Finite Group Databasebeta

← #29571697

Subgroups — #29571697

Order 15616.

1034 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic122C211yesmaximal, characteristic
2#21445753C212yesmaximal, complemented
3#21444390C212yesmaximal, complemented
4#21444137C211yesmaximal, characteristic
5C61×#1799C211yesmaximal, characteristic
6C23×Dic122C411yescharacteristic, complemented
7C23×Dic122C2211yescharacteristic
8D4×Dic122—24no—
9D4×Dic122—24no—
10C2×#14519777C411yescharacteristic, complemented
11C2×#14519776—21no—
12C2×#14519756C411yescharacteristic, complemented
13C2×#14519756C2211yescharacteristic
14C2×#14519754—21no—
15C2×#14519754—21no—
16C2×C4×Dic122C411yescharacteristic, complemented
17C2×D4×Dic61—21no—
18C2×#14519700—24no—
19C2×C22⋊#92418C2212yes—
20#16839498—24no—
21#16839488—21no—
22#16839486—21no—
23C2×C4×D4×C61C2211yescharacteristic
24C2×#99×C61—21no—
25C61×#380C2212yes—
26C61×#342—21no—
27C22×Dic122—21no—
28C22×Dic122—21no—
29C22×Dic122—22no—
30C22×Dic122—22no—
31C22×Dic122—22no—
32C22×Dic122—44no—
33C22×Dic122—44no—
34C22×Dic122C2×C411yescharacteristic
35C22×Dic122C2×C411yescharacteristic
36C22×Dic122C2×C411yescharacteristic
37C22×Dic122D414yes—
38C23×Dic61—21no—
39C23×Dic61—21no—
40C22×#92418—22no—
41#14519777—22no—
42#14519777—22no—
43#14519776—28no—
44#14519756—22no—
45#14519756—22no—
46#14519756—22no—
47#14519756D414yes—
48#14519754—28no—
49#14519754—28no—
50C4×Dic122—22no—
51C4×Dic122—22no—
52D4×Dic61—28no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×C4×Dic61—21no—
62#14519700—48no—
63C22⋊#92418—22no—
64C22⋊#92418—22no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61C2×C411yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C4×D4×C61—22no—
69C4×D4×C61—22no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61D411yescharacteristic
74C2×C22⋊C4×C61D411yescharacteristic
75C2×C42×C61C2×C411yescharacteristic
76#99×C61—42no—
77#95×C61—22no—
78C24×C61C22⋊C411yescharacteristic
79C24×C61C2×D411yescharacteristic
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61SD1612yes—
83C2×D4×C61Q1612yes—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61—42no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—24no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—24no—
102C22×Dic61—44no—
103C22×Dic61—44no—
104C2×Dic122—21no—
105C2×Dic122—24no—
106C2×Dic122—28no—
107C2×Dic122—24no—
108C2×Dic122—21no—
109C2×Dic122—21no—
110C2×Dic122—24no—
111C2×Dic122—42no—
112C2×Dic122—44no—
113C2×Dic122—48no—
114C2×Dic122—48no—
115C2×Dic122—44no—
116C2×Dic122—44no—
117C2×Dic122—44no—
118C2×Dic122—44no—
119C2×Dic122—42no—
120C2×Dic122C22⋊C412yes—
121C2×Dic122C22⋊C412yes—
122C2×Dic122C22⋊C412yes—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—44no—
125C22⋊C4×C61—22no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—42no—
128C22⋊C4×C61—42no—
129C42×C61—21no—
130C42×C61—21no—
131C22⋊Dic61—24no—
132C22⋊Dic61—24no—
133C4⋊Dic61—24no—
134C4⋊Dic61—24no—
135C4⋊Dic61—21no—
136C4⋊Dic61C22⋊C412yes—
137#11758196—24no—
138#11758196—24no—
139#11758196—24no—
140C4×Dic61—24no—
141C2×#92418—44no—
142C2×#92418—42no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61#10312yescomplemented
