Finite Group Databasebeta

← C2×#11729454 #13380592

Subgroups — C2×#11729454 #13380592

Order 1792.

2182 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2C23×#1144C211yesmaximal, characteristic, complemented
3C22×#6487C211yesmaximal, characteristic
4C22×#6409C211yesmaximal, characteristic, complemented
5C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic
6C22×#6384C211yesmaximal, characteristic, complemented
7#11729454C218yesmaximal, direct product factor
8C2×#89324C211yesmaximal, characteristic, complemented
9C23×C7×Q8C2211yescharacteristic
10C24×C4×C7C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C2×C7×C22⋊Q8—24no—
13C2×C7×C22⋊Q8—22no—
14C2×C7×C22⋊Q8—24no—
15C2×C7×C22⋊Q8C2214yes—
16C22×C7×C4⋊C4—21no—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4—21no—
19C22×C22⋊Dic7—21no—
20C22×#1144—24no—
21C22×#1144—24no—
22C22×#1144—24no—
23C22×#1144C2214yescomplemented
24C22×C4⋊Dic7C2211yescharacteristic
25C2×#6487—22no—
26C2×#6487—24no—
27C2×#6487—24no—
28C2×#6487C2214yes—
29C22×Dic7⋊C4—21no—
30C23×C7⋊C8C2211yescharacteristic
31C23×C7⋊C8C2211yescharacteristic
32C2×#6409—24no—
33C2×#6409C2214yescomplemented
34C2×C22⋊(C7⋊C8)—24no—
35C2×C22⋊(C7⋊C8)C2214yes—
36C2×#6384—24no—
37C2×#6384C2214yescomplemented
38#89324C2214yescomplemented
39C2×#2001—71nomaximal, Sylow
40C25×C7D411yescharacteristic
41C22×C7×Q8—22no—
42C22×C7×Q8—21no—
43C22×C7×Q8—22no—
44C22×C7×Q8—22no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8D414yes—
48C22×C7×Q8C2312yes—
49C23×C4×C7—21no—
50C23×C4×C7—21no—
51C23×C4×C7—22no—
52C23×C4×C7—22no—
53C23×C4×C7—22no—
54C23×C4×C7—22no—
55C23×C4×C7—22no—
56C23×C4×C7—22no—
57C23×C4×C7D411yescharacteristic
58C23×C4×C7D412yes—
59C23×C4×C7C2312yes—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C23×Dic7—21no—
63C22×Dic14—22no—
64C22×Dic14—22no—
65C22×Dic14—22no—
66C22×Dic14—22no—
67C22×Dic14—21no—
68C22×Dic14—22no—
69C22×Dic14D414yes—
70C22×Dic14C2312yes—
71C7×C22⋊Q8—216no—
72C7×C22⋊Q8—28no—
73C7×C22⋊Q8—28no—
74C2×C7×C4⋊C4—28no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4D414yes—
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C22⋊C4—28no—
81C2×C7×C22⋊C4—24no—
82C2×C22⋊Dic7—28no—
83C2×C22⋊Dic7—24no—
84C2×#1144—232no—
85C2×#1144—28no—
86C2×#1144—28no—
87C2×#1144—28no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7D414yes—
91C2×C4⋊Dic7C2312yes—
92#6487—216no—
93#6487—28no—
94#6487—28no—
95C2×Dic7⋊C4—28no—
96C2×Dic7⋊C4—24no—
97C22×C7⋊C8—22no—
98C22×C7⋊C8—22no—
99C22×C7⋊C8—22no—
100C22×C7⋊C8—22no—
101C22×C7⋊C8—22no—
102C22×C7⋊C8D414yes—
103C22×C7⋊C8C2312yes—
104C22×C7⋊C8C2312yes—
105#6409—28no—
106C22⋊(C7⋊C8)—28no—
107#6384—28no—
108C23×Q16—71no—
109C22×C22⋊Q8—71no—
110C22×C22⋊Q8D711yescharacteristic
111C22×#103—71no—
112C22×#103—71no—
113C22×C22⋊C8—71no—
114#2001—78no—
115C2×#340—71no—
116C24×C7—22no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7—22no—
122C24×C7Q1614yescomplemented
123C24×C7C2×D412yes—
124C24×C7C2×D411yescharacteristic
125C2×C7×Q8—24no—
126C2×C7×Q8—22no—
127C2×C7×Q8—28no—
128C2×C7×Q8—28no—
