Finite Group Databasebeta

← C2×#20254335 #27087468

Subgroups — C2×#20254335 #27087468

Order 12032.

2134 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689677C211yesmaximal, characteristic
2C2×C4⋊Q8×D47C211yesmaximal, characteristic
3#20254335C2116yesmaximal, direct product factor
4C2×#15689501C212yesmaximal, complemented
5C2×#15689472C212yesmaximal, complemented
6C2×#15689334C211yesmaximal, characteristic, complemented
7C2×#15689333C211yesmaximal, characteristic, complemented
8C2×#15689330C211yesmaximal, characteristic, complemented
9C2×#15689280C212yesmaximal
10C2×#15689277C212yesmaximal
11C2×#15689175C211yesmaximal, characteristic
12C2×C47×#462C211yesmaximal, characteristic, complemented
13C22×Q8×D47C2212yes—
14C22×Dic188C2212yes—
15C22×#10625337C2212yes—
16C22×#10625337C2212yes—
17#15689677—24no—
18#15689677C2218yes—
19C2×#13094024C2211yescharacteristic
20C2×#13094023—21no—
21C2×#13094023—21no—
22C2×#13094023C2211yescharacteristic
23C2×#13094022C2212yes—
24C2×#13094022C2211yescharacteristic
25C4⋊Q8×D47—24no—
26C4⋊Q8×D47C2218yes—
27C2×#13094014—21no—
28C2×#13094004C2211yescharacteristic
29C2×#13094002—21no—
30C2×#13094000C2211yescharacteristic
31C2×C4×Dic94—21no—
32C2×C4×Dic94C2211yescharacteristic
33C2×#13093981—21no—
34C2×C4⋊C4×D47—21no—
35C2×C4⋊C4×D47—21no—
36C2×C42×D47C2211yescharacteristic
37#15689501C22116yescomplemented
38#15689472C22116yescomplemented
39#15689334C2218yescomplemented
40#15689333C2218yescomplemented
41#15689330C2218yescomplemented
42#15689280C22116yes—
43#15689277C22116yes—
44#15689175C2218yes—
45C2×C4⋊(C47⋊C8)C2211yescharacteristic
46C2×#13093949C2212yes—
47C2×#13093948C2212yes—
48C2×#13093947C2212yes—
49C2×(C2×D47)⋊C8C2212yes—
50C2×C8×Dic47C2212yes—
51C22×Q16×C47C2212yes—
52C2×C4⋊Q8×C47C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C47×#462C2218yescomplemented
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×#103×C47C2212yes—
57C22×Dic94—21no—
58C22×Dic94—22no—
59C22×Dic94C2311yescharacteristic
60C22×Dic94C2312yes—
61C2×Q8×D47—24no—
62C2×Q8×D47—44no—
63C2×Q8×D47—44no—
64C2×Q8×D47C2318yes—
65C22×C4×D47—22no—
66C22×C4×D47D411yescharacteristic
67C22×C4×D47D411yescharacteristic
68C22×C4×D47C2311yescharacteristic
69C2×Dic188—44no—
70C2×Dic188C2318yes—
71C2×#10625337—44no—
72C2×#10625337—44no—
73C2×#10625337C2318yes—
74C2×#10625337C2318yes—
75C22×C47⋊C8C2312yes—
76#13094024—22no—
77#13094024C2314yes—
78#13094023—28no—
79#13094023—28no—
80#13094023—22no—
81#13094023C2314yes—
82#13094022—24no—
83#13094022—22no—
84#13094022C2318yes—
85#13094022C2314yes—
86#13094014—28no—
87#13094004—22no—
88#13094004C2314yes—
89#13094002—28no—
90#13094000—22no—
91#13094000C2314yes—
92C4×Dic94—22no—
93C4×Dic94—28no—
