Finite Group Databasebeta

← #29698490

Subgroups — #29698490

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839877C211yesmaximal, characteristic
2#21454690C212yesmaximal
3#21454414C211yesmaximal, characteristic, complemented
4#21454410C211yesmaximal, characteristic, complemented
5#21446603C212yesmaximal, complemented
6#21446055C212yesmaximal, complemented
7#21446050C212yesmaximal, complemented
8#21445980C211yesmaximal, characteristic, complemented
9#21445875C211yesmaximal, characteristic, complemented
10#21445701C211yesmaximal, characteristic, complemented
11C61×#1815C211yesmaximal, characteristic, complemented
12C2×#14519797C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840245C2212yes—
15#16840243C2212yes—
16#16839877C418yescomplemented
17#16839865—22no—
18#16839859—22no—
19#16839854—22no—
20#16839842—22no—
21C2×#14519712C2211yescharacteristic
22#16839804—22no—
23#16839804C2212yes—
24#16839802C2212yes—
25#16839801—22no—
26#16839800—21no—
27#16839800C2211yescharacteristic
28#16839797—21no—
29C2×#14519710C2212yes—
30C2×#14519709—21no—
31C2×#14519709C2211yescharacteristic
32#106×D61—21no—
33C2×#14519704C2211yescharacteristic
34C2×#14519703C2211yescharacteristic
35#16839764C2212yes—
36#16839720C2212yes—
37C2×C8×Dic61C2211yescharacteristic
38#16839627C2212yes—
39#16839530C2212yes—
40#16839523C2212yes—
41C2×#131×C61C2211yescharacteristic
42C2×#108×C61C2211yescharacteristic
43C2×#106×C61C2211yescharacteristic
44C61×#417C2212yes—
45C61×#343C2212yes—
46C2×#11758222C2311yescharacteristic
47#14519797C2×C414yes—
48#14519797Q814yes—
49C2×#11758202D411yescharacteristic
50C2×#11758200D411yescharacteristic
51C2×#11758201D411yescharacteristic
52C2×#11758199D411yescharacteristic
53#14519783—22no—
54#14519783C2×C414yescomplemented
55C2×#11758194C2311yescharacteristic
56C2×#11758189—21no—
57C22×#92418C2311yescharacteristic
58C2×C8×D61—21no—
59Q8×Dic61—22no—
60#14519742—22no—
61D4×Dic61—22no—
62D4×Dic61—42no—
63#14519739—22no—
64C4×C22⋊D61—42no—
65#14519736—22no—
66#14519736C2312yes—
67#14519735—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61C2312yes—
70C4⋊C4×D61—22no—
71C2×C4×Dic61C2311yescharacteristic
72#14519712C2×C414yescomplemented
73#14519711—22no—
74#14519711D412yes—
75#14519710—22no—
76#14519710—22no—
77#14519710—22no—
78#14519709—21no—
79#14519709—21no—
80#14519709—22no—
81#14519709D411yescharacteristic
82#14519709D411yescharacteristic
83#14519708—22no—
84#14519704C2×C414yescomplemented
85#14519703C2×C414yescomplemented
86#14519702—22no—
87#14519699—22no—
88#14519687—22no—
89C8×Dic61—22no—
90C8×Dic61C2×C414yescomplemented
91#14519684—42no—
92C2×#41×C61C2311yescharacteristic
93#131×C61—22no—
94#131×C61C2×C414yescomplemented
95C2×M16×C61—21no—
96C2×M16×C61C2312yes—
97C2×M16×C61C2311yescharacteristic
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—
106#41×C61—21no—
107#41×C61—21no—
108#41×C61C22×C412yes—
109#41×C61C2×Q812yes—
110C2×Q8×C61C2×D411yescharacteristic
111C2×D4×C61—21no—
112C2×D4×C61C2×D411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—21no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—42no—
120#11758222—21no—
121#11758222—21no—
122#11758222C22×C412yes—
123#11758222C2×Q812yes—
124C2×D244C2×D411yescharacteristic
125C2×C4×D61—21no—
126C2×C4×D61—42no—
127C2×Dic122C2×D411yescharacteristic
128M16×C61—21no—
129M16×C61—24no—
130M16×C61—22no—
131M16×C61—22no—
132M16×C61C22×C412yes—
133C2×C8×C61—21no—
134C2×C8×C61—24no—
135C2×C8×C61—22no—
136C2×C8×C61C22×C412yes—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61#4112yes—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C22⋊Dic61—22no—
