Finite Group Databasebeta

← #29730155

Subgroups — #29730155

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840048C211yesmaximal, characteristic, complemented
2C2×#16839827C211yesmaximal, characteristic
3#21450814C211yesmaximal, characteristic
4#21450742C211yesmaximal, characteristic
5#21447254C212yesmaximal, complemented
6#21447017C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446241C212yesmaximal, complemented
9#21445841C212yesmaximal, complemented
10C4×#14519700C211yesmaximal, characteristic, complemented
11C61×#1948C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519765—21no—
15C2×#14519757C2212yes—
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16840048—24no—
19#16839827—24no—
20C2×#14519709C2212yes—
21C2×#14519702C2212yes—
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519689C2212yes—
25C2×C4×#92418C2211yescharacteristic
26#16839689—22no—
27#16839684—22no—
28#16839668—21no—
29#16839663—21no—
30#16839661—21no—
31#16839619C2212yes—
32#16839619C2212yes—
33#16839619C2212yes—
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2212yes—
37#16839524C2212yes—
38C2×#106×C61C2212yes—
39C2×C4⋊C8×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122—21no—
44C22×Dic122C2311yescharacteristic
45C22×D244D411yescharacteristic
46C22×C4×D61C2312yes—
47C22×#92418C2311yescharacteristic
48C22×#92418C2311yescharacteristic
49C2×C8×D61—24no—
50C2×C8×D61—24no—
51C2×C8×D61—42no—
52#14519777—22no—
53#14519777—22no—
54#14519765—44no—
55#14519757—24no—
56#14519757—42no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×D244—22no—
60C4×D244—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—21no—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198D412yes—
72C2×C4×Dic61—21no—
73#14519709—24no—
74#14519702—24no—
75#14519700—22no—
76#14519700—22no—
77#14519689—24no—
78C4×#92418—22no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C42×C61—21no—
89C2×C42×C61C2311yescharacteristic
90#106×C61—42no—
91C4⋊C8×C61D414yes—
92#95×C61—21no—
93C23×D61C2×D412yes—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—22no—
102C22×Dic61—21no—
103C22×Dic61C2×D412yes—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244SD1614yes—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×Dic122—21no—
114C2×Dic122—21no—
115C2×Dic122—21no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—42no—
119C2×Dic122—42no—
120C2×Dic122—42no—
121C2×Dic122#4112yes—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61C2×D414yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—42no—
132C4⋊C4×C61—81no—
133C42×C61—42no—
134C42×C61C2×D412yes—
135C42×C61#4112yes—
136#11758198—24no—
137#11758198—44no—
138#11758198—44no—
139#11758198—44no—
140#11758198—82no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—44no—
147C4⋊Dic61—42no—
148#11758196—42no—
149C4×Dic61—42no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418#4112yes—
156C8×D61—48no—
157C23×C61C22×D411yescharacteristic
158C22×D61—22no—
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—22no—
163C22×D61—22no—
164C22×D61C2×SD1614yes—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—42no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61#12112yes—
177C2×C4×C61#12112yes—
178C2×C4×C61C2×SD1612yes—
179C2×C4×C61#13512yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C2×#4111yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×C4×C61H(5,𝔽2)11yescharacteristic
185C2×C4×C612−1+411yescharacteristic
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198D244—28no—
199C4×D61—42no—
200C4×D61—42no—
201C4×D61—42no—
202C4×D61—42no—
203Dic122—22no—
204Dic122—22no—
205Dic122—41no—
206Dic122—41no—
207Dic122—84no—
208C8×C61—24no—
209#92418—22no—
210#92418—41no—
211#25341—611nomaximal, Sylow
212C22×C61C2×#12111yescharacteristic
213C22×C61#54611yescharacteristic
214C22×C61#54711yescharacteristic
215C22×C61C22×SD1611yescharacteristic
216C22×C61C2×#13511yescharacteristic
217C22×C61#57711yescharacteristic
218C22×C61#57811yescharacteristic
219C2×D61—24no—
220C2×D61—24no—
221C2×D61—24no—
222C2×D61—42no—
223C2×D61—42no—
224C2×D61—42no—
225C2×D61—42no—
226C4×C61—22no—
227C4×C61—81no—
228C4×C61—81no—
229C4×C61—82no—
230C4×C61#46514yes—
231C4×C61#52911yescharacteristic
232C4×C61#54211yescharacteristic
233C4×C61#54611yescharacteristic
234C4×C61#54811yescharacteristic
235Dic61—42no—
236Dic61—42no—
237Dic61—82no—
238Dic61—81no—
239C2×#475—611no—
240C2×C8×D4—611no—
241#2547—611no—
242#2538—611no—
243#2077—612no—
244#2035—612no—
245#1990—611no—
246#1948D6111yescharacteristic, complemented
247#1892—612no—
248#1807—612no—
249C4×#103—611no—
250C2×C61#242711yescharacteristic
251C2×C61C2×#46511yescharacteristic
252C2×C61#306311yescharacteristic
253C2×C61#329811yescharacteristic
254C2×C61#330411yescharacteristic
255C2×C61#333511yescharacteristic
