Finite Group Databasebeta

← #29698498

Subgroups — #29698498

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839877C211yesmaximal, characteristic
2#21454698C212yesmaximal
3#21454419C211yesmaximal, characteristic, complemented
4#21454415C211yesmaximal, characteristic, complemented
5#21446598C212yesmaximal, complemented
6#21446055C212yesmaximal, complemented
7#21446050C212yesmaximal, complemented
8#21445982C211yesmaximal, characteristic, complemented
9#21445880C211yesmaximal, characteristic, complemented
10#21445701C211yesmaximal, characteristic, complemented
11C61×#1816C211yesmaximal, 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#16839803—22no—
25#16839802C2212yes—
26#16839800—21no—
27#16839800C2211yescharacteristic
28#16839799—21no—
29C2×#14519711—21no—
30C2×#14519711C2211yescharacteristic
31#107×D61—21no—
32C2×#14519708C2212yes—
33C2×#14519704C2211yescharacteristic
34C2×#14519703C2211yescharacteristic
35#16839764C2212yes—
36#16839720C2212yes—
37C2×C8×Dic61C2211yescharacteristic
38#16839627C2212yes—
39#16839530C2212yes—
40#16839523C2212yes—
41C2×#131×C61C2211yescharacteristic
42C2×#108×C61C2211yescharacteristic
43C2×#107×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—21no—
74#14519711—21no—
75#14519711—22no—
76#14519711D411yescharacteristic
77#14519711D411yescharacteristic
78#14519710—22no—
79#14519709—22no—
80#14519709D412yes—
81#14519708—22no—
82#14519708—22no—
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#107×C61—21no—
103#107×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—41no—
196C2×Dic61—41no—
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#22691—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—22no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61#52011yescharacteristic
221C22×C61D4211yescharacteristic
222C22×C61#55411yescharacteristic
223C2×D61—22no—
224C2×D61—41no—
225C2×D61—41no—
226C4×C61—22no—
227C4×C61—82no—
228C4×C61C22×C4⋊C411yescharacteristic
229C4×C61C2×C4×D411yescharacteristic
230C4×C61#54711yescharacteristic
231C4×C61D4×Q811yescharacteristic
232Dic61—22no—
233Dic61—42no—
234Dic61—42no—
235Dic61—42no—
236Dic61—82no—
237C2×#443—611no—
238#2952—612no—
239#2913—611no—
240#2913—611no—
241#1951—612no—
242#1860—612no—
243#1860—612no—
244#1846—611no—
245#1816—611no—
246#1816D6111yescharacteristic, complemented
247#1772—611no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—42no—
251C2×C61D4×C4⋊C411yescharacteristic
252C2×C61#329011yescharacteristic
253C2×C61#329011yescharacteristic
254D61—42no—
255C2×#135—611no—
256C2×#131—611no—
257C2×#131C2×D6111yescharacteristic
258#520—612no—
259#520—612no—
260#443—618no—
261#442—1222no—
262#441—1222no—
263C4×SD16—1222no—
264C4×SD16—1222no—
265C2×#108—611no—
266C2×#108C2×D6111yescharacteristic
267#428—612no—
268#428—612no—
269#428—1222no—
270#428—1222no—
271#427—611no—
272#427—1221no—
273#427—1221no—
274C2×#107—611no—
275C2×#107—1221no—
276C2×#107—1221no—
277C2×#107C2×D6111yescharacteristic
278C2×#106—612no—
279C2×#104—611no—
280C2×#104—611no—
281#417—612no—
282#417C2×D6112yes—
283#405—612no—
284C2×C4×C8—611no—
285#382—612no—
286#343—612no—
287#343C2×D6112yes—
288#339—612no—
289C61#2269111yescharacteristic, complemented, Sylow
290C2×#41—611no—
291C2×#41C22×D6111yescharacteristic
292#135—614no—
293#135—614no—
294C2×Q16—611no—
295C2×SD16—611no—
296C2×SD16—611no—
297C2×D8—611no—
298#131—22no—
299#131—614no—
300#131—1222no—
301#131C4×D6114yescomplemented
302C2×M16—21no—
303C2×M16—611no—
304C2×M16—1221no—
305C2×M16C22×D6112yes—
306C2×M16C22×D6111yescharacteristic
307C22×C8—21no—
308C22×C8—611no—
309C22×C8—1221no—
310C22×C8C22×D6111yescharacteristic
311C4×Q8—1222no—
312C4×Q8—1222no—
313C4×D4—1222no—
314C4×D4—1222no—
315C4×D4—2442no—
316C4×D4—2442no—
317#117—612no—
318#117—1222no—
319#117—1222no—