94C4×Dic94C2314yes—
95#13093981—28no—
96C2×C4⋊Dic47—21no—
97C2×C4⋊Dic47—21no—
98C2×C4⋊Dic47—21no—
99C2×C4⋊Dic47C2311yescharacteristic
100C2×Dic47⋊C4—21no—
101C2×Dic47⋊C4—22no—
102C2×Dic47⋊C4—22no—
103C2×Dic47⋊C4—21no—
104C2×Dic47⋊C4—21no—
105C2×Dic47⋊C4—21no—
106C2×Dic47⋊C4—21no—
107C4⋊C4×D47—28no—
108C4⋊C4×D47—28no—
109C2×(C2×D47)⋊C4—21no—
110C2×(C2×D47)⋊C4—21no—
111C2×C4×Dic47—21no—
112C2×C4×Dic47D411yescharacteristic
113C2×C4×Dic47D411yescharacteristic
114C2×C4×Dic47C2311yescharacteristic
115C42×D47D414yes—
116C42×D47C2314yes—
117C4⋊(C47⋊C8)C2314yes—
118#13093949C2318yes—
119#13093948C2318yes—
120#13093947C2318yes—
121(C2×D47)⋊C8C2318yes—
122C8×Dic47C2318yes—
123C22×Q8×C47C2311yescharacteristic
124C22×Q8×C47C2312yes—
125C2×Q16×C47—44no—
126C2×Q16×C47C2318yes—
127C22×C8×C47C2312yes—
128C4⋊Q8×C47—22no—
129C4⋊Q8×C47C2314yes—
130C4×Q8×C47—22no—
131C4×Q8×C47C2314yes—
132C2×C4⋊C4×C47—21no—
133C2×C4⋊C4×C47—21no—
134C2×C4⋊C4×C47—21no—
135C2×C4⋊C4×C47C2311yescharacteristic
136C2×C42×C47—21no—
137C2×C42×C47C2311yescharacteristic
138C4⋊C8×C47C2314yes—
139#103×C47C2318yes—
140C23×D47C2×D411yescharacteristic
141C2×Q8×C47—21no—
142C2×Q8×C47—22no—
143C2×Q8×C47—22no—
144C2×Q8×C47—24no—
145C2×Q8×C47—22no—
146C2×Q8×C47—24no—
147C2×Q8×C47C2414yes—
148C2×Q8×C47C2412yes—
149C22×C4×C47—21no—
150C22×C4×C47—21no—
151C22×C4×C47—22no—
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—22no—
158C22×Dic47C2×D411yescharacteristic
159C22×Dic47C2×D411yescharacteristic
160C22×Dic47C2×D411yescharacteristic
161Q8×D47—48no—
162Q8×D47—48no—
163Q8×D47—416no—
164C2×C4×D47—22no—
165C2×C4×D47—22no—
166C2×C4×D47—24no—
167C2×C4×D47—28no—
168C2×C4×D47—48no—
169C2×C4×D47Q1618yescomplemented
170C2×C4×D47C2×D412yes—
171C2×C4×D47C2×D414yes—
172C2×C4×D47C2×D414yes—
173C2×C4×D47C2412yes—
174C2×Dic94—22no—
175C2×Dic94—24no—
176C2×Dic94—22no—
177C2×Dic94—24no—
178C2×Dic94—21no—
179C2×Dic94—22no—
180C2×Dic94—24no—
181C2×Dic94—28no—
182C2×Dic94—48no—
183C2×Dic94—44no—
184C2×Dic94C2414yes—
185C2×Dic94C2412yes—
186Q16×C47—48no—
187C2×C8×C47—24no—
188C2×C8×C47C2414yes—
189C4⋊C4×C47—24no—
190C4⋊C4×C47—24no—
191C4⋊C4×C47—24no—
192C4⋊C4×C47—21no—
193C4⋊C4×C47C2412yes—
194C42×C47—24no—
195C42×C47C2×D412yes—
196C42×C47C2412yes—
197#10625337—48no—
198#10625337—48no—
199(C2×D47)⋊C4—24no—
200(C2×D47)⋊C4—24no—
201C4⋊Dic47—24no—
202C4⋊Dic47—24no—
203C4⋊Dic47—21no—
204C4⋊Dic47—24no—
205C4⋊Dic47C2412yes—
206Dic47⋊C4—24no—
207Dic47⋊C4—28no—
208Dic47⋊C4—28no—
209Dic47⋊C4—24no—
210Dic47⋊C4—24no—
211Dic47⋊C4—24no—
212Dic47⋊C4—24no—
213C4×Dic47—24no—
214C4×Dic47C2×D412yes—
215C4×Dic47C2×D414yes—
216C4×Dic47C2×D414yes—
217C4×Dic47C2412yes—
218C2×C47⋊C8—24no—
219C2×C47⋊C8C2414yes—
220Dic188—48no—
