Finite Group Databasebeta

← #13276321

Subgroups — #13276321

Order 1792.

1862 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6489C211yesmaximal, characteristic
2C2×#89423C211yesmaximal, characteristic, complemented
3C2×#89421C211yesmaximal, characteristic, complemented
4C7×C22⋊#95C211yesmaximal, characteristic
5#11729881C211yesmaximal, characteristic
6#11729301C212yesmaximal
7#11729299C212yesmaximal
8#11727165C212yesmaximal, complemented
9#11727161C212yesmaximal, complemented
10#11727156C211yesmaximal, characteristic, complemented
11#11727156C211yesmaximal, characteristic, complemented
12C24×C4×C7C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C22⋊Dic7C2211yescharacteristic
17C2×#6489—24no—
18C2×#6489—24no—
19C2×#6489C418yescomplemented
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2211yescharacteristic
22C22×Dic7⋊C4C2212yes—
23C22×C4×Dic7C2211yescharacteristic
24C22×C4×Dic7C2212yes—
25C22×C4×Dic7C2211yescharacteristic
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89423—24no—
39#89421—24no—
40#13938—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7—24no—
43C23×C4×C7—22no—
44C23×C4×C7—22no—
45C23×C4×C7—24no—
46C23×C4×C7C2×C414yes—
47C23×C4×C7C2312yes—
48C23×C4×C7C2312yes—
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×Dic7C2312yes—
52C23×Dic7C2311yescharacteristic
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2312yes—
55C23×Dic7C2311yescharacteristic
56C23×Dic7C2311yescharacteristic
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—42no—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—22no—
64C2×C22⋊Dic7—24no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—42no—
67C2×C22⋊Dic7C2×C414yes—
68C2×C22⋊Dic7C2×C414yes—
69#6489—232no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7C2×C414yes—
75C2×C4⋊Dic7C2×C414yes—
76C2×Dic7⋊C4—24no—
77C2×Dic7⋊C4—24no—
78C2×Dic7⋊C4—24no—
79C2×C4×Dic7—22no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7—42no—
83C2×C4×Dic7—42no—
84C2×C4×Dic7C2×C414yes—
85C2×C4×Dic7C2×C414yes—
86C2×C4×Dic7D414yes—
87C2×C4×Dic7D414yes—
88C7×#95—22no—
89C7×#95—22no—
90C7×#95—24no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95—24no—
96C7⋊#95—24no—
97C7⋊#95—24no—
98C7⋊#95—42no—
99C7⋊#95—42no—
100C22×#117—71no—
101C2×#383—71no—
102C2×C4×C4⋊C4—71no—
103#1441—72no—
104#1439—72no—
105#1437—72no—
106#1436—72no—
107#1434—71no—
108#1434—71no—
109C22⋊#95—71no—
110C22⋊#95D711yescharacteristic
111C24×C7—21no—
112C24×C7—21no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—21no—
117C24×C7—21no—
118C24×C7—21no—
119C24×C7—21no—
120C24×C7C22×C412yes—
121C24×C7#4114yes—
122C24×C7C2411yescharacteristic
123C22×C4×C7—216no—
124C22×C4×C7—216no—
125C22×C4×C7—28no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—28no—
132C22×C4×C7—28no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7C22×C412yes—
147C22×C4×C7C22×C412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D412yes—
150C22×C4×C7#4114yes—
151C22×C4×C7#4114yes—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—22no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—22no—
180C22×Dic7—22no—
181C22×Dic7C22×C412yes—
182C22×Dic7C22×C412yes—
183C22×Dic7C22×C412yes—
184C22×Dic7C22×C412yes—
185C22×Dic7C2×D412yes—
186C22×Dic7C2×D412yes—
187C22×Dic7C2×D412yes—
188C22×Dic7C2×D412yes—
189C22×Dic7#4114yes—
190C7×C22⋊C4—48no—
191C22⋊Dic7—28no—
192C22⋊Dic7—28no—
193C22⋊Dic7—48no—
194C4⋊Dic7—28no—
195C4⋊Dic7—28no—
196Dic7⋊C4—216no—
197C4×Dic7—28no—
198C4×Dic7—48no—
199C4×Dic7C22⋊C4116yescomplemented
200C24×C4C2×D711yescharacteristic
201C2×#117—78no—
202C2×#117—144no—
203C2×#117—144no—
204C22×C4⋊C4—71no—
205C22×C4⋊C4—71no—
206C22×C4⋊C4—72no—
207C22×C22⋊C4—71no—
208C22×C22⋊C4—71no—
209C22×C22⋊C4—72no—
210C22×C22⋊C4C2×D712yes—
211C22×C42—71no—
212C22×C42—71no—
213C22×C42—72no—
214#383—144no—
215C4×C4⋊C4—144no—
216C2×#95—71no—
217C2×#95—71no—
218C2×#95—72no—
219C2×#95—72no—
220C2×#95—72no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—72no—
