Finite Group Databasebeta

← #13285306

Subgroups — #13285306

Order 1792.

2190 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic7C211yesmaximal, characteristic
2C2×#89718C211yesmaximal, characteristic
3C2×#89718C211yesmaximal, characteristic
4C2×#89714C211yesmaximal, characteristic
5C2×#89429C212yesmaximal, complemented
6C7×C22⋊#95C211yesmaximal, characteristic
7#11729881C211yesmaximal, characteristic
8#11729298C211yesmaximal, characteristic
9#11727174C212yesmaximal, complemented
10#11727166C212yesmaximal, complemented
11#11727165C212yesmaximal, complemented
12C24×C4×C7C2211yescharacteristic
13C24×Dic7C2211yescharacteristic
14C24×Dic7C2211yescharacteristic
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7—24no—
20C22×C4⋊Dic7—28no—
21C22×C4⋊Dic7C2212yes—
22C22×C4⋊Dic7C2212yes—
23C22×Dic7⋊C4C2211yescharacteristic
24C22×Dic7⋊C4C2212yes—
25C22×Dic7⋊C4C2212yes—
26C22×Dic7⋊C4C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37#89718—24no—
38#89718—24no—
39#89714—24no—
40#89429—28no—
41#14728—71nomaximal, Sylow
42C25×C7C2311yescharacteristic
43C23×C4×C7—22no—
44C23×C4×C7—22no—
45C23×C4×C7—24no—
46C23×C4×C7—24no—
47C23×C4×C7Q814yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2312yes—
52C23×Dic7—22no—
53C23×Dic7—22no—
54C23×Dic7—22no—
55C23×Dic7—24no—
56C23×Dic7—24no—
57C23×Dic7—22no—
58C23×Dic7—22no—
59C23×Dic7—42no—
60C23×Dic7—42no—
61C23×Dic7D414yes—
62C23×Dic7C2312yes—
63C23×Dic7C2311yescharacteristic
64C23×Dic7C2312yes—
65C23×Dic7C2311yescharacteristic
66C23×Dic7C2311yescharacteristic
67C23×Dic7C2311yescharacteristic
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—22no—
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—22no—
78C2×C22⋊Dic7D414yes—
79C2×C22⋊Dic7Q814yes—
80C2×C4⋊Dic7—232no—
81C2×C4⋊Dic7—24no—
82C2×C4⋊Dic7—24no—
83C2×C4⋊Dic7—24no—
84C2×C4⋊Dic7—24no—
85C2×C4⋊Dic7—24no—
86C2×C4⋊Dic7D418yes—
87C2×Dic7⋊C4—24no—
88C2×Dic7⋊C4—24no—
89C2×Dic7⋊C4—24no—
90C2×Dic7⋊C4—24no—
91C2×Dic7⋊C4—24no—
92C2×Dic7⋊C4—22no—
93C2×Dic7⋊C4—22no—
94C2×Dic7⋊C4—42no—
95C2×Dic7⋊C4—42no—
96C2×Dic7⋊C4—42no—
97C7×#95—24no—
98C7×#95—24no—
99C7⋊#95—24no—
100C7⋊#95—24no—
101C7⋊#95—24no—
102C7⋊#95—24no—
103C7⋊#95—22no—
104C7⋊#95—22no—
105C7⋊#95—24no—
106C7⋊#95—42no—
107C23×C4⋊C4—71no—
108C2×#398—72no—
109C2×#385—71no—
110C2×#385—71no—
111C2×#380—71no—
112#1444—72no—
113#1442—72no—
114#1441—72no—
115C22⋊#95—71no—
116C22⋊#95—71no—
117C22⋊#95D711yescharacteristic
118C24×C7—22no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7—22no—
122C24×C7C2×D412yes—
123C24×C7C2×D412yes—
124C24×C7C2×Q812yes—
125C24×C7C2×Q812yes—
126C24×C7#4112yes—
127C24×C7#4112yes—
128C24×C7#4112yes—
129C24×C7C2411yescharacteristic
130C22×C4×C7—28no—
131C22×C4×C7—216no—
132C22×C4×C7—28no—
133C22×C4×C7—28no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—216no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×Q814yes—
158C22×C4×C7#4114yes—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—22no—
162C22×Dic7—28no—
163C22×Dic7—28no—
164C22×Dic7—22no—
165C22×Dic7—21no—
166C22×Dic7—22no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—21no—
180C22×Dic7—22no—