221C23×C47C22×D411yescharacteristic
222C22×D47—22no—
223C22×D47—22no—
224C22×D47C2×Q1614yes—
225C22×D47C22×D412yes—
226Q8×C47—22no—
227Q8×C47—24no—
228Q8×C47—22no—
229Q8×C47—24no—
230Q8×C47—44no—
231Q8×C47—42no—
232C2×C4×C47—22no—
233C2×C4×C47—22no—
234C2×C4×C47—24no—
235C2×C4×C47—22no—
236C2×C4×C47—44no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47C2×Q1614yes—
240C2×C4×C47C22×D412yes—
241C2×C4×C47C22×D411yescharacteristic
242C2×C4×C47C22×D412yes—
243C2×C4×C47H(5,𝔽2)12yes—
244C2×C4×C47H(5,𝔽2)12yes—
245C2×C4×C47C2511yescharacteristic
246C2×Dic47—22no—
247C2×Dic47—22no—
248C2×Dic47—22no—
249C2×Dic47—24no—
250C2×Dic47—44no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47C2×Q1614yes—
254C2×Dic47#13714yes—
255C2×Dic47C22×D412yes—
256C2×Dic47C22×D412yes—
257C2×Dic47C22×D412yes—
258C4×D47—24no—
259C4×D47—22no—
260C4×D47—22no—
261C4×D47—28no—
262C4×D47—416no—
263C4×D47C2×Q16116yescomplemented
264Dic94—22no—
265Dic94—24no—
266Dic94—22no—
267Dic94—24no—
268Dic94—416no—
269Dic94—42no—
270Dic94—44no—
271Dic94—48no—
272C8×C47—24no—
273C47⋊C8—24no—
274C2×#3209—471nomaximal, Sylow
275C22×C47#53512yes—
276C22×C47C22×Q1612yes—
277C22×C47C2×#13712yes—
278C22×C47C23×D411yescharacteristic
279C2×D47—22no—
280C2×D47—28no—
281C2×D47—22no—
282C2×D47C22×Q1614yes—
283C4×C47—22no—
284C4×C47—44no—
285C4×C47—42no—
286C4×C47—42no—
287C4×C47#53512yes—
288C4×C47C22×Q1614yes—
289C4×C47C2×H(5,𝔽2)11yescharacteristic
290C4×C47C2×H(5,𝔽2)11yescharacteristic
291Dic47—22no—
292Dic47—44no—
293Dic47—42no—
294Dic47—42no—
295Dic47C22×Q1614yes—
296Dic47C2×#13714yes—
297C2×#533—471no—
298C22×C4⋊Q8—471no—
299#3209—4716no—
300C2×#494—472no—
301C2×#487—472no—
302C2×#462—471no—
303C2×#462—471no—
304C2×#462—471no—
305C2×#462D4711yescharacteristic, complemented
306C2×#451—472no—
307C2×#451—472no—
308C2×#431—471no—
309C2×C47#320914yescomplemented
310C2×C47C2×#53511yescharacteristic
311C2×C47C23×Q1611yescharacteristic
312C2×C47C22×#13711yescharacteristic
313D47—24no—
314C23×Q8—472no—
315C22×Q16—472no—
316C22×Q16—472no—
317C22×Q16—472no—
318C22×Q16C2×D4712yes—
319#533—478no—
320#533—944no—
321C2×C4⋊Q8—471no—
322C2×C4⋊Q8—472no—
323C2×C4⋊Q8—471no—
324C2×C4⋊Q8—478no—
325C2×C4⋊Q8—471no—
326C2×C4⋊Q8—941no—
327C2×C4⋊Q8—941no—
328C2×C4⋊Q8—944no—
329C2×C4⋊Q8C2×D4711yescharacteristic
330C2×#125—941no—
331C2×C22⋊Q8—471no—
332C2×C22⋊Q8—471no—
333C2×C22⋊Q8—941no—
334C2×C4×Q8—471no—
335C2×C4×Q8—941no—
336C2×C4×Q8C2×D4711yescharacteristic
337C2×#117—941no—
338C22×C4⋊C4—941no—
339C22×C4⋊C4—941no—
340C22×C42—471no—
341#494—4716no—
342#487—4716no—
343#462—478no—
344#462—478no—
345#462—478no—
346#462C2×D4718yescomplemented
347#451—4716no—
348#451—4716no—
349#431—478no—
350C2×C4⋊C8—471no—