320C2×C4⋊C4—612no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4C22×D6112yes—
324C2×C42—611no—
325#108—22no—
326#108—614no—
327#107—21no—
328#107—21no—
329#107—611no—
330#107—611no—
331#107—1222no—
332#107—1222no—
333#107—1221no—
334#107—1221no—
335#106—612no—
336#106—1222no—
337#106—1222no—
338#106—1222no—
339#106—1222no—
340#104—614no—
341#104—614no—
342#103—22no—
343#103—1222no—
344#102—22no—
345#102—1222no—
346#97—1222no—
347C4×C8—614no—
348C4×C8—1222no—
349#95—2442no—
350#41—21no—
351#41—21no—
352#41—612no—
353#41—612no—
354#41—1221no—
355#41—1221no—
356#41C2×C4×D6112yes—
357#41Q8×D6112yes—
358C2×Q8—611no—
359C2×Q8D4×D6111yescharacteristic
360C2×D4—21no—
361C2×D4—611no—
362C2×D4—1221no—
363C2×D4D4×D6111yescharacteristic
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4—2442no—
370C22×C4—2442no—
371C22×C4C23×D6111yescharacteristic
372Q16—614no—
373SD16—614no—
374SD16—614no—
375D8—614no—
376M16—21no—
377M16—24no—
378M16—22no—
379M16—22no—
380M16—612no—
381M16—1221no—
382M16—1224no—
383M16C2×C4×D6112yes—
384C2×C8—21no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—612no—
388C2×C8—611no—
389C2×C8—611no—
390C2×C8—612no—
391C2×C8—1221no—
392C2×C8—1224no—
393C2×C8C2×C4×D6112yes—
394C2×C8D4×D6111yescharacteristic
395C2×C8D4×D6111yescharacteristic
396C2×C8#1175822412yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—612no—
400C4⋊C4—612no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—2442no—
404C4⋊C4—2442no—
405C22⋊C4—1222no—
406C22⋊C4—2442no—
407C22⋊C4—2442no—
408C42—612no—
409C42—1222no—
410C42—1221no—
411C42—2442no—
412C42—2442no—
413C23—21no—
414C23—1221no—
415C23#1451982311yescharacteristic
416Q8—21no—
417Q8—612no—
418Q8—1221no—
419Q8C4⋊C4×D6112yes—
420D4—22no—
421D4—22no—
422D4—21no—
423D4—612no—
424D4—1222no—
425D4—1222no—
426D4—1221no—
427D4C4⋊C4×D6112yes—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—1221no—
434C2×C4—1222no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—9762no—
440C2×C4—9762no—
441C2×C4—4882no—
442C2×C4—2442no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4C22×C4×D6111yescharacteristic
446C2×C4C2×D4×D6111yescharacteristic
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4C2×Q8×D6111yescharacteristic
449C2×C4C2×#1175822411yescharacteristic
450C2×C4#1451982411yescharacteristic
451C8—22no—
452C8—42no—
453C8—612no—
454C8—612no—
455C8—1222no—
456C8#1451973712yes—
457C8#1451973712yes—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—1222no—
464C22—2441no—
465C22—2441no—
466C22#1684024211yescharacteristic
467C22#1684039611yescharacteristic
468C22#1684047011yescharacteristic
469C4—22no—
470C4—82no—
471C4—1222no—
472C4—4882no—
473C4—2442no—
474C4—2442no—
475C4—2442no—
476C4C2×C4⋊C4×D6111yescharacteristic
477C4C2×#1451973711yescharacteristic
478C4#1684041211yescharacteristic
479C4#1684042511yescharacteristic
480C2—21no—
481C2—21no—
482C2—42no—
483C2—2442no—
484C2#2144871511yescharacteristic
485C2#2145760011yescharacteristic
486C2#2145760811yescharacteristic
subgroupG/H
centerC22#16840396
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1816C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1816D61
hypocenterC61#22691
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
3. #21454419C2semidirect
4. #21454415C2semidirect
5. #21446598C2semidirect
6. #21446055C2semidirect
7. #21446050C2semidirect
8. #21445982C2semidirect
9. #21445880C2semidirect
10. #21445701C2semidirect
11. C61×#1816C2semidirect
16. #16839877C4semidirect
54. #14519783C2×C4semidirect
72. #14519712C2×C4semidirect
84. #14519704C2×C4semidirect
85. #14519703C2×C4semidirect
90. C8×Dic61C2×C4semidirect
94. #131×C61C2×C4semidirect
246. #1816D61semidirect
289. C61#22691semidirect
301. #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
#214544192
#214544152
#214465982
#214460552
#214460502
#214459822
#214458802
#214457012
C61×#18162
#2269161