Finite Group Databasebeta

← #13285714

Subgroups — #13285714

Order 1792.

2242 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C22×C7×#95C211yesmaximal, characteristic
3C2×#89982C211yesmaximal, characteristic
4C2×#89984C212yesmaximal
5C2×#89428C211yesmaximal, characteristic, complemented
6#11729881C212yesmaximal
7#11727178C211yesmaximal, characteristic, complemented
8#11727174C212yesmaximal, complemented
9#11727173C211yesmaximal, characteristic, complemented
10#11727165C212yesmaximal, complemented
11#11727164C211yesmaximal, characteristic, complemented
12C24×C4×C7C2211yescharacteristic
13C24×C4×C7C2212yes—
14C23×Dic14C2211yescharacteristic
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2212yes—
20C22×C4⋊Dic7C2212yes—
21C2×#6487—24no—
22C2×#6487—28no—
23C22×Dic7⋊C4C2212yes—
24C22×Dic7⋊C4C2211yescharacteristic
25C22×Dic7⋊C4C2211yescharacteristic
26C2×C7×#95—24no—
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2212yes—
29C2×C7×#95C2211yescharacteristic
30#89984—28no—
31#89982—24no—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35C2×C7⋊#95C2211yescharacteristic
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C2×C7⋊#95C2212yes—
39C2×C7⋊#95C2212yes—
40#89428—24no—
41#14758—71nomaximal, Sylow
42C25×C7C2311yescharacteristic
43C23×C4×C7—24no—
44C23×C4×C7—24no—
45C23×C4×C7—24no—
46C23×C4×C7—22no—
47C23×C4×C7—24no—
48C23×C4×C7—22no—
49C23×C4×C7—24no—
50C23×C4×C7—42no—
51C23×C4×C7—42no—
52C23×C4×C7—42no—
53C23×C4×C7D414yes—
54C23×C4×C7Q818yescomplemented
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2312yes—
59C23×Dic7C2311yescharacteristic
60C23×Dic7C2311yescharacteristic
61C23×Dic7C2312yes—
62C23×Dic7C2312yes—
63C23×Dic7C2311yescharacteristic
64C23×Dic7C2311yescharacteristic
65C22×Dic14—22no—
66C22×Dic14—24no—
67C22×Dic14—22no—
68C22×Dic14—22no—
69C22×Dic14—42no—
70C2×C22⋊Dic7—24no—
71C2×C22⋊Dic7—24no—
72C2×C22⋊Dic7—22no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—24no—
76C2×C22⋊Dic7D418yes—
77C2×C22⋊Dic7Q814yescomplemented
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—24no—
80C2×C4⋊Dic7—24no—
81C2×C4⋊Dic7—24no—
82C2×C4⋊Dic7—24no—
83C2×C4⋊Dic7—24no—
84C2×C4⋊Dic7—42no—
85C2×C4⋊Dic7—42no—
86#6487—232no—
87C2×Dic7⋊C4—24no—
88C2×Dic7⋊C4—24no—
89C2×Dic7⋊C4—24no—
90C2×Dic7⋊C4—22no—
91C2×Dic7⋊C4—22no—
92C2×Dic7⋊C4—22no—
93C2×Dic7⋊C4—22no—
94C2×Dic7⋊C4—42no—
95C7×#95—22no—
96C7×#95—24no—
97C7×#95D414yes—
98C7⋊#95—24no—
99C7⋊#95—22no—
100C7⋊#95—22no—
101C7⋊#95—24no—
102C7⋊#95—24no—
103C7⋊#95—24no—
104C7⋊#95—42no—
105C7⋊#95—42no—
106C22×C22⋊Q8—71no—
107C2×#395—71no—
108C2×#385—71no—
109C2×#380—72no—
110C22×#95D711yescharacteristic
111#1446—71no—
112#1444—71no—
113#1444—72no—
114#1441—71no—
115#1441—72no—
116C22⋊#95—72no—
117C24×C7—21no—
