Finite Group Databasebeta

← #29718409

Subgroups — #29718409

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840075C211yesmaximal, characteristic
2#21449473C211yesmaximal, characteristic
3#21448688C211yesmaximal, characteristic, complemented
4#21447210C212yesmaximal, complemented
5#21447057C212yesmaximal, complemented
6#21446748C212yesmaximal, complemented
7#21446463C212yesmaximal, complemented
8#21446247C211yesmaximal, characteristic
9#21445997C211yesmaximal, characteristic, complemented
10#21445754C211yesmaximal, characteristic, complemented
11C22×#103×C61C211yesmaximal, characteristic, complemented
12C2×#14519766C2212yes—
13C2×#14519751C2211yescharacteristic
14C2×#14519736—21no—
15#16840075—24no—
16C2×#14519710C2211yescharacteristic
17C2×#14519710C2211yescharacteristic
18C2×#14519706C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519699C2212yes—
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25#16839673—22no—
26#16839669C2212yes—
27#16839661C2212yes—
28#16839648—22no—
29#16839641—21no—
30#16839641—21no—
31#16839625—21no—
32#16839622C2211yescharacteristic
33C4⋊C4×Dic61C2212yes—
34C4×C22⋊Dic61C2211yescharacteristic
35#16839525C2212yes—
36#16839516C2212yes—
37C23×Q8×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C22×C4⋊C4×C61C2211yescharacteristic
40C2×#103×C61—24no—
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C22×#92418C2311yescharacteristic
46C22×#92418C2311yescharacteristic
47#14519766—24no—
48#14519766—42no—
49#14519751—22no—
50#14519751—22no—
51#14519736—44no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2312yes—
55C2×C22⋊Dic61—21no—
56C2×C22⋊Dic61D411yescharacteristic
57C2×C22⋊Dic61D411yescharacteristic
58C2×#11758198—21no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×C4×Dic61—22no—
62C2×C4×Dic61C2312yes—
63#14519710—22no—
64#14519710—22no—
65#14519706—42no—
66#14519701—24no—
67#14519701—42no—
68#14519699—42no—
69#14519697—22no—
70#14519697—22no—
71C22⋊#92418—22no—
72#14519684—21no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76#14519684—22no—
77#14519684D412yes—
78C22×Q8×C61—21no—
79C22×Q8×C61—24no—
80C22×Q8×C61—22no—
81C22×Q8×C61—22no—
82C22×Q8×C61—42no—
83C22×Q8×C61C2312yes—
84C23×C4×C61—21no—
85C23×C4×C61—21no—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—22no—
88C22×C8×C61—42no—
89C22×C8×C61C2312yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61D414yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95#103×C61—24no—
96#103×C61—24no—
97C24×C61C2×D411yescharacteristic
98C23×D61—21no—
99C2×Q8×C61—22no—
100C2×Q8×C61—28no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—44no—
106C2×Q8×C61—48no—
107C2×Q8×C61—44no—
108C2×Q8×C61—42no—
109C2×Q8×C61#4114yes—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—24no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—44no—
117C22×C4×C61—42no—
118C22×C4×C61—42no—
119C22×C4×C61—42no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61#4112yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—42no—
125C2×C22⋊D61—42no—
126C2×C22⋊D61—42no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61C2×D412yes—
130C22×Dic61C2×D412yes—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—22no—
134C2×D244—22no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—44no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D412yes—
143C4⋊C4×C61C2×D412yes—
144C22⋊Dic61—22no—
145C22⋊Dic61—22no—
146C22⋊Dic61—42no—
147#11758198—44no—
148#11758198—44no—
149#11758198—42no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—44no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C4×Dic61—44no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—22no—
161C2×#92418—22no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61#12112yes—
165C23×C61#13512yes—
166C23×C61AGL(1,ℤ/8)12yes—
167C23×C61C22×D411yescharacteristic
168C22×D61—41no—
169C22×D61—42no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173Q8×C61—24no—
174Q8×C61—24no—
175Q8×C61—48no—
176Q8×C61—42no—
177Q8×C61—42no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—24no—
185C2×C4×C61—48no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—48no—
193C2×C4×C61#12112yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4112yes—
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)12yes—
198C22⋊D61—84no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211D244—41no—
212D244—41no—
213D244—42no—
214C8×C61—42no—
215#92418—41no—
216#92418—41no—
217#24481—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—24no—
221C22×C61—24no—
222C22×C61#46314yes—
223C22×C61C2×#12111yescharacteristic
224C22×C61#54612yes—
225C22×C61C2×#13511yescharacteristic
226C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C22×C61#57812yes—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—82no—
232C2×D61—82no—
233C4×C61—22no—
234C4×C61—44no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61#52511yescharacteristic
238C4×C61#53511yescharacteristic
239C4×C61#55112yes—
240Dic61—42no—
241Dic61—42no—
242Dic61—81no—
243Dic61—82no—
244C2×#480—611no—
