Finite Group Databasebeta

← #29683685

Subgroups — #29683685

Order 15616.

1638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic
2C2×#16839762C214yesmaximal, complemented
3C2×#16839714C211yesmaximal, characteristic, complemented
4#382×D61C211yesmaximal, characteristic, complemented
5#21446643C211yesmaximal, characteristic, complemented
6#21446642C211yesmaximal, characteristic, complemented
7#21446640C211yesmaximal, characteristic, complemented
8#21446627C214yesmaximal, complemented
9C61×#1960C211yesmaximal, characteristic, complemented
10C22×#11758226C2212yes—
11C22×#11758189C2212yes—
12#16840313—24no—
13#16840313C418yescomplemented
14C2×#14519774C2211yescharacteristic
15C2×#14519766C2212yes—
16C2×#14519752—22no—
17C2×#14519739—22no—
18C2×C4⋊C4×D61C2212yes—
19C2×C42×D61C2211yescharacteristic
20#16839762—216no—
21C2×#14519701C2214yes—
22C2×#14519699C2214yes—
23C2×#14519697C2214yes—
24#16839714—24no—
25C2×#14519688C2211yescharacteristic
26C2×C8×Dic61C2212yes—
27#16839628C2211yescharacteristic
28#16839627C2212yes—
29#16839626—22no—
30#95×D61—22no—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C2×#103×C61C2214yes—
33C2×#97×C61C2211yescharacteristic
34C61×#382C2211yescharacteristic
35C2×#11758226—24no—
36C2×#11758226—28no—
37C2×#11758226—28no—
38C2×#11758226C2×C418yes—
39C22×D244—22no—
40C22×D244—22no—
41C22×D244C2312yes—
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61D412yes—
45C22×C4×D61C2311yescharacteristic
46C2×#11758189—24no—
47C2×#11758189—24no—
48C2×#11758189D418yescomplemented
49C22×#92418C2312yes—
50#14519774—22no—
51#14519774C2×C414yescomplemented
52#14519766—24no—
53#14519766C2×C418yescomplemented
54#14519752—216no—
55#14519739—216no—
56C2×C4⋊Dic61C2312yes—
57C2×#11758196—22no—
58C4⋊C4×D61—28no—
59C2×#11758198—24no—
60C2×C4×Dic61D412yes—
61C2×C4×Dic61C2311yescharacteristic
62C42×D61C2×C414yes—
63C42×D61D414yes—
64#14519701—28no—
65#14519699—28no—
66#14519697—28no—
67#14519688—22no—
68C8×Dic61—24no—
69#14519684—22no—
70#14519684—24no—
71C22×Q8×C61C2312yes—
72C22×C8×C61C2312yes—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61C2×C414yescomplemented
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61C2312yes—
77C2×C42×C61C2311yescharacteristic
78#103×C61—28no—
79#97×C61—22no—
80#95×C61—22no—
81C23×D61—22no—
82C23×D61C2×D411yescharacteristic
83C2×Q8×C61—22no—
84C2×Q8×C61—24no—
85C2×Q8×C61—24no—
86C2×Q8×C61C22×C414yes—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2411yescharacteristic
91C22×Dic61—22no—
92C22×Dic61C2×D412yes—
93C22×Dic61C2×D411yescharacteristic
94#11758226—232no—
95#11758226—416no—
96C2×D244—24no—
97C2×D244—24no—
98C2×D244—28no—
99C2×D244—28no—
100C2×D244—22no—
101C2×D244—48no—
102C2×D244—44no—
103C2×D244—44no—
104C2×D244C22×C414yes—
105C2×C4×D61—28no—
106C2×C4×D61—44no—
107C2×C4×D61—44no—
108C2×C4×D61—44no—
109C2×C4×D61—48no—
110C2×C4×D61—44no—
111C2×C4×D61C22⋊C418yes—
112C2×C4×D61C22×C412yes—
113C2×C4×D61C2×D414yes—
114C2×C4×D61C2×D412yes—
115C2×C4×D61#4114yes—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61C2×D414yes—
119C4⋊C4×C61—28no—