118C24×C7—21no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7—21no—
122C24×C7—21no—
123C24×C7C2×D412yes—
124C24×C7C2×D414yes—
125C24×C7C2×Q814yes—
126C24×C7C2×Q812yes—
127C24×C7#4112yes—
128C24×C7C2411yescharacteristic
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—28no—
136C22×C4×C7—216no—
137C22×C4×C7—22no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—28no—
148C22×C4×C7—216no—
149C22×C4×C7—44no—
150C22×C4×C7—48no—
151C22×C4×C7—48no—
152C22×C4×C7—48no—
153C22×C4×C7—48no—
154C22×C4×C7—48no—
155C22×C4×C7—48no—
156C22×C4×C7—48no—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D414yes—
161C22×C4×C7C2×Q814yes—
162C22×C4×C7C2×Q814yes—
163C22×C4×C7#4112yes—
164C22×C4×C7#4114yes—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—21no—
176C22×Dic7—22no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—22no—
180C22×Dic7—21no—
181C22×Dic7—22no—
182C22×Dic7—22no—
183C22×Dic7—22no—
184C22×Dic7—22no—
185C22×Dic7—22no—
186C22×Dic7—21no—
187C22×Dic7—21no—
188C22×Dic7—22no—
189C22×Dic7—22no—
190C22×Dic7—21no—
191C22×Dic7—21no—
192C22×Dic7—21no—
193C22×Dic7—22no—
194C22×Dic7—22no—
195C22×Dic7—21no—
196C22×Dic7—21no—
197C22×Dic7C2×D414yes—
198C22×Dic7C2×D414yes—
199C22×Dic7C2×Q812yes—
200C22×Dic7C2×Q812yes—
201C2×Dic14—28no—
202C2×Dic14—48no—
203C2×Dic14—48no—
204C2×Dic14—44no—
205C2×Dic14—44no—
206C22⋊Dic7—216no—
207C22⋊Dic7—28no—
208C22⋊Dic7—44no—
209C4⋊Dic7—28no—
210C4⋊Dic7—48no—
211C4⋊Dic7—48no—
212Dic7⋊C4—216no—
213Dic7⋊C4—44no—
214Dic7⋊C4—44no—
215C23×Q8—71no—
216C24×C4C2×D712yes—
217C24×C4C2×D711yescharacteristic
218C2×C22⋊Q8—148no—
219C2×C22⋊Q8—144no—
220C22×C4⋊C4—71no—
221C22×C4⋊C4—72no—
222C22×C4⋊C4—71no—
223C22×C4⋊C4—72no—
224C22×C4⋊C4—72no—
225C22×C4⋊C4—71no—
226C22×C22⋊C4—72no—
227C22×C22⋊C4—71no—
228C22×C22⋊C4—71no—
229#395—144no—
230#385—144no—
231#380—148no—
232C2×#95—24no—
233C2×#95—72no—
234C2×#95—72no—
235C2×#95—71no—
236C2×#95—72no—
237C2×#95—72no—
238C2×#95—71no—
239C2×#95—72no—
240C2×#95—72no—
241C2×#95C2×D712yes—
242C2×#95C2×D711yescharacteristic
243C2×#95C2×D711yescharacteristic
244C23×C7—22no—
245C23×C7—24no—
246C23×C7—22no—
247C23×C7—24no—
248C23×C7—22no—
249C23×C7—22no—
250C23×C7—24no—
251C23×C7—24no—
252C23×C7—24no—
253C23×C7—22no—
254C23×C7—24no—
255C23×C7—24no—
256C23×C7—24no—
257C23×C7—22no—
258C23×C7—24no—
259C23×C7—22no—
260C23×C7—24no—
261C23×C7—22no—
262C23×C7#12014yes—
263C23×C7C22⋊Q818yes—
264C23×C7C22⋊Q814yes—
265C23×C7C4⋊Q814yes—
266C23×C7C4⋊Q818yes—
267C23×C7C22×D411yescharacteristic
268C23×C7C22×D412yes—
269C23×C7C22×D411yescharacteristic
270C23×C7C22×Q812yes—
271C23×C7C22×Q811yescharacteristic
272C23×C7C2×#4111yescharacteristic
273C23×C7C2×#4112yes—
274C23×C7C2×#4111yescharacteristic
275C23×C7H(5,𝔽2)12yes—
