Finite Group Databasebeta

← #29718405

Subgroups — #29718405

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×#16840073C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21448686C211yesmaximal, characteristic, complemented
4#21447214C212yesmaximal, complemented
5#21447055C212yesmaximal, complemented
6#21446752C212yesmaximal, complemented
7#21446457C212yesmaximal, complemented
8#21446248C211yesmaximal, characteristic
9#21445999C211yesmaximal, characteristic, complemented
10#21445754C211yesmaximal, characteristic, complemented
11C61×#2436C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519733—21no—
14C22×C4⋊Dic61C2211yescharacteristic
15#16840073—24no—
16C2×#14519710C2211yescharacteristic
17C2×#14519710C2211yescharacteristic
18C2×C4⋊#92418C2212yes—
19C2×#14519701—24no—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2212yes—
27C2×C22⋊#92418C2211yescharacteristic
28#16839660C2212yes—
29#16839647—22no—
30#16839623—21no—
31#16839622C2211yescharacteristic
32C4×C4⋊Dic61C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C23×Q8×C61C2211yescharacteristic
36C2×#124×C61C2212yes—
37C2×#121×C61C2211yescharacteristic
38C61×#393—22no—
39C61×#391C2212yes—
40C61×#386—21no—
41C61×#386—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C23×Dic61—21no—
44C22×#92418—22no—
45C22×#92418—42no—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48C22×#92418C2312yes—
49#14519733—44no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61D414yes—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—22no—
57C2×C22⋊Dic61—21no—
58C2×C4×Dic61—22no—
59#14519710—22no—
60#14519710—22no—
61C4⋊#92418—42no—
62#14519701—24no—
63#14519701—24no—
64#14519701—24no—
65#14519701—42no—
66#14519698—22no—
67#14519698—22no—
68#14519698—42no—
69C22⋊#92418—22no—
70#14519684—22no—
71#14519684—21no—
72#14519684—22no—
73C22×Q8×C61—21no—
74C22×Q8×C61—24no—
75C22×Q8×C61—22no—
76C22×Q8×C61—22no—
77C22×Q8×C61—42no—
78C22×Q8×C61C2312yes—
79C22×D4×C61—21no—
80C22×D4×C61C2311yescharacteristic
81C23×C4×C61—21no—
82C23×C4×C61C2311yescharacteristic
83#124×C61—24no—
84#124×C61—42no—
85#121×C61—22no—
86#121×C61—22no—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C22⋊C4×C61D411yescharacteristic
93C2×C42×C61C2312yes—
94#95×C61—22no—
95#95×C61—22no—
96#95×C61D412yes—
97C24×C61—21no—
98C24×C61C2×D411yescharacteristic
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—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—44no—
122C22×C4×C61—42no—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61#4112yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—24no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—44no—
140C22⋊C4×C61—44no—
141C22⋊C4×C61—42no—
142C42×C61—22no—
143C42×C61—22no—
144C22⋊Dic61—42no—
145C4⋊Dic61—24no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61C2×D412yes—
151C4⋊Dic61C2×D412yes—
152#11758196—44no—
153C4×Dic61—44no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—44no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—41no—
165C23×C61—42no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61—42no—
169C23×C61#12112yes—
170C23×C61#13512yes—
171C23×C61AGL(1,ℤ/8)12yes—
172C23×C61C22×D411yescharacteristic
173Q8×C61—24no—
174Q8×C61—24no—
175Q8×C61—48no—
176Q8×C61—42no—
177Q8×C61—42no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—42no—
181D4×C61—84no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—48no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61#12112yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4112yes—
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61H(5,𝔽2)12yes—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—48no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214#92418—41no—
215#92418—41no—
216#92418—42no—
217#24481—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—24no—
221C22×C61—24no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—82no—
226C22×C61—82no—
227C22×C61#46314yes—
228C22×C61C2×#12111yescharacteristic
229C22×C61#54612yes—
230C22×C61C2×#13511yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61#57812yes—
233C4×C61—22no—
234C4×C61—44no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—81no—
238C4×C61#52511yescharacteristic
239C4×C61#53511yescharacteristic
240C4×C61#55112yes—
241Dic61—42no—
242Dic61—42no—
243Dic61—82no—
244C2×#480—611no—