150C23×C61C2×C22⋊C411yescharacteristic
151C23×C61#12012yes—
152C23×C61C2×SD1611yescharacteristic
153C23×C61C2×Q1611yescharacteristic
154C23×C61#13712yes—
155D4×C61—24no—
156D4×C61—24no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—24no—
160C2×C4×C61—22no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—82no—
170C2×C4×C61—82no—
171C2×C4×C61—82no—
172C2×C4×C61#10312yescomplemented
173C2×C4×C61C2×C22⋊C411yescharacteristic
174C2×C4×C61C2×C22⋊C411yescharacteristic
175C2×C4×C61#12012yes—
176C2×Dic61—24no—
177C2×Dic61—22no—
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—48no—
182C2×Dic61—48no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186Dic122—24no—
187Dic122—22no—
188Dic122—24no—
189Dic122—48no—
190Dic122—48no—
191Dic122—42no—
192Dic122—42no—
193Dic122—42no—
194Dic122#9914yescomplemented
195#92418—82no—
196#8379—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—22no—
199C22×C61—21no—
200C22×C61—22no—
201C22×C61—22no—
202C22×C61—41no—
203C22×C61—41no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61—41no—
208C22×C61—41no—
209C22×C61#37911yescharacteristic
210C22×C61C2×#10311yescharacteristic
211C22×C61#41911yescharacteristic
212C22×C61#44812yes—
213C22×C61#45112yes—
214C4×C61—22no—
215C4×C61—21no—
216C4×C61—22no—
217C4×C61—41no—
218C4×C61—41no—
219C4×C61—162no—
220C4×C61C2×#9912yes—
221Dic61—24no—
222Dic61—42no—
223Dic61—42no—
224Dic61—42no—
225C2×D4×Q8—611no—
226#1799D6111yescharacteristic
227#1787—612no—
228#1520—612no—
229#1459—611no—
230C2×C61—22no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—22no—
234C2×C61—42no—
235C2×C61#178111yescharacteristic
236C2×C61#178711yescharacteristic
237C2×C61#179611yescharacteristic
238C23×Q8—611no—
239C23×Q8—611no—
240D4×Q8—1224no—
241D4×Q8—1224no—
242C2×C4⋊Q8—611no—
243C2×C4⋊Q8—1221no—
244C2×C22⋊Q8—611no—
245C2×C22⋊Q8—611no—
246C2×C22⋊Q8—1221no—
247C2×C22⋊Q8—1221no—
248C2×C4×Q8—611no—
249C2×C4×D4—1221no—
250C2×C4×D4C2×D6111yescharacteristic
251C2×#103—1224no—
252C2×#99—21no—
253C2×C22⋊C8—612no—
254#380C2×D6112yes—
255#342—21no—
256#328—1224no—
257#324—1221no—
258#323—1221no—
259C61#837911yescharacteristic, complemented, Sylow
260C22×Q8—611no—
261C22×Q8—611no—
262C22×Q8—611no—
263C22×Q8—614no—
264C22×Q8—1222no—
265C22×Q8—1222no—
266C22×Q8—1221no—
267C22×Q8—1221no—
268C22×Q8—1222no—
269C22×Q8—2444no—
270C22×Q8—2444no—
271C22×D4C22⋊D6111yescharacteristic
272C23×C4—1221no—
273C23×C4—1221no—
274C23×C4C4×D6111yescharacteristic
275C23×C4C22×D6111yescharacteristic
276C22×C8—1222no—
277C4⋊Q8—1222no—
278C4⋊Q8—1222no—
279C4⋊Q8—1228no—
280C22⋊Q8—614no—
281C22⋊Q8—1228no—
282C22⋊Q8—1228no—
283C22⋊Q8—1222no—
284C22⋊Q8—1222no—
285C22⋊Q8—1222no—
286C4×Q8—1222no—
287C4×Q8—1222no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—1228no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4D24411yescharacteristic
299C2×C22⋊C4—22no—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4D24411yescharacteristic
303C2×C22⋊C4C22⋊D6111yescharacteristic
304C2×C42—1221no—
305C2×C42C4×D6111yescharacteristic
306#103—2448no—
307#99—42no—
308C22⋊C8—1222no—
