Finite Group Databasebeta

← C2×#13118244 #16031623

Subgroups — C2×#13118244 #16031623

Order 3328.

3570 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88189C211yesmaximal, characteristic
2C22×#88180C211yesmaximal, characteristic, complemented
3C23×#6101C211yesmaximal, characteristic, complemented
4#13118244C2116yesmaximal, direct product factor
5C2×#11718645C212yesmaximal, complemented
6C2×#11718609C212yesmaximal, complemented
7C2×#11718633C212yesmaximal, complemented
8C2×#11718606C212yesmaximal, complemented
9C2×#11718881C211yesmaximal, characteristic
10C2×#11718607C211yesmaximal, characteristic, complemented
11C2×#11718604C211yesmaximal, characteristic, complemented
12C22×C13×#123C211yesmaximal, characteristic
13C23×D4×C13C2211yescharacteristic
14C24×C4×C13C2211yescharacteristic
15C24×Dic13C2211yescharacteristic
16C24×Dic13C2211yescharacteristic
17C2×C13×#123—28no—
18C2×C13×#123C2218yes—
19C22×C13×C4⋊C4C2211yescharacteristic
20C22×C13×C4⋊C4C2211yescharacteristic
21C22×C13×C22⋊C4C2211yescharacteristic
22C22×C13×C22⋊C4C2211yescharacteristic
23C22×C13×C22⋊C4C2211yescharacteristic
24C22×C22⋊Dic13C2211yescharacteristic
25C22×C22⋊Dic13C2212yes—
26C22×C22⋊Dic13C2212yes—
27C22×C22⋊Dic13C2211yescharacteristic
28C2×#88189—28no—
29C2×#88189C2218yes—
30C2×#88180—28no—
31C2×#88180C2218yescomplemented
32C22×C4⋊Dic13C2212yes—
33C22×#6101—28no—
34C22×#6101C2212yes—
35C22×#6101C2212yes—
36C22×#6101C2212yes—
37C22×#6101C2212yes—
38C22×#6101C2218yescomplemented
39C22×C4×Dic13C2212yes—
40#11718881C2218yes—
41C2×#88071C2212yes—
42C2×#88071C2212yes—
43C2×#88071C2212yes—
44C2×#88071C2212yes—
45#11718645C22116yescomplemented
46#11718633C22116yescomplemented
47#11718609C22116yescomplemented
48#11718607C2218yescomplemented
49#11718606C22116yescomplemented
50#11718604C2218yescomplemented
51C25×C13C2311yescharacteristic
52C25×C13C2311yescharacteristic
53C22×D4×C13—24no—
54C22×D4×C13—24no—
55C22×D4×C13—24no—
56C22×D4×C13C2314yes—
57C23×C4×C13—24no—
58C23×C4×C13—24no—
59C23×C4×C13D418yes—
60C23×C4×C13C2311yescharacteristic
61C23×C4×C13C2311yescharacteristic
62C23×C4×C13C2311yescharacteristic
63C23×C4×C13C2311yescharacteristic
64C23×C4×C13C2311yescharacteristic
65C23×C4×C13C2314yes—
66C23×Dic13—24no—
67C23×Dic13—28no—
68C23×Dic13—24no—
69C23×Dic13—24no—
70C23×Dic13—24no—
71C23×Dic13C2312yes—
72C23×Dic13C2312yes—
73C23×Dic13C2314yes—
74C23×Dic13C2314yes—
75C23×Dic13C2312yes—
76C23×Dic13C2312yes—
77C13×#123—232no—
78C2×C13×C4⋊C4—24no—
79C2×C13×C4⋊C4—24no—
80C2×C13×C4⋊C4C2314yes—
81C2×C13×C4⋊C4C2314yes—
82C2×C13×C22⋊C4—24no—
83C2×C13×C22⋊C4—24no—
