Finite Group Databasebeta

← C2×#13119356 #16032164

Subgroups — C2×#13119356 #16032164

Order 3328.

2054 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13119356C2116yesmaximal, direct product factor
2C2×#11718318C211yesmaximal, characteristic
3C2×#11718645C211yesmaximal, characteristic
4C2×#11718644C211yesmaximal, characteristic
5C2×#11718646C211yesmaximal, characteristic
6C2×#11718308C211yesmaximal, characteristic
7C2×#11718308C211yesmaximal, characteristic
8C2×#11718633C211yesmaximal, characteristic
9C2×#11718633C211yesmaximal, characteristic
10C2×#11718633C211yesmaximal, characteristic
11C2×#11718314C211yesmaximal, characteristic
12C2×#11718315C211yesmaximal, characteristic
13C2×#11718637C211yesmaximal, characteristic
14C2×C4⋊C4×Dic13C211yesmaximal, characteristic
15C2×#11718310C211yesmaximal, characteristic
16C2×C13×#398C211yesmaximal, characteristic
17C22×C13×C4⋊C4C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C22×C13×C4⋊C4C2211yescharacteristic
20C22×C13×C4⋊C4C2211yescharacteristic
21C22×C4⋊Dic13C2211yescharacteristic
22C22×C4⋊Dic13C2211yescharacteristic
23C22×C4⋊Dic13C2211yescharacteristic
24C22×C4⋊Dic13C2211yescharacteristic
25C22×#6101C2211yescharacteristic
26C22×#6101C2211yescharacteristic
27C22×#6101C2211yescharacteristic
28C22×#6101C2211yescharacteristic
29C22×#6101C2211yescharacteristic
30C22×#6101C2211yescharacteristic
31C22×#6101C2211yescharacteristic
32C22×#6101C2211yescharacteristic
33C22×#6101C2211yescharacteristic
34C22×#6101C2211yescharacteristic
35C22×C4×Dic13C2211yescharacteristic
36C22×C4×Dic13C2211yescharacteristic
37C22×C4×Dic13C2211yescharacteristic
38C22×C4×Dic13C2211yescharacteristic
39C22×C4×Dic13C2211yescharacteristic
40C13×#398C2218yes—
41C2×C13×#95C2211yescharacteristic
42C2×C13×#95C2211yescharacteristic
43C2×C13×#95C2211yescharacteristic
44C2×#88071C2211yescharacteristic
45C2×#88071C2211yescharacteristic
46C2×#88071C2211yescharacteristic
47C2×#88071C2211yescharacteristic
48C2×#88071C2211yescharacteristic
49C2×#88071C2211yescharacteristic
50C2×#88071C2211yescharacteristic
51C2×#88071C2211yescharacteristic
52C2×#88071C2211yescharacteristic
53#11718646C2218yes—
54#11718645C2218yes—
55#11718644C2218yes—
56#11718637C2218yes—
57C4⋊C4×Dic13C2218yes—
58#11718633C2218yes—
59#11718633C2218yes—
60#11718633C2218yes—
61#11718318C2218yes—
62#11718315C2218yes—
63#11718314C2218yes—
64#11718310C2218yes—
65#11718308C2218yes—
66#11718308C2218yes—
67C23×C4×C13C2311yescharacteristic
68C23×C4×C13C2311yescharacteristic
69C23×C4×C13C2311yescharacteristic
70C23×C4×C13C2311yescharacteristic
71C23×C4×C13C2311yescharacteristic
72C23×C4×C13C2311yescharacteristic
73C23×C4×C13C2311yescharacteristic
74C23×Dic13C2311yescharacteristic
75C23×Dic13C2311yescharacteristic
76C23×Dic13C2311yescharacteristic
77C23×Dic13C2311yescharacteristic
78C23×Dic13C2311yescharacteristic
79C23×Dic13C2311yescharacteristic
80C23×Dic13C2311yescharacteristic