129C2×C7×Q8—22no—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—22no—
133C2×C7×Q8—42no—
134C2×C7×Q8—42no—
135C2×C7×Q8—44no—
136C2×C7×Q8—44no—
137C2×C7×Q8C2×D414yes—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7—24no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—44no—
159C22×C4×C7—44no—
160C22×C4×C7—44no—
161C22×C4×C7—44no—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7#4112yes—
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—42no—
176C22×Dic7—42no—
177C22×Dic7—42no—
178C22×Dic7—42no—
179C2×Dic14—28no—
180C2×Dic14—28no—
181C2×Dic14—22no—
182C2×Dic14—22no—
183C2×Dic14—22no—
184C2×Dic14—22no—
185C2×Dic14—24no—
186C2×Dic14—22no—
187C2×Dic14—42no—
188C2×Dic14—44no—
189C2×Dic14—42no—
190C2×Dic14—44no—
191C2×Dic14C2×D414yes—
192C7×C4⋊C4—216no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4C2×D414yes—
196C7×C22⋊C4—216no—
197C22⋊Dic7—216no—
198#1144—232no—
199C4⋊Dic7—24no—
200C4⋊Dic7—22no—
201C4⋊Dic7C2×D414yes—
202Dic7⋊C4—216no—
203C2×C7⋊C8—28no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—44no—
210C2×C7⋊C8C2×D414yes—
211C23×Q8—71no—
212C23×Q8C2×D711yescharacteristic
213C24×C4C2×D711yescharacteristic
214C22×Q16—74no—
215C22×Q16—144no—
216C22×Q16—144no—
217C22×Q16—144no—
218C23×C8—71no—
219C23×C8—71no—
220C2×C22⋊Q8—24no—
221C2×C22⋊Q8—22no—
222C2×C22⋊Q8—24no—
223C2×C22⋊Q8—74no—
224C2×C22⋊Q8—144no—
225C2×C22⋊Q8—144no—
226C2×C22⋊Q8—142no—
227C2×C22⋊Q8C2×D714yes—
228C22×C4⋊C4—21no—
229C22×C4⋊C4—71no—
230C22×C4⋊C4—141no—
231C22×C4⋊C4C2×D711yescharacteristic
232C22×C22⋊C4—21no—
233C22×C22⋊C4—141no—
234C2×#103—74no—
235C2×#103—74no—
236C2×#103—144no—
237C2×#103—144no—
238C2×C22⋊C8—74no—
239C2×C22⋊C8—144no—
240#340—74no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—28no—
248C23×C7—28no—
249C23×C7—28no—
250C23×C7—44no—
251C23×C7—44no—
252C23×C7—44no—
253C23×C7—44no—
254C23×C7#12012yes—
255C23×C7#12012yes—
256C23×C7#12112yes—
257C23×C7C2×Q1612yes—
258C23×C7C2×Q1614yescomplemented
259C23×C7#13512yes—
260C23×C7#13712yes—
261C23×C7#13712yes—
262C23×C7C22×D411yescharacteristic
263C7×Q8—24no—
264C7×Q8—24no—
265C7×Q8—42no—
266C7×Q8—42no—
267C7×Q8—48no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—44no—
278C2×C4×C7—48no—
279C2×C4×C7—48no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—48no—
285C2×C4×C7#12014yes—
286C2×C4×C7#12012yes—
287C2×C4×C7#12012yes—
288C2×C4×C7#12114yes—
289C2×C4×C7#12114yes—
290C2×C4×C7#12112yes—
291C2×C4×C7#12714yes—
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×Dic7—24no—
299C2×Dic7—24no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—48no—
305C2×Dic7—48no—
306Dic14—24no—
307Dic14—24no—
308Dic14—42no—
309Dic14—42no—
310Dic14—48no—
311C7⋊C8—24no—
312C7⋊C8—42no—
313C25C22⋊D711yescharacteristic
314C22×Q8—22no—
315C22×Q8—21no—
316C22×Q8—22no—
317C22×Q8—22no—
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8—74no—
321C22×Q8—72no—
322C22×Q8—142no—
323C22×Q8—141no—
324C22×Q8—142no—
325C22×Q8—142no—
326C22×Q8—142no—
327C22×Q8—142no—
328C22×Q8C22⋊D714yes—
329C22×Q8C22×D712yes—
330C23×C4—21no—