276C23×C7H(5,𝔽2)11yescharacteristic
277C23×C72−1+411yescharacteristic
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—22no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—24no—
285C2×C4×C7—24no—
286C2×C4×C7—24no—
287C2×C4×C7—24no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—24no—
291C2×C4×C7—24no—
292C2×C4×C7—22no—
293C2×C4×C7—24no—
294C2×C4×C7—416no—
295C2×C4×C7—432no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7#12114yes—
303C2×C4×C7C22⋊Q818yescomplemented
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—24no—
309C2×Dic7—24no—
310C2×Dic7—24no—
311C2×Dic7—24no—
312C2×Dic7—22no—
313C2×Dic7—24no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—42no—
317C2×Dic7—41no—
318C2×Dic7—42no—
319C2×Dic7—41no—
320C2×Dic7—42no—
321C2×Dic7—41no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—41no—
327C2×Dic7—42no—
328C2×Dic7—42no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—42no—
337C2×Dic7—42no—
338C2×Dic7—42no—
339C2×Dic7—42no—
340Dic14—416no—
341C25C22×D711yescharacteristic
342C22×Q8—144no—
343C22×Q8—142no—
344C22×Q8—142no—
345C22×Q8—142no—
346C22×Q8—282no—
347C23×C4—24no—
348C23×C4—24no—
349C23×C4—24no—
350C23×C4—24no—
351C23×C4—22no—
352C23×C4—24no—
353C23×C4—22no—
354C23×C4—42no—
355C23×C4—42no—
356C23×C4—42no—
357C23×C4—72no—
358C23×C4—72no—
359C23×C4—71no—
360C23×C4—71no—
361C23×C4—71no—
362C23×C4—71no—
363C23×C4Dic1418yescomplemented
364C23×C4D2814yes—
365C23×C4C22×D712yes—
366C23×C4C22×D711yescharacteristic
367C23×C4C22×D712yes—
368C23×C4C22×D711yescharacteristic
369C22⋊Q8—1432no—
370C2×C4⋊C4—144no—
371C2×C4⋊C4—144no—
372C2×C4⋊C4—144no—
373C2×C4⋊C4—144no—
374C2×C4⋊C4—142no—
375C2×C4⋊C4—144no—
376C2×C4⋊C4—144no—
377C2×C4⋊C4—142no—
378C2×C4⋊C4—142no—
379C2×C4⋊C4—142no—
380C2×C4⋊C4—144no—
381C2×C4⋊C4—142no—
382C2×C4⋊C4—144no—
383C2×C4⋊C4—282no—
384C2×C4⋊C4—282no—
385C2×C4⋊C4—282no—
386C2×C22⋊C4—78no—
387C2×C22⋊C4—74no—
388C2×C22⋊C4—144no—
389C2×C22⋊C4—142no—
390C2×C22⋊C4—144no—
391C2×C22⋊C4—142no—
392C2×C22⋊C4—142no—
393C2×C22⋊C4—144no—
394#95—24no—
395#95—22no—
396#95—144no—
397#95—144no—
398#95—142no—
399#95—142no—
400#95—144no—
401#95—144no—
402#95—282no—
403#95—282no—
404#95C22⋊D714yes—
405C22×C7—24no—
406C22×C7—24no—
407C22×C7—24no—
408C22×C7—24no—
409C22×C7—24no—
410C22×C7—24no—
411C22×C7—28no—
412C22×C7—28no—
413C22×C7—28no—
414C22×C7—28no—
415C22×C7#39518yescomplemented
416C22×C7C2×#12011yescharacteristic
417C22×C7C2×#12111yescharacteristic
418C22×C7C2×C22⋊Q812yes—
419C22×C7C2×C22⋊Q811yescharacteristic
420C22×C7C2×C22⋊Q812yes—
421C22×C7C2×C4⋊Q811yescharacteristic
422C22×C7C2×C4⋊Q812yes—
423C22×C7#53211yescharacteristic
424C22×C7#53312yes—
425C22×C7#53411yescharacteristic
426C22×C7#53711yescharacteristic
427C22×C7#54312yes—
