Finite Group Databasebeta

← C2×#21447173 #29708144

Subgroups — C2×#21447173 #29708144

Order 15616.

1614 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840388C211yesmaximal, characteristic
2C2×#16839637C212yesmaximal
3C2×#16839634C211yesmaximal, characteristic
4#21447173C218yesmaximal, direct product factor
5C2×#16839528C212yesmaximal
6C2×C61×#342C211yesmaximal, characteristic
7C24×Dic61C2211yescharacteristic
8#16840388—24no—
9#16840388—22no—
10#16840388C2214yes—
11C2×#14519756—21no—
12C2×#14519756—21no—
13C2×#14519754—22no—
14C2×#14519754—22no—
15C22×C4⋊Dic61—22no—
16C22×#11758196—21no—
17C22×C22⋊Dic61C2211yescharacteristic
18C22×C22⋊Dic61C2212yes—
19#16839637—24no—
20#16839637C2218yes—
21#16839634—22no—
22#16839634C2214yes—
23C2×#14519684—21no—
24C2×#14519684—22no—
25#16839528C2218yes—
26C22×C22⋊C4×C61C2211yescharacteristic
27C22×C22⋊C4×C61C2212yes—
28C61×#342C2214yes—
29C22×Dic122—21no—
30C22×Dic122—21no—
31C23×Dic61—21no—
32C23×Dic61—21no—
33C23×Dic61—22no—
34C23×Dic61—21no—
35C23×Dic61—22no—
36C23×Dic61—44no—
37C23×Dic61D412yes—
38C23×Dic61D412yes—
39C23×Dic61C2312yes—
40#14519756—24no—
41#14519756—24no—
42#14519756—24no—
43#14519756—24no—
44#14519754—28no—
45#14519754—28no—
46#14519754—28no—
47#14519754—28no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—28no—
51C2×C4⋊Dic61—48no—
52C2×#11758196—22no—
53C2×#11758196—22no—
54C2×#11758196—24no—
55C2×#11758196—44no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—22no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61D414yes—
63C2×C22⋊Dic61Q814yescomplemented
64C2×C22⋊Dic61C2314yes—
65C2×C22⋊Dic61C2312yes—
66#14519684—24no—
67#14519684—28no—
68C25×C61C2311yescharacteristic
69C23×C4×C61—22no—
70C23×C4×C61D411yescharacteristic
71C23×C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61D412yes—
77C2×C22⋊C4×C61Q818yescomplemented
78C2×C22⋊C4×C61C2312yes—
79C2×C22⋊C4×C61C2314yes—
80C24×C61—42no—
81C24×C61—42no—
82C24×C61C2×D411yescharacteristic
83C24×C61C2×D412yes—
84C24×C61C2×D411yescharacteristic
85C24×C61C2×D412yes—
86C24×C61C2×Q814yes—
87C24×C61C2×Q812yes—
88C24×C61#4112yes—
89C24×C61C2411yescharacteristic
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—28no—
93C22×C4×C61—24no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—44no—
97C22×C4×C61—44no—
98C22×C4×C61—42no—
99C22×C4×C61—42no—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D412yes—
102C22×Dic61—24no—
103C22×Dic61—28no—
104C22×Dic61—22no—
105C22×Dic61—22no—
106C22×Dic61—24no—
107C22×Dic61—22no—
108C22×Dic61—24no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61—44no—
113C22×Dic61—44no—
114C22×Dic61—48no—
115C22×Dic61—42no—
116C22×Dic61—42no—
117C22×Dic61—44no—
118C22×Dic61—42no—
119C22×Dic61—41no—
120C22×Dic61—42no—
121C22×Dic61—41no—
122C22×Dic61—44no—
123C22×Dic61—44no—
124C22×Dic61C2×D414yes—
125C22×Dic61C2×D414yes—
126C2×Dic122—24no—
127C2×Dic122—24no—
128C2×Dic122—82no—
129C2×Dic122—82no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—28no—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61C2×Q818yescomplemented
135C22⋊Dic61—24no—
136C22⋊Dic61—24no—
137C22⋊Dic61—24no—
138C22⋊Dic61—24no—
139C22⋊Dic61—24no—
140C22⋊Dic61—22no—
141C22⋊Dic61C2×Q814yescomplemented
142C4⋊Dic61—24no—
143C4⋊Dic61—24no—
144C4⋊Dic61—416no—
145#11758196—28no—
146#11758196—28no—
147#11758196—48no—
148C23×C61—22no—
149C23×C61—24no—
150C23×C61—22no—
151C23×C61—24no—
152C23×C61—41no—
153C23×C61—42no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—42no—
157C23×C61—44no—
158C23×C61—48no—
159C23×C61—44no—
160C23×C61—41no—
161C23×C61#12011yescharacteristic
162C23×C61#12012yes—
