Finite Group Databasebeta

← #29722572

Subgroups — #29722572

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455432C212yesmaximal, complemented
2C2×#16840012C211yesmaximal, characteristic, complemented
3#21454520C211yesmaximal, characteristic, complemented
4C2×#16839809C211yesmaximal, characteristic, complemented
5#21447175C212yesmaximal, complemented
6#21447108C212yesmaximal, complemented
7#21446730C211yesmaximal, characteristic, complemented
8#21446600C211yesmaximal, characteristic, complemented
9#21446279C212yesmaximal, complemented
10#21445878C211yesmaximal, characteristic, complemented
11C2×C61×#472C211yesmaximal, characteristic, complemented
12C22×#11758189C2211yescharacteristic
13C2×M16×D61C2212yes—
14#16840269C2211yescharacteristic
15C2×#14519748C2212yes—
16#16840012—22no—
17#16840012—22no—
18#16840012—22no—
19#16840005—22no—
20#16839995—22no—
21#16839988—22no—
22#16839838—22no—
23#16839833—22no—
24#16839809—24no—
25C2×#14519713C2211yescharacteristic
26C2×#14519712C2212yes—
27#16839804C2212yes—
28C2×#14519709C2211yescharacteristic
29#16839790C2211yescharacteristic
30#16839767C2212yes—
31#16839763C2212yes—
32C2×#14519698C2212yes—
33C2×#14519694C2212yes—
34#16839729C2211yescharacteristic
35#16839725C2211yescharacteristic
36#16839717C2211yescharacteristic
37#16839530C2212yes—
38#16839530C2212yes—
39C2×AGL(1,ℤ/8)×C61C2212yes—
40C22×D8×C61C2211yescharacteristic
41C2×#131×C61C2211yescharacteristic
42C61×#472—24no—
43C2×#108×C61C2211yescharacteristic
44C2×#100×C61C2212yes—
45C22×C22⋊D61—22no—
46C22×C4×D61C2311yescharacteristic
47C2×#11758194C2311yescharacteristic
48C2×#11758194C2312yes—
49C2×#11758189—22no—
50C2×#11758189—22no—
51C2×#11758189—22no—
52C2×#11758189D412yes—
53C2×#11758189D412yes—
54M16×D61—24no—
55M16×D61—42no—
56C22×#92418C2311yescharacteristic
57C2×C8×D61—22no—
58#14519758—21no—
59#14519753D412yes—
60#14519748—22no—
61#14519748—22no—
62#14519748—22no—
63#14519748—22no—
64#14519748D412yes—
65#14519745—21no—
66C4×C22⋊D61—22no—
67#14519736C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C22⋊Dic61—22no—
70#14519713—22no—
71#14519712—24no—
72#14519711—22no—
73#14519709—21no—
74#14519709—21no—
75#14519709—22no—
76#14519706—22no—
77#14519701—22no—
78#14519698—22no—
79#14519698—22no—
80#14519698—22no—
81#14519698—24no—
82#14519694—42no—
83C22⋊#92418—22no—
84#14519689—22no—
85#14519687—21no—
86C8×Dic61—21no—
87C2×#41×C61C2311yescharacteristic
88C22×D4×C61C2312yes—
89AGL(1,ℤ/8)×C61—24no—
90AGL(1,ℤ/8)×C61—42no—
91C2×SD16×C61—22no—
92C2×D8×C61—22no—
93C2×D8×C61—21no—
94C2×D8×C61—21no—
95C2×D8×C61—22no—
96C2×D8×C61—22no—
97#131×C61—22no—
98#131×C61—22no—
99C2×M16×C61—21no—
100C2×M16×C61C2311yescharacteristic
101C2×M16×C61C2312yes—
102C22×C8×C61—21no—
103C22×C8×C61C2311yescharacteristic
104#108×C61D414yescomplemented
105#100×C61—24no—
106C24×C61—22no—
107C23×D61#4111yescharacteristic
108#41×C61—21no—
109#41×C61—21no—
110#41×C61—21no—
111#41×C61—21no—
112C2×Q8×C61C2×D411yescharacteristic
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—22no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—84no—
127C22×Dic61—22no—
128C22×Dic61#4111yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61C2×D412yes—
134C2×C4×D61C2×D412yes—
135SD16×C61—44no—
136D8×C61—44no—
137D8×C61—42no—
138D8×C61—42no—
139M16×C61—22no—
140M16×C61—21no—
141M16×C61—21no—
142M16×C61—42no—
143M16×C61C2×D414yes—
144C2×C8×C61—22no—
145C2×C8×C61—21no—
146C2×C8×C61—42no—
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61C2×D412yes—
150C2×C8×C61#4112yes—
151C22⋊Dic61—21no—