331C23×C4—21no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—22no—
337C23×C4—22no—
338C23×C4—141no—
339C23×C4—141no—
340C23×C4C22⋊D712yes—
341C23×C4C22⋊D711yescharacteristic
342C23×C4C22×D712yes—
343C23×C4C22×D711yescharacteristic
344C2×Q16—148no—
345C2×Q16—148no—
346C2×Q16—1432no—
347C2×Q16—148no—
348C22×C8—74no—
349C22×C8—72no—
350C22×C8—72no—
351C22×C8—142no—
352C22×C8—142no—
353C22×C8—142no—
354C22×C8—142no—
355C22×C8—142no—
356C22⋊Q8—216no—
357C22⋊Q8—28no—
358C22⋊Q8—28no—
359C22⋊Q8—148no—
360C22⋊Q8—148no—
361C22⋊Q8—1416no—
362C2×C4⋊C4—28no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—74no—
367C2×C4⋊C4—72no—
368C2×C4⋊C4—148no—
369C2×C4⋊C4—142no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—144no—
372C2×C4⋊C4C22⋊D714yes—
373C2×C4⋊C4C22×D712yes—
374C2×C22⋊C4—28no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—148no—
377C2×C22⋊C4—144no—
378#103—148no—
379#103—148no—
380C22⋊C8—148no—
381C22×C7—24no—
382C22×C7—24no—
383C22×C7—24no—
384C22×C7—42no—
385C22×C7—42no—
386C22×C7—42no—
387C22×C7—416no—
388C22×C7—42no—
389C22×C7#39214yes—
390C22×C7#45114yes—
391C22×C7#45114yes—
392C22×C7#45214yes—
393C22×C7#45214yes—
394C22×C7#46714yes—
395C22×C7#47014yes—
396C22×C7C2×#12011yescharacteristic
397C22×C7C2×#12011yescharacteristic
398C22×C7C2×#12111yescharacteristic
399C22×C7C22×Q1611yescharacteristic
400C22×C7C2×#13511yescharacteristic
401C22×C7C2×#13711yescharacteristic
402C22×C7C2×#13711yescharacteristic
403C4×C7—42no—
404C4×C7—44no—
405C4×C7—44no—
406C4×C7#39214yes—
407C4×C7C2×#12011yescharacteristic
408C4×C7C2×#12111yescharacteristic
409C4×C7C2×#12111yescharacteristic
410C4×C7C2×#12711yescharacteristic
411Dic7—44no—
412Dic7—44no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24#114414yescomplemented
420C24C2×C22⋊D712yes—
421C24C2×C22⋊D711yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—28no—
425C2×Q8—28no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—42no—
431C2×Q8—44no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—74no—
435C2×Q8—144no—
436C2×Q8—142no—
437C2×Q8—148no—
438C2×Q8—148no—
439C2×Q8—142no—
440C2×Q8—142no—
441C2×Q8—142no—
442C2×Q8—142no—
443C2×Q8—282no—
444C2×Q8—284no—
445C2×Q8—284no—
446C2×Q8—282no—
447C2×Q8C2×C22⋊D714yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—144no—
473C22×C4—144no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—282no—
477C22×C4—282no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4D4×D712yes—
481C22×C4D4×D712yes—
482C22×C4D4×D712yes—
483C22×C4#115912yes—
484C22×C4C2×C22⋊D712yes—
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C2×C22⋊D712yes—
487C22×C4C2×C22⋊D712yes—
488C22×C4C23×D711yescharacteristic
489Q16—1432no—
490C2×C8—74no—
491C2×C8—148no—
492C2×C8—142no—
493C2×C8—142no—
494C2×C8—142no—
495C2×C8—142no—
496C2×C8—142no—
497C2×C8—284no—
498C4⋊C4—216no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—74no—
502C4⋊C4—1416no—
503C4⋊C4—144no—
504C4⋊C4—142no—
505C4⋊C4C2×C22⋊D714yes—
506C22⋊C4—216no—
507C22⋊C4—1416no—
508C2×C7—44no—
509C2×C7#200118yescomplemented