84C2×C13×C22⋊C4—24no—
85C2×C13×C22⋊C4C2314yes—
86C2×C13×C22⋊C4C2314yes—
87C2×C13×C22⋊C4C2314yes—
88C2×C22⋊Dic13—24no—
89C2×C22⋊Dic13—28no—
90C2×C22⋊Dic13—28no—
91C2×C22⋊Dic13D418yes—
92C2×C22⋊Dic13C2314yes—
93C2×C22⋊Dic13C2318yes—
94C2×C22⋊Dic13C2318yes—
95C2×C22⋊Dic13C2314yes—
96#88189—232no—
97#88180—232no—
98C2×C4⋊Dic13—28no—
99C2×C4⋊Dic13C2318yes—
100C2×#6101—28no—
101C2×#6101—28no—
102C2×#6101—232no—
103C2×#6101—28no—
104C2×#6101—28no—
105C2×#6101C2318yes—
106C2×#6101C2318yes—
107C2×#6101C2318yes—
108C2×#6101C2318yes—
109C2×C4×Dic13—28no—
110C2×C4×Dic13C2318yes—
111#88071C2318yes—
112#88071C2318yes—
113#88071C2318yes—
114#88071C2318yes—
115C2×#2413—131nomaximal, Sylow
116C24×C13—24no—
117C24×C13—24no—
118C24×C13—22no—
119C24×C13—22no—
120C24×C13—24no—
121C24×C13—22no—
122C24×C13—24no—
123C24×C13C2×D414yes—
124C24×C13C2×D414yes—
125C24×C13#4114yes—
126C24×C13C2412yes—
127C24×C13C2412yes—
128C24×C13C2411yescharacteristic
129C2×D4×C13—28no—
130C2×D4×C13—28no—
131C2×D4×C13—28no—
132C2×D4×C13—416no—
133C22×C4×C13—232no—
134C22×C4×C13—22no—
135C22×C4×C13—22no—
136C22×C4×C13—28no—
137C22×C4×C13—28no—
138C22×C4×C13—22no—
139C22×C4×C13—22no—
140C22×C4×C13—22no—
141C22×C4×C13—22no—
142C22×C4×C13—22no—
143C22×C4×C13—22no—
144C22×C4×C13—24no—
145C22×C4×C13—22no—
146C22×C4×C13—24no—
147C22×C4×C13C2×D414yes—
148C22×C4×C13C2×D414yes—
149C22×C4×C13C2×D4116yes—
150C22×C4×C13C2412yes—
151C22×C4×C13C2412yes—
152C22×C4×C13C2412yes—
153C22×C4×C13C2412yes—
154C22×C4×C13C2412yes—
155C22×Dic13—232no—
156C22×Dic13—28no—
157C22×Dic13—24no—
158C22×Dic13—24no—
159C22×Dic13—24no—
160C22×Dic13—24no—
161C22×Dic13—216no—
162C22×Dic13—24no—
163C22×Dic13—28no—
164C22×Dic13—24no—
165C22×Dic13—24no—
166C22×Dic13—28no—
167C22×Dic13—24no—
168C22×Dic13—28no—
169C22×Dic13—24no—
170C22×Dic13—24no—
171C22×Dic13—416no—
172C22×Dic13C2×D418yes—
173C22×Dic13#4118yes—
174C22×Dic13C2414yes—
175C22×Dic13C2414yes—
176C22×Dic13C2414yes—
177C22×Dic13C2414yes—
178C13×C4⋊C4—28no—
179C13×C4⋊C4—28no—
180C13×C22⋊C4—28no—
181C13×C22⋊C4—28no—
182C13×C22⋊C4—28no—
183C22⋊Dic13—28no—
184C22⋊Dic13—216no—
185C22⋊Dic13—216no—
186C22⋊Dic13C2×D4116yes—
187C4⋊Dic13—216no—
188#6101—216no—
189#6101—216no—
190#6101—216no—
191#6101—216no—
192C4×Dic13—216no—
193C22×#123—131no—
194C22×#123D1311yescharacteristic
195C22×#117—131no—
196C23×C4⋊C4—131no—
197#2413—1316no—
198C2×#398—132no—
199C2×#396—132no—
200C2×#389—132no—
201C2×#388—132no—
202C2×#386—131no—
203C2×#385—131no—
204C2×#381—131no—
205C23×C13—28no—
206C23×C13—28no—
207C23×C13—216no—
208C23×C13—216no—
209C23×C13—24no—
210C23×C13—24no—