152C22⋊Dic61—22no—
153C22⋊Dic61—42no—
154C22⋊Dic61—42no—
155#11758198—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61C2×D412yes—
159#11758196—22no—
160C4×Dic61—21no—
161#11758194—22no—
162#11758194—21no—
163#11758194—22no—
164#11758194—21no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418C2×D412yes—
170#11758189—24no—
171#11758189—24no—
172#11758189—44no—
173C8×D61—44no—
174C23×C61—21no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—82no—
179C23×C61—82no—
180C23×C61C2×#4111yescharacteristic
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—42no—
184C22×D61#12112yes—
185Q8×C61—21no—
186Q8×C61—21no—
187D4×C61—21no—
188D4×C61—21no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61H(5,𝔽2)11yescharacteristic
203C2×C4×C612−1+411yescharacteristic
204C22⋊D61—82no—
205C22⋊D61—84no—
206C22⋊D61—84no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—22no—
210C2×Dic61—42no—
211C2×Dic61—82no—
212C2×Dic61#12112yes—
213C4×D61—22no—
214C4×D61—22no—
215C4×D61—22no—
216C4×D61—22no—
217C8×C61—24no—
218C8×C61—41no—
219C8×C61#12112yes—
220C8×C61#12112yes—
221#92418—22no—
222#92418—41no—
223#92418—42no—
224#24806—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—41no—
227C22×C61—42no—
228C22×C61—82no—
229C22×C61—82no—
230C22×C61—82no—
231C22×C61—162no—
232C22×C61—162no—
233C22×C61C2×#12111yescharacteristic
234C22×C61#54811yescharacteristic
235C22×C61#55311yescharacteristic
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—81no—
240C2×D61—81no—
241C4×C61—41no—
242C4×C61C2×#12111yescharacteristic
243C4×C61#53711yescharacteristic
244C4×C61D4211yescharacteristic
245C4×C61#54711yescharacteristic
246Dic61—22no—
247Dic61—42no—
248Dic61—82no—
249C2×#472D6111yescharacteristic, complemented
250#3055—612no—
251C2×#469—611no—
252#2930—611no—
253C2×#430—611no—
254#2063—612no—
255#2051—612no—
256#1978—611no—
257#1951—611no—
258#1900—612no—
259#1815—611no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61—81no—
263C2×C61—162no—
264C2×C61#241611yescharacteristic
265C2×C61#306812yes—
266D61—42no—
267C2×AGL(1,ℤ/8)C2×D6112yes—
268C22×D8C2×D6111yescharacteristic
269C2×#131C2×D6111yescharacteristic
270C22×M16—611no—
271C22×M16—612no—
272#525—611no—
273C2×#121—612no—
274#472—24no—
275#469—1222no—
276#469—1222no—
277#469—1222no—
278#468—1222no—
279#466—1222no—
280#465—1222no—
281#436—1222no—
282#435—1222no—
283#430—1224no—
284C2×#108—612no—
285C2×#108—611no—
286C2×#108C2×D6111yescharacteristic
287#428—612no—
288C2×#106—611no—
289#424—611no—
290#418—612no—
291#417—612no—
292C2×#102—612no—
293C2×#100—612no—
294C2×#100C2×D6112yes—
295#408—611no—
296#407—611no—
297#405—611no—
298#343—612no—
299#343—612no—
300C61#2480611yescharacteristic, complemented, Sylow
301C2×#41C22×D6111yescharacteristic
302C22×D4—1222no—
303C22×D4C22×D6112yes—
304C23×C4—611no—
305AGL(1,ℤ/8)—24no—
306AGL(1,ℤ/8)—42no—
307C2×SD16—22no—
308C2×D8—22no—
309C2×D8—21no—
310C2×D8—21no—
311C2×D8—22no—
312C2×D8—22no—
313#131—22no—
314#131—22no—
315C2×M16—21no—
316C2×M16—612no—
317C2×M16—612no—
318C2×M16—612no—
319C2×M16—611no—
320C2×M16—1222no—
321C2×M16—1222no—
322C2×M16—1222no—
323C2×M16—1224no—
324C2×M16—2442no—
325C2×M16C22×D6111yescharacteristic
326C2×M16C22×D6112yes—
327C22×C8—21no—
328C22×C8—611no—
329C22×C8—1222no—
330C22×C8C22×D6111yescharacteristic
331#123—1221no—
332C22⋊Q8—612no—
333#121—612no—
334#121—1222no—
335#121—1222no—
336#121—1222no—
337#121—1222no—
338#120—1221no—
339C4×D4—1222no—
340#117—611no—
341C2×C4⋊C4—612no—
342C2×C22⋊C4—1222no—
343#108—1224no—
344#108—1222no—
