Finite Group Databasebeta

← #29707749

Subgroups — #29707749

Order 15616.

1414 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840338C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic
3C2×#16839770C211yesmaximal, characteristic, complemented
4C2×#16839756C211yesmaximal, characteristic
5C22×#14519700C211yesmaximal, characteristic, complemented
6#21447065C214yesmaximal, complemented
7#21447060C214yesmaximal, complemented
8#21445516C211yesmaximal, characteristic, complemented
9C61×#2317C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×Dic122C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840338—24no—
15#16840338—24no—
16C2×#14519777C2212yes—
17C2×#14519765C2212yes—
18C2×#14519761—22no—
19C2×#14519757—21no—
20C2×#14519757—21no—
21#16839770—24no—
22#16839761—24no—
23#16839756—24no—
24C2×#14519700—24no—
25C2×#14519700C2214yes—
26C2×#14519700C2214yes—
27C2×#14519700C2212yes—
28C2×#14519700C2212yes—
29C2×#14519697C2212yes—
30C2×#14519697C2212yes—
31#16839517C2214yes—
32C2×#117×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#389C2214yes—
36C61×#385—22no—
37C61×#384—21no—
38C61×#382—21no—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222D414yes—
45C22×Dic122—22no—
46C22×Dic122—22no—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122—21no—
50C22×Dic122—22no—
51C22×Dic122D414yes—
52C22×Dic122C2312yes—
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61—21no—
57C22×C4×D61—21no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C22×#92418—22no—
62C22×#92418—22no—
63C22×#92418—22no—
64C22×#92418C2312yes—
65C22×#92418C2312yes—
66#14519777—24no—
67#14519777—24no—
68#14519765—24no—
69#14519765—24no—
70#14519761—216no—
71#14519757—28no—
72#14519757—28no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—22no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77#14519700—28no—
78#14519700—24no—
79#14519700—24no—
80#14519700—28no—
81#14519697—24no—
82#14519697—24no—
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85#117×C61—22no—
86#117×C61—22no—
87#117×C61—22no—
88#117×C61D414yes—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—24no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61D411yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C4⋊C4×C61C2312yes—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61D412yes—
99C2×C42×C61—21no—
100C2×C42×C61C2312yes—
101#95×C61—24no—
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—42no—
107C22×C4×C61—82no—
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61#4112yes—
117C22×C4×C61C2411yescharacteristic
118C2×C22⋊D61—24no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C22×Dic61—24no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126#11758222—28no—
127#11758222—28no—
128#11758222—44no—
129#11758222—44no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244C2×D412yes—
136C2×C4×D61—24no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×Dic122—28no—
140C2×Dic122—28no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—24no—
150C2×Dic122—22no—
151C2×Dic122—21no—
152C2×Dic122—44no—
153C2×Dic122—42no—
154C2×Dic122—44no—
155C2×Dic122—42no—
156C2×Dic122C2×D414yes—
157C2×Dic122C2×D412yes—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—84no—
165C4⋊C4×C61C2×D414yes—
166C22⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C42×C61—24no—
169C42×C61—22no—
170C42×C61—22no—
171#11758198—28no—
172#11758198—28no—
173C4⋊Dic61—28no—
174C4⋊Dic61—28no—
175#11758194—44no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—44no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61#12012yes—
