Finite Group Databasebeta

← #30620415

Subgroups — #30620415

Order 15616.

1354 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊#6923C212yesmaximal, complemented
2#21462010C214yesmaximal, complemented
3#21462009C212yesmaximal, complemented
4#21461681C212yesmaximal
5#21461679C211yesmaximal, characteristic
6#21461566C212yesmaximal
7C4×#14519835C211yesmaximal, characteristic
8C2×#16840425C211yesmaximal, characteristic
9C2×Q8×#6923—22no—
10C2×#14519835C2211yescharacteristic
11C2×#14519835C2212yes—
12C2×#14519835C2211yescharacteristic
13(C2×#6923)⋊Q8—24no—
14(C2×#6923)⋊Q8—22no—
15#16840704C2212yes—
16#16840704C2214yes—
17C4×C22⋊#6923C2212yes—
18C4×C22⋊#6923C2214yes—
19#16840640C2212yes—
20#16840640C2214yes—
21#16840640C2212yes—
22#16840640C2214yes—
23C23×Dic122C411yescharacteristic, complemented
24C22×Q8×D61C2211yescharacteristic
25#16840425—22no—
26#16840425—24no—
27#16840425—22no—
28C2×#14519776C412yescomplemented
29C2×#14519775C411yescharacteristic, complemented
30C2×#14519757C2211yescharacteristic
31C2×#14519755C2212yes—
32C2×#14519754C412yescomplemented
33C2×#14519742C411yescharacteristic, complemented
34C2×C4×C22⋊D61C2211yescharacteristic
35C2×#14519737C2212yes—
36C2×C22⋊Q8×C61C411yescharacteristic, complemented
37Q8×#6923—48no—
38C2×C4⋊#6923—24no—
39C2×C4⋊#6923—22no—
40C2×C22⋊#6923C2314yes—
41C2×C22⋊#6923C2312yes—
42C2×C22⋊#6923C2312yes—
43C2×C4×#6923—24no—
44C2×C4×#6923—22no—
45#14519835—22no—
46#14519835—24no—
47#14519835Q814yescomplemented
48(C2×#6923)⋊C4—24no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—22no—
52C22×Dic122—22no—
53C22×Dic122—21no—
54C22×Dic122—21no—
55C22×Dic122—44no—
56C22×Dic122C2×C411yescharacteristic
57C22×Dic122C2×C412yes—
58C22×Dic122C2×C411yescharacteristic
59C22×Dic122C2×C412yes—
60C2×Q8×D61—44no—
61C2×Q8×D61D412yes—
62C2×Q8×D61D414yes—
63C2×Q8×D61D412yes—
64C22×C22⋊D61C2311yescharacteristic
65C23×Dic61C2×C412yes—
66C22×C4×D61C2312yes—
67C22×C4×D61C2311yescharacteristic
68#14519776—24no—
69#14519776—24no—
70#14519775—22no—
71#14519775—22no—
72#14519757—22no—
73#14519757—22no—
74#14519755—24no—
75#14519755—24no—
76#14519754—24no—
77#14519754—24no—
78#14519742—22no—
79#14519742—21no—
80#14519742—21no—
81C4×C22⋊D61—22no—
82C4×C22⋊D61—22no—
83#14519737—24no—
84#14519737—24no—
85C2×C4⋊Dic61C2×C412yes—
86C2×#11758196C2×C412yes—
87C2×#11758196C2×C411yescharacteristic
88C2×#11758196C2×C412yes—
89C2×#11758196C2×C412yes—
90C2×C22⋊Dic61C2×C411yescharacteristic
91C2×#11758198C2312yes—
92C2×#11758198C2311yescharacteristic
93C2×C4×Dic61C2×C412yes—
94C2×C4×Dic61C2×C411yescharacteristic
95C22×Q8×C61C2×C411yescharacteristic
96C23×C4×C61C2×C411yescharacteristic
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C4⋊C4×C61C2×C412yes—
101C2×C22⋊C4×C61C2×C412yes—
102C24×C61C22×C411yescharacteristic
103C23×D61C2411yescharacteristic
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—24no—
107C2×Q8×C61—42no—
108C2×Q8×C61C22⋊C412yes—
