Finite Group Databasebeta

← C2×#21444791 #29652195

Subgroups — C2×#21444791 #29652195

Order 15616.

1200 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840391C211yesmaximal, characteristic
2C2×#16839918C211yesmaximal, characteristic
3C2×#16839681C211yesmaximal, characteristic
4#21444791C218yesmaximal, direct product factor
5C2×#16839500C211yesmaximal, characteristic
6C2×#16839499C211yesmaximal, characteristic
7C2×#16839487C211yesmaximal, characteristic
8C2×C61×#451C211yesmaximal, characteristic
9C22×#11758202—21no—
10#16840391—22no—
11#16840391—22no—
12#16840391—22no—
13#16840391C2214yes—
14C2×#14519776—21no—
15C2×#14519775—21no—
16C2×#14519767—21no—
17C2×#14519767—21no—
18C2×#14519756C2211yescharacteristic
19C2×#14519754—21no—
20C2×#14519733—21no—
21C22×#11758196—21no—
22C22×#11758196C2211yescharacteristic
23#16839918—22no—
24#16839918C2214yes—
25C2×#14519701—21no—
26C2×C22⋊#92418C2211yescharacteristic
27#16839681—22no—
28#16839681C2214yes—
29C2×#14519684—21no—
30C2×#14519684C2211yescharacteristic
31#16839500C2214yes—
32#16839499C2214yes—
33#16839487C2214yes—
34C23×Q8×C61C2211yescharacteristic
35C22×Q16×C61—21no—
36C2×C22⋊Q8×C61C2211yescharacteristic
37C61×#451—22no—
38C61×#451C2214yes—
39C2×#103×C61—21no—
40C2×C22⋊C8×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122D411yescharacteristic
43C23×Dic61—21no—
44C23×Dic61D411yescharacteristic
45C23×Dic61D411yescharacteristic
46C2×#11758202—24no—
47C2×#11758202—44no—
48C22×#92418—21no—
49#14519776—24no—
50#14519776—24no—
51#14519775—24no—
52#14519775—24no—
53#14519767—24no—
54#14519767—24no—
55#14519767—24no—
56#14519767—24no—
57#14519756—21no—
58#14519756—21no—
59#14519756D412yes—
60#14519756C2312yes—
61#14519754—24no—
62#14519754—24no—
63#14519733—24no—
64#14519733—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—24no—
77C2×#11758196—44no—
78C2×#11758196D412yes—
79C2×#11758196C2312yes—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83#14519701—24no—
84C22⋊#92418—21no—
85C22⋊#92418C2312yes—
86#14519684—21no—
87#14519684—24no—
88#14519684C2312yes—
89C22×Q8×C61—21no—
90C22×Q8×C61—21no—
91C22×Q8×C61—21no—
92C22×Q8×C61—21no—
93C22×Q8×C61—21no—
94C22×Q8×C61—44no—
95C22×Q8×C61D411yescharacteristic
96C22×Q8×C61C2312yes—
97C23×C4×C61—21no—
98C23×C4×C61C2311yescharacteristic
99C2×Q16×C61—24no—
100C2×Q16×C61—44no—
101C22×C8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61D412yes—
105C22⋊Q8×C61C2312yes—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C22⋊C4×C61—21no—
109#103×C61—24no—
110C22⋊C8×C61—21no—
111C22⋊C8×C61C2312yes—
112C24×C61C2×D411yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—24no—
115C2×Q8×C61—24no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—44no—
119C2×Q8×C61—42no—
120C2×Q8×C61—42no—
121C2×Q8×C61—44no—
122C2×Q8×C61—42no—
123C2×Q8×C61—44no—
124C2×Q8×C61—44no—
125C2×Q8×C61—41no—
126C2×Q8×C61—41no—
127C2×Q8×C61C2×D412yes—
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—41no—
135C22×C4×C61—41no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—24no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61Q1614yescomplemented
155C22×Dic61C2×D412yes—
156C22×Dic61C2×D412yes—
157C2×Dic122—24no—
158C2×Dic122—21no—
159C2×Dic122—42no—
160C2×Dic122—82no—