120C4⋊C4×C61—24no—
121C42×C61C22×C412yes—
122C42×C61C2×D412yes—
123#11758198—216no—
124C4⋊Dic61—24no—
125#11758196—44no—
126C4×Dic61C22⋊C418yes—
127C4×Dic61C22×C412yes—
128C4×Dic61C2×D412yes—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418C2×D414yes—
132#11758189—216no—
133C23×C61C22×D411yescharacteristic
134C22×D61—24no—
135C22×D61—24no—
136C22×D61—42no—
137C22×D61—42no—
138C22×D61—48no—
139C22×D61C2×C22⋊C412yes—
140C22×D61#12412yes—
141C22×D61#12712yes—
142Q8×C61—28no—
143Q8×C61—44no—
144C2×C4×C61—24no—
145C2×C4×C61—24no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—44no—
150C2×C4×C61C2×C22⋊C414yes—
151C2×C4×C61C23×C411yescharacteristic
152C2×C4×C61C22×D412yes—
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C2×#4112yes—
155C2×Dic61—24no—
156C2×Dic61—42no—
157C2×Dic61—44no—
158C2×Dic61C2×C22⋊C414yes—
159C2×Dic61C2×C22⋊C412yes—
160C2×Dic61#12412yes—
161C2×Dic61#12712yes—
162D244—28no—
163D244—416no—
164D244—416no—
165D244—44no—
166C4×D61—48no—
167C4×D61—416no—
168C4×D61—88no—
169C4×D61C4×D418yes—
170C4×D61#12118yes—
171C8×C61—24no—
172#92418—24no—
173#21201—611nomaximal, Sylow
174C22×C61C22×C22⋊C411yescharacteristic
175C22×C61C2×#12411yescharacteristic
176C22×C61C2×#12711yescharacteristic
177C22×C61#57514yes—
178C2×D61—24no—
179C2×D61—24no—
180C2×D61—24no—
181C2×D61—48no—
182C2×D61—44no—
183C2×D61—84no—
184C2×D61—84no—
185C2×D61#39014yes—
186C4×C61—24no—
187C4×C61—44no—
188C4×C61—82no—
189C4×C61C2×C4×D412yes—
190C4×C61C2×#12112yes—
191Dic61—24no—
192Dic61—82no—
193Dic61#39014yes—
194C2×#532—611no—
195C2×#417—614no—
196C2×C4×M16—611no—
197C2×#382—611no—
198#1961—611no—
199#1961—611no—
200#1960—611no—
201#1960D6111yescharacteristic, complemented
202#1958—614no—
203C2×C61C2×#39011yescharacteristic
204C2×C61#289612yes—
205C2×C61#314212yes—
206C2×C61#315312yes—
207D61—24no—
208D61—84no—
209C22×#41—612no—
210C22×M16—612no—
211#532—618no—
212#532—1224no—
213C2×C4⋊Q8C2×D6111yescharacteristic
214C2×#127—611no—
215C2×#124—612no—
216C2×#121—1222no—
217C2×C4×D4—1222no—
218C22×C4⋊C4—612no—
219C22×C42—611no—
220#417—12216no—
221C2×#103—614no—
222C2×#103C2×D6114yes—
223C2×#102—614no—
224C2×#102—614no—
225C4×M16—1224no—
226C2×#97—611no—
227C2×#97C2×D6111yescharacteristic
228C2×C4×C8—612no—
229#382—612no—
230#382—611no—
231#382—1222no—
232#382C2×D6111yescharacteristic
233C2×#95—1222no—
234C61#2120111yescharacteristic, complemented, Sylow
235C2×#41—618no—
236C2×#41—1224no—
237C2×#41—1228no—
238C2×#41—1228no—
239C22×Q8C22×D6112yes—
240C22×D4—612no—
241C22×D4—1222no—
242C22×D4—1222no—
243C23×C4—611no—
244C23×C4—612no—
245C23×C4—1222no—
246C23×C4—1222no—
247C2×M16—618no—
248C2×M16—1224no—
249C2×M16—1224no—
250C22×C8—612no—
251C22×C8C22×D6112yes—
252C4⋊Q8—22no—
253C4⋊Q8C4×D6114yescomplemented
254#127—614no—
255#127—1222no—
256#124—618no—
257#124—1224no—
258#121—12216no—
259C4×D4—12216no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—612no—
262C2×C4⋊C4—1228no—
263C2×C4⋊C4—1222no—
264C2×C4⋊C4C22×D6112yes—
265C2×C22⋊C4—1224no—
266C2×C42—612no—
267C2×C42—611no—
268C2×C42—614no—
