Finite Group Databasebeta

← C2×#21444742 #29652089

Subgroups — C2×#21444742 #29652089

Order 15616.

1696 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839912C212yesmaximal, complemented
2C2×#16839671C211yesmaximal, characteristic
3#21444742C218yesmaximal, direct product factor
4C2×#16839499C212yesmaximal, complemented
5C2×#16839486C211yesmaximal, characteristic, complemented
6C2×C61×#543C211yesmaximal, characteristic, complemented
7C23×D244C2211yescharacteristic
8C23×C22⋊D61—21no—
9C22×#11758201—22no—
10C22×#11758198—21no—
11#16839912—24no—
12#16839912C2218yescomplemented
13C2×#14519697—22no—
14C2×C22⋊#92418C2212yes—
15#16839671—22no—
16#16839671C2214yes—
17C2×#14519684C2211yescharacteristic
18#16839499C2218yescomplemented
19#16839486C2214yescomplemented
20C61×#543—24no—
21C61×#543—22no—
22C61×#543C2214yescomplemented
23C2×C22⋊Q8×C61—22no—
24C2×C22⋊Q8×C61—21no—
25C2×C22⋊Q8×C61—22no—
26C2×C22⋊Q8×C61C2212yes—
27C22×C22⋊C4×C61—21no—
28C22×C22⋊C4×C61C2211yescharacteristic
29C24×D61—21no—
30C22×D244—22no—
31C22×D244—22no—
32C22×D244—21no—
33C22×D244—44no—
34C22×D244C2312yes—
35C22×C22⋊D61—24no—
36C22×C22⋊D61—44no—
37C22×C22⋊D61—44no—
38C22×C22⋊D61—44no—
39C23×Dic61—21no—
40C2×#11758201—28no—
41C2×#11758201—48no—
42C22×#92418—22no—
43C2×#11758198—24no—
44C2×#11758198—44no—
45#14519697—28no—
46C22⋊#92418—22no—
47C22⋊#92418C2314yes—
48#14519684—21no—
49#14519684C2312yes—
50C25×C61D411yescharacteristic
51C22×Q8×C61—21no—
52C22×Q8×C61D412yescomplemented
53C23×C4×C61—21no—
54C23×C4×C61—21no—
55C23×C4×C61D411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C22⋊Q8×C61—28no—
58C22⋊Q8×C61—28no—
59C22⋊Q8×C61—24no—
60C22⋊Q8×C61—28no—
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61—24no—
64C22⋊Q8×C61—28no—
65C22⋊Q8×C61D414yescomplemented
66C22⋊Q8×C61C2314yes—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61D412yescomplemented
71C2×C22⋊C4×C61—24no—
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—22no—
77C2×C22⋊C4×C61D412yes—
78C2×C22⋊C4×C61C2312yes—
79C24×C61—21no—
80C24×C61—21no—
81C24×C61—22no—
82C24×C61—42no—
83C24×C61—42no—
84C24×C61SD1614yescomplemented
85C24×C61C2×D411yescharacteristic
86C24×C61C2×D412yes—
87C23×D61—24no—
88C23×D61—22no—
89C23×D61—42no—
90C23×D61—42no—
91C23×D61—42no—
92C23×D61—41no—
93C23×D61—41no—
94C23×D61—42no—
95C23×D61—42no—
96C2×Q8×C61—24no—
97C2×Q8×C61—22no—
98C2×Q8×C61—44no—
99C2×Q8×C61—82no—
100C2×Q8×C61C2×D414yescomplemented
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—24no—
105C22×C4×C61—24no—
106C22×C4×C61—21no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—82no—
110C22×C4×C61—82no—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—48no—
117C2×C22⋊D61—48no—
118C2×C22⋊D61—432no—
119C2×C22⋊D61—48no—
120C22×Dic61—24no—
121C22×Dic61—22no—
122C22×Dic61—42no—
123C22×Dic61—42no—
124C2×D244—28no—
125C2×D244—24no—
126C2×D244—44no—
127C2×D244—48no—
128C2×D244—44no—
129C2×D244—44no—
130C2×D244—42no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—28no—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61C2×D414yescomplemented
137C22⋊C4×C61—28no—
138C22⋊C4×C61—24no—
139C22⋊C4×C61—24no—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—24no—
142#11758201—416no—
143#11758198—48no—
144C2×#92418—24no—
145C2×#92418—44no—
146C23×C61—22no—
147C23×C61—22no—
148C23×C61—22no—
149C23×C61—24no—
150C23×C61—22no—
151C23×C61—24no—
152C23×C61—48no—
153C23×C61—42no—
154C23×C61—42no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—84no—
158C23×C61—81no—
159C23×C61—81no—
160C23×C61#12011yescharacteristic