161C2×Dic122C2×D412yes—
162Q16×C61—48no—
163C2×C8×C61—22no—
164C2×C8×C61—42no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C2×D412yes—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊Dic61—22no—
172C22⋊Dic61—22no—
173#11758202—48no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61C2×D412yes—
177#11758196—24no—
178#11758196—24no—
179#11758196—24no—
180#11758196—24no—
181#11758196—22no—
182#11758196—22no—
183#11758196—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196—22no—
187#11758196—48no—
188C4×Dic61—24no—
189C4×Dic61—24no—
190C2×#92418—22no—
191C2×#92418—42no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61#12011yescharacteristic
195C23×C61#12011yescharacteristic
196C23×C61C2×Q1612yes—
197C23×C61#13712yes—
198C23×C61C22×D411yescharacteristic
199Q8×C61—24no—
200Q8×C61—24no—
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—44no—
204Q8×C61—44no—
205Q8×C61—42no—
206Q8×C61—81no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—44no—
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—42no—
235C2×Dic61—48no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—81no—
242C2×Dic61C2×Q1614yescomplemented
243Dic122—42no—
244Dic122—81no—
245Dic122—84no—
246C8×C61—42no—
247#92418—42no—
248C2×#1605—611nomaximal, Sylow
249C22×C61—24no—
250C22×C61—24no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61#45112yes—
256C22×C61#45112yes—
257C22×C61#45612yes—
258C22×C61B(4,𝔽2)12yes—
259C22×C61C2×#12011yescharacteristic
260C22×C61C22×Q1611yescharacteristic
261C22×C61C2×#13711yescharacteristic
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—44no—
269Dic61—24no—
270Dic61—44no—
271Dic61—42no—
272Dic61—42no—
273Dic61—42no—
274C2×#544—611no—
275C2×#451—611no—
276C2×#451D6111yescharacteristic
277C2×#395—611no—
278#1605—618no—
279C2×#328—611no—
280C2×#328—611no—
281C2×#324—611no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—22no—
285C2×C61#160514yescomplemented
286C2×C61C2×#45111yescharacteristic
287C2×C61C2×#45611yescharacteristic
288C2×C61C2×B(4,𝔽2)11yescharacteristic
289C23×Q8C2×D6111yescharacteristic
290C22×Q16—21no—
291C22×Q16—1221no—
292#544—614no—
293#544—1222no—
294#544—1222no—
295#544—1222no—
296C2×C4⋊Q8—1221no—
297C2×C4⋊Q8—1221no—
298C2×#125—1221no—
299C2×#125—1221no—
300C2×C22⋊Q8—611no—
301C2×C22⋊Q8—1221no—
302C2×C22⋊Q8C2×D6111yescharacteristic
303C2×#117—1221no—
304C22×C4⋊C4—611no—
305C22×C4⋊C4—1221no—
306#451—22no—
307#451—614no—
308#451—1222no—
309#451C2×D6114yes—
310C2×#103—21no—
311C2×#103—1221no—
312C2×C22⋊C8—611no—
313C2×C22⋊C8C2×D6111yescharacteristic
314#395—614no—
315#395—1222no—
316C2×#95—611no—
317C2×#95—1221no—
318#328—614no—
319#328—614no—
320#324—614no—
321C61C2×#160511yescharacteristic, complemented, Sylow
322C22×Q8—21no—
323C22×Q8—21no—
324C22×Q8—21no—
325C22×Q8—21no—
326C22×Q8—21no—
327C22×Q8—44no—
328C22×Q8—611no—
329C22×Q8—1221no—
330C22×Q8C22⋊D6111yescharacteristic
331C22×Q8C22×D6112yes—
332C23×C4—21no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—1221no—
336C23×C4C22×D6111yescharacteristic
337C2×Q16—24no—
338C2×Q16—44no—
339C2×Q16—1224no—
340C2×Q16—2444no—
341C22×C8—21no—
342C22×C8—1221no—
343C4⋊Q8—1224no—
344C4⋊Q8—1224no—
345C4⋊Q8—1224no—
346C4⋊Q8—1224no—
347#125—1224no—
348#125—1224no—