211C23×C13—22no—
212C23×C13—22no—
213C23×C13—24no—
214C23×C13—22no—
215C23×C13—24no—
216C23×C13—44no—
217C23×C13—44no—
218C23×C13—44no—
219C23×C13—48no—
220C23×C13—48no—
221C23×C13#121116yes—
222C23×C13C22×D414yes—
223C23×C13C22×D414yes—
224C23×C13C22×D412yes—
225C23×C13C22×D412yes—
226C23×C13C2×#4114yes—
227C23×C13C2×#4112yes—
228C23×C13C2×#4114yes—
229C23×C132−1+414yes—
230C23×C13C2511yescharacteristic
231D4×C13—416no—
232C2×C4×C13—232no—
233C2×C4×C13—28no—
234C2×C4×C13—28no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—24no—
244C2×C4×C13—22no—
245C2×C4×C13—24no—
246C2×C4×C13—44no—
247C2×C4×C13—44no—
248C2×C4×C13—44no—
249C2×C4×C13C22×D414yes—
250C2×C4×C13C22×D414yes—
251C2×Dic13—232no—
252C2×Dic13—216no—
253C2×Dic13—216no—
254C2×Dic13—24no—
255C2×Dic13—24no—
256C2×Dic13—24no—
257C2×Dic13—24no—
258C2×Dic13—24no—
259C2×Dic13—24no—
260C2×Dic13—24no—
261C2×Dic13—24no—
262C2×Dic13—24no—
263C2×Dic13—24no—
264C2×Dic13—416no—
265C2×Dic13—48no—
266C2×Dic13—48no—
267C2×Dic13C22×D418yes—
268C2×Dic13C2×#4118yes—
269C23×D4C2×D1311yescharacteristic
270C24×C4—131no—
271C24×C4—131no—
272C24×C4C2×D1311yescharacteristic
273C2×#123—28no—
274C2×#123—138no—
275C2×#123—268no—
276C2×#123C2×D1318yes—
277C2×#117—138no—
278C2×#117—268no—
279C22×C4⋊C4—132no—
280C22×C4⋊C4—132no—
281C22×C4⋊C4—132no—
282C22×C4⋊C4—132no—
283C22×C4⋊C4—138no—
284C22×C4⋊C4—132no—
285C22×C4⋊C4—268no—
286C22×C4⋊C4C2×D1311yescharacteristic
287C22×C4⋊C4C2×D1311yescharacteristic
288C22×C22⋊C4—131no—
289C22×C22⋊C4—132no—
290C22×C22⋊C4—132no—
291C22×C22⋊C4—131no—
292C22×C22⋊C4C2×D1311yescharacteristic
293C22×C22⋊C4C2×D1311yescharacteristic
294C22×C22⋊C4C2×D1311yescharacteristic
295C22×C42—132no—
296#398—1316no—
297#396—1316no—
298#389—1316no—
299#388—1316no—
300#386—138no—
301#385—138no—
302#381—138no—
303C2×#95—132no—
304C2×#95—132no—
305C2×#95—132no—
306C2×#95—132no—
307C22×C13—24no—
308C22×C13—232no—
309C22×C13—24no—
310C22×C13—28no—
311C22×C13—28no—
312C22×C13—42no—
313C22×C13—42no—
314C22×C13—42no—
315C22×C13—44no—
316C22×C13—44no—
317C22×C13—88no—
318C22×C13C2×#12118yes—
319C22×C13C2×#12114yes—
320C22×C13#52514yes—
321C22×C13#53614yes—
322C22×C13#54718yes—
323C22×C13#55218yes—
324C22×C13C23×D411yescharacteristic
325C22×C13C23×D411yescharacteristic
326C22×C13C22×#4111yescharacteristic
327C22×C13C22×#4112yes—
328C22×C13C2×2−1+412yes—
329C4×C13—24no—
330C4×C13—24no—
331C4×C13—42no—
332C4×C13—42no—
333C4×C13—42no—
334Dic13—28no—
335Dic13—28no—
336Dic13—44no—
337Dic13—44no—
338C25C22×D1311yescharacteristic
339C25C22×D1311yescharacteristic
340C22×D4—24no—