163C23×C61#12011yescharacteristic
164C23×C61C22⋊Q814yes—
165C23×C61C22⋊Q812yes—
166C23×C61C22⋊Q814yes—
167C23×C61C22⋊Q812yes—
168C23×C61C4⋊Q814yes—
169C23×C61C4⋊Q812yes—
170C23×C61C4⋊Q814yes—
171C23×C61C4⋊Q812yes—
172C23×C61C22×D412yes—
173C23×C61C22×D411yescharacteristic
174C23×C61C22×Q811yescharacteristic
175C23×C61C22×Q812yes—
176C23×C61C2×#4111yescharacteristic
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—28no—
180C2×C4×C61—44no—
181C2×C4×C61—416no—
182C2×C4×C61—44no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—81no—
188C2×C4×C61—81no—
189C2×Dic61—24no—
190C2×Dic61—28no—
191C2×Dic61—24no—
192C2×Dic61—24no—
193C2×Dic61—24no—
194C2×Dic61—42no—
195C2×Dic61—48no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—44no—
199C2×Dic61—48no—
200C2×Dic61—44no—
201C2×Dic61—42no—
202C2×Dic61—44no—
203C2×Dic61—44no—
204C2×Dic61—44no—
205C2×Dic61—44no—
206C2×Dic61—44no—
207C2×Dic61—82no—
208C2×Dic61—82no—
209Dic122—84no—
210Dic122—84no—
211C2×#2062—611nomaximal, Sylow
212C22×C61—24no—
213C22×C61—28no—
214C22×C61—28no—
215C22×C61—24no—
216C22×C61—22no—
217C22×C61—21no—
218C22×C61—22no—
219C22×C61—21no—
220C22×C61—22no—
221C22×C61—21no—
222C22×C61—22no—
223C22×C61—21no—
224C22×C61—44no—
225C22×C61—42no—
226C22×C61—44no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—42no—
230C22×C61—41no—
231C22×C61#39512yes—
232C22×C61#39514yes—
233C22×C61#39512yes—
234C22×C61B(4,𝔽2)12yes—
235C22×C61#45812yes—
236C22×C61C2×#12011yescharacteristic
237C22×C61C2×C22⋊Q812yes—
238C22×C61C2×C22⋊Q811yescharacteristic
239C22×C61C2×C4⋊Q812yes—
240C22×C61C2×C4⋊Q811yescharacteristic
241C4×C61—48no—
242C4×C61—42no—
243C4×C61—42no—
244Dic61—44no—
245Dic61—44no—
246Dic61—44no—
247Dic61—44no—
248C2×#543—611no—
249C2×#385—611no—
250C2×#385—612no—
251#2062—618no—
252C2×#342—612no—
253C2×#342D6111yescharacteristic
254C2×C61—21no—
255C2×C61—22no—
256C2×C61—22no—
257C2×C61—21no—
258C2×C61—22no—
259C2×C61—24no—
260C2×C61#206214yescomplemented
261C2×C61C2×#39511yescharacteristic
262C2×C61C2×B(4,𝔽2)11yescharacteristic
263C2×C61C2×#45811yescharacteristic
264C24×C4—611no—
265#543—614no—
266#543—1224no—
267#543—1222no—
268C2×C22⋊Q8—1222no—
269C2×C22⋊Q8—1222no—
270C2×C22⋊Q8—1221no—
271C2×C22⋊Q8—1221no—
272C22×C4⋊C4—1222no—
273C22×C4⋊C4—1221no—
274C22×C22⋊C4—611no—
275C22×C22⋊C4—612no—
276C22×C22⋊C4C2×D6111yescharacteristic
277C22×C22⋊C4C2×D6112yes—
278#385—614no—
279#385—618no—
280#385—1224no—
281#385—1222no—
282C2×#95—1222no—
283C2×#95—1221no—
284#342—618no—
285#342C2×D6114yes—
286C61C2×#206211yescharacteristic, complemented, Sylow
287C25C22×D6111yescharacteristic
288C22×Q8—1221no—
289C22×Q8—1221no—
290C23×C4—22no—
291C23×C4—612no—
292C23×C4—612no—
293C23×C4—612no—
294C23×C4—1221no—
295C23×C4—1221no—
296C23×C4—1222no—
297C23×C4—1221no—
298C23×C4—1222no—
299C23×C4—2444no—
300C23×C4D24411yescharacteristic
301C23×C4D24411yescharacteristic
302C22⋊Q8—1228no—
303C22⋊Q8—1228no—
304C22⋊Q8—1228no—
305C22⋊Q8—1228no—
306C22⋊Q8—1224no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309C22⋊Q8—1224no—
310C2×C4⋊C4—1222no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1228no—
315C2×C4⋊C4—1224no—
316C2×C4⋊C4—2444no—
317C2×C4⋊C4—2448no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—614no—
323C2×C22⋊C4—612no—
324C2×C22⋊C4—614no—
325C2×C22⋊C4—614no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4Dic12218yescomplemented
333C2×C22⋊C4D24412yes—
334C2×C22⋊C4C22×D6112yes—
335C2×C22⋊C4C22×D6114yes—
336#95—1228no—
337#95—1224no—
338C24—42no—
339C24—42no—
340C24C2×Dic12214yes—
341C24C2×D24411yescharacteristic
342C24C2×D24411yescharacteristic
343C24D4×D6112yes—
344C24D4×D6112yes—