81C23×Dic13C2311yescharacteristic
82C2×C13×C4⋊C4—24no—
83C2×C13×C4⋊C4—24no—
84C2×C13×C4⋊C4—42no—
85C2×C13×C4⋊C4—42no—
86C2×C13×C4⋊C4C2314yes—
87C2×C13×C4⋊C4C2314yes—
88C2×C13×C4⋊C4C2314yes—
89C2×C13×C4⋊C4C2314yes—
90C2×C4⋊Dic13—42no—
91C2×C4⋊Dic13—42no—
92C2×C4⋊Dic13—42no—
93C2×C4⋊Dic13Q818yes—
94C2×C4⋊Dic13C2314yes—
95C2×C4⋊Dic13C2314yes—
96C2×C4⋊Dic13C2314yes—
97C2×C4⋊Dic13C2314yes—
98C2×#6101—24no—
99C2×#6101—24no—
100C2×#6101—24no—
101C2×#6101—24no—
102C2×#6101—42no—
103C2×#6101—42no—
104C2×#6101—42no—
105C2×#6101—42no—
106C2×#6101—42no—
107C2×#6101Q818yes—
108C2×#6101C2314yes—
109C2×#6101C2314yes—
110C2×#6101C2314yes—
111C2×#6101C2314yes—
112C2×#6101C2314yes—
113C2×#6101C2314yes—
114C2×#6101C2314yes—
115C2×#6101C2314yes—
116C2×#6101C2314yes—
117C2×#6101C2314yes—
118C2×C4×Dic13—24no—
119C2×C4×Dic13—24no—
120C2×C4×Dic13—24no—
121C2×C4×Dic13—42no—
122C2×C4×Dic13—42no—
123C2×C4×Dic13C2314yes—
124C2×C4×Dic13C2314yes—
125C2×C4×Dic13C2314yes—
126C2×C4×Dic13C2314yes—
127C2×C4×Dic13C2314yes—
128C13×#95C2314yes—
129C13×#95C2314yes—
130C13×#95C2314yes—
131#88071C2314yes—
132#88071C2314yes—
133#88071C2314yes—
134#88071C2314yes—
135#88071C2314yes—
136#88071C2314yes—
137#88071C2314yes—
138#88071C2314yes—
139#88071C2314yes—
140C2×#2507—131nomaximal, Sylow
141C24×C13C2411yescharacteristic
142C22×C4×C13—22no—
143C22×C4×C13—22no—
144C22×C4×C13—22no—
145C22×C4×C13—22no—
146C22×C4×C13—22no—
147C22×C4×C13—22no—
148C22×C4×C13—22no—
149C22×C4×C13—22no—
150C22×C4×C13—22no—
151C22×C4×C13—22no—
152C22×C4×C13—22no—
153C22×C4×C13—22no—
154C22×C4×C13—22no—
155C22×C4×C13—22no—
156C22×C4×C13—22no—
157C22×C4×C13—22no—
158C22×C4×C13—22no—
159C22×C4×C13—22no—
160C22×C4×C13C2×Q814yes—
161C22×C4×C13C2×Q814yes—
162C22×C4×C13#4114yes—
163C22×C4×C13C2412yes—
164C22×C4×C13C2412yes—
165C22×C4×C13C2412yes—
166C22×C4×C13C2412yes—
167C22×C4×C13C2412yes—
168C22×C4×C13C2412yes—
169C22×C4×C13C2412yes—
170C22×Dic13—22no—
171C22×Dic13—22no—
172C22×Dic13—22no—
173C22×Dic13—22no—
174C22×Dic13—22no—
175C22×Dic13—22no—
176C22×Dic13—22no—
177C22×Dic13—22no—
178C22×Dic13—22no—
179C22×Dic13—22no—
180C22×Dic13—22no—
181C22×Dic13—22no—
182C22×Dic13—22no—
183C22×Dic13—22no—
184C22×Dic13—22no—
185C22×Dic13—22no—
186C22×Dic13—22no—
187C22×Dic13—22no—
188C22×Dic13C2×Q814yes—
189C22×Dic13C2×Q814yes—
190C22×Dic13C2×Q814yes—
191C22×Dic13C2×Q814yes—
192C22×Dic13#4114yes—
193C22×Dic13#4114yes—
194C22×Dic13C2412yes—
195C22×Dic13C2412yes—
196C22×Dic13C2412yes—
197C22×Dic13C2412yes—
198C22×Dic13C2412yes—
199C22×Dic13C2412yes—
200C22×Dic13C2412yes—
201C22×Dic13C2412yes—
202C13×C4⋊C4—28no—
203C13×C4⋊C4—28no—
204C13×C4⋊C4—44no—
205C13×C4⋊C4—44no—
206C4⋊Dic13—44no—
207C4⋊Dic13—44no—
208C4⋊Dic13—44no—
209C4⋊Dic13C2×Q8116yes—
210#6101—28no—
211#6101—28no—
212#6101—28no—
213#6101—28no—
214#6101—44no—
215#6101—44no—
216#6101—44no—
217#6101—44no—
218#6101—44no—
219#6101C2×Q8116yes—
220C4×Dic13—28no—