428C22×C7#54412yes—
429C22×C7D4212yes—
430C22×C7#54612yes—
431C22×C7#54612yes—
432C22×C7#54611yescharacteristic
433C22×C7#54712yes—
434C22×C7D4×Q811yescharacteristic
435C22×C7#55112yes—
436C22×C7#55211yescharacteristic
437C22×C7#55412yes—
438C22×C7#55411yescharacteristic
439C4×C7—24no—
440C4×C7—28no—
441C4×C7—42no—
442C4×C7—44no—
443Dic7—42no—
444Dic7—44no—
445Dic7—42no—
446Dic7—44no—
447Dic7—81no—
448Dic7—81no—
449C24—21no—
450C24—22no—
451C24—21no—
452C24—22no—
453C24—21no—
454C24—21no—
455C24C2×Dic1414yes—
456C24C2×D2812yes—
457C24D4×D714yes—
458C24#115912yes—
459C24Q8×D712yes—
460C24C23×D711yescharacteristic
461C2×Q8—148no—
462C2×Q8—288no—
463C2×Q8—288no—
464C2×Q8—284no—
465C2×Q8—284no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—216no—
473C22×C4—28no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—216no—
485C22×C4—28no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—74no—
495C22×C4—74no—
496C22×C4—72no—
497C22×C4—72no—
498C22×C4—141no—
499C22×C4—141no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—141no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—142no—
509C22×C4—141no—
510C22×C4—141no—
511C22×C4—141no—
512C22×C4—142no—
513C22×C4—142no—
514C22×C4—142no—
515C22×C4—142no—
516C22×C4—142no—
517C22×C4—141no—
518C22×C4—141no—
519C22×C4—142no—
520C22×C4—141no—
521C22×C4—141no—
522C22×C4—142no—
523C22×C4—141no—
524C22×C4—142no—
525C22×C4—142no—
526C22×C4—141no—
527C22×C4—142no—
528C22×C4—141no—
529C22×C4—141no—
530C22×C4C2×Dic1414yes—
531C22×C4C2×Dic1414yes—
532C22×C4C2×D2812yes—
533C22×C4C2×D2812yes—
534C22×C4#115714yes—
535C22×C4#115712yes—
536C22×C4C2×C22⋊D714yes—
537C22×C4C2×C22⋊D712yes—
538C4⋊C4—1416no—
539C4⋊C4—148no—
540C4⋊C4—288no—
541C4⋊C4—288no—
542C4⋊C4—284no—
543C4⋊C4—284no—
544C22⋊C4—1416no—
545C22⋊C4—148no—
546C22⋊C4—284no—
547C2×C7—28no—
548C2×C7C2×#39511yescharacteristic
549C2×C7#241011yescharacteristic
550C2×C7#241711yescharacteristic
551C2×C7#243011yescharacteristic
552C2×C7#243712yes—
553C2×C7#243812yes—
554C2×C7#244012yes—
555C2×C7#254912yes—
556C2×C7#256411yescharacteristic
557C2×C7#265911yescharacteristic
558C2×C7#285711yescharacteristic
559C23—24no—
560C23—22no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—22no—
572C23—24no—
573C23—22no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23#643214yes—
578C23#644118yes—
579C23#644514yes—
580C23#645118yes—
581C23#646014yes—
582C23C22×Dic1412yes—
583C23C22×D2811yescharacteristic
584C23C2×#115711yescharacteristic
585C23C2×#115712yes—
586C23C2×D4×D712yes—
587C23C2×#115911yescharacteristic
588C23#654811yescharacteristic
589C23#654812yes—
590C23C2×Q8×D711yescharacteristic
591C23#655111yescharacteristic
592C23C22×C22⋊D711yescharacteristic