351C2×C4⋊C8C2×D4711yescharacteristic
352C2×#103—472no—
353C2×#103—472no—
354C2×#103—472no—
355C2×#103C2×D4712yes—
356C2×C22⋊C8—472no—
357C2×C4×C8—472no—
358C47C2×#320911yescharacteristic, complemented, Sylow
359C22×Q8—478no—
360C22×Q8—471no—
361C22×Q8—472no—
362C22×Q8—941no—
363C22×Q8—942no—
364C22×Q8—944no—
365C22×Q8—1884no—
366C22×Q8—1884no—
367C22×Q8C22×D4711yescharacteristic
368C22×Q8C22×D4712yes—
369C23×C4—471no—
370C23×C4—471no—
371C23×C4—471no—
372C23×C4—942no—
373C2×Q16—44no—
374C2×Q16—478no—
375C2×Q16—478no—
376C2×Q16—478no—
377C2×Q16—1884no—
378C2×Q16—1884no—
379C2×Q16—1884no—
380C2×Q16C22×D4718yes—
381C22×C8—472no—
382C22×C8C22×D4712yes—
383C4⋊Q8—22no—
384C4⋊Q8—474no—
385C4⋊Q8—478no—
386C4⋊Q8—474no—
387C4⋊Q8—474no—
388C4⋊Q8—948no—
389C4⋊Q8—948no—
390C4⋊Q8—942no—
391C4⋊Q8—944no—
392C4⋊Q8—942no—
393C4⋊Q8—942no—
394C4⋊Q8C22×D4714yes—
395#125—948no—
396C22⋊Q8—474no—
397C22⋊Q8—474no—
398C22⋊Q8—942no—
399C22⋊Q8—948no—
400C22⋊Q8—942no—
401C4×Q8—22no—
402C4×Q8—474no—
403C4×Q8—948no—
404C4×Q8—942no—
405C4×Q8C22×D4714yes—
406#117—948no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—471no—
411C2×C4⋊C4—948no—
412C2×C4⋊C4—948no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—941no—
416C2×C4⋊C4—941no—
417C2×C4⋊C4—941no—
418C2×C4⋊C4—941no—
419C2×C4⋊C4—941no—
420C2×C4⋊C4—941no—
421C2×C4⋊C4—941no—
422C2×C4⋊C4—941no—
423C2×C4⋊C4C22×D4711yescharacteristic
424C2×C22⋊C4—941no—
425C2×C22⋊C4—941no—
426C2×C42—21no—
427C2×C42—471no—
428C2×C42—471no—
429C2×C42—471no—
430C2×C42—474no—
431C2×C42—474no—
432C2×C42—941no—
433C2×C42C22×D4711yescharacteristic
434C4⋊C8—474no—
435C4⋊C8C22×D4714yes—
436#103—478no—
437#103—478no—
438#103—478no—
439#103C22×D4718yes—
440C22⋊C8—478no—
441C4×C8—478no—
442C24—471no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—474no—
450C2×Q8—472no—
451C2×Q8—944no—
452C2×Q8—948no—
453C2×Q8—941no—
454C2×Q8—942no—
455C2×Q8—942no—
456C2×Q8—944no—
457C2×Q8—942no—
458C2×Q8—944no—
459C2×Q8—1884no—
460C2×Q8—1888no—
461C2×Q8—1888no—
462C2×Q8—1888no—
463C2×Q8—18816no—
464C2×Q8C23×D4714yes—
465C2×Q8C23×D4712yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—471no—
470C22×C4—471no—
471C22×C4—478no—
472C22×C4—474no—
473C22×C4—471no—
474C22×C4—474no—
475C22×C4—472no—
476C22×C4—472no—
477C22×C4—944no—
478C22×C4—948no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—941no—
482C22×C4—941no—
483C22×C4—942no—
484C22×C4—1888no—
485C22×C4D4×D4711yescharacteristic
486C22×C4D4×D4711yescharacteristic
487C22×C4C23×D4711yescharacteristic
488Q16—48no—
489Q16—1888no—
490Q16—1888no—
491Q16—1888no—
492C2×C8—24no—
493C2×C8—474no—
494C2×C8—944no—
495C2×C8C23×D4714yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—472no—
501C4⋊C4—948no—