181C22×Dic7—22no—
182C22×Dic7—22no—
183C22×Dic7—22no—
184C22×Dic7—22no—
185C22×Dic7—22no—
186C22×Dic7—22no—
187C22×Dic7—22no—
188C22×Dic7—28no—
189C22×Dic7—44no—
190C22×Dic7—48no—
191C22×Dic7—48no—
192C22×Dic7—48no—
193C22×Dic7—48no—
194C22×Dic7—44no—
195C22×Dic7—44no—
196C22×Dic7C2×D414yes—
197C22×Dic7C2×D412yes—
198C22×Dic7C2×D414yes—
199C22×Dic7C2×D412yes—
200C22×Dic7C2×D414yes—
201C22×Dic7C2×Q812yes—
202C22×Dic7C2×Q812yes—
203C7×C22⋊C4—44no—
204C7×C22⋊C4—44no—
205C22⋊Dic7—28no—
206C22⋊Dic7—28no—
207C4⋊Dic7—216no—
208C4⋊Dic7—216no—
209Dic7⋊C4—48no—
210Dic7⋊C4—48no—
211Dic7⋊C4—44no—
212Dic7⋊C4—44no—
213C24×C4—71no—
214C24×C4—71no—
215C24×C4C2×D711yescharacteristic
216C22×C4⋊C4—72no—
217C22×C4⋊C4—71no—
218C22×C4⋊C4—72no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—72no—
221C22×C4⋊C4—71no—
222C22×C4⋊C4—148no—
223C22×C4⋊C4—144no—
224C22×C22⋊C4—71no—
225C22×C22⋊C4—71no—
226C22×C22⋊C4C2×D711yescharacteristic
227C22×C22⋊C4C2×D711yescharacteristic
228#398—148no—
229#385—144no—
230#385—144no—
231#380—144no—
232C2×#95—72no—
233C2×#95—72no—
234C2×#95—72no—
235C2×#95—71no—
236C2×#95—72no—
237C2×#95—71no—
238C2×#95—72no—
239C2×#95—72no—
240C2×#95C2×D712yes—
241C2×#95C2×D712yes—
242C23×C7—22no—
243C23×C7—24no—
244C23×C7—22no—
245C23×C7—24no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—24no—
249C23×C7—24no—
250C23×C7—24no—
251C23×C7—24no—
252C23×C7—24no—
253C23×C7—24no—
254C23×C7—24no—
255C23×C7—24no—
256C23×C7—24no—
257C23×C7—24no—
258C23×C7#12114yes—
259C23×C7C22⋊Q814yes—
260C23×C7C22⋊Q814yes—
261C23×C7#12314yes—
262C23×C7#12514yes—
263C23×C7#12514yes—
264C23×C7C4⋊Q814yes—
265C23×C7C22×D411yescharacteristic
266C23×C7C22×D412yes—
267C23×C7C22×D411yescharacteristic
268C23×C7C22×Q811yescharacteristic
269C23×C7C22×Q811yescharacteristic
270C23×C7C2×#4111yescharacteristic
271C23×C7C2×#4111yescharacteristic
272C23×C7C2×#4111yescharacteristic
273C23×C7C2×#4112yes—
274C23×C7H(5,𝔽2)12yes—
275C23×C72−1+412yes—
276C2×C4×C7—232no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—42no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7C22⋊Q818yes—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—24no—
304C2×Dic7—24no—
305C2×Dic7—22no—
306C2×Dic7—24no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—24no—
311C2×Dic7—22no—
312C2×Dic7—24no—
313C2×Dic7—24no—
314C2×Dic7—24no—
315C2×Dic7—24no—
316C2×Dic7—24no—
317C2×Dic7—24no—
318C2×Dic7—416no—
319C2×Dic7—416no—
320C2×Dic7—41no—
321C2×Dic7—42no—
322C2×Dic7—41no—
323C2×Dic7—42no—
324C2×Dic7—41no—
325C2×Dic7—42no—
326C2×Dic7—41no—
327C2×Dic7—42no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7—42no—
332C2×Dic7—42no—
333C2×Dic7—42no—
334C2×Dic7—42no—
335C2×Dic7—42no—
336C2×Dic7#12014yes—
337C25C22×D711yescharacteristic
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—24no—
341C23×C4—24no—
342C23×C4—71no—
343C23×C4—72no—
344C23×C4—71no—
345C23×C4—71no—
346C23×C4—72no—
347C23×C4—71no—
348C23×C4—74no—
349C23×C4—144no—
350C23×C4—144no—
351C23×C4—142no—
352C23×C4—142no—
353C23×C4—142no—
354C23×C4—142no—
355C23×C4—142no—
356C23×C4—282no—
357C23×C4—282no—
358C23×C4Dic1414yes—
359C23×C4C22×D711yescharacteristic