221C4×Dic13—28no—
222C4×Dic13—28no—
223C4×Dic13—44no—
224C4×Dic13—44no—
225#2507—1316no—
226C2×#398—131no—
227C2×#398—131no—
228C2×#398—131no—
229C2×#398D1311yescharacteristic
230C2×#389—131no—
231C2×#389—131no—
232C2×#389—131no—
233C2×#389—131no—
234C2×#389—131no—
235C2×#389—131no—
236C2×#387—131no—
237C2×#387—131no—
238C2×#384—131no—
239C2×C4×C4⋊C4—131no—
240C2×C4×C4⋊C4—131no—
241C23×C13C22×Q812yes—
242C23×C13C22×Q812yes—
243C23×C13C2×#4112yes—
244C23×C13C2×#4112yes—
245C23×C13C2×#4112yes—
246C23×C13H(5,𝔽2)12yes—
247C23×C13H(5,𝔽2)12yes—
248C23×C13C2511yescharacteristic
249C2×C4×C13—28no—
250C2×C4×C13—28no—
251C2×C4×C13—28no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—22no—
257C2×C4×C13—22no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—22no—
263C2×C4×C13—22no—
264C2×C4×C13—22no—
265C2×C4×C13—22no—
266C2×C4×C13—22no—
267C2×C4×C13—22no—
268C2×C4×C13—22no—
269C2×C4×C13—22no—
270C2×C4×C13—44no—
271C2×C4×C13—44no—
272C2×C4×C13—44no—
273C2×C4×C13—44no—
274C2×C4×C13C22×Q814yes—
275C2×C4×C13C22×Q814yes—
276C2×C4×C13C2×#4114yes—
277C2×Dic13—28no—
278C2×Dic13—28no—
279C2×Dic13—28no—
280C2×Dic13—22no—
281C2×Dic13—22no—
282C2×Dic13—22no—
283C2×Dic13—22no—
284C2×Dic13—22no—
285C2×Dic13—22no—
286C2×Dic13—22no—
287C2×Dic13—22no—
288C2×Dic13—22no—
289C2×Dic13—22no—
290C2×Dic13—22no—
291C2×Dic13—22no—
292C2×Dic13—22no—
293C2×Dic13—22no—
294C2×Dic13—22no—
295C2×Dic13—22no—
296C2×Dic13—22no—
297C2×Dic13—22no—
298C2×Dic13—44no—
299C2×Dic13—44no—
300C2×Dic13—44no—
301C2×Dic13—44no—
302C2×Dic13#125116yes—
303C2×Dic13C22×Q814yes—
304C2×Dic13C22×Q814yes—
305C2×Dic13C22×Q814yes—
306C2×Dic13C22×Q814yes—
307C2×Dic13C2×#4114yes—
308C2×Dic13C2×#4114yes—
309C22×C4⋊C4—131no—
310C22×C4⋊C4—131no—
311C22×C4⋊C4—131no—
312C22×C4⋊C4—131no—
313C22×C4⋊C4—131no—
314C22×C4⋊C4—131no—
315C22×C4⋊C4—131no—
316C22×C4⋊C4—131no—
317C22×C4⋊C4—131no—
318C22×C4⋊C4—131no—
319C22×C4⋊C4—131no—
320C22×C4⋊C4—131no—
321C22×C4⋊C4—131no—
322C22×C4⋊C4—131no—
323C22×C4⋊C4C2×D1311yescharacteristic
324C22×C4⋊C4C2×D1311yescharacteristic
325C22×C4⋊C4C2×D1311yescharacteristic
326C22×C4⋊C4C2×D1311yescharacteristic
327C22×C42—131no—
328C22×C42—131no—
329C22×C42—131no—
330C22×C42—131no—
331C22×C42—131no—
332#398—138no—
333#398—138no—
334#398—138no—
335#398C2×D1318yes—
336#389—138no—
337#389—138no—
338#389—138no—
339#389—138no—
340#389—138no—
341#389—138no—
342#387—138no—
343#387—138no—
344#384—138no—
345C4×C4⋊C4—138no—
346C4×C4⋊C4—138no—
347C2×#95—131no—
348C2×#95—131no—
349C2×#95—131no—
350C2×#95—131no—
351C2×#95—131no—
352C2×#95—131no—
353C2×#95—131no—
354C2×#95—131no—
355C2×#95—131no—
356C2×#95C2×D1311yescharacteristic
357C2×#95C2×D1311yescharacteristic
358C2×#95C2×D1311yescharacteristic
359C22×C13C2×#12514yes—
360C22×C13#53314yes—
361C22×C13#54014yes—
362C22×C13#55314yes—
363C22×C13#55414yes—
364C22×C13#55414yes—
365C22×C13Q8214yes—
366C22×C13C23×Q811yescharacteristic
367C22×C13C23×Q811yescharacteristic
