4—1222no—
333#108—22no—
334#108—614no—
335#108—1224no—
336#107—1222no—
337#106—21no—
338#106—21no—
339#106—1222no—
340C4⋊C8—1222no—
341#103—22no—
342#103—1224no—
343#103—1222no—
344#103—1222no—
345#102—22no—
346#101—1224no—
347#101—2442no—
348C22⋊C8—1222no—
349C22⋊C8—1222no—
350#97—1221no—
351C4×C8—1221no—
352C24—611no—
353#41—21no—
354#41—21no—
355#41—21no—
356#41—21no—
357C2×Q8—1222no—
358C2×Q8—1222no—
359C2×Q8—1222no—
360C2×Q8—1222no—
361C2×Q8—1222no—
362C2×Q8—1222no—
363C2×Q8D4×D6111yescharacteristic
364C2×D4—21no—
365C2×D4—1222no—
366C2×D4D4×D6111yescharacteristic
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—611no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—1222no—
377C22×C4C23×D6111yescharacteristic
378Q16—2444no—
379Q16—2442no—
380Q16—2442no—
381SD16—2444no—
382M16—21no—
383M16—22no—
384M16—21no—
385M16—22no—
386M16—42no—
387M16—614no—
388M16—1222no—
389M16—1224no—
390M16—1224no—
391M16—1221no—
392M16—1221no—
393M16—2444no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—42no—
397C2×C8—612no—
398C2×C8—1222no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—2444no—
403C2×C8#1175822212yes—
404C2×C8D4×D6111yescharacteristic
405C2×C8D4×D6111yescharacteristic
406C2×C8D4×D6112yes—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—612no—
410C4⋊C4—1222no—
411C4⋊C4—2442no—
412C4⋊C4—2442no—
413C22⋊C4—1222no—
414C22⋊C4—1222no—
415C22⋊C4—1221no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C42—1221no—
419C23—21no—
420C23—612no—
421C23—1221no—
422C23—1221no—
423C23—2442no—
424C23#41×D6111yescharacteristic
425Q8—21no—
426Q8—21no—
427Q8—2442no—
428Q8—2442no—
429Q8—2442no—
430Q8—2442no—
431D4—21no—
432D4—21no—
433D4—41no—
434D4—41no—
435D4—4882no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—612no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4C2×D4×D6111yescharacteristic
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C2×D4×D6111yescharacteristic
455C2×C4C2×#1175822211yescharacteristic
456C2×C4#1451982311yescharacteristic
457C2×C4#1451982511yescharacteristic
458C8—41no—
459C8—42no—
460C8—1222no—
461C8—1224no—
462C8—2441no—
463C8#1451975012yes—
464C8#1451975012yes—
465C22—22no—
466C22—41no—
467C22—42no—
468C22—1222no—
469C22—1222no—
470C22—1222no—
471C22—4881no—
472C22—4881no—
473C22#121×D6111yescharacteristic
474C22#1684042211yescharacteristic
475C22#1684045611yescharacteristic
476C4—41no—
477C4—82no—
478C4—1222no—
479C4—4882no—
480C4—2442no—
481C4C2×#1451975011yescharacteristic
482C4#1684034211yescharacteristic
483C4#1684039611yescharacteristic
484C4#1684041311yescharacteristic
485C2—21no—
486C2—21no—
487C2—81no—
488C2—2442no—
489C2#2144926111yescharacteristic
490C2#2145555112yes—
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1815C2
socleC22×C61C2×#121
prime radical#16840269C22
hypercenter#1815D61
hypocenterC61#24807
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1815

Complements

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

NH ≅ G/Ntype
1. C2×#16840037C2semidirect
3. C2×#16840021C2semidirect
5. C2×#16839809C2semidirect
6. #21447187C2semidirect
7. #21447116C2semidirect
8. #21446732C2semidirect
9. #21446597C2semidirect
10. #21446286C2semidirect
11. C61×#1815C2semidirect
78. #14519713D4semidirect
250. #1815D61semidirect
290. C61#24807semidirect

Maximal subgroups

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

subgroupindex
C2×#168400372
#214554522
C2×#168400212
#214545202
C2×#168398092
#214471872
#214471162
#214467322
#214465972
#214462862
C61×#18152
#2480761