502C4⋊C4—948no—
503C4⋊C4—944no—
504C4⋊C4—944no—
505C4⋊C4—944no—
506C4⋊C4—944no—
507C4⋊C4—944no—
508C4⋊C4—944no—
509C4⋊C4—941no—
510C4⋊C4—944no—
511C4⋊C4—944no—
512C4⋊C4C23×D4712yes—
513C22⋊C4—944no—
514C22⋊C4—944no—
515C42—24no—
516C42—474no—
517C42—474no—
518C42—472no—
519C42—472no—
520C42—944no—
521C42D4×D4712yes—
522C42C23×D4712yes—
523C23—474no—
524C23—472no—
525C23—942no—
526C23—942no—
527C23C2×D4×D4711yescharacteristic
528Q8—22no—
529Q8—24no—
530Q8—22no—
531Q8—24no—
532Q8—42no—
533Q8—44no—
534Q8—942no—
535Q8—944no—
536Q8—942no—
537Q8—944no—
538Q8—1888no—
539Q8—18816no—
540Q8—1882no—
541Q8—1884no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—474no—
550C2×C4—474no—
551C2×C4—472no—
552C2×C4—472no—
553C2×C4—4716no—
554C2×C4—472no—
555C2×C4—948no—
556C2×C4—942no—
557C2×C4—944no—
558C2×C4—942no—
559C2×C4—942no—
560C2×C4—942no—
561C2×C4—942no—
562C2×C4—944no—
563C2×C4—18816no—
564C2×C4—1882no—
565C2×C4—1884no—
566C2×C4—1882no—
567C2×C4Q16×D4714yes—
568C2×C4C2×D4×D4712yes—
569C2×C4C2×D4×D4712yes—
570C2×C4C2×D4×D4711yescharacteristic
571C2×C4#1309406912yes—
572C2×C4#1309407012yes—
573C2×C4C24×D4711yescharacteristic
574C8—24no—
575C8—944no—
576C22—474no—
577C22—942no—
578C22—948no—
579C22—942no—
580C22#1568968912yes—
581C22C2×Q16×D4712yes—
582C22#137×D4712yes—
583C22C22×D4×D4711yescharacteristic
584C4—22no—
585C4—42no—
586C4—44no—
587C4—42no—
588C4—474no—
589C4—474no—
590C4—942no—
591C4—1882no—
592C4—1884no—
593C4—1882no—
594C4#1568968912yes—
595C4C2×Q16×D4714yes—
596C4C2×#1309406911yescharacteristic
597C4C2×#1309407011yescharacteristic
598C2—944no—
599C2#2025433514yesdirect product factor
600C2C2×#1568968911yescharacteristic
601C2C22×Q16×D4711yescharacteristic
602C2C2×#137×D4711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#462C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
hypercenterC2×#462D47
hypocenterC47C2×#3209
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #20254335C2direct (N × H)
4. C2×#15689501C2semidirect
5. C2×#15689472C2semidirect
6. C2×#15689334C2semidirect
7. C2×#15689333C2semidirect
8. C2×#15689330C2semidirect
12. C2×C47×#462C2semidirect
37. #15689501C22semidirect
38. #15689472C22semidirect
39. #15689334C22semidirect
40. #15689333C22semidirect
41. #15689330C22semidirect
54. C47×#462C22semidirect
169. C2×C4×D47Q16semidirect
263. C4×D47C2×Q16semidirect
305. C2×#462D47semidirect
309. C2×C47#3209semidirect
346. #462C2×D47semidirect
358. C47C2×#3209semidirect
599. C2#20254335direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156896772
C2×C4⋊Q8×D472
#202543352
C2×#156895012
C2×#156894722
C2×#156893342
C2×#156893332
C2×#156893302
C2×#156892802
C2×#156892772
C2×#156891752
C2×C47×#4622
C2×#320947