109C2×Q8×C61C22⋊C412yes—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—42no—
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C412yes—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—22no—
119C2×C22⋊D61C2×Q812yes—
120C2×C22⋊D61#4112yes—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—44no—
124C22×Dic61C22×C412yes—
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C412yes—
127C22×Dic61C22×C411yescharacteristic
128C22×Dic61C22×C411yescharacteristic
129Q8×D61—28no—
130Q8×D61—48no—
131Q8×D61—44no—
132C2×C4×D61—44no—
133C2×C4×D61—42no—
134C2×C4×D61C2×D414yes—
135C2×C4×D61C2×D412yes—
136C2×C4×D61C2×D414yes—
137C2×C4×D61C2×D412yes—
138C2×Dic122—24no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—44no—
144C2×Dic122—44no—
145C2×Dic122—48no—
146C2×Dic122—44no—
147C2×Dic122—44no—
148C2×Dic122—42no—
149C2×Dic122C22⋊C411yescharacteristic
150C2×Dic122C22⋊C412yes—
151C2×Dic122C22⋊C414yes—
152C2×Dic122C22⋊C414yes—
153C2×Dic122C22⋊C411yescharacteristic
154C22⋊#6923—22no—
155C22⋊#6923—22no—
156C22⋊#6923—24no—
157C22⋊#6923—24no—
158C22⋊#6923—22no—
159C22⋊#6923C2×Q814yescomplemented
160C4⋊#6923—44no—
161C4⋊#6923—48no—
162C4×#6923—44no—
163C4×#6923—48no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—24no—
167C22⋊C4×C61—24no—
168C22⋊Dic61—22no—
169#11758198—24no—
170#11758198—22no—
171C4⋊Dic61—24no—
172#11758196—24no—
173#11758196—24no—
174#11758196—22no—
175#11758196—22no—
176#11758196—22no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61C4×Q812yescomplemented
186C23×C61C23×C411yescharacteristic
187C22×D61—41no—
188C22×D61—41no—
189C22×D61C22×D412yes—
190C22×D61C22×Q811yescharacteristic
191C22×D61C2×#4111yescharacteristic
192C22×D612−1+412yes—
193C2×#6923—24no—
194C2×#6923—42no—
195C2×#6923—42no—
196C2×#6923—42no—
197C2×#6923—44no—
198C2×#6923—42no—
199C2×#6923—44no—
200C2×#6923—42no—
201C2×#6923—44no—
202Q8×C61—24no—
203Q8×C61—42no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C22⋊C414yes—
214C2×C4×C61C2×C22⋊C411yescharacteristic
215C22⋊D61—24no—
216C22⋊D61—24no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—24no—
220C2×Dic61—21no—
221C2×Dic61—22no—
222C2×Dic61—41no—
223C2×Dic61—44no—
224C2×Dic61—44no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C2×C22⋊C412yes—
229C2×Dic61C2×C22⋊C412yes—
230C2×Dic61C4×Q812yescomplemented
231C4×D61—24no—
232C4×D61—28no—
233C4×D61—44no—
234C4×D61—42no—
235Dic122—22no—
236Dic122—24no—
237Dic122—28no—
238Dic122—44no—
239Dic122—44no—
240Dic122—42no—
241Dic122—42no—
242Dic122—44no—
243Dic122#9914yescomplemented
244Q8×#99—611nomaximal, Sylow
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—22no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61C22×C22⋊C411yescharacteristic
253C22×C61C2×C4×Q811yescharacteristic
254C22×C61#51911yescharacteristic
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—42no—
260C2×D61#54812yes—
261C2×D61D4×Q812yes—
262#6923—44no—
263#6923—82no—
264#6923—84no—
265C4×C61—21no—