360C23×C4C22×D712yes—
361C23×C4C22×D711yescharacteristic
362C23×C4C22×D712yes—
363C2×C4⋊C4—78no—
364C2×C4⋊C4—144no—
365C2×C4⋊C4—144no—
366C2×C4⋊C4—144no—
367C2×C4⋊C4—144no—
368C2×C4⋊C4—1432no—
369C2×C4⋊C4—144no—
370C2×C4⋊C4—144no—
371C2×C4⋊C4—142no—
372C2×C4⋊C4—144no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—144no—
375C2×C4⋊C4—144no—
376C2×C4⋊C4—144no—
377C2×C4⋊C4—282no—
378C2×C4⋊C4—282no—
379C2×C4⋊C4—282no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—74no—
387C2×C22⋊C4—74no—
388C2×C22⋊C4—142no—
389C2×C22⋊C4—142no—
390C2×C22⋊C4—142no—
391C2×C22⋊C4—142no—
392#95—24no—
393#95—24no—
394#95—144no—
395#95—142no—
396#95—144no—
397#95—142no—
398#95—144no—
399#95—144no—
400#95—144no—
401#95—282no—
402C22×C7—24no—
403C22×C7—24no—
404C22×C7—24no—
405C22×C7—24no—
406C22×C7—24no—
407C22×C7—24no—
408C22×C7—28no—
409C22×C7—28no—
410C22×C7—28no—
411C22×C7—28no—
412C22×C7#39818yescomplemented
413C22×C7C2×#12011yescharacteristic
414C22×C7C2×#12111yescharacteristic
415C22×C7C2×C22⋊Q811yescharacteristic
416C22×C7C2×C22⋊Q812yes—
417C22×C7C2×C22⋊Q811yescharacteristic
418C22×C7C2×#12311yescharacteristic
419C22×C7#52511yescharacteristic
420C22×C7#52512yes—
421C22×C7C2×#12511yescharacteristic
422C22×C7C2×#12511yescharacteristic
423C22×C7C2×C4⋊Q811yescharacteristic
424C22×C7#53411yescharacteristic
425C22×C7#53611yescharacteristic
426C22×C7#53912yes—
427C22×C7#54412yes—
428C22×C7D4212yes—
429C22×C7#54612yes—
430C22×C7#54712yes—
431C22×C7#54812yes—
432C22×C7D4×Q812yes—
433C22×C7#55212yes—
434C22×C7#55312yes—
435C22×C7#55412yes—
436C4×C7—28no—
437C4×C7—81no—
438C4×C7—81no—
439C4×C7—82no—
440Dic7—24no—
441Dic7—42no—
442Dic7—42no—
443Dic7—42no—
444Dic7—44no—
445Dic7—44no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×Dic1412yes—
451C24#115712yes—
452C24#115712yes—
453C24D4×D712yes—
454C24D4×D712yes—
455C24#115912yes—
456C24Q8×D712yes—
457C24C23×D711yescharacteristic
458C22×C4—28no—
459C22×C4—216no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—216no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—72no—
485C22×C4—74no—
486C22×C4—74no—
487C22×C4—74no—
488C22×C4—72no—
489C22×C4—72no—
490C22×C4—72no—
491C22×C4—148no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4—142no—
495C22×C4—148no—
496C22×C4—148no—
497C22×C4—142no—
498C22×C4—141no—
499C22×C4—141no—
500C22×C4—141no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—141no—
504C22×C4—141no—
505C22×C4—141no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—142no—
509C22×C4—142no—
510C22×C4—142no—
511C22×C4—142no—
512C22×C4—142no—
513C22×C4—142no—
514C22×C4—142no—
515C22×C4—142no—
516C22×C4—142no—
517C22×C4—142no—
518C22×C4—142no—
519C22×C4—142no—
520C22×C4—142no—
521C22×C4—284no—
522C22×C4—288no—
523C22×C4—288no—
524C22×C4—288no—
525C22×C4—288no—
526C22×C4—284no—
527C22×C4—284no—
528C22×C4C2×Dic1414yes—
529C22×C4D4×D714yes—
530C22×C4#115914yes—
531C4⋊C4—1416no—
532C4⋊C4—1416no—
533C4⋊C4—284no—
534C4⋊C4—288no—
535C4⋊C4—288no—
536C4⋊C4—284no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—148no—
540C22⋊C4—148no—
541C2×C7—28no—
542C2×C7C2×#39811yescharacteristic
