Finite Group Databasebeta

← #29698482

Subgroups — #29698482

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839877C211yesmaximal, characteristic
2#21454698C212yesmaximal
3#21454414C211yesmaximal, characteristic, complemented
4#21454410C211yesmaximal, characteristic, complemented
5#21446598C212yesmaximal, complemented
6#21446048C212yesmaximal, complemented
7#21446043C212yesmaximal, complemented
8#21445974C211yesmaximal, 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#16839871—22no—
18#16839866—22no—
19#16839853—22no—
20#16839847—22no—
21C2×#14519712C2211yescharacteristic
22#16839804—22no—
23#16839804C2212yes—
24#16839802C2212yes—
25#16839801—22no—
26#16839800—21no—
27#16839800C2211yescharacteristic
28#16839797—21no—
29C2×#14519709—21no—
30C2×#14519709C2211yescharacteristic
31C2×#14519708C2212yes—
32#106×D61—21no—
33C2×#14519704C2211yescharacteristic
34C2×#14519703C2211yescharacteristic
35#16839764C2212yes—
36#16839720C2212yes—
37C2×C8×Dic61C2211yescharacteristic
38#16839631C2212yes—
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—
70C2×#11758196—42no—
71C4⋊C4×D61—22no—
72C2×C4×Dic61C2311yescharacteristic
73#14519712C2×C414yescomplemented
74#14519711—22no—
75#14519711D412yes—
76#14519710—22no—
77#14519709—21no—
78#14519709—21no—
79#14519709—22no—
80#14519709D411yescharacteristic
81#14519709D411yescharacteristic
82#14519708—22no—
83#14519708—22no—
84#14519708—22no—
85#14519704C2×C414yescomplemented
86#14519703C2×C414yescomplemented
87#14519702—22no—
88#14519699—22no—
89#14519687—22no—
90C8×Dic61—22no—
91C8×Dic61C2×C414yescomplemented
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⋊C414yescomplemented
145#11758201C4⋊C414yescomplemented
146#11758200C4⋊C414yescomplemented
147#11758199C4⋊C414yescomplemented
148#11758198—42no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61C2×D412yes—
153C4⋊Dic61C2×D412yes—
154#11758196—42no—
155#11758196—84no—
156C4×Dic61—22no—
157C4×Dic61—21no—
158C4×Dic61—42no—
159C4×Dic61—42no—
160C4×Dic61C22×C412yes—
161#11758194—21no—
162#11758194C22×C412yes—
163C2×#92418—21no—
164C2×#92418C22×C412yes—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×Q812yes—
168#11758189—24no—
169C8×D61—24no—
170C23×C61—21no—
171C23×C61H(5,𝔽2)11yescharacteristic
172C22×D61—21no—
173Q8×C61—21no—
174Q8×C61C2×C4⋊C412yes—
175D4×C61—22no—
176D4×C61—22no—
177D4×C61—21no—
178D4×C61C2×C4⋊C412yes—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—82no—
184C2×C4×C61C23×C411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×Q811yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C612−1+411yescharacteristic
190C22⋊D61—22no—
191C22⋊D61—22no—
192C2×Dic61—24no—
193C2×Dic61—22no—
194C2×Dic61—21no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—42no—
198C2×Dic61—82no—
199C2×Dic61—162no—
200D244—21no—
201D244C2×C4⋊C412yes—
202C4×D61—22no—
203C4×D61—22no—
204C4×D61—162no—
205Dic122—21no—
206Dic122C2×C4⋊C412yes—
207C8×C61—22no—
208C8×C61—42no—
209C8×C61C4×D412yes—
210C8×C61C4×D412yes—