143C22⋊Dic61—42no—
144#11758202C4⋊C414yes—
145#11758201C4⋊C414yes—
146#11758200C4⋊C414yes—
147#11758199C4⋊C414yes—
148#11758198—42no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61C2×D412yes—
153C4⋊Dic61C2×D412yes—
154#11758196—42no—
155C4×Dic61—22no—
156C4×Dic61—21no—
157C4×Dic61—42no—
158C4×Dic61—42no—
159C4×Dic61C22×C412yes—
160#11758194—21no—
161#11758194C22×C412yes—
162C2×#92418—21no—
163C2×#92418C22×C412yes—
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×Q812yes—
167#11758189—24no—
168C8×D61—24no—
169C23×C61—21no—
170C23×C61H(5,𝔽2)11yescharacteristic
171C22×D61—21no—
172Q8×C61—21no—
173Q8×C61C2×C4⋊C412yes—
174D4×C61—22no—
175D4×C61—22no—
176D4×C61—21no—
177D4×C61C2×C4⋊C412yes—
178C2×C4×C61—21no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—42no—
182C2×C4×C61—82no—
183C2×C4×C61C23×C411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×Q811yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C612−1+411yescharacteristic
189C22⋊D61—22no—
190C22⋊D61—22no—
191C2×Dic61—21no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—82no—
198C2×Dic61—162no—
199D244—21no—
200D244C2×C4⋊C412yes—
201C4×D61—22no—
202C4×D61—22no—
203C4×D61—162no—
204Dic122—21no—
205Dic122C2×C4⋊C412yes—
206C8×C61—22no—
207C8×C61—42no—
208C8×C61C4×D412yes—
209C8×C61C4×D412yes—
210#92418—22no—
211#92418C2×C4⋊C412yes—
212#92418C2×C4⋊C412yes—
213#22690—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—42no—
217C22×C61#52011yescharacteristic
218C22×C61D4211yescharacteristic
219C22×C61#55411yescharacteristic
220C2×D61—22no—
221C2×D61—42no—
222C4×C61—22no—
223C4×C61—82no—
224C4×C61C22×C4⋊C411yescharacteristic
225C4×C61C2×C4×D411yescharacteristic
226C4×C61#54711yescharacteristic
227C4×C61D4×Q811yescharacteristic
228Dic61—22no—
229Dic61—42no—
230Dic61—42no—
231Dic61—42no—
232Dic61—82no—
233C2×#443—611no—
234#2951—612no—
235#2912—611no—
236#2912—611no—
237#1952—612no—
238#1860—612no—
239#1860—612no—
240#1846—611no—
241#1815—611no—
242#1815D6111yescharacteristic, complemented
243#1772—611no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—42no—
247C2×C61D4×C4⋊C411yescharacteristic
248C2×C61#329012yes—
249D61—42no—
250C2×#135—611no—
251C2×#131—611no—
252C2×#131C2×D6111yescharacteristic
253#520—612no—
254#520—612no—
255#443—618no—
256#440—1222no—
257#440—1222no—
258C4×Q16—1222no—
259C4×D8—1222no—
260C2×#108—611no—
261C2×#108C2×D6111yescharacteristic
262#428—612no—
263#428—612no—
264#428—1222no—
265#428—1222no—
266#427—611no—
267#427—1221no—
268#427—1221no—
269C2×#107—612no—
270C2×#106—611no—
271C2×#106—1221no—
272C2×#106—1221no—
273C2×#106C2×D6111yescharacteristic
274C2×#104—611no—
275C2×#104—611no—
276#417—612no—
277#417C2×D6112yes—
278#405—612no—
279C2×C4×C8—611no—
280#382—612no—
281#343—612no—
282#343C2×D6112yes—
283#339—612no—
284C61#2269011yescharacteristic, complemented, Sylow
285C2×#41—611no—
286C2×#41C22×D6111yescharacteristic
287#135—614no—
288#135—614no—
289C2×Q16—611no—
290C2×SD16—611no—
291C2×SD16—611no—
292C2×D8—611no—
293#131—22no—
294#131—614no—
295#131—1222no—
296#131C4×D6114yescomplemented
297C2×M16—21no—
298C2×M16—611no—
299C2×M16—1221no—
300C2×M16C22×D6112yes—
301C2×M16C22×D6111yescharacteristic
302C22×C8—21no—
303C22×C8—611no—
304C22×C8—1221no—
305C22×C8C22×D6111yescharacteristic
306C4×Q8—1222no—
307C4×Q8—1222no—
308C4×D4—1222no—
309C4×D4—1222no—
310C4×D4—2442no—
311C4×D4—2442no—
312#117—612no—
313#117—1222no—
314#117—1222no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4C22×D6112yes—
319C2×C42—611no—
320#108—22no—
321#108—614no—
322#107—612no—
323#107—1222no—
324#107—1222no—
