Finite Group Databasebeta

← #29575922

Subgroups — #29575922

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448729C211yesmaximal, characteristic
2#21444383C211yesmaximal, characteristic, complemented
3#21444318C211yesmaximal, characteristic, complemented
4#21444210C211yesmaximal, characteristic
5#21443720C211yesmaximal, characteristic, complemented
6#21443679C211yesmaximal, characteristic, complemented
7C61×#3077C211yesmaximal, characteristic
8C2×D4×Dic61—21no—
9#16839815C2211yescharacteristic
10#16839815C2211yescharacteristic
11#16839661—21no—
12#16839637—22no—
13#16839628C2211yescharacteristic
14#16839622—21no—
15C4×C4⋊Dic61C2211yescharacteristic
16C2×#14519684—21no—
17#16839499—22no—
18#16839499—21no—
19#16839499—21no—
20D4⋊#92418—22no—
21C61×#533C2211yescharacteristic
22C2×C4×D4×C61C2211yescharacteristic
23C61×#478—22no—
24C61×#476C412yescomplemented
25C61×#474C412yescomplemented
26C61×#428—21no—
27C61×#423C2211yescharacteristic
28C61×#418—21no—
29C23×Dic61—22no—
30D4×Dic61—42no—
31D4×Dic61—42no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—22no—
35C2×C22⋊Dic61—22no—
36C2×C4×Dic61—21no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C22⋊#92418—21no—
40C22⋊#92418—21no—
41C4×#92418—21no—
42C4×#92418—21no—
43#14519684—22no—
44#14519684—21no—
45#14519684D411yescharacteristic
46#14519684D411yescharacteristic
47#14519684D412yes—
48C22×D4×C61D411yescharacteristic
49C23×C4×C61—21no—
50C2×M16×C61—21no—
51C4⋊Q8×C61C2×C411yescharacteristic, complemented
52C4⋊Q8×C61C2×C411yescharacteristic, complemented
53#125×C61—21no—
54C22⋊Q8×C61—21no—
55C22⋊Q8×C61—21no—
56C4×Q8×C61—21no—
57C4×Q8×C61—21no—
58C4×D4×C61—21no—
59C4×D4×C61—21no—
60C4×D4×C61C2×C412yes—
61C4×D4×C61D412yes—
62#117×C61—21no—
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C2×C42×C61C2311yescharacteristic
66#107×C61—22no—
67#106×C61—22no—
68C4⋊C8×C61—21no—
69C4⋊C8×C61C2×C412yescomplemented
70#102×C61—22no—
71#102×C61—22no—
72C24×C61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—21no—
77C2×D4×C61—21no—
78C2×D4×C61C22⋊C412yes—
79C2×D4×C61SD1612yes—
80C2×D4×C61Q1612yes—
81C22×C4×C61—21no—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—82no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×Dic61—21no—
89C22×Dic61—22no—
90C22×Dic61—21no—
91C22×Dic61—44no—
92C22×Dic61—44no—
93C22×Dic61—82no—
94M16×C61—41no—
95M16×C61—41no—
96C2×C8×C61—22no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61C22⋊C412yes—
105C22⋊C4×C61—22no—
106C22⋊C4×C61—22no—
107C22⋊C4×C61—41no—
108C42×C61—41no—
109C42×C61C22×C411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61#4111yescharacteristic
112C42×C61#4111yescharacteristic
113C22⋊Dic61—44no—
114C4⋊Dic61—42no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117C4⋊Dic61—82no—
118C4×Dic61—41no—
119C4×Dic61—41no—
120C2×#92418—41no—
121C2×#92418—41no—
122C23×C61—22no—
123C23×C61—22no—
124C23×C61—22no—
125C23×C61—42no—
126C23×C61—42no—
127C23×C61#12011yescharacteristic
128Q8×C61—81no—
129Q8×C61—81no—
130D4×C61—22no—
131D4×C61—22no—
132D4×C61—22no—
133D4×C61#10314yes—
134C2×C4×C61—22no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—42no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—82no—
141C2×C4×C61—82no—
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×C4×C61C4×D411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12311yescharacteristic
148C2×C4×C61C2×SD1611yescharacteristic
149C2×C4×C61C2×Q1611yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—42no—
158C2×Dic61—42no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—88no—
162C8×C61—42no—
163#92418—81no—
164#92418—81no—
165#9175—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—24no—
169C22×C61—44no—
170C22×C61—44no—
171C22×C61—42no—
172C22×C61—42no—
173C22×C61#38611yescharacteristic
174C22×C61#45011yescharacteristic
175C22×C61#45111yescharacteristic
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—42no—
181C4×C61—81no—
182C4×C61—81no—
183C4×C61C2×#10311yescharacteristic
184C4×C61#41711yescharacteristic
185C4×C61#48311yescharacteristic
186C4×C61#48411yescharacteristic
187Dic61—42no—
188Dic61—81no—
189Dic61—82no—
190#3077D6111yescharacteristic
191#2349—611no—
192#1518—611no—
193#1503—611no—
194#1478—611no—
195#1303—611no—
196#1286—611no—
197C2×C61—21no—
198C2×C61—21no—