309C22⋊C8—1222no—
310#95—22no—
311C24#1175819811yescharacteristic
312C24C2×C22⋊D6111yescharacteristic
313C2×Q8—612no—
314C2×Q8—612no—
315C2×Q8—612no—
316C2×Q8—1224no—
317C2×Q8—1224no—
318C2×Q8—1228no—
319C2×Q8—1221no—
320C2×Q8—1221no—
321C2×Q8—1221no—
322C2×Q8—1224no—
323C2×Q8—2444no—
324C2×Q8—2442no—
325C2×Q8—2442no—
326C2×Q8—2444no—
327C2×Q8—2444no—
328C2×Q8—2444no—
329C2×Q8—2448no—
330C2×Q8—2448no—
331C2×Q8—2444no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4#1175820012yes—
335C2×D4#1175820212yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—1221no—
342C22×C4—1224no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1224no—
347C22×C4—2444no—
348C22×C4—2444no—
349C22×C4#1175819811yescharacteristic
350C22×C4#1175819811yescharacteristic
351C22×C4#1175819811yescharacteristic
352C22×C4C2×C4×D6111yescharacteristic
353C22×C4C2×D24411yescharacteristic
354C22×C4D4×D6112yes—
355C22×C4D4×D6112yes—
356C2×C8—2444no—
357C2×C8—2442no—
358C4⋊C4—22no—
359C4⋊C4—44no—
360C4⋊C4—612no—
361C4⋊C4—1224no—
362C4⋊C4—1224no—
363C4⋊C4—1224no—
364C4⋊C4—1224no—
365C4⋊C4—1221no—
366C4⋊C4—1224no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—1224no—
372C22⋊C4—1224no—
373C42—21no—
374C42—21no—
375C42—1224no—
376C23—21no—
377C23—21no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23#1451970012yescomplemented
383C23C2×#1175819811yescharacteristic
384C23#1451974512yes—
385C23C2×#1175820011yescharacteristic
386C23C2×#1175820211yescharacteristic
387C23#1451980712yes—
388Q8—614no—
389Q8—1224no—
390Q8—1222no—
391Q8—1224no—
392Q8—2448no—
393Q8—2442no—
394Q8—2442no—
395Q8—2448no—
396Q8—2442no—
397D4—24no—
398D4—24no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—1224no—
415C2×C4—1222no—
416C2×C4—1222no—
417C2×C4—1222no—
418C2×C4—1222no—
419C2×C4—2448no—
420C2×C4—2448no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4#1451970012yescomplemented
425C2×C4C22⋊C4×D6111yescharacteristic
426C2×C4C22⋊C4×D6111yescharacteristic
427C2×C4#1451974712yes—
428C8—4882no—
429C22—21no—
430C22—22no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22#1683961711yescharacteristic
442C22C2×#1451970011yescharacteristic
443C22#1683977011yescharacteristic
444C22#1683990012yes—
445C22#1683991712yes—
446C4—22no—
447C4—21no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—162no—
452C4—1224no—
453C4—2442no—
454C4—2442no—
455C4—2442no—
456C4#99×D6112yes—
457C2—22no—
458C2—21no—
459C2—21no—
460C2—22no—
461C2—42no—
462C2#2144573211yescharacteristic
463C2#2144575311yescharacteristic
464C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#379
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#8379
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21445753C2semidirect
3. #21444390C2semidirect
6. C23×Dic122C4semidirect
10. C2×#14519777C4semidirect
12. C2×#14519756C4semidirect
16. C2×C4×Dic122C4semidirect
149. C23×C61#103semidirect
172. C2×C4×C61#103semidirect
194. Dic122#99semidirect
259. C61#8379semidirect
382. C23#14519700semidirect
424. C2×C4#14519700semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
#214457532
#214443902
#214441372
C61×#17992
#837961