Finite Group Databasebeta

← C2×#324×D61 #29645800

Subgroups — C2×#324×D61 #29645800

Order 15616.

1806 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C2×C22⋊C8×D61C212yesmaximal
3C2×#16839488C211yesmaximal, characteristic, complemented
4C2×#16839487C212yesmaximal, complemented
5#324×D61C218yesmaximal, direct product factor
6C2×C61×#324C211yesmaximal, characteristic, complemented
7C23×C4×D61C411yescharacteristic
8C23×C4×D61C411yescharacteristic
9C23×C4×D61C2211yescharacteristic
10C22×C8×D61—22no—
11C22×C4⋊Dic61C411yescharacteristic
12C22×#11758196C411yescharacteristic
13C22×#11758196C2211yescharacteristic
14C2×C4⋊C4×D61—22no—
15C2×C4⋊C4×D61—24no—
16C2×C4⋊C4×D61—22no—
17C2×C4⋊C4×D61C414yes—
18C2×C4⋊C4×D61C2214yes—
19C2×C22⋊#92418C2212yes—
20C2×#14519689—22no—
21C22⋊C8×D61—24no—
22C22⋊C8×D61C2218yes—
23#16839488C2214yescomplemented
24#16839487C2218yescomplemented
25C22×C4⋊C4×C61C2211yescharacteristic
26C2×C22⋊C8×C61C2212yes—
27C61×#324C2214yescomplemented
28C24×D61C2×C411yescharacteristic
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C23×Dic61C2×C411yescharacteristic
32C22×C4×D61—22no—
33C22×C4×D61—21no—
34C22×C4×D61—21no—
35C22×C4×D61—24no—
36C22×C4×D61—22no—
37C22×C4×D61—21no—
38C22×C4×D61—21no—
39C22×C4×D61—44no—
40C22×C4×D61—44no—
41C22×C4×D61C818yescomplemented
42C22×C4×D61C2×C414yes—
43C22×C4×D61C2×C414yes—
44C22×C4×D61C2×C412yes—
45C22×C4×D61D412yes—
46C22×C4×D61D414yes—
47C22×C4×D61D412yes—
48C22×C4×D61C2312yes—
49C22×#92418—22no—
50C2×C8×D61—28no—
51C2×C8×D61—48no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61C2×C414yes—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×#11758196C2×C412yes—
60C2×#11758196C2×C414yes—
61C2×#11758196C2312yes—
62C4⋊C4×D61—28no—
63C4⋊C4×D61—28no—
64C4⋊C4×D61—28no—
65C4⋊C4×D61—28no—
66C22⋊#92418—22no—
67C22⋊#92418C2314yes—
68#14519689—28no—
69C23×C4×C61C2×C411yescharacteristic
70C23×C4×C61C2×C411yescharacteristic
71C23×C4×C61C2311yescharacteristic
72C22×C8×C61—22no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61C2×C412yes—
76C2×C4⋊C4×C61C2312yes—
77C22⋊C8×C61—22no—
78C22⋊C8×C61C2314yes—
79C24×C61C22×C411yescharacteristic
80C23×D61—24no—
81C23×D61—24no—
82C23×D61C22⋊C411yescharacteristic
83C23×D61C22⋊C412yes—
84C23×D61C22⋊C411yescharacteristic
85C23×D61C2×C814yes—
86C23×D61M1614yes—
87C23×D61C22×C412yes—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61C2×C814yescomplemented
96C22×C4×C61C22×C412yes—
97C22×C4×C61C22×C412yes—
98C22×C4×C61C22×C411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—42no—
108C22×Dic61—42no—
109C22×Dic61C22⋊C412yes—
110C22×Dic61C22⋊C412yes—
111C22×Dic61C2×C814yescomplemented
112C22×Dic61C22×C412yes—
113C22×Dic61C22×C412yes—
114C22×Dic61C22×C412yes—
115C2×C4×D61—28no—
116C2×C4×D61—216no—
117C2×C4×D61—28no—
118C2×C4×D61—28no—
119C2×C4×D61—24no—
120C2×C4×D61—24no—
121C2×C4×D61—28no—
122C2×C4×D61—24no—
123C2×C4×D61—24no—
124C2×C4×D61—44no—
125C2×C4×D61—48no—
126C2×C4×D61—44no—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—48no—
130C2×C4×D61—44no—
131C2×C4×D61—48no—
132C2×C4×D61C22⋊C414yes—
133C2×C4×D61C22⋊C414yes—
134C2×C4×D61C2×C8116yescomplemented
135C2×C4×D61C2×D414yes—
136C2×C4×D61C2×D414yes—
137C2×C8×C61—24no—
138C2×C8×C61—44no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊Dic61—28no—
142#11758196—24no—
143#11758196—24no—
144#11758196—28no—
145C2×#92418—24no—
146C2×#92418—44no—
147C8×D61—416no—
148C23×C61—22no—
149C23×C61—22no—
150C23×C61C2×C22⋊C411yescharacteristic
151C23×C61C2×C22⋊C411yescharacteristic
152C23×C61C22×C812yes—
153C23×C61C2×M1612yes—
154C23×C61C23×C411yescharacteristic
155C22×D61—22no—
156C22×D61—22no—
157C22×D61—216no—