325#107—1222no—
326#107—1222no—
327#106—21no—
328#106—21no—
329#106—611no—
330#106—611no—
331#106—1222no—
332#106—1222no—
333#106—1221no—
334#106—1221no—
335#104—614no—
336#104—614no—
337#103—22no—
338#103—1222no—
339#102—22no—
340#102—1222no—
341#97—1222no—
342C4×C8—614no—
343C4×C8—1222no—
344#95—2442no—
345#41—21no—
346#41—21no—
347#41—612no—
348#41—612no—
349#41—1221no—
350#41—1221no—
351#41C2×C4×D6112yes—
352#41Q8×D6112yes—
353C2×Q8—611no—
354C2×Q8D4×D6111yescharacteristic
355C2×D4—21no—
356C2×D4—611no—
357C2×D4—1221no—
358C2×D4D4×D6111yescharacteristic
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4C23×D6111yescharacteristic
367Q16—614no—
368SD16—614no—
369SD16—614no—
370D8—614no—
371M16—21no—
372M16—24no—
373M16—22no—
374M16—22no—
375M16—612no—
376M16—1221no—
377M16—1224no—
378M16C2×C4×D6112yes—
379C2×C8—21no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—612no—
383C2×C8—611no—
384C2×C8—611no—
385C2×C8—612no—
386C2×C8—1221no—
387C2×C8—1224no—
388C2×C8C2×C4×D6112yes—
389C2×C8D4×D6111yescharacteristic
390C2×C8D4×D6111yescharacteristic
391C2×C8#1175822412yes—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—612no—
395C4⋊C4—612no—
396C4⋊C4—1222no—
397C4⋊C4—1222no—
398C4⋊C4—2442no—
399C4⋊C4—2442no—
400C22⋊C4—1222no—
401C22⋊C4—2442no—
402C22⋊C4—2442no—
403C42—612no—
404C42—1222no—
405C42—1221no—
406C42—2442no—
407C42—2442no—
408C23—21no—
409C23—1221no—
410C23#1451982311yescharacteristic
411Q8—21no—
412Q8—612no—
413Q8—1221no—
414Q8C4⋊C4×D6112yes—
415D4—22no—
416D4—22no—
417D4—21no—
418D4—612no—
419D4—1222no—
420D4—1222no—
421D4—1221no—
422D4C4⋊C4×D6112yes—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—1221no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—1222no—
433C2×C4—1222no—
434C2×C4—9762no—
435C2×C4—9762no—
436C2×C4—4882no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4C22×C4×D6111yescharacteristic
440C2×C4C2×D4×D6111yescharacteristic
441C2×C4C2×D4×D6111yescharacteristic
442C2×C4C2×Q8×D6111yescharacteristic
443C2×C4C2×#1175822411yescharacteristic
444C2×C4#1451982411yescharacteristic
445C8—22no—
446C8—42no—
447C8—612no—
448C8—612no—
449C8—1222no—
450C8#1451973712yes—
451C8#1451973712yes—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—1222no—
456C22—2442no—
457C22#1684024211yescharacteristic
458C22#1684039611yescharacteristic
459C22#1684047011yescharacteristic
460C4—22no—
461C4—82no—
462C4—1222no—
463C4—4882no—
464C4—2442no—
465C4—2442no—
466C4—2442no—
467C4C2×C4⋊C4×D6111yescharacteristic
468C4C2×#1451973711yescharacteristic
469C4#1684041211yescharacteristic
470C4#1684042511yescharacteristic
471C2—21no—
472C2—21no—
473C2—42no—
474C2—2442no—
475C2#2144871511yescharacteristic
476C2#2145760212yes—
subgroupG/H
centerC22#16840396
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1815C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1815D61
hypocenterC61#22690
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
3. #21454414C2semidirect
4. #21454410C2semidirect
5. #21446603C2semidirect
6. #21446055C2semidirect
7. #21446050C2semidirect
8. #21445980C2semidirect
9. #21445875C2semidirect
10. #21445701C2semidirect
11. C61×#1815C2semidirect
16. #16839877C4semidirect
54. #14519783C2×C4semidirect
72. #14519712C2×C4semidirect
84. #14519704C2×C4semidirect
85. #14519703C2×C4semidirect
90. C8×Dic61C2×C4semidirect
94. #131×C61C2×C4semidirect
242. #1815D61semidirect
284. C61#22690semidirect
296. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398772
#214546902
#214544142
#214544102
#214466032
#214460552
#214460502
#214459802
#214458752
#214457012
C61×#18152
#2269061