345C24#1175822412yes—
346C24Q8×D6112yes—
347C24C23×D6111yescharacteristic
348C2×Q8—1224no—
349C2×Q8—1224no—
350C2×Q8—4882no—
351C2×Q8—4882no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—28no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—614no—
363C22×C4—614no—
364C22×C4—1222no—
365C22×C4—1222no—
366C22×C4—1224no—
367C22×C4—1228no—
368C22×C4—1224no—
369C22×C4—1222no—
370C22×C4—1222no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—1224no—
374C22×C4—2444no—
375C22×C4—2444no—
376C22×C4—2448no—
377C22×C4—2444no—
378C22×C4—2444no—
379C22×C4—2442no—
380C22×C4—2442no—
381C22×C4—2444no—
382C22×C4—2442no—
383C22×C4—2442no—
384C22×C4—2441no—
385C22×C4—2441no—
386C22×C4C2×D24412yes—
387C22×C4C2×D24412yes—
388C4⋊C4—1228no—
389C4⋊C4—1228no—
390C4⋊C4—1224no—
391C4⋊C4—1224no—
392C4⋊C4—2448no—
393C4⋊C4—24416no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—28no—
398C22⋊C4—614no—
399C22⋊C4—1224no—
400C22⋊C4—1224no—
401C22⋊C4—1222no—
402C22⋊C4—1224no—
403C22⋊C4—1224no—
404C22⋊C4—1224no—
405C22⋊C4C2×Dic12218yescomplemented
406C23—22no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—41no—
411C23—42no—
412C23—41no—
413C23—48no—
414C23—41no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—41no—
419C23#1451974712yes—
420C23#1451974711yescharacteristic
421C23#1451974711yescharacteristic
422C23#1451975414yes—
423C23#1451975414yes—
424C23#1451975712yes—
425C23#1451975712yes—
426C23#1451977614yes—
427C23#1451977614yes—
428C23#1451977712yes—
429C23#1451977712yes—
430C23C2×D4×D6112yes—
431C23C22×D24411yescharacteristic
432C23C2×Q8×D6111yescharacteristic
433C23C22×Dic12212yes—
434C23C2×#1175822411yescharacteristic
435Q8—4884no—
436Q8—4884no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—28no—
440C2×C4—44no—
441C2×C4—416no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—1228no—
450C2×C4—1224no—
451C2×C4—1224no—
452C2×C4—1224no—
453C2×C4—1224no—
454C2×C4—4882no—
455C2×C4—4882no—
456C2×C4—2442no—
457C2×C4—2448no—
458C2×C4—2442no—
459C2×C4—2444no—
460C2×C4—2442no—
461C2×C4—2444no—
462C2×C4—2448no—
463C2×C4—2444no—
464C2×C4—2444no—
465C2×C4—2444no—
466C2×C4—2442no—
467C2×C4—2444no—
468C2×C4—2444no—
469C22—28no—
470C22—24no—
471C22—28no—
472C22—24no—
473C22—22no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—22no—
478C22—21no—
479C22—21no—
480C22—22no—
481C22—41no—
482C22—42no—
483C22—41no—
484C22—44no—
485C22—42no—
486C22—44no—
487C22—42no—
488C22#1683968314yes—
489C22#1683968312yes—
490C22#1683968312yes—
491C22#1683995012yes—
492C22#1683995312yes—
493C22C2×#1451974711yescharacteristic
494C22C2×#1451975412yes—
495C22C2×#1451975711yescharacteristic
496C22C2×#1451977612yes—
497C22C2×#1451977711yescharacteristic
498C4—48no—
499C4—42no—
500C4—42no—
501C4—2444no—
502C4—2444no—
503C4—2444no—
504C4—2444no—
505C2—22no—
506C2—21no—
507C2—24no—
508C2—22no—
509C2—22no—
510C2—21no—
511C2#2144717314yesdirect product factor
512C2C2×#1683968311yescharacteristic
513C2C2×#1683995011yescharacteristic
514C2C2×#1683995311yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC2×C61×#342C2
socleC23×C61#120
prime radicalC25×C61C23
hypercenterC2×#342D61
hypocenterC61C2×#2062
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. #21447173C2direct (N × H)
63. C2×C22⋊Dic61Q8semidirect
77. C2×C22⋊C4×C61Q8semidirect
134. C22⋊C4×C61C2×Q8semidirect
141. C22⋊Dic61C2×Q8semidirect
260. C2×C61#2062semidirect
286. C61C2×#2062semidirect
332. C2×C22⋊C4Dic122semidirect
405. C22⋊C4C2×Dic122semidirect
511. C2#21447173direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403882
C2×#168396372
C2×#168396342
#214471732
C2×#168395282
C2×C61×#3422
C2×#206261