Finite Group Databasebeta

← #29645902

Subgroups — #29645902

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519742C211yesmaximal, characteristic
2#21454448C212yesmaximal
3#21444161C214yesmaximal, complemented
4#21444149C211yesmaximal, characteristic
5#21444148C212yesmaximal
6#21444141C214yesmaximal, complemented
7C61×#1463C211yesmaximal, characteristic
8C23×Dic122C411yescharacteristic
9C2×#14519742—22no—
10C2×#14519742—22no—
11C2×#14519742—22no—
12C2×#14519742—22no—
13C2×#14519742C412yes—
14C2×#14519742C412yes—
15C2×#14519742C2212yes—
16C2×#14519742C2212yes—
17C22×#11758196C411yescharacteristic
18C22×#11758196C2211yescharacteristic
19C22×C4×Dic61C411yescharacteristic
20C22×C4×Dic61C411yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22#16839816—22no—
23#16839816C2214yes—
24#16839812—22no—
25#16839812C2214yes—
26#16839785—22no—
27#16839726C2212yes—
28M16×Dic61—22no—
29#16839488C2214yes—
30#16839487C2218yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C61×#407C2212yes—
33C61×#324C2214yes—
34C22×Dic122—22no—
35C22×Dic122—22no—
36C22×Dic122—28no—
37C22×Dic122C2×C412yes—
38C22×Dic122C2×C412yes—
39C23×Dic61C2×C411yescharacteristic
40C23×Dic61C2×C411yescharacteristic
41C23×Dic61C2×C411yescharacteristic
42C23×Dic61C2×C411yescharacteristic
43C2×#11758194—22no—
44#14519742—24no—
45#14519742—24no—
46#14519742—24no—
47#14519742—24no—
48#14519742—24no—
49#14519742—24no—
50#14519742—24no—
51#14519742—24no—
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×#11758196C2×C412yes—
57C2×#11758196C2×C412yes—
58C2×#11758196C2×C412yes—
59C2×#11758196C2312yes—
60C2×C4×Dic61—21no—
61C2×C4×Dic61—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2×C412yes—
69C2×C4×Dic61C2×C412yes—
70C2×C4×Dic61C2×C412yes—
71C2×C4×Dic61C2×C412yes—
72C2×C4×Dic61C2×C411yescharacteristic
73C2×C4×Dic61C2×C411yescharacteristic
74C2×C4×Dic61D412yes—
75C2×C4×Dic61D412yes—
76C2×C4×Dic61D412yes—
77C2×C4×Dic61D412yes—
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80#14519706—24no—
81#14519706—24no—
82C22⋊#92418—22no—
83C22⋊#92418C2314yes—
84#14519687—24no—
85C8×Dic61—24no—
86C23×C4×C61C2×C411yescharacteristic
87C23×C4×C61C2×C411yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C2×M16×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61C2×C412yes—
93C2×C4⋊C4×C61C2312yes—
94C22⋊C8×C61—22no—
95C22⋊C8×C61C2314yes—
96C24×C61C22×C411yescharacteristic
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—24no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—42no—
103C22×C4×C61—42no—
104C22×C4×C61C22×C412yes—
105C22×C4×C61C22×C412yes—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—24no—
113C22×Dic61—24no—
114C22×Dic61—24no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—42no—
118C22×Dic61C22⋊C412yes—
119C22×Dic61C22⋊C412yes—
120C22×Dic61C22⋊C411yescharacteristic
121C22×Dic61C22⋊C411yescharacteristic
122C22×Dic61C22⋊C411yescharacteristic
123C22×Dic61C22⋊C411yescharacteristic
124C22×Dic61C22×C411yescharacteristic
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C412yes—
129C22×Dic61C22×C411yescharacteristic
130C2×Dic122—216no—
131C2×Dic122—216no—
132C2×Dic122—24no—
133C2×Dic122—24no—
134C2×Dic122—24no—
135C2×Dic122—24no—
136C2×Dic122—24no—
137C2×Dic122M1618yescomplemented