266C4×C61—24no—
267C4×C61—41no—
268C4×C61—42no—
269C4×C61C2×#9912yes—
270Dic61—22no—
271Dic61—22no—
272Dic61—22no—
273Dic61—42no—
274Dic61—41no—
275Dic61C2×#9914yes—
276C2×D4×Q8—611no—
277Q8×C22⋊C4—612no—
278#1996—612no—
279#1996—614no—
280#1799—611no—
281#1799—612no—
282C4×#99—611no—
283C4×#99—612no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—42no—
287C2×C61Q8×C22⋊C411yescharacteristic
288C2×C61C22×#9911yescharacteristic
289C2×C61#288411yescharacteristic
290D61—22no—
291D61—22no—
292C23×Q8—611no—
293C23×Q8—611no—
294D4×Q8—1222no—
295D4×Q8—1224no—
296D4×Q8—1222no—
297C2×C4⋊Q8—611no—
298C2×C4⋊Q8—612no—
299C2×C22⋊Q8—612no—
300C2×C22⋊Q8—611no—
301C2×C22⋊Q8—612no—
302C2×C22⋊Q8#692311yescharacteristic, complemented
303C2×C4×Q8—611no—
304C2×C4×Q8—1222no—
305C2×C4×D4—611no—
306C2×C4×D4—612no—
307C2×#99—611no—
308C2×#99—612no—
309C2×#99—611no—
310#391—1224no—
311#391—1222no—
312#380—612no—
313#380—614no—
314C4×C22⋊C4—612no—
315C4×C22⋊C4—614no—
316#342—612no—
317#342—614no—
318#342—612no—
319#342—614no—
320C61Q8×#9911yescharacteristic, complemented, Sylow
321C22×Q8—612no—
322C22×Q8—614no—
323C22×Q8—612no—
324C22×Q8—611no—
325C22×Q8—612no—
326C22×Q8—612no—
327C22×Q8—611no—
328C22×Q8—1221no—
329C22×Q8—1222no—
330C22×Q8—1221no—
331C22×Q8—2444no—
332C22×Q8—2444no—
333C22×Q8C2×#692311yescharacteristic
334C22×D4—611no—
335C23×C4—612no—
336C23×C4—611no—
337C23×C4—612no—
338C23×C4C2×#692311yescharacteristic
339C4⋊Q8—1224no—
340C4⋊Q8—1224no—
341C4⋊Q8—1222no—
342C4⋊Q8—1222no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—1224no—
346C22⋊Q8—1224no—
347C22⋊Q8—1224no—
348C22⋊Q8—1224no—
349C22⋊Q8—1222no—
350C22⋊Q8—1222no—
351C4×Q8—1221no—
352C4×Q8—1222no—
353C4×Q8—1221no—
354C4×Q8—2448no—
355C4×D4—1224no—
356C4×D4—1224no—
357C4×D4—1222no—
358C4×D4—1222no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4C2×#692311yescharacteristic
367C2×C4⋊C4C2×#692312yes—
368C2×C22⋊C4—612no—
369C2×C22⋊C4—611no—
370C2×C22⋊C4—611no—
371C2×C22⋊C4—612no—
372C2×C22⋊C4—614no—
373C2×C22⋊C4—612no—
374C2×C22⋊C4C2×#692312yes—
375C2×C42—612no—
376C2×C42—611no—
377C2×C42—1224no—
378C2×C42—1222no—
379#99—614no—
380#99—1222no—
381#99—1224no—
382#95—1224no—
383#95—1222no—
384#95—1224no—
385#95—1222no—
386C24—611no—
387C24C22×#692311yescharacteristic
388C2×Q8—42no—
389C2×Q8—611no—
390C2×Q8—612no—
391C2×Q8—611no—
392C2×Q8—614no—
393C2×Q8—614no—
394C2×Q8—1222no—
395C2×Q8—1222no—
396C2×Q8—1224no—
397C2×Q8—1228no—
398C2×Q8—1221no—
399C2×Q8—1221no—
400C2×Q8—2442no—
401C2×Q8—2444no—
402C2×Q8—2444no—
403C2×Q8—2444no—
404C2×Q8—2448no—
405C2×Q8—2444no—
406C2×Q8—2444no—
407C2×Q8—2448no—
408C2×Q8C22⋊#692312yes—
409C2×Q8C22⋊#692312yes—
410C2×D4—612no—
411C2×D4—612no—
412C2×D4—1222no—
413C2×D4—1222no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—611no—
418C22×C4—612no—
419C22×C4—611no—
420C22×C4—614no—
421C22×C4—612no—
422C22×C4—614no—
423C22×C4—612no—
424C22×C4—611no—
425C22×C4—612no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—2442no—
