Finite Group Databasebeta

← D61×#1271 #29567945

Subgroups — D61×#1271 #29567945

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#389×D61C211yesmaximal, characteristic
2#336×D61C211yesmaximal, characteristic
3#336×D61C211yesmaximal, characteristic
4#21443643C211yesmaximal, characteristic, complemented
5#21443642C211yesmaximal, characteristic, complemented
6#21443641C211yesmaximal, characteristic, complemented
7C61×#1271C211yesmaximal, characteristic, complemented
8C22×C8×D61—21no—
9C22×C8×D61—21no—
10C2×C4⋊C4×D61—21no—
11C2×C4⋊C4×D61C411yescharacteristic
12C2×C4⋊C4×D61C411yescharacteristic
13C2×C42×D61C2211yescharacteristic
14#16839661C411yescharacteristic
15#16839660C2211yescharacteristic
16#16839659—21no—
17#16839658C411yescharacteristic
18#16839657—21no—
19#95×D61—21no—
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22#16839516—21no—
23#16839516—21no—
24C61×#389C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×C4×D61—21no—
28C22×C4×D61—21no—
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C22×#92418—21no—
33C22×#92418—21no—
34C2×C8×D61—24no—
35C2×C8×D61—24no—
36C2×C8×D61—42no—
37C2×C8×D61—42no—
38C2×C8×D61—42no—
39C2×C8×D61—42no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C4⋊C4×D61—22no—
48C4⋊C4×D61—42no—
49C4⋊C4×D61—42no—
50C4⋊C4×D61—42no—
51C4⋊C4×D61C814yescomplemented
52C2×C4×Dic61C2×C411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C42×D61—42no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59C22×C8×C61—21no—
60C22×C8×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C4⋊C4×C61C2×C411yescharacteristic
64C2×C42×C61C2311yescharacteristic
65#95×C61—21no—
66C23×D61C22⋊C411yescharacteristic
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—24no—
84C2×C4×D61—42no—
85C2×C4×D61—42no—
86C2×C4×D61—42no—
87C2×C4×D61—42no—
88C2×C4×D61—42no—
89C2×C4×D61C2×C812yes—
90C2×C4×D61M1612yes—
91C2×C4×D61D812yes—
92C2×C4×D61SD1612yes—
93C2×C4×D61SD1612yes—
94C2×C4×D61Q1612yes—
95C2×C8×C61—22no—
96C2×C8×C61—22no—
97C2×C8×C61—41no—
98C2×C8×C61—41no—
99C2×C8×C61—41no—
100C2×C8×C61—41no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61C2×C812yescomplemented
106C42×C61—41no—
107C4⋊Dic61—21no—
108C4⋊Dic61—42no—
109C4⋊Dic61—41no—
110C4⋊Dic61C2×C812yescomplemented
111#11758196—24no—
112#11758196—42no—
113#11758196—41no—
114#11758196—41no—
115#11758196—42no—
116C4×Dic61—41no—
117C4×Dic61—41no—
118C4×Dic61—41no—
119C2×#92418—22no—
120C2×#92418—22no—
121C2×#92418—41no—
122C2×#92418—41no—
123C2×#92418—41no—
124C2×#92418—41no—
125C8×D61—48no—
126C8×D61—48no—
127C23×C61C2×C22⋊C411yescharacteristic
128C22×D61C22⋊C812yes—
129C22×D61#9912yes—
130C22×D61#10112yes—
131C22×D61#10212yes—
132C22×D61#10312yes—
133C22×D61#10412yes—
134C22×D61#10412yes—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61C22×C811yescharacteristic
148C2×C4×C61C2×M1611yescharacteristic
149C2×C4×C61C2×D811yescharacteristic
150C2×C4×C61C2×SD1611yescharacteristic
151C2×C4×C61C2×SD1611yescharacteristic
152C2×C4×C61C2×Q1611yescharacteristic
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61C22⋊C812yes—
166C2×Dic61#10212yes—
167C2×Dic61#10312yes—
168C4×D61—24no—
169C4×D61—24no—
170C4×D61—24no—
171C4×D61—24no—
172C4×D61—24no—
173C4×D61—24no—
174C4×D61—48no—
175C4×D61—84no—
176C8×C61—42no—
177C8×C61—42no—
178#92418—42no—
179#92418—42no—
180C2×#1271—611nomaximal, Sylow
181C22×C61C2×C22⋊C811yescharacteristic
182C22×C61C2×#9911yescharacteristic
183C22×C61C2×#10111yescharacteristic
184C22×C61C2×#10211yescharacteristic
185C22×C61C2×#10311yescharacteristic
186C22×C61C2×#10411yescharacteristic
187C22×C61C2×#10411yescharacteristic
188C2×D61#32414yes—
189C2×D61D4⋊C814yes—
190C2×D61Q8⋊C814yes—
191C2×D61#32714yes—
192C2×D61#32814yes—
193C2×D61#33014yes—
194C2×D61#33214yes—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—42no—
202C4×C61—81no—
203Dic61—22no—
204Dic61—22no—
205Dic61—22no—
206Dic61—42no—
207Dic61—81no—
208C2×#389—611no—
209C2×#336—611no—
210C2×#336—611no—
211#1271—611no—
212#1271—611no—
213#1271—611no—
214#1271D6111yescharacteristic, direct product factor
215C2×C61C2×#32411yescharacteristic
216C2×C61C2×D4⋊C811yescharacteristic
217C2×C61C2×Q8⋊C811yescharacteristic
218C2×C61C2×#32711yescharacteristic
219C2×C61C2×#32811yescharacteristic