368C22×C13C22×#4111yescharacteristic
369C22×C13C22×#4111yescharacteristic
370C22×C13C22×#4111yescharacteristic
371C22×C13C2×H(5,𝔽2)11yescharacteristic
372C22×C13C2×H(5,𝔽2)11yescharacteristic
373C4×C13—24no—
374C4×C13—24no—
375C4×C13—24no—
376C4×C13—42no—
377C4×C13—42no—
378C4×C13—42no—
379C4×C13—42no—
380Dic13—24no—
381Dic13—24no—
382Dic13—24no—
383Dic13—42no—
384Dic13—42no—
385Dic13—42no—
386Dic13—42no—
387Dic13C2×#12518yes—
388C23×C4—131no—
389C23×C4—131no—
390C23×C4—131no—
391C23×C4—131no—
392C23×C4—131no—
393C23×C4—131no—
394C23×C4—131no—
395C23×C4—131no—
396C23×C4C22×D1311yescharacteristic
397C23×C4C22×D1311yescharacteristic
398C23×C4C22×D1311yescharacteristic
399C23×C4C22×D1311yescharacteristic
400C23×C4C22×D1311yescharacteristic
401C23×C4C22×D1311yescharacteristic
402C23×C4C22×D1311yescharacteristic
403C2×C4⋊C4—24no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—42no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4—138no—
408C2×C4⋊C4—138no—
409C2×C4⋊C4—134no—
410C2×C4⋊C4—134no—
411C2×C4⋊C4—134no—
412C2×C4⋊C4—134no—
413C2×C4⋊C4—134no—
414C2×C4⋊C4—134no—
415C2×C4⋊C4—134no—
416C2×C4⋊C4—134no—
417C2×C4⋊C4—134no—
418C2×C4⋊C4—134no—
419C2×C4⋊C4—134no—
420C2×C4⋊C4—134no—
421C2×C4⋊C4—134no—
422C2×C4⋊C4—134no—
423C2×C4⋊C4—264no—
424C2×C4⋊C4—264no—
425C2×C4⋊C4—264no—
426C2×C4⋊C4—264no—
427C2×C4⋊C4—522no—
428C2×C4⋊C4—522no—
429C2×C4⋊C4—522no—
430C2×C4⋊C4—522no—
431C2×C4⋊C4—522no—
432C2×C4⋊C4—522no—
433C2×C4⋊C4—522no—
434C2×C4⋊C4—522no—
435C2×C4⋊C4C22×D1314yes—
436C2×C4⋊C4C22×D1314yes—
437C2×C4⋊C4C22×D1314yes—
438C2×C4⋊C4C22×D1314yes—
439C2×C42—134no—
440C2×C42—134no—
441C2×C42—134no—
442C2×C42—134no—
443C2×C42—134no—
444C2×C42—264no—
445C2×C42—264no—
446C2×C42—264no—
447C2×C42—522no—
448C2×C42—522no—
449#95—134no—
450#95—134no—
451#95—134no—
452#95—134no—
453#95—134no—
454#95—134no—
455#95—134no—
456#95—134no—
457#95—134no—
458#95C22×D1314yes—
459#95C22×D1314yes—
460#95C22×D1314yes—
461C2×C13#250718yescomplemented
462C2×C13C22×#12511yescharacteristic
463C2×C13C2×#53311yescharacteristic
464C2×C13C2×#54011yescharacteristic
465C2×C13C2×#55311yescharacteristic
466C2×C13C2×#55411yescharacteristic
467C2×C13C2×#55411yescharacteristic
468C2×C13C2×Q8211yescharacteristic
469C24C23×D1311yescharacteristic
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—134no—
489C22×C4—134no—
490C22×C4—134no—
491C22×C4—132no—
492C22×C4—134no—
493C22×C4—134no—
494C22×C4—134no—
495C22×C4—132no—
496C22×C4—132no—
497C22×C4—132no—
498C22×C4—132no—
499C22×C4—132no—
500C22×C4—132no—
501C22×C4—132no—
502C22×C4—262no—
503C22×C4—262no—
504C22×C4—262no—
505C22×C4—262no—
506C22×C4—262no—
507C22×C4—262no—
508C22×C4—262no—
509C22×C4—262no—
510C22×C4—262no—
511C22×C4—262no—
512C22×C4—262no—
513C22×C4—262no—
514C22×C4—262no—
515C22×C4—262no—
516C22×C4—262no—
517C22×C4—262no—
518C22×C4—262no—
519C22×C4—262no—
520C22×C4#612914yes—
521C22×C4Q8×D1314yes—
522C22×C4Q8×D1314yes—