161C23×C61#12011yescharacteristic
162C23×C61C2×SD1612yes—
163C23×C61C2×SD1614yescomplemented
164C23×C61#13712yes—
165C23×C61C22×D411yescharacteristic
166C22×D61—24no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61—44no—
170C22×D61—48no—
171C22×D61—42no—
172C22×D61—48no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—44no—
177C22×D61—84no—
178C22×D61—84no—
179C22×D61—82no—
180C22×D61—82no—
181C22×D61—82no—
182C22×D61—82no—
183Q8×C61—44no—
184Q8×C61—82no—
185Q8×C61—84no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—44no—
195C2×C4×C61—44no—
196C2×C4×C61—42no—
197C2×C4×C61—81no—
198C2×C4×C61—84no—
199C2×C4×C61—84no—
200C2×C4×C61#12012yes—
201C2×C4×C61C22×D412yes—
202C22⋊D61—432no—
203C2×Dic61—24no—
204C2×Dic61—48no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207D244—28no—
208D244—44no—
209D244—48no—
210#92418—44no—
211C2×#1596—611nomaximal, Sylow
212C22×C61—24no—
213C22×C61—24no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—24no—
219C22×C61—44no—
220C22×C61—48no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—82no—
224C22×C61—84no—
225C22×C61#44812yes—
226C22×C61#44812yes—
227C22×C61#45312yes—
228C22×C61B(4,𝔽2)12yes—
229C22×C61C2×#12011yescharacteristic
230C22×C61C22×SD1611yescharacteristic
231C22×C61C2×#13711yescharacteristic
232C2×D61—44no—
233C2×D61—44no—
234C2×D61—44no—
235C2×D61—82no—
236C2×D61—82no—
237C2×D61—88no—
238C2×D61—88no—
239C2×D61—82no—
240C2×D61—82no—
241C4×C61—22no—
242C4×C61—22no—
243C4×C61—42no—
244C4×C61—44no—
245C4×C61—44no—
246Dic61—44no—
247C2×#543D6111yescharacteristic, complemented
248C2×#450—612no—
249C2×#392—611no—
250#1596—618no—
251C2×#328—612no—
252C2×#323—611no—
253C2×C61—21no—
254C2×C61—21no—
255C2×C61—22no—
256C2×C61—44no—
257C2×C61#159614yescomplemented
258C2×C61C2×#44811yescharacteristic
259C2×C61C2×#45311yescharacteristic
260C2×C61C2×B(4,𝔽2)11yescharacteristic
261D61—84no—
262C23×D4—611no—
263C23×D4—1221no—
264C22×SD16—1222no—
265#543—24no—
266#543—22no—
267#543C2×D6114yescomplemented
268C2×C22⋊Q8—22no—
269C2×C22⋊Q8—21no—
270C2×C22⋊Q8—22no—
271C2×C22⋊Q8C2×D6112yes—
272C22×C22⋊C4—21no—
273C22×C22⋊C4—1221no—
274C22×C22⋊C4C2×D6111yescharacteristic
275#450—618no—
276#450—1224no—
277C2×#102—1222no—
278C2×C22⋊C8—612no—
279#392—614no—
280#392—1222no—
281C2×#95—611no—
282#328—618no—
283#323—614no—
284C61C2×#159611yescharacteristic, complemented, Sylow
285C25—1221no—
286C25C22⋊D6111yescharacteristic
287C22×Q8—21no—
288C22×Q8C22⋊D6112yescomplemented
289C22×D4—612no—
290C22×D4—1224no—
291C22×D4—1222no—
292C22×D4—1222no—
293C22×D4—1221no—
294C22×D4—2444no—
295C22×D4—2444no—
296C22×D4—2444no—
297C22×D4—2444no—
298C23×C4—21no—
299C23×C4—21no—
300C23×C4—1221no—
301C23×C4C22⋊D6111yescharacteristic
302C23×C4C22×D6111yescharacteristic
303C2×SD16—1228no—
304C2×SD16—2448no—
305C22×C8—1222no—
306C22⋊Q8—28no—
307C22⋊Q8—28no—
308C22⋊Q8—24no—
309C22⋊Q8—28no—
310C22⋊Q8—24no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—28no—
314C22⋊Q8C22⋊D6114yescomplemented
315C22⋊Q8C22×D6114yes—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4C22⋊D6112yescomplemented
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—1224no—
327C2×C22⋊C4—2444no—
328C2×C22⋊C4C22⋊D6112yes—
329C2×C22⋊C4C22×D6112yes—
330#102—1228no—
331C22⋊C8—614no—
332C22⋊C8—1222no—
333#95—612no—
334#95—1221no—
335C24—21no—
336C24—22no—
337C24—21no—
338C24—42no—
339C24—42no—
340C24—1224no—
341C24—1222no—
342C24—2442no—
343C24—2442no—
344C24—2442no—
345C24—2442no—
346C24—2441no—
347C24—2441no—
348C24—2442no—
349C24#1175820114yescomplemented
350C24C2×C22⋊D6111yescharacteristic