345#108C22⋊D6114yescomplemented
346#107—1222no—
347#106—1222no—
348#106—1221no—
349#106—1221no—
350C4⋊C8—1222no—
351#103—1222no—
352#102—1222no—
353#102—1222no—
354#102—1224no—
355#102—1222no—
356#100—24no—
357#100—2442no—
358C22⋊C8—1222no—
359C22⋊C8—1222no—
360#97—1221no—
361C4×C8—1221no—
362C24—22no—
363C24—611no—
364#41—21no—
365#41—21no—
366#41—21no—
367#41—21no—
368C2×Q8D4×D6111yescharacteristic
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—1222no—
377C2×D4—4884no—
378C2×D4—2442no—
379C2×D4—2442no—
380C2×D4D4×D6111yescharacteristic
381C22×C4—21no—
382C22×C4—611no—
383C22×C4—612no—
384C22×C4—612no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1222no—
390C22×C4C23×D6111yescharacteristic
391SD16—44no—
392D8—44no—
393D8—42no—
394D8—42no—
395M16—22no—
396M16—21no—
397M16—21no—
398M16—42no—
399M16—1224no—
400M16—1224no—
401M16—1222no—
402M16—1221no—
403M16—1222no—
404M16—1221no—
405M16—2444no—
406M16C2×C22⋊D6114yes—
407C2×C8—22no—
408C2×C8—21no—
409C2×C8—42no—
410C2×C8—612no—
411C2×C8—1221no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1222no—
415C2×C8—2444no—
416C2×C8D4×D6111yescharacteristic
417C2×C8D4×D6111yescharacteristic
418C2×C8#1175822412yes—
419C2×C8C2×C22⋊D6112yes—
420C4⋊C4—612no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C22⋊C4—1222no—
425C22⋊C4—1222no—
426C22⋊C4—1221no—
427C22⋊C4—2442no—
428C22⋊C4—2442no—
429C42—1221no—
430C23—21no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—82no—
435C23—82no—
436C23—612no—
437C23—1221no—
438C23—1221no—
439C23—2442no—
440C23#41×D6111yescharacteristic
441Q8—21no—
442Q8—21no—
443D4—21no—
444D4—21no—
445D4—41no—
446D4—41no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—4884no—
452D4—4884no—
453D4—4882no—
454C2×C4—21no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—612no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—4882no—
466C2×C4—2442no—
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C22×C22⋊D6111yescharacteristic
470C2×C4C2×#1175822411yescharacteristic
471C2×C4#1451982211yescharacteristic
472C2×C4#1451982411yescharacteristic
473C8—24no—
474C8—41no—
475C8—1222no—
476C8—2441no—
477C8—2442no—
478C8#1451974812yes—
479C8#1451974812yes—
480C22—22no—
481C22—41no—
482C22—42no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—162no—
487C22—162no—
488C22—1222no—
489C22—1222no—
490C22—1222no—
491C22—4881no—
492C22—4881no—
493C22#121×D6111yescharacteristic
494C22#1684041911yescharacteristic
495C22#1684045911yescharacteristic
496C4—41no—
497C4—1222no—
498C4—4882no—
499C4—2442no—
500C4C2×#1451974811yescharacteristic
501C4#1684034111yescharacteristic
502C4D4×C22⋊D6111yescharacteristic
503C4#1684041211yescharacteristic
504C2—21no—
505C2—21no—
506C2—81no—
507C2—162no—
508C2—2442no—
509C2#2144925511yescharacteristic
510C2#2145554512yes—
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#472C2
socleC22×C61C2×#121
hypercenterC2×#472D61
hypocenterC61#24806
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. #21455432C2semidirect
2. C2×#16840012C2semidirect
3. #21454520C2semidirect
4. C2×#16839809C2semidirect
5. #21447175C2semidirect
6. #21447108C2semidirect
7. #21446730C2semidirect
8. #21446600C2semidirect
9. #21446279C2semidirect
10. #21445878C2semidirect
11. C2×C61×#472C2semidirect
104. #108×C61D4semidirect
249. C2×#472D61semidirect
300. C61#24806semidirect
345. #108C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214554322
C2×#168400122
#214545202
C2×#168398092
#214471752
#214471082
#214467302
#214466002
#214462792
#214458782
C2×C61×#4722
#2480661