138M16×C61—44no—
139C2×C8×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142#11758196—24no—
143#11758196—24no—
144#11758196—24no—
145#11758196—24no—
146C4×Dic61—22no—
147C4×Dic61—22no—
148C4×Dic61—22no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C4×Dic61—22no—
152C4×Dic61—22no—
153C4×Dic61—22no—
154C4×Dic61C22⋊C412yes—
155C4×Dic61C22⋊C412yes—
156C4×Dic61C22⋊C412yes—
157C4×Dic61C22⋊C412yes—
158C4×Dic61C2×D412yes—
159C4×Dic61C2×D412yes—
160C4×Dic61C2×D412yes—
161C4×Dic61C2×D412yes—
162#11758194—44no—
163C2×#92418—24no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61C2×C22⋊C411yescharacteristic
168C23×C61C2×C22⋊C411yescharacteristic
169C23×C61C23×C411yescharacteristic
170C2×C4×C61—24no—
171C2×C4×C61—24no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61C2×C22⋊C412yes—
183C2×C4×C61C2×M1614yes—
184C2×C4×C61C22×D412yes—
185C2×Dic61—24no—
186C2×Dic61—24no—
187C2×Dic61—22no—
188C2×Dic61—22no—
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—24no—
192C2×Dic61—24no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61#10114yes—
202C2×Dic61C2×C22⋊C411yescharacteristic
203C2×Dic61C2×C22⋊C411yescharacteristic
204C2×Dic61C2×C22⋊C412yes—
205C2×Dic61C2×C22⋊C412yes—
206C2×Dic61C2×C22⋊C411yescharacteristic
207C2×Dic61C2×C22⋊C411yescharacteristic
208C2×Dic61C2×M1614yes—
209C2×Dic61C2×M1614yes—
210Dic122—216no—
211Dic122—216no—
212C8×C61—44no—
213#92418—44no—
214#17365—611nomaximal, Sylow
215C22×C61—28no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61#41012yes—
223C22×C61C2×#10112yes—
224C22×C61C22×C22⋊C411yescharacteristic
225C22×C61C22×M1612yes—
226C4×C61—24no—
227C4×C61—22no—
228C4×C61—22no—
229C4×C61—42no—
230Dic61—22no—
231Dic61—24no—
232Dic61—21no—
233Dic61—21no—
234Dic61—42no—
235Dic61#40714yes—
236Dic61C2×#10112yes—
237Dic61C2×#10112yes—
238C22×C4×Q8—611no—
239#2920—612no—
240#1467—614no—
241#1463—612no—
242#1463—611no—
243#1463D6111yescharacteristic
244#1461—614no—
245C2×C61—22no—
246C2×C61—22no—
247C2×C61—22no—
248C2×C61C2×#40711yescharacteristic
249C2×C61C2×#41011yescharacteristic
250C2×C61C22×#10111yescharacteristic
251C23×Q8—611no—
252C2×C4×Q8—612no—
253C2×C4×Q8—612no—
254C2×C4×Q8—612no—
255C2×C4×Q8—612no—
256C2×C4×Q8—1222no—
257C2×C4×Q8—1222no—
258C2×C4×Q8—1222no—
259C2×C4×Q8—1222no—
260C22×C4⋊C4—611no—
261C22×C4⋊C4—611no—
262C22×C4⋊C4C2×D6111yescharacteristic
263C22×C42—611no—
264C22×C42—611no—
265C22×C42—611no—
266#432—614no—
267#432—1222no—
268#431—614no—
269#431—1222no—
270#423—1222no—
271#407—612no—
272#407C2×D6112yes—
273C4×M16—1222no—
274#324—614no—
275#324—618no—
276#324C2×D6114yes—
277C61#1736511yescharacteristic, complemented, Sylow
278C22×Q8—612no—
279C22×Q8—612no—
280C22×Q8—1222no—
281C22×Q8—1222no—
282C22×Q8—1228no—
283C23×C4—611no—
284C23×C4—611no—
285C23×C4—611no—
286C23×C4—611no—
287C23×C4C4×D6111yescharacteristic
288C23×C4C4×D6111yescharacteristic
289C23×C4C22×D6111yescharacteristic
290C2×M16—22no—
291C2×M16—1222no—
292C4×Q8—1224no—
293C4×Q8—1224no—
294C4×Q8—1224no—
295C4×Q8—1224no—
296C4×Q8—1224no—
297C4×Q8—1224no—
298C4×Q8—1224no—
299C4×Q8—1224no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—612no—
303C2×C4⋊C4—612no—