220C2×C61C2×#33011yescharacteristic
221C2×C61C2×#33211yescharacteristic
222D61#127118yesdirect product factor
223C23×C8—1221no—
224C23×C8—1221no—
225C22×C4⋊C4—611no—
226C22×C4⋊C4—611no—
227C22×C4⋊C4—1221no—
228C22×C42—611no—
229#389—611no—
230#389—611no—
231#389—611no—
232#389—1221no—
233#389—1221no—
234#389C2×D6111yescharacteristic
235C2×#95—1221no—
236#336—611no—
237#336—611no—
238#336—1221no—
239#336—1221no—
240#336C2×D6111yescharacteristic
241#336C2×D6111yescharacteristic
242C61C2×#127111yescharacteristic, complemented, Sylow
243C23×C4—611no—
244C23×C4—611no—
245C23×C4—611no—
246C23×C4—1221no—
247C23×C4—1221no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—1224no—
251C22×C8—1224no—
252C22×C8—1221no—
253C22×C8—1221no—
254C22×C8—2442no—
255C22×C8—2442no—
256C22×C8—2442no—
257C22×C8—2442no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4—614no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—1222no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4—2442no—
269C2×C4⋊C4—2442no—
270C2×C4⋊C4—2442no—
271C2×C4⋊C4C4×D6111yescharacteristic
272C2×C4⋊C4C4×D6111yescharacteristic
273C2×C42—611no—
274C2×C42—611no—
275C2×C42—611no—
276C2×C42—2442no—
277C2×C42C22×D6111yescharacteristic
278#95—21no—
279#95—1221no—
280#95—1221no—
281#95—1221no—
282C24—611no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—611no—
286C22×C4—611no—
287C22×C4—611no—
288C22×C4—612no—
289C22×C4—612no—
290C22×C4—612no—
291C22×C4—612no—
292C22×C4—612no—
293C22×C4—612no—
294C22×C4—1222no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—1222no—
300C22×C4—1224no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—2442no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4—2442no—
307C22×C4—2442no—
308C22×C4C2×C4×D6111yescharacteristic
309C22×C4D4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8—2448no—
320C2×C8—2448no—
321C2×C8—2441no—
322C2×C8—2441no—
323C2×C8—2441no—
324C2×C8—2441no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—612no—
330C4⋊C4—1224no—
331C4⋊C4—1221no—
332C4⋊C4—2441no—
333C4⋊C4—2442no—
334C4⋊C4—2442no—
335C4⋊C4—2442no—
336C4⋊C4—2441no—
337C4⋊C4—2441no—
338C4⋊C4C8×D6112yescomplemented
339C42—41no—
340C42—2441no—
341C42—2441no—
342C42—2441no—
343C23—612no—
344C23—612no—
345C23—612no—
346C23—612no—
347C23—612no—
348C23—612no—
349C23—612no—
350C23C22⋊C4×D6111yescharacteristic
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—612no—
364C2×C4—612no—
365C2×C4—612no—
366C2×C4—1224no—
367C2×C4—1224no—
368C2×C4—1224no—
369C2×C4—1224no—
370C2×C4—1224no—
371C2×C4—1224no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1222no—
379C2×C4—4884no—
380C2×C4—2448no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4C2×C8×D6111yescharacteristic
387C2×C4M16×D6111yescharacteristic
388C2×C4D8×D6111yescharacteristic
389C2×C4SD16×D6111yescharacteristic
390C2×C4SD16×D6111yescharacteristic
391C2×C4Q16×D6111yescharacteristic
392C8—42no—
393C8—42no—
394C8—2442no—
395C8—2442no—
396C22—614no—
397C22—614no—
398C22—614no—
399C22—614no—
400C22—614no—
401C22—614no—
402C22—614no—
403C22C22⋊C8×D6111yescharacteristic
404C22#99×D6111yescharacteristic
405C22#101×D6111yescharacteristic
406C22#102×D6111yescharacteristic
407C22#103×D6111yescharacteristic
408C22#104×D6111yescharacteristic
409C22#104×D6111yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—1222no—
419C4—1222no—
420C4—1222no—
421C4—4881no—
422C4—2442no—
423C2—618no—
424C2#324×D6111yescharacteristic
425C2D4⋊C8×D6111yescharacteristic
426C2Q8⋊C8×D6111yescharacteristic
427C2#327×D6111yescharacteristic
428C2#328×D6111yescharacteristic
429C2#330×D6111yescharacteristic
430C2#332×D6111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#1271C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1271D61
hypocenterC61C2×#1271
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #21443643C2semidirect
5. #21443642C2semidirect
6. #21443641C2semidirect
7. C61×#1271C2semidirect
51. C4⋊C4×D61C8semidirect
105. C4⋊C4×C61C2×C8semidirect
110. C4⋊Dic61C2×C8semidirect
214. #1271D61direct (N × H)
222. D61#1271direct (N × H)
242. C61C2×#1271semidirect
338. C4⋊C4C8×D61semidirect

Maximal subgroups

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

subgroupindex
#389×D612
#336×D612
#214436432
#214436422
#214436412
C61×#12712
C2×#127161