341C22×D4—24no—
342C22×D4—24no—
343C22×D4C22×D1314yes—
344C23×C4—24no—
345C23×C4—24no—
346C23×C4—132no—
347C23×C4—132no—
348C23×C4—134no—
349C23×C4—134no—
350C23×C4—132no—
351C23×C4—132no—
352C23×C4—268no—
353C23×C4—264no—
354C23×C4—264no—
355C23×C4—264no—
356C23×C4—264no—
357C23×C4C22⋊D1318yes—
358C23×C4C22×D1311yescharacteristic
359C23×C4C22×D1311yescharacteristic
360C23×C4C22×D1311yescharacteristic
361C23×C4C22×D1314yes—
362C23×C4C22×D1311yescharacteristic
363C23×C4C22×D1311yescharacteristic
364#123—232no—
365#123—2632no—
366#117—2632no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—138no—
370C2×C4⋊C4—138no—
371C2×C4⋊C4—138no—
372C2×C4⋊C4—138no—
373C2×C4⋊C4—138no—
374C2×C4⋊C4—268no—
375C2×C4⋊C4—268no—
376C2×C4⋊C4—268no—
377C2×C4⋊C4—268no—
378C2×C4⋊C4—268no—
379C2×C4⋊C4—2632no—
380C2×C4⋊C4C22×D1314yes—
381C2×C4⋊C4C22×D1314yes—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—138no—
386C2×C22⋊C4—134no—
387C2×C22⋊C4—134no—
388C2×C22⋊C4—138no—
389C2×C22⋊C4—138no—
390C2×C22⋊C4—268no—
391C2×C22⋊C4—268no—
392C2×C22⋊C4—264no—
393C2×C22⋊C4C22×D1314yes—
394C2×C22⋊C4C22×D1314yes—
395C2×C22⋊C4C22×D1314yes—
396C2×C42—138no—
397C2×C42—268no—
398#95—138no—
399#95—138no—
400#95—138no—
401#95—138no—
402C2×C13—28no—
403C2×C13—82no—
404C2×C13#241318yescomplemented
405C2×C13C22×#12111yescharacteristic
406C2×C13C2×#52511yescharacteristic
407C2×C13C2×#53611yescharacteristic
408C2×C13C2×#54712yes—
409C2×C13C2×#55212yes—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—22no—
414C24—24no—
415C24—24no—
416C24—22no—
417C24D4×D1314yes—
418C24#612914yes—
419C24C2×C22⋊D1314yes—
420C24C23×D1312yes—
421C24C23×D1312yes—
422C24C23×D1311yescharacteristic
423C2×D4—28no—
424C2×D4—28no—
425C2×D4—28no—
426C2×D4—416no—
427C22×C4—232no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—138no—
442C22×C4—138no—
443C22×C4—134no—
444C22×C4—134no—
445C22×C4—134no—
446C22×C4—134no—
447C22×C4—2616no—
448C22×C4—2632no—
449C22×C4—264no—
450C22×C4—264no—
451C22×C4—264no—
452C22×C4—268no—
453C22×C4—264no—
454C22×C4—264no—
455C22×C4—264no—
456C22×C4—268no—
457C22×C4—268no—
458C22×C4—264no—
459C22×C4—264no—
460C22×C4—264no—
461C22×C4—268no—
462C22×C4—264no—
463C22×C4—5216no—
464C22×C4C2×C22⋊D1314yes—
465C22×C4C2×C22⋊D1314yes—
466C22×C4C2×C22⋊D13116yes—
467C22×C4C23×D1312yes—
468C22×C4C23×D1312yes—
469C22×C4C23×D1312yes—
470C22×C4C23×D1312yes—
471C22×C4C23×D1312yes—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—2616no—
475C4⋊C4—2616no—
476C4⋊C4—2616no—
477C4⋊C4—2616no—
478C4⋊C4—2616no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—1316no—
483C22⋊C4—2616no—