256C2×C61#334211yescharacteristic
257D61—44no—
258C23×C8—612no—
259C2×C4⋊Q8—611no—
260C2×#124—1221no—
261C2×C22⋊Q8—612no—
262C2×C4×Q8—611no—
263C2×C4×D4—611no—
264#475—1224no—
265C8×D4—1224no—
266C2×#106—612no—
267C2×#106C2×D6112yes—
268C2×C4⋊C8C2×D6111yescharacteristic
269C2×#103—612no—
270C2×#103—611no—
271C2×#103—611no—
272C2×C22⋊C8—612no—
273C2×C4×C8—611no—
274#397—1222no—
275#396—1222no—
276#391—1221no—
277#390—1221no—
278#389—1221no—
279#384C2×D6111yescharacteristic
280#380—612no—
281#380—612no—
282#380—612no—
283C4×C4⋊C4C2×D6111yescharacteristic
284#340—611no—
285#340—612no—
286#340—611no—
287#340—612no—
288#340C2×D6112yes—
289C61#2534111yescharacteristic, complemented, Sylow
290C22×Q8—611no—
291C22×Q8—1221no—
292C22×D4—611no—
293C23×C4—612no—
294C22×C8—611no—
295C22×C8—611no—
296C22×C8—1224no—
297C22×C8—1224no—
298C22×C8—2442no—
299C22×C8C22×D6112yes—
300C4⋊Q8—1222no—
301C4⋊Q8—1222no—
302#124—2444no—
303C22⋊Q8—1224no—
304C22⋊Q8—2442no—
305C4×Q8—1222no—
306C4×Q8—1222no—
307C4×D4—1222no—
308C4×D4—1222no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—612no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4C22×D6112yes—
318C2×C4⋊C4C22×D6111yescharacteristic
319C2×C4⋊C4C22×D6111yescharacteristic
320C2×C22⋊C4—612no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C42—21no—
326C2×C42—1221no—
327C2×C42C22×D6111yescharacteristic
328#106—42no—
329#106—1224no—
330C4⋊C8D24414yes—
331#103—1222no—
332#103—1222no—
333#103—1224no—
334C22⋊C8—1224no—
335C4×C8—1222no—
336#95—21no—
337#95—1222no—
338#95—1222no—
339#95—1222no—
340C24—612no—
341C2×Q8—612no—
342C2×Q8—1221no—
343C2×Q8—1221no—
344C2×Q8—1221no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×Q8—2442no—
348C2×Q8—2442no—
349C2×Q8—2442no—
350C2×D4—614no—
351C2×D4—1222no—
352C2×D4—1222no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—612no—
357C22×C4—1221no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1221no—
365C22×C4—1222no—
366C22×C4D4×D6111yescharacteristic
367C22×C4D4×D6111yescharacteristic
368C22×C4C23×D6111yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—612no—
372C2×C8—1221no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8—1221no—
376C2×C8—1221no—
377C2×C8—2448no—
378C2×C8C2×D24414yes—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—81no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1222no—
392C4⋊C4—2444no—
393C4⋊C4—2442no—
394C4⋊C4—2442no—
395C22⋊C4—1224no—
396C22⋊C4—4882no—
397C22⋊C4—2444no—
398C22⋊C4—2444no—
399C22⋊C4—2444no—
400C42—42no—
401C42—2442no—
402C42C2×D24412yes—
403C42#1175822612yes—
404C23—614no—
405C23—1222no—
406C23—1222no—
407C23—1222no—
408C23—1222no—
409C23—1222no—
410C23—1222no—
411C23C2×D4×D6111yescharacteristic
412Q8—1222no—
413Q8—1222no—
414Q8—4884no—
415Q8—2441no—
416Q8—2441no—
417D4—1228no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—1222no—
433C2×C4—2441no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4#1451975212yes—
446C2×C4#1451975212yes—
447C2×C4SD16×D6112yes—
448C2×C4#1451979812yes—
449C2×C4C22×D24411yescharacteristic
450C2×C4#41×D6111yescharacteristic
451C2×C4#41×D6111yescharacteristic
452C2×C4C2×#1175822611yescharacteristic
453C2×C4#1451982211yescharacteristic
454C2×C4#1451982511yescharacteristic
455C8—24no—
456C8—1222no—
457C8—2441no—
458C22—1224no—
459C22—1224no—
460C22—1224no—
461C22—2442no—
462C22—2442no—
463C22—2442no—
464C22—2442no—
465C22C2×#1451975211yescharacteristic
466C22#1684040911yescharacteristic
467C22#1684041411yescharacteristic
468C22C2×SD16×D6111yescharacteristic
469C22C2×#1451979811yescharacteristic
470C22#1684057211yescharacteristic
471C22#1684057811yescharacteristic
472C4—22no—
473C4—81no—
474C4—82no—
475C4—81no—
476C4—4882no—
477C4—4881no—
478C4—2442no—
479C4—2442no—
480C4#1683999214yes—
481C4#1684029711yescharacteristic
482C4#1684038611yescharacteristic
483C4#1684040711yescharacteristic
484C4#1684042411yescharacteristic
485C2—2444no—
486C2#2144939411yescharacteristic
487C2C2×#1683999211yescharacteristic
488C2#2145551111yescharacteristic
489C2#2145768711yescharacteristic
490C2#2145776111yescharacteristic
491C2#2145817311yescharacteristic
492C2#2145824811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1948C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1948D61
hypocenterC61#25341
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
1. C2×#16840048C2semidirect
5. #21447254C2semidirect
6. #21447017C2semidirect
7. #21446779C2semidirect
8. #21446241C2semidirect
9. #21445841C2semidirect
10. C4×#14519700C2semidirect
11. C61×#1948C2semidirect
246. #1948D61semidirect
289. C61#25341semidirect

Maximal subgroups

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

subgroupindex
C2×#168400482
C2×#168398272
#214508142
#214507422
#214472542
#214470172
#214467792
#214462412
#214458412
C4×#145197002
C61×#19482
#2534161