269C2×C42—614no—
270C2×C42C22×D6111yescharacteristic
271#103—28no—
272#103—1228no—
273#102—1228no—
274#102—1228no—
275#97—22no—
276#97—1222no—
277C4×C8—1224no—
278#95—22no—
279#95—1224no—
280#95—1222no—
281C24—611no—
282C24—1222no—
283#41—12232no—
284#41—24416no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8C2×C4×D6114yes—
289C2×D4—614no—
290C2×D4—1228no—
291C2×D4—1224no—
292C2×D4—1222no—
293C2×D4—1224no—
294C2×D4—1228no—
295C2×D4—2448no—
296C2×D4—2444no—
297C2×D4—2444no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—611no—
301C22×C4—612no—
302C22×C4—612no—
303C22×C4—614no—
304C22×C4—614no—
305C22×C4—612no—
306C22×C4—618no—
307C22×C4—1228no—
308C22×C4—1222no—
309C22×C4—2444no—
310C22×C4—2444no—
311C22×C4—2444no—
312C22×C4—2444no—
313C22×C4—2448no—
314C22×C4D4×D6112yes—
315C22×C4C23×D6111yescharacteristic
316M16—12216no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—614no—
320C2×C8—1222no—
321C2×C8—1222no—
322C2×C8D4×D6114yes—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—1224no—
326C4⋊C4—2444no—
327C22⋊C4—12216no—
328C42—612no—
329C42—612no—
330C42—618no—
331C42C2×C4×D6112yes—
332C42D4×D6112yes—
333C23—612no—
334C23—612no—
335C23—612no—
336C23—1224no—
337C23—1224no—
338C23—2442no—
339C23—2442no—
340C23—2448no—
341C23C2×D4×D6111yescharacteristic
342Q8—28no—
343Q8—44no—
344D4—1228no—
345D4—24416no—
346D4—24416no—
347D4—2444no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—612no—
355C2×C4—612no—
356C2×C4—612no—
357C2×C4—618no—
358C2×C4—618no—
359C2×C4—614no—
360C2×C4—1224no—
361C2×C4—4888no—
362C2×C4—2448no—
363C2×C4—24416no—
364C2×C4—2442no—
365C2×C4—2444no—
366C2×C4C22⋊C4×D6114yes—
367C2×C4C22×C4×D6111yescharacteristic
368C2×C4C2×D4×D6112yes—
369C2×C4C2×D4×D6111yescharacteristic
370C2×C4#41×D6112yes—
371C8—24no—
372C8—1224no—
373C22—614no—
374C22—1224no—
375C22—1224no—
376C22—1224no—
377C22—4884no—
378C22—4884no—
379C22—2448no—
380C22—2444no—
381C22C2×C22⋊C4×D6111yescharacteristic
382C22#124×D6111yescharacteristic
383C22#127×D6111yescharacteristic
384C22#1684056114yes—
385C4—24no—
386C4—44no—
387C4—82no—
388C4—614no—
389C4—1224no—
390C4—4882no—
391C4C4×D4×D6112yes—
392C4#121×D6112yes—
393C2—1224no—
394C2—4884no—
395C2#390×D6111yescharacteristic
396C2#2145433712yes—
397C2#2145622112yes—
398C2#2145628212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1960C2
socleC23×C61C22×D4
prime radicalC2×#16840313C2
cosocleC22×C4×C61C24
hypercenter#1960D61
hypocenterC61#21201
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
2. C2×#16839762C2semidirect
3. C2×#16839714C2semidirect
4. #382×D61C2semidirect
5. #21446643C2semidirect
6. #21446642C2semidirect
7. #21446640C2semidirect
8. #21446627C2semidirect
9. C61×#1960C2semidirect
13. #16840313C4semidirect
48. C2×#11758189D4semidirect
51. #14519774C2×C4semidirect
53. #14519766C2×C4semidirect
74. C4⋊Q8×C61C2×C4semidirect
201. #1960D61semidirect
234. C61#21201semidirect
253. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403132
C2×#168397622
C2×#168397142
#382×D612
#214466432
#214466422
#214466402
#214466272
C61×#19602
#2120161