484C22⋊C4—2616no—
485C22⋊C4—268no—
486C42—2616no—
487C13C2×#241311yescharacteristic, complemented, Sylow
488C23—28no—
489C23—28no—
490C23—216no—
491C23—216no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—48no—
503C23—48no—
504C23#88193116yes—
505C23C2×D4×D1314yes—
506C23C2×D4×D1312yes—
507C23C2×#612914yes—
508C23C2×#612912yes—
509C23#41×D1314yes—
510C23#8825714yes—
511C23C22×C22⋊D1314yes—
512C23C22×C22⋊D1312yes—
513C23C24×D1311yescharacteristic
514D4—416no—
515C2×C4—232no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—138no—
533C2×C4—138no—
534C2×C4—2632no—
535C2×C4—2616no—
536C2×C4—2616no—
537C2×C4—264no—
538C2×C4—264no—
539C2×C4—264no—
540C2×C4—264no—
541C2×C4—264no—
542C2×C4—264no—
543C2×C4—264no—
544C2×C4—264no—
545C2×C4—264no—
546C2×C4—264no—
547C2×C4—5216no—
548C2×C4—528no—
549C2×C4—528no—
550C2×C4C22×C22⋊D1314yes—
551C2×C4C22×C22⋊D1314yes—
552C22—232no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—88no—
563C22#1171935518yes—
564C22#1171936018yes—
565C22C2×#8819318yes—
566C22C2×#8819314yes—
567C22#1171953014yes—
568C22#1171953314yes—
569C22C22×D4×D1311yescharacteristic
570C22C22×#612911yescharacteristic
571C22C2×#41×D1312yes—
572C22C2×#8825712yes—
573C22C23×C22⋊D1311yescharacteristic
574C4—24no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—268no—
580C4—268no—
581C4—524no—
582C4—524no—
583C2—28no—
584C2—82no—
585C2#1311824418yesdirect product factor
586C2C22×#8819311yescharacteristic
587C2C2×#1171953311yescharacteristic
588C2C2×#1171953011yescharacteristic
589C2C2×#1171936012yes—
590C2C2×#1171935512yes—
subgroupG/H
centerC24C23×D13
commutator, cosocleC23×C13C25
FrattiniC23C24×D13
FittingC22×C13×#123C2
socleC24×C13C24
prime radicalC23×D4×C13C22
hypercenterC22×#123D13
hypocenterC13C2×#2413
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. C22×#88180C2semidirect
3. C23×#6101C2semidirect
4. #13118244C2direct (N × H)
5. C2×#11718645C2semidirect
6. C2×#11718609C2semidirect
7. C2×#11718633C2semidirect
8. C2×#11718606C2semidirect
10. C2×#11718607C2semidirect
11. C2×#11718604C2semidirect
31. C2×#88180C22semidirect
38. C22×#6101C22semidirect
45. #11718645C22semidirect
46. #11718633C22semidirect
47. #11718609C22semidirect
48. #11718607C22semidirect
49. #11718606C22semidirect
50. #11718604C22semidirect
404. C2×C13#2413semidirect
487. C13C2×#2413semidirect
585. C2#13118244direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#881892
C22×#881802
C23×#61012
#131182442
C2×#117186452
C2×#117186092
C2×#117186332
C2×#117186062
C2×#117188812
C2×#117186072
C2×#117186042
C22×C13×#1232
C2×#241313