349#125—1224no—
350#125—1224no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8—612no—
354C22⋊Q8—612no—
355C22⋊Q8—1224no—
356C22⋊Q8—1224no—
357C22⋊Q8—1221no—
358C22⋊Q8—1221no—
359C22⋊Q8C22⋊D6112yes—
360C22⋊Q8C22×D6112yes—
361#117—1224no—
362#117—1224no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—2444no—
379C2×C4⋊C4C22⋊D6111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1221no—
382C2×C42—1221no—
383C2×C42—1221no—
384#103—24no—
385#103—1224no—
386C22⋊C8—21no—
387C22⋊C8—612no—
388C22⋊C8—1221no—
389C22⋊C8C22×D6112yes—
390#95—612no—
391#95—1224no—
392#95—1221no—
393C24D4×D6111yescharacteristic
394C2×Q8—21no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—41no—
407C2×Q8—41no—
408C2×Q8—612no—
409C2×Q8—1224no—
410C2×Q8—1221no—
411C2×Q8—4882no—
412C2×Q8—2442no—
413C2×Q8C2×C22⋊D6112yes—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—612no—
423C22×C4—614no—
424C22×C4—612no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4D4×D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443Q16—48no—
444Q16—2448no—
445C2×C8—22no—
446C2×C8—42no—
447C2×C8—1222no—
448C2×C8—2442no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—612no—
453C4⋊C4—1224no—
454C4⋊C4—1224no—
455C4⋊C4—1224no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1221no—
464C4⋊C4—1222no—
465C4⋊C4—2448no—
466C4⋊C4C2×C22⋊D6112yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—1222no—
470C22⋊C4—1222no—
471C42—1224no—
472C42—1224no—
473C23—22no—
474C23—22no—
475C23#1451974511yescharacteristic
476C23#1451974511yescharacteristic
477C23Q16×D6112yes—
478C23#1451980912yes—
479C23C2×D4×D6111yescharacteristic
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—44no—
487Q8—81no—
488Q8—4881no—
489Q8—4884no—
490Q8—2442no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—614no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—4881no—
516C2×C4—2442no—
517C2×C4—2448no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4#1451974511yescharacteristic
524C2×C4#1451974511yescharacteristic
525C2×C4C2×D4×D6111yescharacteristic
526C2×C4C22×C22⋊D6111yescharacteristic
527C8—42no—
528C8—2442no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—24no—
534C22—24no—
535C22#1683991612yes—
536C22#1683991612yes—
537C22#1683994312yes—
538C22#1683994812yes—
539C22C2×#1451974511yescharacteristic
540C22C2×Q16×D6111yescharacteristic
541C22C2×#1451980911yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—42no—
547C4—44no—
548C4—42no—
549C4—1224no—
550C4—2444no—
551C4—2442no—
552C4—2442no—
553C4—2442no—
554C2—21no—
555C2—21no—
556C2—22no—
557C2#2144479114yesdirect product factor
558C2C2×#1683991611yescharacteristic
559C2C2×#1683994311yescharacteristic
560C2C2×#1683994811yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#451C2
socleC23×C61#120
prime radicalC24×C61C2×D4
hypercenterC2×#451D61
hypocenterC61C2×#1605
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
4. #21444791C2direct (N × H)
154. C22×Dic61Q16semidirect
242. C2×Dic61C2×Q16semidirect
285. C2×C61#1605semidirect
321. C61C2×#1605semidirect
557. C2#21444791direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403912
C2×#168399182
C2×#168396812
#214447912
C2×#168395002
C2×#168394992
C2×#168394872
C2×C61×#4512
C2×#160561