523C22×C4C23×D1312yes—
524C22×C4C23×D1312yes—
525C22×C4C23×D1312yes—
526C22×C4C23×D1312yes—
527C22×C4C23×D1312yes—
528C22×C4C23×D1312yes—
529C22×C4C23×D1312yes—
530C4⋊C4—28no—
531C4⋊C4—28no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—1316no—
535C4⋊C4—1316no—
536C4⋊C4—268no—
537C4⋊C4—268no—
538C4⋊C4—268no—
539C4⋊C4—268no—
540C4⋊C4—524no—
541C4⋊C4—524no—
542C4⋊C4—524no—
543C4⋊C4—524no—
544C4⋊C4—524no—
545C4⋊C4—524no—
546C4⋊C4—524no—
547C4⋊C4—524no—
548C42—268no—
549C42—268no—
550C42—268no—
551C42—524no—
552C42—524no—
553C13C2×#250711yescharacteristic, complemented, Sylow
554C23C2×#612912yes—
555C23#8825112yes—
556C23C2×Q8×D1312yes—
557C23C2×Q8×D1312yes—
558C23#41×D1312yes—
559C23#41×D1312yes—
560C23#8825612yes—
561C23C24×D1311yescharacteristic
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—134no—
588C2×C4—134no—
589C2×C4—134no—
590C2×C4—1316no—
591C2×C4—134no—
592C2×C4—134no—
593C2×C4—134no—
594C2×C4—268no—
595C2×C4—262no—
596C2×C4—262no—
597C2×C4—262no—
598C2×C4—262no—
599C2×C4—262no—
600C2×C4—262no—
601C2×C4—268no—
602C2×C4—268no—
603C2×C4—262no—
604C2×C4—262no—
605C2×C4—262no—
606C2×C4—262no—
607C2×C4—262no—
608C2×C4—262no—
609C2×C4—262no—
610C2×C4—262no—
611C2×C4—262no—
612C2×C4—262no—
613C2×C4—262no—
614C2×C4—262no—
615C2×C4—524no—
616C2×C4—524no—
617C2×C4—524no—
618C2×C4—524no—
619C2×C4C2×#612914yes—
620C2×C4C2×Q8×D1314yes—
621C2×C4C2×Q8×D1314yes—
622C22#1171930614yes—
623C22#1171932714yes—
624C22#1171934014yes—
625C22#1171936814yes—
626C22D13×#12514yes—
627C22#1171939514yes—
628C22#1171942714yes—
629C22C22×#612911yescharacteristic
630C22C2×#8825111yescharacteristic
631C22C22×Q8×D1311yescharacteristic
632C22C22×Q8×D1311yescharacteristic
633C22C2×#41×D1311yescharacteristic
634C22C2×#41×D1311yescharacteristic
635C22C2×#8825611yescharacteristic
636C4—24no—
637C4—24no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—138no—
644C4—264no—
645C4—264no—
646C4—264no—
647C4—522no—
648C4—522no—
649C4—522no—
650C4—522no—
651C2#1311935618yesdirect product factor
652C2C2×D13×#12511yescharacteristic
653C2C2×#1171932711yescharacteristic
654C2C2×#1171936811yescharacteristic
655C2C2×#1171930611yescharacteristic
656C2C2×#1171939511yescharacteristic
657C2C2×#1171934011yescharacteristic
658C2C2×#1171942711yescharacteristic
subgroupG/H
centerC24C23×D13
commutator, cosocleC23×C13C25
FrattiniC23C24×D13
FittingC2×C13×#398C2
socle, prime radicalC24×C13C24
hypercenterC2×#398D13
hypocenterC13C2×#2507
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
1. #13119356C2direct (N × H)
461. C2×C13#2507semidirect
553. C13C2×#2507semidirect
651. C2#13119356direct (N × H)

Maximal subgroups

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

subgroupindex
#131193562
C2×#117183182
C2×#117186452
C2×#117186442
C2×#117186462
C2×#117183082
C2×#117186332
C2×#117183142
C2×#117183152
C2×#117186372
C2×C4⋊C4×Dic132
C2×#117183102
C2×C13×#3982
C2×#250713