543C2×C7#240911yescharacteristic
544C2×C7#241011yescharacteristic
545C2×C7#241311yescharacteristic
546C2×C7#244012yes—
547C2×C7#244212yes—
548C2×C7#244312yes—
549C2×C7#247312yes—
550C2×C7#257911yescharacteristic
551C2×C7#262711yescharacteristic
552C2×C7#284811yescharacteristic
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23#643314yes—
570C23#644114yes—
571C23#644614yes—
572C23#644714yes—
573C23#645114yes—
574C23#645214yes—
575C23#645914yes—
576C23C22×Dic1411yescharacteristic
577C23C2×#115711yescharacteristic
578C23C2×#115711yescharacteristic
579C23C2×D4×D711yescharacteristic
580C23C2×D4×D712yes—
581C23C2×D4×D711yescharacteristic
582C23C2×#115911yescharacteristic
583C23C2×#115912yes—
584C23C2×Q8×D711yescharacteristic
585C23#655312yes—
586C23#655412yes—
587C2×C4—232no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—74no—
612C2×C4—142no—
613C2×C4—144no—
614C2×C4—144no—
615C2×C4—142no—
616C2×C4—144no—
617C2×C4—142no—
618C2×C4—142no—
619C2×C4—142no—
620C2×C4—144no—
621C2×C4—144no—
622C2×C4—142no—
623C2×C4—144no—
624C2×C4—144no—
625C2×C4—144no—
626C2×C4—144no—
627C2×C4—144no—
628C2×C4—142no—
629C2×C4—2816no—
630C2×C4—2816no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4—282no—
639C2×C4—281no—
640C2×C4—281no—
641C2×C4—282no—
642C2×C4—282no—
643C2×C4—281no—
644C2×C4—281no—
645C2×C4—282no—
646C2×C4—282no—
647C2×C4#644118yes—
648C7#1472811yescharacteristic, complemented, Sylow
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—28no—
654C22—24no—
655C22—28no—
656C22—28no—
657C22—24no—
658C22—28no—
659C22#8942918yescomplemented
660C22C2×#643311yescharacteristic
661C22C2×#644111yescharacteristic
662C22C2×#644112yes—
663C22C2×#644611yescharacteristic
664C22C2×#644711yescharacteristic
665C22C2×#645111yescharacteristic
666C22C2×#645211yescharacteristic
667C22C2×#645911yescharacteristic
668C22#9021012yes—
669C22#9023212yes—
670C22#9024112yes—
671C22#9024312yes—
672C22#9025612yes—
673C22#9027711yescharacteristic
674C22#9027712yes—
675C22D7×#12011yescharacteristic
676C22#9030012yes—
677C22#9030212yes—
678C22#9032312yes—
679C22#9032612yes—
680C22#9033911yescharacteristic
681C22#9034411yescharacteristic
682C22#9035612yes—
683C4—28no—
684C4—81no—
685C4—81no—
686C4—82no—
687C4—144no—
688C4—282no—
689C4—284no—
690C4—282no—
691C4—284no—
692C4—282no—
693C2—28no—
694C2C2×#8942911yescharacteristic
695C2#1173094012yes—
696C2#1173129111yescharacteristic
697C2#1173134412yes—
698C2#1173153511yescharacteristic
699C2#1173185011yescharacteristic
700C2#1173188612yes—
701C2#1173193011yescharacteristic
702C2#1173213611yescharacteristic
703C2#1173214912yes—
704C2#1173233211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC7×C22⋊#95C2
prime radicalC25×C7C23
hypercenterC22⋊#95D7
hypocenterC7#14728
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
5. C2×#89429C2semidirect
9. #11727174C2semidirect
10. #11727166C2semidirect
11. #11727165C2semidirect
412. C22×C7#398semidirect
648. C7#14728semidirect
659. C22#89429semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic72
C2×#897182
C2×#897142
C2×#894292
C7×C22⋊#952
#117298812
#117292982
#117271742
#117271662
#117271652
#147287