593Q8—2816no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—432no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—144no—
619C2×C4—144no—
620C2×C4—144no—
621C2×C4—144no—
622C2×C4—144no—
623C2×C4—142no—
624C2×C4—142no—
625C2×C4—144no—
626C2×C4—142no—
627C2×C4—144no—
628C2×C4—142no—
629C2×C4—281no—
630C2×C4—282no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—281no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4—281no—
639C2×C4—281no—
640C2×C4—281no—
641C2×C4—282no—
642C2×C4—282no—
643C2×C4—281no—
644C2×C4—281no—
645C2×C4—281no—
646C2×C4—281no—
647C2×C4—282no—
648C2×C4—282no—
649C2×C4—281no—
650C2×C4—282no—
651C2×C4—281no—
652C2×C4—281no—
653C2×C4—282no—
654C2×C4#648718yescomplemented
655C2×C4#649314yes—
656C7#1475811yescharacteristic, complemented, Sylow
657C22—28no—
658C22—28no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—28no—
664C22—24no—
665C22—28no—
666C22—24no—
667C22#8942818yescomplemented
668C22C2×#643211yescharacteristic
669C22#9016112yes—
670C22#9016811yescharacteristic
671C22C2×#644112yes—
672C22#9017412yes—
673C22C2×#644511yescharacteristic
674C22#9018112yes—
675C22#9018112yes—
676C22#9018411yescharacteristic
677C22#9018612yes—
678C22C2×#645112yes—
679C22#9019112yes—
680C22#9019812yes—
681C22C2×#646011yescharacteristic
682C22#9020111yescharacteristic
683C22#9020211yescharacteristic
684C22#9020212yes—
685C22#9020411yescharacteristic
686C22C2×#648712yes—
687C22C2×#649311yescharacteristic
688C22D4×C22⋊D712yes—
689C22#9049711yescharacteristic
690C22Q8×C22⋊D711yescharacteristic
691C4—28no—
692C4—24no—
693C4—44no—
694C4—42no—
695C4—282no—
696C4—284no—
697C4—284no—
698C4—282no—
699C4—561no—
700C4—561no—
701C2—28no—
702C2C2×#8942811yescharacteristic
703C2#1173079512yes—
704C2#1173080212yes—
705C2#1173080311yescharacteristic
706C2#1173080611yescharacteristic
707C2#1173082111yescharacteristic
708C2#1173952111yescharacteristic
709C2#1173955512yes—
710C2#1173957211yescharacteristic
711C2#1173961612yes—
712C2#1173963911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC22×C7×#95C2
prime radicalC25×C7C23
hypercenterC22×#95D7
hypocenterC7#14758
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
5. C2×#89428C2semidirect
7. #11727178C2semidirect
8. #11727174C2semidirect
9. #11727173C2semidirect
10. #11727165C2semidirect
11. #11727164C2semidirect
54. C23×C4×C7Q8semidirect
77. C2×C22⋊Dic7Q8semidirect
303. C2×C4×C7C22⋊Q8semidirect
363. C23×C4Dic14semidirect
415. C22×C7#395semidirect
654. C2×C4#6487semidirect
656. C7#14758semidirect
667. C22#89428semidirect

Maximal subgroups

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

subgroupindex
C22×#64872
C22×C7×#952
C2×#899822
C2×#899842
C2×#894282
#117298812
#117271782
#117271742
#117271732
#117271652
#117271642
#147587