304C2×C4⋊C4—612no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1222no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4C4×D6112yes—
311C2×C4⋊C4C22×D6112yes—
312C2×C42—611no—
313C2×C42—611no—
314C2×C42—612no—
315C2×C42—612no—
316C2×C42—612no—
317C2×C42—612no—
318C2×C42—611no—
319C2×C42—611no—
320C2×C42—612no—
321C2×C42—612no—
322C2×C42—612no—
323C2×C42—612no—
324C2×C42—1221no—
325C2×C42—1221no—
326C2×C42—1221no—
327C2×C42—1221no—
328C2×C42—1221no—
329C2×C42—1221no—
330C2×C42—1221no—
331C2×C42—1221no—
332C4⋊C8—1224no—
333C4⋊C8—1224no—
334C22⋊C8—22no—
335C22⋊C8—614no—
336C22⋊C8—1222no—
337C22⋊C8C22×D6114yes—
338#97—1224no—
339C4×C8—1224no—
340C24C2×C4×D6111yescharacteristic
341C2×Q8—618no—
342C2×Q8—1224no—
343C2×Q8—1224no—
344C2×Q8—1224no—
345C2×Q8—1224no—
346C2×Q8—12216no—
347C2×Q8—12216no—
348C2×Q8—1224no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—611no—
360C22×C4—611no—
361C22×C4—612no—
362C22×C4—611no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—612no—
366C22×C4—612no—
367C22×C4—612no—
368C22×C4—1224no—
369C22×C4—1224no—
370C22×C4—1224no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—2442no—
376C22×C4C2×C4×D6112yes—
377C22×C4C2×C4×D6112yes—
378C22×C4C2×C4×D6111yescharacteristic
379C22×C4D4×D6112yes—
380C22×C4D4×D6112yes—
381C22×C4C23×D6111yescharacteristic
382M16—44no—
383M16—2444no—
384C2×C8—24no—
385C2×C8—1224no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—1224no—
389C4⋊C4—1224no—
390C4⋊C4—1224no—
391C4⋊C4—1224no—
392C42—612no—
393C42—612no—
394C42—612no—
395C42—612no—
396C42—612no—
397C42—612no—
398C42—612no—
399C42—612no—
400C42—1222no—
401C42—1222no—
402C42—1222no—
403C42—1222no—
404C42—1222no—
405C42—1222no—
406C42—1222no—
407C42—1222no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23C22⋊C4×D6111yescharacteristic
412C23C22⋊C4×D6111yescharacteristic
413C23C22×C4×D6111yescharacteristic
414Q8—12216no—
415Q8—12216no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—611no—
429C2×C4—611no—
430C2×C4—612no—
431C2×C4—614no—
432C2×C4—614no—
433C2×C4—611no—
434C2×C4—611no—
435C2×C4—612no—
436C2×C4—614no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1224no—
442C2×C4—1224no—
443C2×C4—1224no—
444C2×C4—1224no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4C22⋊C4×D6112yes—
454C2×C4M16×D6114yes—
455C2×C4C2×D4×D6112yes—
456C8—44no—
457C8—2444no—
458C22—28no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22#1683973412yes—
466C22#101×D6112yes—
467C22C2×C22⋊C4×D6111yescharacteristic
468C22C2×M16×D6112yes—
469C4—24no—
470C4—22no—
471C4—22no—
472C4—42no—
473C4—614no—
474C4—612no—
475C4—612no—
476C4—1222no—
477C4—1224no—
478C4—1221no—
479C4—1221no—
480C4—2442no—
481C2—22no—
482C2—22no—
483C2—22no—
484C2#407×D6111yescharacteristic
485C2C2×#1683973411yescharacteristic
486C2C2×#101×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1463C2
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC22×C4×C61C24
hypercenter#1463D61
hypocenterC61#17365
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #21444161C2semidirect
6. #21444141C2semidirect
137. C2×Dic122M16semidirect
277. C61#17365semidirect

Maximal subgroups

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

subgroupindex
C22×#145197422
#214544482
#214441612
#214441492
#214441482
#214441412
C61×#14632
#1736561