199C2×C61—44no—
200C2×C61#187211yescharacteristic
201C2×C61#190411yescharacteristic
202C2×C61#190611yescharacteristic
203#533C2×D6111yescharacteristic
204C2×C4×D4—1221no—
205C2×C4×D4C2×D6111yescharacteristic
206#478—22no—
207#476Dic6112yescomplemented
208#474Dic6112yescomplemented
209#431—611no—
210#431—611no—
211#428—21no—
212#423C2×D6111yescharacteristic
213#418—21no—
214#389—1221no—
215#385—1222no—
216#382—611no—
217#380—1221no—
218C4×C4⋊C4—611no—
219C2×#95—1221no—
220#328—1221no—
221#328—1221no—
222#328—1222no—
223D4⋊C8—1222no—
224C61#917511yescharacteristic, complemented, Sylow
225C22×D4C22⋊D6111yescharacteristic
226C23×C4—21no—
227C23×C4—1222no—
228C2×M16—21no—
229C4⋊Q8C2×Dic6111yescharacteristic, complemented
230C4⋊Q8C2×Dic6111yescharacteristic, complemented
231#125—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—2442no—
239C4×D4—2442no—
240C4×D4C2×Dic6112yes—
241C4×D4C22⋊D6112yes—
242#117—21no—
243C2×C4⋊C4—1222no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4C22⋊D6111yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—1222no—
249C2×C42—1221no—
250C2×C42C22×D6111yescharacteristic
251#107—22no—
252#106—22no—
253C4⋊C8—21no—
254C4⋊C8—1221no—
255C4⋊C8—1221no—
256C4⋊C8C2×Dic6112yescomplemented
257#102—22no—
258#102—22no—
259C22⋊C8—1221no—
260C22⋊C8—1221no—
261C4×C8—1221no—
262C4×C8—1221no—
263#95—611no—
264#95—611no—
265#95—612no—
266#95—1222no—
267#95—1221no—
268C24—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#1175820112yes—
275C2×D4#1175820212yes—
276C2×D4C22⋊Dic6112yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—82no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1221no—
284C22×C4—4882no—
285C22×C4—2444no—
286C22×C4—2444no—
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289C22×C4C2×C22⋊D6111yescharacteristic
290M16—41no—
291M16—41no—
292C2×C8—22no—
293C2×C8—2441no—
294C2×C8—2441no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—4882no—
303C4⋊C4—2442no—
304C4⋊C4—2441no—
305C4⋊C4—2441no—
306C4⋊C4C22⋊Dic6112yes—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—2444no—
311C42—41no—
312C42—2441no—
313C42—2441no—
314C42#1175822411yescharacteristic
315C42#1175822411yescharacteristic
316C42C22×Dic6111yescharacteristic
317C42C2×C22⋊D6111yescharacteristic
318C23—22no—
319C23—22no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23#1451974511yescharacteristic
324Q8—81no—
325Q8—81no—
326D4—22no—
327D4—22no—
328D4—22no—
329D4#1451970114yes—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—1222no—
339C2×C4—4888no—
340C2×C4—2441no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2442no—
344C2×C4—2441no—
345C2×C4—2442no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4C2×C22⋊Dic6111yescharacteristic
349C2×C4D4×Dic6111yescharacteristic
350C2×C4D4×Dic6111yescharacteristic
351C2×C4#1451974911yescharacteristic
352C2×C4#1451974911yescharacteristic
353C2×C4#1451975811yescharacteristic
354C2×C4C2×#1175820111yescharacteristic
355C2×C4C2×#1175820211yescharacteristic
356C2×C4#1451980211yescharacteristic
357C2×C4#1451980711yescharacteristic
358C8—42no—
359C8—4881no—
360C8—4881no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#1683964211yescharacteristic
369C22#1683991211yescharacteristic
370C22#1683991711yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—4882no—
379C4—4881no—
380C4—2442no—
381C4C2×#1451970111yescharacteristic
382C4#1683976311yescharacteristic
383C4#1684009011yescharacteristic
384C4#1684009511yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#2144612711yescharacteristic
389C2#2144630811yescharacteristic
390C2#2144632211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3077C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#3077D61
hypocenterC61#9175
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #21444383C2semidirect
3. #21444318C2semidirect
5. #21443720C2semidirect
6. #21443679C2semidirect
24. C61×#476C4semidirect
25. C61×#474C4semidirect
51. C4⋊Q8×C61C2×C4semidirect
52. C4⋊Q8×C61C2×C4semidirect
69. C4⋊C8×C61C2×C4semidirect
207. #476Dic61semidirect
208. #474Dic61semidirect
224. C61#9175semidirect
229. C4⋊Q8C2×Dic61semidirect
230. C4⋊Q8C2×Dic61semidirect
256. C4⋊C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214487292
#214443832
#214443182
#214442102
#214437202
#214436792
C61×#30772
#917561