351C24C2×C22⋊D6112yes—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—44no—
355C2×Q8—82no—
356C2×Q8C2×C22⋊D6114yescomplemented
357C2×D4—1228no—
358C2×D4—1224no—
359C2×D4—2444no—
360C2×D4—2444no—
361C2×D4—2448no—
362C2×D4—2444no—
363C2×D4—2448no—
364C2×D4—2448no—
365C2×D4—24432no—
366C2×D4—2448no—
367C2×D4—2442no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—1224no—
379C22×C4—1222no—
380C22×C4—2442no—
381C22×C4—2442no—
382C22×C4C2×C22⋊D6112yes—
383C22×C4C2×C22⋊D6112yes—
384C22×C4C2×C22⋊D6112yes—
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C23×D6111yescharacteristic
387SD16—24416no—
388C2×C8—1224no—
389C2×C8—2444no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4C2×C22⋊D6114yescomplemented
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—2448no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—24no—
408C23—48no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—81no—
414C23—81no—
415C23—84no—
416C23—1224no—
417C23—4882no—
418C23—4884no—
419C23—4884no—
420C23—4882no—
421C23—4882no—
422C23—4882no—
423C23—2442no—
424C23—2442no—
425C23—2442no—
426C23—2444no—
427C23—2448no—
428C23—2448no—
429C23—2444no—
430C23—2442no—
431C23—2442no—
432C23—2442no—
433C23#1451974611yescharacteristic
434C23#1451974611yescharacteristic
435C23C2×#1175820112yes—
436C23C2×#1175820114yescomplemented
437C23#1451980812yes—
438C23C22×C22⋊D6111yescharacteristic
439Q8—44no—
440Q8—82no—
441Q8—84no—
442D4—1228no—
443D4—2444no—
444D4—2448no—
445D4—24432no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—81no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—1224no—
461C2×C4—2442no—
462C2×C4—2448no—
463C2×C4—2442no—
464C2×C4#1451974612yes—
465C2×C4C22×C22⋊D6112yes—
466C8—2444no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—48no—
478C22—84no—
479C22—82no—
480C22—4882no—
481C22—4888no—
482C22—4882no—
483C22—4882no—
484C22—4888no—
485C22—4882no—
486C22—2444no—
487C22—2444no—
488C22—2444no—
489C22#1683990112yes—
490C22#1683990112yes—
491C22#1683993012yes—
492C22#1683994912yes—
493C22C2×#1451974611yescharacteristic
494C22C22×#1175820111yescharacteristic
495C22C2×#1451980811yescharacteristic
496C4—22no—
497C4—22no—
498C4—42no—
499C4—44no—
500C4—44no—
501C4—2444no—
502C2—21no—
503C2—22no—
504C2—21no—
505C2—44no—
506C2—4884no—
507C2#2144474214yesdirect product factor
508C2C2×#1683990111yescharacteristic
509C2C2×#1683993011yescharacteristic
510C2C2×#1683994911yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#543C2
socleC23×C61#120
prime radicalC2×#16839671C2
hypercenterC2×#543D61
hypocenterC61C2×#1596
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. C2×#16839912C2semidirect
3. #21444742C2direct (N × H)
4. C2×#16839499C2semidirect
5. C2×#16839486C2semidirect
6. C2×C61×#543C2semidirect
12. #16839912C22semidirect
18. #16839499C22semidirect
19. #16839486C22semidirect
22. C61×#543C22semidirect
52. C22×Q8×C61D4semidirect
65. C22⋊Q8×C61D4semidirect
70. C2×C4⋊C4×C61D4semidirect
84. C24×C61SD16semidirect
100. C2×Q8×C61C2×D4semidirect
136. C4⋊C4×C61C2×D4semidirect
163. C23×C61C2×SD16semidirect
247. C2×#543D61semidirect
257. C2×C61#1596semidirect
267. #543C2×D61semidirect
284. C61C2×#1596semidirect
288. C22×Q8C22⋊D61semidirect
314. C22⋊Q8C22⋊D61semidirect
319. C2×C4⋊C4C22⋊D61semidirect
349. C24#11758201semidirect
356. C2×Q8C2×C22⋊D61semidirect
395. C4⋊C4C2×C22⋊D61semidirect
436. C23C2×#11758201semidirect
507. C2#21444742direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168399122
C2×#168396712
#214447422
C2×#168394992
C2×#168394862
C2×C61×#5432
C2×#159661