245C22×#103D6111yescharacteristic, complemented
246#2436—611no—
247#2344—611no—
248#2068—612no—
249#2042—612no—
250#1983—612no—
251#1928—612no—
252#1893—611no—
253#1850—611no—
254#1787—611no—
255C2×C61—24no—
256C2×C61#241111yescharacteristic
257C2×C61C2×#46311yescharacteristic
258C2×C61#304411yescharacteristic
259C2×C61#330012yes—
260C2×C61#330812yes—
261D61—161no—
262C23×Q8C2×D6111yescharacteristic
263C23×C8C2×D6111yescharacteristic
264C2×#124—612no—
265C2×#121—611no—
266C2×#117—1221no—
267C22×C4⋊C4C2×D6111yescharacteristic
268#480—1224no—
269C2×#107—611no—
270C2×#107—611no—
271C2×C4⋊C8—612no—
272C2×#103—24no—
273C2×#103—612no—
274C2×#103—612no—
275C2×#103C2×D6112yes—
276C2×#103C2×D6112yes—
277C2×#102—611no—
278C2×#102—611no—
279C2×#102—612no—
280C2×C22⋊C8—611no—
281#393—1222no—
282#391—612no—
283#389—612no—
284#387—1222no—
285#386—1221no—
286#386—1221no—
287#381—1221no—
288#380—611no—
289C4×C4⋊C4—612no—
290C4×C22⋊C4—611no—
291#340—612no—
292#336—612no—
293C61#2448111yescharacteristic, complemented, Sylow
294C22×Q8—21no—
295C22×Q8—24no—
296C22×Q8—22no—
297C22×Q8—22no—
298C22×Q8—42no—
299C22×Q8C22×D6112yes—
300C22×D4—611no—
301C22×D4—1221no—
302C23×C4—21no—
303C23×C4—21no—
304C23×C4C22×D6111yescharacteristic
305C22×C8—22no—
306C22×C8—42no—
307C22×C8—611no—
308C22×C8—611no—
309C22×C8C22×D6112yes—
310#124—1224no—
311#124—2442no—
312#121—1222no—
313#121—1222no—
314#117—2444no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4C22⋊D6114yes—
321C2×C4⋊C4C22×D6111yescharacteristic
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C22⋊C4—611no—
324C2×C22⋊C4—611no—
325C2×C22⋊C4—1222no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1221no—
329C2×C42—612no—
330C2×C42—1222no—
331#107—1222no—
332#107—1222no—
333C4⋊C8—2442no—
334#103—24no—
335#103—24no—
336#103—1224no—
337#103—2442no—
338#102—1222no—
339#102—1222no—
340#102—2442no—
341C22⋊C8—1222no—
342#95—612no—
343#95—1222no—
344#95—1222no—
345#95—1222no—
346#95—1222no—
347#95—1221no—
348C24—1221no—
349C24D4×D6111yescharacteristic
350C2×Q8—22no—
351C2×Q8—28no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—42no—
360C2×Q8#1175822214yes—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1222no—
364C2×D4—1222no—
365C2×D4—2442no—
366C2×D4—2442no—
367C2×D4—2442no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—612no—
379C22×C4—612no—
380C22×C4—1221no—
381C22×C4—1222no—
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822412yes—
384C22×C4C2×C22⋊D6112yes—
385C22×C4C23×D6111yescharacteristic
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—44no—
390C2×C8—1221no—
391C2×C8—1221no—
392C2×C8—1222no—
393C2×C8—1222no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—24no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1222no—
400C4⋊C4—1222no—
401C4⋊C4—2444no—
402C4⋊C4C2×C22⋊D6112yes—
403C4⋊C4C2×C22⋊D6112yes—
404C22⋊C4—1222no—
405C22⋊C4—1222no—
406C22⋊C4—2444no—
407C22⋊C4—2444no—
408C22⋊C4—2442no—
409C22⋊C4—2442no—
410C42—1222no—
411C42—1222no—
412C42—2444no—
413C23—22no—
414C23—22no—
415C23—2441no—
416C23—2441no—
417C23—2441no—
418C23—2442no—
419C23—2442no—
420C23#1451974912yes—
421C23#1451979812yes—
422C23#1451980412yes—
423C23C2×D4×D6111yescharacteristic
424Q8—24no—
425Q8—24no—
426Q8—48no—
427Q8—42no—
428Q8—42no—
429D4—4884no—
430D4—2441no—
431D4—2441no—
432D4—2442no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—48no—
441C2×C4—48no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2442no—
457C2×C4—2441no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4#1451974912yes—
461C2×C4C22×C22⋊D6111yescharacteristic
462C2×C4C2×#1175822411yescharacteristic
463C2×C4C2×#1175822212yes—
464C2×C4#1451982212yes—
465C8—42no—
466C8—2441no—
467C8—2441no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—22no—
472C22—4882no—
473C22—4881no—
474C22—4882no—
475C22—4881no—
476C22—4881no—
477C22#1683997914yes—
478C22C2×#1451974911yescharacteristic
479C22#1684040212yes—
480C22C2×#1451979811yescharacteristic
481C22C2×#1451980411yescharacteristic
482C22#1684057512yes—
483C4—22no—
484C4—42no—
485C4—42no—
486C4—44no—
487C4—4881no—
488C4—4882no—
489C4—2442no—
490C4—2442no—
491C4#1684027311yescharacteristic
492C4#1684032911yescharacteristic
493C4#1684043812yes—
494C2—24no—
495C2—9761no—
496C2#2144920411yescharacteristic
497C2C2×#1683997911yescharacteristic
498C2#2145534611yescharacteristic
499C2#2145770212yes—
500C2#2145780612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC22×#103D61
hypocenterC61#24481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. #21448688C2semidirect
4. #21447210C2semidirect
5. #21447057C2semidirect
6. #21446748C2semidirect
7. #21446463C2semidirect
9. #21445997C2semidirect
10. #21445754C2semidirect
11. C22×#103×C61C2semidirect
245. C22×#103D61semidirect
293. C61#24481semidirect

Maximal subgroups

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

subgroupindex
C2×#168400752
#214494732
#214486882
#214472102
#214470572
#214467482
#214464632
#214462472
#214459972
#214457542
C22×#103×C612
#2448161