211#92418—22no—
212#92418C2×C4⋊C412yes—
213#92418C2×C4⋊C412yes—
214#22689—611nomaximal, Sylow
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—42no—
218C22×C61#52011yescharacteristic
219C22×C61D4211yescharacteristic
220C22×C61#55411yescharacteristic
221C2×D61—22no—
222C2×D61—42no—
223C4×C61—22no—
224C4×C61—82no—
225C4×C61C22×C4⋊C411yescharacteristic
226C4×C61C2×C4×D411yescharacteristic
227C4×C61#54711yescharacteristic
228C4×C61D4×Q811yescharacteristic
229Dic61—24no—
230Dic61—22no—
231Dic61—42no—
232Dic61—42no—
233Dic61—82no—
234C2×#443—611no—
235#2952—612no—
236#2912—611no—
237#2912—611no—
238#1951—612no—
239#1859—612no—
240#1859—612no—
241#1844—611no—
242#1815—611no—
243#1815D6111yescharacteristic, complemented
244#1772—611no—
245C2×C61—21no—
246C2×C61—21no—
247C2×C61—42no—
248C2×C61D4×C4⋊C411yescharacteristic
249C2×C61#329112yes—
250D61—42no—
251C2×#135—611no—
252C2×#131—611no—
253C2×#131C2×D6111yescharacteristic
254#520—612no—
255#520—612no—
256#443—618no—
257#442—1222no—
258#441—1222no—
259C4×SD16—1222no—
260C4×SD16—1222no—
261C2×#108—611no—
262C2×#108C2×D6111yescharacteristic
263#428—612no—
264#428—612no—
265#428—1222no—
266#428—1222no—
267#427—611no—
268#427—1221no—
269#427—1221no—
270C2×#106—611no—
271C2×#106—612no—
272C2×#106—1221no—
273C2×#106—1221no—
274C2×#106C2×D6111yescharacteristic
275C2×#104—611no—
276C2×#104—611no—
277#417—612no—
278#417C2×D6112yes—
279#405—612no—
280C2×C4×C8—611no—
281#384—612no—
282#343—612no—
283#343C2×D6112yes—
284#339—612no—
285C61#2268911yescharacteristic, complemented, Sylow
286C2×#41—611no—
287C2×#41C22×D6111yescharacteristic
288#135—614no—
289#135—614no—
290C2×Q16—611no—
291C2×SD16—611no—
292C2×SD16—611no—
293C2×D8—611no—
294#131—22no—
295#131—614no—
296#131—1222no—
297#131C4×D6114yescomplemented
298C2×M16—21no—
299C2×M16—611no—
300C2×M16—1221no—
301C2×M16C22×D6112yes—
302C2×M16C22×D6111yescharacteristic
303C22×C8—21no—
304C22×C8—611no—
305C22×C8—1221no—
306C22×C8C22×D6111yescharacteristic
307C4×Q8—1222no—
308C4×Q8—1222no—
309C4×D4—1222no—
310C4×D4—1222no—
311C4×D4—2442no—
312C4×D4—2442no—
313#117—612no—
314#117—1222no—
315#117—1222no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—2442no—
320C2×C4⋊C4C22×D6112yes—
321C2×C42—611no—
322#108—22no—
323#108—614no—
324#107—612no—
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—1222no—
334#106—1222no—
335#106—1221no—
336#106—1221no—
337#104—614no—
338#104—614no—
339#103—22no—
340#103—1222no—
341#102—22no—
342#102—1222no—
343#97—1222no—
344C4×C8—614no—
345C4×C8—1222no—
346#41—21no—
347#41—21no—
348#41—612no—
349#41—612no—
350#41—1221no—
351#41—1221no—
352#41C2×C4×D6112yes—
353#41Q8×D6112yes—
354C2×Q8—611no—
355C2×Q8D4×D6111yescharacteristic
356C2×D4—21no—
357C2×D4—611no—
358C2×D4—1221no—
359C2×D4D4×D6111yescharacteristic
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1222no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4C23×D6111yescharacteristic
368Q16—614no—
369SD16—614no—
370SD16—614no—
371D8—614no—