432C22×C4—2444no—
433C22×C4—2444no—
434C22×C4C22×#692311yescharacteristic
435C22×C4C22×#692312yes—
436C22×C4C22×#692311yescharacteristic
437C22×C4C22×#692312yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—1222no—
442C4⋊C4—1222no—
443C4⋊C4—1222no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1224no—
447C4⋊C4—2448no—
448C4⋊C4—2444no—
449C22⋊C4—24no—
450C22⋊C4—614no—
451C22⋊C4—1224no—
452C22⋊C4—1222no—
453C22⋊C4—1222no—
454C22⋊C4—1222no—
455C22⋊C4—1222no—
456C22⋊C4—1224no—
457C22⋊C4—1224no—
458C22⋊C4—1222no—
459C42—1222no—
460C42—1222no—
461C42—1221no—
462C42—1221no—
463C42—2448no—
464C42—2444no—
465C23—21no—
466C23—21no—
467C23—41no—
468C23—41no—
469C23—612no—
470C23—611no—
471C23—611no—
472C23—612no—
473C23—2441no—
474C23—2441no—
475C23Q8×#692312yescomplemented
476C23C23×#692311yescharacteristic
477Q8—24no—
478Q8—42no—
479Q8—614no—
480Q8—1228no—
481Q8—1224no—
482Q8—1222no—
483Q8—2444no—
484Q8—2442no—
485Q8—2444no—
486Q8—2444no—
487Q8—2442no—
488D4—1224no—
489D4—1224no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—1228no—
503C2×C4—1224no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1221no—
509C2×C4—1222no—
510C2×C4—2442no—
511C2×C4—2444no—
512C2×C4—2444no—
513C2×C4—2441no—
514C2×C4—2444no—
515C2×C4—2444no—
516C2×C4—2442no—
517C2×C4—2444no—
518C2×C4—2442no—
519C2×C4—2444no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4C2×C22⋊#692311yescharacteristic
526C2×C4C2×C22⋊#692314yes—
527C2×C4C2×C22⋊#692311yescharacteristic
528C22—21no—
529C22—21no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—612no—
536C22—612no—
537C22—1222no—
538C22—1222no—
539C22—2441no—
540C22—2441no—
541C22—2442no—
542C22C22×C22⋊#692311yescharacteristic
543C22C2×Q8×#692311yescharacteristic
544C22#1684083011yescharacteristic
545C4—21no—
546C4—24no—
547C4—41no—
548C4—42no—
549C4—614no—
550C4—1222no—
551C4—1222no—
552C4—1222no—
553C4—4884no—
554C4—4882no—
555C4—2444no—
556C4—2442no—
557C4—2441no—
558C4D61⋊#9912yes—
559C2—21no—
560C2—21no—
561C2—42no—
562C2—1222no—
563C2—1222no—
564C2Q8×C22⋊#692311yescharacteristic
565C2C2×D61⋊#9911yescharacteristic
566C2#2146242111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C23
cosocleC23×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61Q8×#99
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊#6923C2semidirect
2. #21462010C2semidirect
3. #21462009C2semidirect
23. C23×Dic122C4semidirect
28. C2×#14519776C4semidirect
29. C2×#14519775C4semidirect
32. C2×#14519754C4semidirect
33. C2×#14519742C4semidirect
36. C2×C22⋊Q8×C61C4semidirect
47. #14519835Q8semidirect
159. C22⋊#6923C2×Q8semidirect
185. C23×C61C4×Q8semidirect
230. C2×Dic61C4×Q8semidirect
243. Dic122#99semidirect
302. C2×C22⋊Q8#6923semidirect
320. C61Q8×#99semidirect
475. C23Q8×#6923semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊#69232
#214620102
#214620092
#214616812
#214616792
#214615662
C4×#145198352
C2×#168404252
Q8×#9961