510C2×C7C2×#39211yescharacteristic
511C2×C7C2×#45111yescharacteristic
512C2×C7C2×#45111yescharacteristic
513C2×C7C2×#45211yescharacteristic
514C2×C7C2×#45211yescharacteristic
515C2×C7C2×#46711yescharacteristic
516C2×C7C2×#47011yescharacteristic
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—28no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23#650012yes—
531C23#650212yes—
532C23#650512yes—
533C23C2×#114412yes—
534C23C2×#114414yescomplemented
535C23#651312yes—
536C23#651412yes—
537C23#651612yes—
538C23C22×C22⋊D711yescharacteristic
539Q8—24no—
540Q8—24no—
541Q8—42no—
542Q8—42no—
543Q8—48no—
544Q8—144no—
545Q8—144no—
546Q8—282no—
547Q8—282no—
548Q8—288no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—144no—
567C2×C4—144no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—288no—
571C2×C4—282no—
572C2×C4—282no—
573C2×C4—288no—
574C2×C4#650014yes—
575C2×C4#650012yes—
576C2×C4#650114yes—
577C2×C4#650214yes—
578C2×C4#650212yes—
579C2×C4#650314yes—
580C2×C4#651612yes—
581C2×C4C2×D4×D711yescharacteristic
582C2×C4C2×D4×D711yescharacteristic
583C2×C4C2×D4×D711yescharacteristic
584C2×C4C2×#115911yescharacteristic
585C2×C4C22×C22⋊D711yescharacteristic
586C2×C4C22×C22⋊D711yescharacteristic
587C8—144no—
588C8—282no—
589C7C2×#200111yescharacteristic, complemented, Sylow
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—416no—
596C22—42no—
597C22—42no—
598C22#8981914yes—
599C22#8982214yes—
600C22#8982314yes—
601C22#8982414yes—
602C22#8999514yes—
603C22#9000014yes—
604C22#9001114yes—
605C22C2×#650011yescharacteristic
606C22C2×#650211yescharacteristic
607C22C2×#650511yescharacteristic
608C22C22×#114411yescharacteristic
609C22C2×#651311yescharacteristic
610C22C2×#651411yescharacteristic
611C22C2×#651611yescharacteristic
612C4—42no—
613C4—44no—
614C4—44no—
615C4—284no—
616C4—284no—
617C4#8999514yes—
618C4C2×#650011yescharacteristic
619C4C2×#650111yescharacteristic
620C4C2×#650211yescharacteristic
621C4C2×#650311yescharacteristic
622C2—44no—
623C2#1172945418yesdirect product factor
624C2C2×#8982311yescharacteristic
625C2C2×#8982411yescharacteristic
626C2C2×#8981911yescharacteristic
627C2C2×#8982211yescharacteristic
628C2C2×#8999511yescharacteristic
629C2C2×#9000011yescharacteristic
630C2C2×#9001111yescharacteristic
subgroupG/H
centerC24C2×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×C22⋊Q8C2
socleC24×C7C2×D4
prime radicalC25×C7D4
hypercenterC22×C22⋊Q8D7
hypocenterC7C2×#2001
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C23×#1144C2semidirect
4. C22×#6409C2semidirect
6. C22×#6384C2semidirect
7. #11729454C2direct (N × H)
8. C2×#89324C2semidirect
23. C22×#1144C22semidirect
33. C2×#6409C22semidirect
37. C2×#6384C22semidirect
38. #89324C22semidirect
122. C24×C7Q16semidirect
258. C23×C7C2×Q16semidirect
419. C24#1144semidirect
509. C2×C7#2001semidirect
534. C23C2×#1144semidirect
589. C7C2×#2001semidirect
623. C2#11729454direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C23×#11442
C22×#64872
C22×#64092
C22×C22⋊(C7⋊C8)2
C22×#63842
#117294542
C2×#893242
C2×#20017