158C22×D61—216no—
159C22×D61—24no—
160C22×D61—24no—
161C22×D61—24no—
162C22×D61—24no—
163C22×D61C22⋊C814yes—
164C22×D61C22⋊C818yes—
165C22×D61C22⋊C814yes—
166C22×D61#9914yes—
167C22×D61#10114yes—
168C22×D61C2×C22⋊C412yes—
169C22×D61C2×C22⋊C412yes—
170C22×D61C22×C814yes—
171C22×D61C2×M1614yes—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—24no—
176C2×C4×C61—24no—
177C2×C4×C61—22no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61C2×C22⋊C412yes—
185C2×C4×C61C22×C814yescomplemented
186C2×C4×C61C22×D412yes—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—24no—
190C2×Dic61—24no—
191C2×Dic61—24no—
192C2×Dic61—22no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61C2×C22⋊C414yes—
199C2×Dic61C22×C814yescomplemented
200C4×D61—216no—
201C4×D61—216no—
202C4×D61—28no—
203C4×D61—28no—
204C4×D61—44no—
205C4×D61—44no—
206C4×D61—48no—
207C4×D61—48no—
208C8×C61—44no—
209#92418—44no—
210C22×#324—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—24no—
216C22×C61—24no—
217C22×C61C2×C22⋊C812yes—
218C22×C61C2×C22⋊C812yes—
219C22×C61C2×#9912yes—
220C22×C61C2×#10112yes—
221C22×C61C22×C22⋊C411yescharacteristic
222C22×C61C23×C811yescharacteristic
223C22×C61C22×M1611yescharacteristic
224C2×D61—22no—
225C2×D61—22no—
226C2×D61—28no—
227C2×D61—216no—
228C2×D61—28no—
229C2×D61—28no—
230C2×D61#324116yesdirect product factor
231C2×D61C2×C22⋊C814yes—
232C2×D61C2×C22⋊C814yes—
233C2×D61C2×#9914yes—
234C2×D61C2×#10114yes—
235C4×C61—24no—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—42no—
239Dic61—24no—
240Dic61—24no—
241Dic61—42no—
242C23×C4⋊C4—611no—
243C22×C22⋊C8—612no—
244C2×#324—612no—
245C2×#324—618no—
246C2×#324—611no—
247C2×#324D6111yescharacteristic, direct product factor
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—22no—
251C2×C61C2×#32414yescomplemented
252C2×C61C22×C22⋊C811yescharacteristic
253C2×C61C22×#9911yescharacteristic
254C2×C61C22×#10111yescharacteristic
255D61—24no—
256D61C2×#32418yesdirect product factor
257C24×C4—611no—
258C24×C4—611no—
259C24×C4—611no—
260C23×C8—1222no—
261C22×C4⋊C4—611no—
262C22×C4⋊C4—611no—
263C22×C4⋊C4—611no—
264C22×C4⋊C4—614no—
265C22×C4⋊C4—614no—
266C22×C4⋊C4—1222no—
267C22×C4⋊C4—1224no—
268C22×C4⋊C4—1222no—
269C22×C4⋊C4C2×D6111yescharacteristic
270C2×C22⋊C8—612no—
271C2×C22⋊C8—618no—
272C2×C22⋊C8—1224no—
273C2×C22⋊C8—1222no—
274C2×C22⋊C8C2×D6112yes—
275#324—614no—
276#324—618no—
277#324C2×D6114yesdirect product factor
278C61C22×#32411yescharacteristic, complemented, Sylow
279C25—611no—
280C23×C4—611no—
281C23×C4—611no—
282C23×C4—618no—
283C23×C4—614no—
284C23×C4—612no—
285C23×C4—612no—
286C23×C4—612no—
287C23×C4—614no—
288C23×C4—612no—
289C23×C4—611no—
290C23×C4—614no—
291C23×C4—1222no—
292C23×C4—1221no—
293C23×C4—1221no—
294C23×C4—1224no—
295C23×C4—1222no—
296C23×C4—1221no—
297C23×C4—1221no—
298C23×C4—2444no—
299C23×C4—2444no—
300C23×C4C4×D6111yescharacteristic
301C23×C4C4×D6111yescharacteristic
302C23×C4C22×D6111yescharacteristic
303C22×C8—22no—
304C22×C8—1228no—
305C22×C8—1222no—
306C22×C8—2448no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—614no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—614no—
312C2×C4⋊C4—612no—
313C2×C4⋊C4—1228no—
314C2×C4⋊C4—1228no—
315C2×C4⋊C4—1228no—
316C2×C4⋊C4—1228no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4C4×D6112yes—
324C2×C4⋊C4C22×D6112yes—
325C22⋊C8—22no—
326C22⋊C8—614no—
327C22⋊C8—1228no—
328C22⋊C8—1222no—
329C22⋊C8C22×D6114yes—
330C24—611no—
331C24—612no—
332C24—612no—
333C24—611no—
334C24—614no—
335C24—614no—
336C24—1224no—
337C24—1224no—
338C24C2×C4×D6111yescharacteristic
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—612no—