186C23×C61#12112yes—
187C23×C61#12412yes—
188C23×C61C22×D411yescharacteristic
189C22×D61—22no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—42no—
193C22×D61—42no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—44no—
203C2×C4×C61—84no—
204C2×C4×C61#12012yes—
205C2×C4×C61#12114yes—
206C2×C4×C61#12114yes—
207C2×C4×C61#12112yes—
208C2×C4×C61#12314yes—
209C2×C4×C61#12314yes—
210C2×C4×C61#12412yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C22⋊D61—48no—
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—48no—
225D244—22no—
226D244—22no—
227D244—41no—
228D244—41no—
229C4×D61—48no—
230Dic122—24no—
231Dic122—22no—
232Dic122—24no—
233Dic122—22no—
234Dic122—42no—
235Dic122—41no—
236Dic122—42no—
237Dic122—41no—
238Dic122—48no—
239#92418—42no—
240#92418—42no—
241#23510—611nomaximal, Sylow
242C22×C61—24no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—24no—
246C22×C61#39414yes—
247C22×C61C2×#12011yescharacteristic
248C22×C61C2×#12111yescharacteristic
249C22×C61C2×#12411yescharacteristic
250C22×C61#57712yes—
251C22×C61#57812yes—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—82no—
256C2×D61—82no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61—44no—
260C4×C61—82no—
261C4×C61#39414yes—
262C4×C61C2×#12111yescharacteristic
263C4×C61C2×#12111yescharacteristic
264C4×C61C2×#12311yescharacteristic
265C4×C61C2×#12311yescharacteristic
266Dic61—44no—
267Dic61—42no—
268C2×#536—611no—
269C2×#417—611no—
270C2×#419—611no—
271C2×#416—611no—
272C22×#103—611no—
273#2317D6111yescharacteristic, complemented
274#2043—614no—
275#2042—614no—
276#1729—611no—
277C2×C61—24no—
278C2×C61C2×#39411yescharacteristic
279C2×C61#301011yescharacteristic
280C2×C61#301111yescharacteristic
281C2×C61#304612yes—
282C2×C61#313712yes—
283D61—82no—
284C22×#41—611no—
285C23×Q8—611no—
286C22×M16—611no—
287C23×C8—611no—
288#536—1224no—
289#536—1224no—
290C2×C4⋊Q8—612no—
291C2×#124—612no—
292C2×#123—1222no—
293C2×C22⋊Q8—1221no—
294C2×C22⋊Q8—1221no—
295C2×#117C2×D6111yescharacteristic
296C2×#117C2×D6111yescharacteristic
297C22×C4⋊C4C2×D6111yescharacteristic
298#419—1224no—
299#417—1224no—
300#416—1224no—
301C2×#103—614no—
302C2×#103—612no—
303C2×#103—614no—
304C2×#103—612no—
305C2×#103—1224no—
306C2×#102—612no—
307C2×#102—612no—
308#389C2×D6114yes—
309#385—22no—
310#384—21no—
311#382—21no—
312#336—614no—
313C61#2351011yescharacteristic, complemented, Sylow
314C2×#41—614no—
315C2×#41—1222no—
316C2×#41—1222no—
317C2×#41—1222no—
318C2×#41—1222no—
319C2×#41—1222no—
320C22×Q8—614no—
321C22×Q8—612no—
322C22×Q8—611no—
323C22×Q8—1222no—
324C22×Q8—1221no—
325C22×Q8—1222no—
326C22×Q8—1222no—
327C22×Q8—1222no—
328C22×Q8—1222no—
329C22×D4—611no—
330C22×D4—1221no—
331C23×C4—21no—
332C23×C4—1221no—
333C23×C4—1221no—
334C23×C4C22×D6111yescharacteristic
335C2×M16—1222no—
336C2×M16—1222no—
337C2×M16—1222no—
338C22×C8—612no—
339C22×C8—612no—
340C22×C8—1222no—
341C22×C8—1222no—
342C22×C8—1222no—
343C4⋊Q8—1224no—
344C4⋊Q8—1224no—
345#124—1224no—
346#124—1224no—
347#123—12216no—
348C22⋊Q8—1228no—
349C22⋊Q8—1228no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117D24414yes—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22×D6112yes—
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4C22⋊D6112yes—
368C2×C42—21no—
369C2×C42C22×D6112yes—
370#103—1228no—
371#103—1224no—
372#103—1224no—
373#103—1228no—
374#102—1224no—
375#102—1224no—
376#95—24no—
377C24—1221no—
378C24C2×C22⋊D6111yescharacteristic
379#41—1228no—
380#41—1228no—
381#41—2444no—