245C22×#103—611no—
246#2436D6111yescharacteristic
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—161no—
257C2×C61#241111yescharacteristic
258C2×C61C2×#46311yescharacteristic
259C2×C61#304411yescharacteristic
260C2×C61#330012yes—
261C2×C61#330812yes—
262C23×Q8C2×D6111yescharacteristic
263C23×C8—611no—
264C2×#124C2×D6112yes—
265C2×#121C2×D6111yescharacteristic
266C2×#117—1221no—
267C22×C4⋊C4—611no—
268#480—1224no—
269C2×#107—611no—
270C2×#107—611no—
271C2×C4⋊C8—612no—
272C2×#103—612no—
273C2×#103—612no—
274C2×#103—612no—
275C2×#103—612no—
276C2×#103—1224no—
277C2×#102—611no—
278C2×#102—611no—
279C2×#102—612no—
280C2×C22⋊C8—611no—
281#393—22no—
282#391C2×D6112yes—
283#389—612no—
284#387—1222no—
285#386—21no—
286#386—21no—
287#381—1221no—
288#380—611no—
289C4×C4⋊C4—612no—
290C4×C22⋊C4C2×D6111yescharacteristic
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—21no—
301C22×D4C22×D6111yescharacteristic
302C23×C4—21no—
303C23×C4—1221no—
304C23×C4C22×D6111yescharacteristic
305C22×C8—612no—
306C22×C8—611no—
307C22×C8—611no—
308C22×C8—1222no—
309C22×C8—2442no—
310#124—24no—
311#124—42no—
312#121—22no—
313#121—22no—
314#117—2444no—
315C2×C4⋊C4—614no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4C22⋊D6111yescharacteristic
328C2×C22⋊C4C22⋊D6111yescharacteristic
329C2×C42—1222no—
330C2×C42C22×D6112yes—
331#107—1222no—
332#107—1222no—
333C4⋊C8—2442no—
334#103—1224no—
335#103—1224no—
336#103—1224no—
337#103—2442no—
338#102—1222no—
339#102—1222no—
340#102—2442no—
341C22⋊C8—1222no—
342#95—22no—
343#95—22no—
344#95—1222no—
345#95—1221no—
346#95—1222no—
347#95C22⋊D6112yes—
348C24—21no—
349C24C2×C22⋊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#1175822414yes—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—1224no—
377C22×C4—1222no—
378C22×C4—2442no—
379C22×C4—2442no—
380C22×C4D4×D6112yes—
381C22×C4#1175822412yes—
382C22×C4C2×C22⋊D6112yes—
383C22×C4C2×C22⋊D6112yes—
384C22×C4C2×C22⋊D6111yescharacteristic
385C22×C4C23×D6111yescharacteristic
386C2×C8—1221no—
387C2×C8—1221no—
388C2×C8—1222no—
389C2×C8—1222no—
390C2×C8—1222no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—2444no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—612no—
397C4⋊C4—612no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1224no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—2444no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4—2442no—
410C42—22no—
411C42—22no—
412C42—2444no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23#1451974812yes—
421C23#1451979712yes—
422C23#1451980212yes—
423C23C22×C22⋊D6111yescharacteristic
424Q8—24no—
425Q8—24no—
426Q8—48no—
427Q8—42no—
428Q8—42no—
429D4—41no—
430D4—41no—
431D4—42no—
432D4—84no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2448no—
456C2×C4—2441no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4#1451974812yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C2×#1175822412yes—
463C2×C4C2×#1175822411yescharacteristic
464C2×C4#1451982212yes—
465C8—2441no—
466C8—2441no—
467C8—2442no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—22no—
472C22—82no—
473C22—81no—
474C22—82no—
475C22—81no—
476C22—81no—
477C22#1683997414yes—
478C22C2×#1451974811yescharacteristic
479C22#1684040612yes—
480C22C2×#1451979711yescharacteristic
481C22C2×#1451980211yescharacteristic
482C22#1684057712yes—
483C4—22no—
484C4—42no—
485C4—42no—
486C4—44no—
487C4—81no—
488C4—4882no—
489C4—2442no—
490C4—2442no—
491C4#1684026811yescharacteristic
492C4#1684032711yescharacteristic
493C4#1684044312yes—
494C2—24no—
495C2—161no—
496C2#2144920011yescharacteristic
497C2C2×#1683997411yescharacteristic
498C2#2145534511yescharacteristic
499C2#2145770612yes—
500C2#2145781012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2436C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2436D61
hypocenterC61#24481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
2. C22×#14519701C2semidirect
3. #21448686C2semidirect
4. #21447214C2semidirect
5. #21447055C2semidirect
6. #21446752C2semidirect
7. #21446457C2semidirect
9. #21445999C2semidirect
10. #21445754C2semidirect
293. C61#24481semidirect

Maximal subgroups

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

subgroupindex
C2×#168400732
C22×#145197012
#214486862
#214472142
#214470552
#214467522
#214464572
#214462482
#214459992
#214457542
C61×#24362
#2448161