372M16—21no—
373M16—24no—
374M16—22no—
375M16—22no—
376M16—612no—
377M16—1221no—
378M16—1224no—
379M16C2×C4×D6112yes—
380C2×C8—21no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—612no—
384C2×C8—611no—
385C2×C8—611no—
386C2×C8—612no—
387C2×C8—1221no—
388C2×C8—1224no—
389C2×C8C2×C4×D6112yes—
390C2×C8D4×D6111yescharacteristic
391C2×C8D4×D6111yescharacteristic
392C2×C8#1175822412yes—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—612no—
396C4⋊C4—612no—
397C4⋊C4—1222no—
398C4⋊C4—1222no—
399C4⋊C4—4884no—
400C4⋊C4—2442no—
401C4⋊C4—2442no—
402C22⋊C4—1222no—
403C22⋊C4—2442no—
404C22⋊C4—2442no—
405C42—612no—
406C42—1222no—
407C42—1221no—
408C42—2442no—
409C42—2442no—
410C23—21no—
411C23—1221no—
412C23#1451982311yescharacteristic
413Q8—21no—
414Q8—612no—
415Q8—1221no—
416Q8C4⋊C4×D6112yes—
417D4—22no—
418D4—22no—
419D4—21no—
420D4—612no—
421D4—1222no—
422D4—1222no—
423D4—1221no—
424D4C4⋊C4×D6112yes—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—82no—
430C2×C4—1224no—
431C2×C4—1222no—
432C2×C4—1221no—
433C2×C4—1222no—
434C2×C4—1222no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—9762no—
438C2×C4—9762no—
439C2×C4—4882no—
440C2×C4—2442no—
441C2×C4C22×C4×D6111yescharacteristic
442C2×C4C2×D4×D6111yescharacteristic
443C2×C4C2×D4×D6111yescharacteristic
444C2×C4C2×Q8×D6111yescharacteristic
445C2×C4C2×#1175822411yescharacteristic
446C2×C4#1451982411yescharacteristic
447C8—22no—
448C8—42no—
449C8—612no—
450C8—612no—
451C8—1222no—
452C8#1451973712yes—
453C8#1451973712yes—
454C22—22no—
455C22—22no—
456C22—42no—
457C22—1222no—
458C22—2442no—
459C22#1684024211yescharacteristic
460C22#1684039611yescharacteristic
461C22#1684047011yescharacteristic
462C4—22no—
463C4—82no—
464C4—1224no—
465C4—1222no—
466C4—4882no—
467C4—2442no—
468C4—2442no—
469C4C2×C4⋊C4×D6111yescharacteristic
470C4C2×#1451973711yescharacteristic
471C4#1684041211yescharacteristic
472C4#1684042511yescharacteristic
473C2—21no—
474C2—21no—
475C2—42no—
476C2—2442no—
477C2#2144871511yescharacteristic
478C2#2145761112yes—
subgroupG/H
centerC22#16840396
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1815C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1815D61
hypocenterC61#22689
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. #21446598C2semidirect
6. #21446048C2semidirect
7. #21446043C2semidirect
8. #21445974C2semidirect
9. #21445875C2semidirect
10. #21445701C2semidirect
11. C61×#1815C2semidirect
16. #16839877C4semidirect
54. #14519783C2×C4semidirect
73. #14519712C2×C4semidirect
85. #14519704C2×C4semidirect
86. #14519703C2×C4semidirect
91. C8×Dic61C2×C4semidirect
94. #131×C61C2×C4semidirect
144. #11758202C4⋊C4semidirect
145. #11758201C4⋊C4semidirect
146. #11758200C4⋊C4semidirect
147. #11758199C4⋊C4semidirect
243. #1815D61semidirect
285. C61#22689semidirect
297. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398772
#214546982
#214544142
#214544102
#214465982
#214460482
#214460432
#214459742
#214458752
#214457012
C61×#18152
#2268961