347C22×C4—612no—
348C22×C4—614no—
349C22×C4—612no—
350C22×C4—612no—
351C22×C4—614no—
352C22×C4—614no—
353C22×C4—6116no—
354C22×C4—614no—
355C22×C4—614no—
356C22×C4—612no—
357C22×C4—1224no—
358C22×C4—1224no—
359C22×C4—1228no—
360C22×C4—1228no—
361C22×C4—12216no—
362C22×C4—1228no—
363C22×C4—1228no—
364C22×C4—1224no—
365C22×C4—1224no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—2444no—
372C22×C4—2444no—
373C22×C4—2448no—
374C22×C4—2444no—
375C22×C4—2444no—
376C22×C4—2448no—
377C22×C4—2444no—
378C22×C4—2448no—
379C22×C4—2442no—
380C22×C4—2442no—
381C22×C4C8×D6114yescomplemented
382C22×C4C2×C4×D6112yes—
383C22×C4C2×C4×D6112yes—
384C22×C4C2×C4×D6111yescharacteristic
385C22×C4D4×D6112yes—
386C22×C4D4×D6112yes—
387C22×C4C23×D6111yescharacteristic
388C2×C8—24no—
389C2×C8—44no—
390C2×C8—1224no—
391C2×C8—24416no—
392C2×C8—2444no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—1228no—
396C4⋊C4—1228no—
397C4⋊C4—1224no—
398C4⋊C4—1224no—
399C23—22no—
400C23—22no—
401C23—612no—
402C23—614no—
403C23—614no—
404C23—614no—
405C23—612no—
406C23—618no—
407C23—614no—
408C23—614no—
409C23—614no—
410C23—1224no—
411C23—1224no—
412C23—12216no—
413C23—12216no—
414C23—1224no—
415C23—1224no—
416C23—1222no—
417C23—1222no—
418C23C22⋊C4×D6111yescharacteristic
419C23C22⋊C4×D6111yescharacteristic
420C23C2×C8×D6112yes—
421C23M16×D6112yes—
422C23C22×C4×D6111yescharacteristic
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—614no—
436C2×C4—614no—
437C2×C4—1228no—
438C2×C4—12216no—
439C2×C4—12216no—
440C2×C4—1228no—
441C2×C4—1222no—
442C2×C4—1224no—
443C2×C4—1224no—
444C2×C4—1224no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—2444no—
448C2×C4—2444no—
449C2×C4—2448no—
450C2×C4—2448no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4C22⋊C4×D6112yes—
457C2×C4C2×C8×D6114yescomplemented
458C2×C4C2×D4×D6112yes—
459C8—44no—
460C8—2444no—
461C22—24no—
462C22—24no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—6116no—
468C22—614no—
469C22—614no—
470C22—614no—
471C22—614no—
472C22—12216no—
473C22—1228no—
474C22—1228no—
475C22—1228no—
476C22—1222no—
477C22—1222no—
478C22C22⋊C8×D6112yes—
479C22C22⋊C8×D6112yes—
480C22#99×D6112yes—
481C22#101×D6112yes—
482C22C2×C22⋊C4×D6111yescharacteristic
483C22C22×C8×D6111yescharacteristic
484C22C2×M16×D6111yescharacteristic
485C4—24no—
486C4—22no—
487C4—22no—
488C4—42no—
489C4—1224no—
490C4—1224no—
491C4—2442no—
492C2—22no—
493C2—21no—
494C2—21no—
495C2—618no—
496C2—1224no—
497C2#324×D6114yesdirect product factor
498C2C2×C22⋊C8×D6111yescharacteristic
499C2C2×#99×D6111yescharacteristic
500C2C2×#101×D6111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C61C23×C8
FrattiniC22×C4C23×D61
FittingC2×C61×#324C2
socleC23×C61C2×C22⋊C4
prime radicalC24×D61C2×C4
cosocleC22×C4×C61C24
hypercenterC2×#324D61
hypocenterC61C22×#324
LCSG ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. C2×#16839488C2semidirect
4. C2×#16839487C2semidirect
5. #324×D61C2direct (N × H)
6. C2×C61×#324C2semidirect
23. #16839488C22semidirect
24. #16839487C22semidirect
27. C61×#324C22semidirect
41. C22×C4×D61C8semidirect
95. C22×C4×C61C2×C8semidirect
111. C22×Dic61C2×C8semidirect
134. C2×C4×D61C2×C8semidirect
185. C2×C4×C61C22×C8semidirect
199. C2×Dic61C22×C8semidirect
230. C2×D61#324direct (N × H)
247. C2×#324D61direct (N × H)
251. C2×C61C2×#324semidirect
256. D61C2×#324direct (N × H)
277. #324C2×D61direct (N × H)
278. C61C22×#324semidirect
381. C22×C4C8×D61semidirect
457. C2×C4C2×C8×D61semidirect
497. C2#324×D61direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D612
C2×C22⋊C8×D612
C2×#168394882
C2×#168394872
#324×D612
C2×C61×#3242
C22×#32461