382#41—2444no—
383C2×Q8—614no—
384C2×Q8—612no—
385C2×Q8—1222no—
386C2×Q8—1221no—
387C2×Q8—1224no—
388C2×Q8—1222no—
389C2×Q8—1221no—
390C2×Q8—1222no—
391C2×Q8—1221no—
392C2×Q8—1222no—
393C2×Q8—1222no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1228no—
397C2×Q8—1228no—
398C2×Q8—2442no—
399C2×Q8—2444no—
400C2×Q8—2444no—
401C2×Q8—2442no—
402C2×D4—612no—
403C2×D4—1224no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—2442no—
410C2×D4—2442no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4—1222no—
416C22×C4—1221no—
417C22×C4—1224no—
418C22×C4—1224no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4C2×D24412yes—
424C22×C4#1175822212yes—
425C22×C4D4×D6112yes—
426C22×C4D4×D6112yes—
427C22×C4#1175822612yes—
428C22×C4#1175822612yes—
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6112yes—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433M16—2444no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C2×C8—2444no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—84no—
448C4⋊C4—1228no—
449C4⋊C4—1228no—
450C4⋊C4C2×D24414yes—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—1228no—
454C22⋊C4—1228no—
455C42—22no—
456C42—22no—
457C42—24no—
458C23—22no—
459C23—22no—
460C23—1222no—
461C23—2441no—
462C23—2441no—
463C23—2442no—
464C23—2442no—
465C23#1451974512yes—
466C23#1451975112yes—
467C23#1451976612yes—
468C23C22×C22⋊D6111yescharacteristic
469Q8—1224no—
470Q8—1222no—
471Q8—1224no—
472Q8—1222no—
473Q8—2442no—
474Q8—2441no—
475Q8—2442no—
476Q8—2441no—
477Q8—2448no—
478D4—1222no—
479D4—1222no—
480D4—2448no—
481D4—2441no—
482D4—2441no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—84no—
493C2×C4—1224no—
494C2×C4—1222no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4—2448no—
500C2×C4—2448no—
501C2×C4#1451974512yes—
502C2×C4#1451975112yes—
503C2×C4#1451975214yes—
504C2×C4#1451975214yes—
505C2×C4#1451976214yes—
506C2×C4#1451976214yes—
507C2×C4#1451976612yes—
508C2×C4C2×D4×D6111yescharacteristic
509C2×C4C2×D4×D6111yescharacteristic
510C2×C4C22×D24411yescharacteristic
511C2×C4C2×#1175822211yescharacteristic
512C2×C4C2×#1175822611yescharacteristic
513C2×C4C2×#1175822611yescharacteristic
514C8—2442no—
515C8—2442no—
516C22—22no—
517C22—24no—
518C22—24no—
519C22—22no—
520C22—4882no—
521C22—4882no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22#1683967914yes—
526C22C2×#1451974511yescharacteristic
527C22C2×#1451975111yescharacteristic
528C22C2×#1451976611yescharacteristic
529C22#1684057312yes—
530C22#1684057712yes—
531C4—42no—
532C4—44no—
533C4—42no—
534C4—82no—
535C4—2442no—
536C4—2444no—
537C4#1683967914yes—
538C4C2×#1451975211yescharacteristic
539C4C2×#1451975211yescharacteristic
540C4C2×#1451976211yescharacteristic
541C4C2×#1451976211yescharacteristic
542C2—24no—
543C2—4882no—
544C2C2×#1683967911yescharacteristic
545C2#2145510311yescharacteristic
546C2#2145511211yescharacteristic
547C2#2145536912yes—
548C2#2145618912yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2317C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#2317D61
hypocenterC61#23510
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. C2×#16839770C2semidirect
5. C22×#14519700C2semidirect
6. #21447065C2semidirect
7. #21447060C2semidirect
8. #21445516C2semidirect
9. C61×#2317C2semidirect
273. #2317D61semidirect
313. C61#23510semidirect

Maximal subgroups

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

subgroupindex
C2×#168403382
C2×#168397612
C2×#168397702
C2×#168397562
C22×#145197002
#214470652
#214470602
#214455162
C61×#23172
#2351061