224C2×#95—72no—
225C2×#95C2×D711yescharacteristic
226C2×#95C2×D711yescharacteristic
227C2×#95C2×D712yes—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—24no—
235C23×C7—24no—
236C23×C7—24no—
237C23×C7—24no—
238C23×C7—24no—
239C23×C7—24no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7#11714yes—
251C23×C7C23×C411yescharacteristic
252C23×C7C22×D411yescharacteristic
253C23×C7C22×D411yescharacteristic
254C23×C7C2×#4112yes—
255C23×C7C2×#4112yes—
256C23×C7C2×#4112yes—
257C23×C7C2×#4112yes—
258C23×C7H(5,𝔽2)11yescharacteristic
259C23×C7H(5,𝔽2)11yescharacteristic
260C23×C72−1+411yescharacteristic
261C23×C72−1+411yescharacteristic
262C2×C4×C7—232no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7C2×C22⋊C414yes—
286C2×C4×C7#11714yes—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—24no—
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7C2×C22⋊C414yes—
324C2×Dic7C2×C22⋊C414yes—
325C25C22×D711yescharacteristic
326C23×C4—24no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—24no—
330C23×C4—71no—
331C23×C4—71no—
332C23×C4—71no—
333C23×C4—71no—
334C23×C4—72no—
335C23×C4—72no—
336C23×C4C4×D714yes—
337C23×C4C22×D712yes—
338C23×C4C22×D712yes—
339C23×C4C22×D711yescharacteristic
340C23×C4C22×D711yescharacteristic
341#117—1432no—
342C2×C4⋊C4—74no—
343C2×C4⋊C4—74no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—144no—
350C2×C4⋊C4—144no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—74no—
355C2×C22⋊C4—74no—
356C2×C22⋊C4—142no—
357C2×C22⋊C4—142no—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—142no—
360C2×C22⋊C4—144no—
361C2×C22⋊C4—144no—
362C2×C22⋊C4—282no—
363C2×C42—74no—
364C2×C42—74no—
365C2×C42—74no—
366C2×C42—74no—
367C2×C42—142no—
368C2×C42—142no—
369C2×C42—144no—
370C2×C42—282no—
371C2×C42—282no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—144no—
376#95—144no—
377#95—144no—
378#95—144no—
379#95—142no—
380#95—142no—
381#95—144no—
382#95—282no—
383#95—282no—
384C22×C7—24no—
385C22×C7—24no—
386C22×C7—24no—
387C22×C7—42no—
388C22×C7—44no—
389C22×C7—44no—
390C22×C7—44no—
391C22×C7—44no—
392C22×C7—42no—
393C22×C7—42no—
394C22×C7—42no—
395C22×C7C22×C22⋊C411yescharacteristic
396C22×C7C2×#11711yescharacteristic
397C22×C7C2×#11711yescharacteristic
398C22×C7C4×#4111yescharacteristic
399C22×C7C4×#4111yescharacteristic
400C22×C7#52011yescharacteristic
401C22×C7#52011yescharacteristic
402C22×C7#52512yes—
403C22×C7#52512yes—
404C22×C7#52512yes—
405C22×C7#53212yes—
406C22×C7#53511yescharacteristic
407C22×C7#53611yescharacteristic
408C22×C7#53711yescharacteristic
409C22×C7#53711yescharacteristic
410C22×C7#55112yes—
411C22×C7#55112yes—
412C22×C7#55212yes—
413C22×C7#55212yes—
414C22×C7#55312yes—
415C22×C7#55312yes—
416C22×C7#55612yes—
417C22×C7#55612yes—
418C4×C7—24no—
419C4×C7—24no—
420C4×C7—44no—
421C4×C7—82no—
422Dic7—24no—
423Dic7—24no—
424Dic7—42no—
425Dic7—44no—
426Dic7—42no—
427Dic7—82no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24—21no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24C2×C4×D712yes—
438C24#115914yes—
439C24C23×D711yescharacteristic
440C22×C4—216no—
441C22×C4—216no—
442C22×C4—28no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—72no—
464C22×C4—72no—
465C22×C4—72no—
466C22×C4—72no—
467C22×C4—72no—
468C22×C4—72no—
469C22×C4—72no—
470C22×C4—72no—
471C22×C4—74no—
472C22×C4—142no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—142no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—141no—
488C22×C4—142no—
489C22×C4—141no—
490C22×C4—142no—
491C22×C4—141no—
492C22×C4—142no—
493C22×C4—141no—
494C22×C4—142no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—142no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4C2×C4×D712yes—
502C22×C4C2×C4×D712yes—
503C22×C4#115714yes—
504C22×C4D4×D712yes—
505C22×C4D4×D712yes—
506C22×C4#115914yes—
507C4⋊C4—148no—
508C4⋊C4—148no—
509C4⋊C4—1416no—
510C22⋊C4—48no—
511C22⋊C4—148no—
512C22⋊C4—148no—
513C22⋊C4—288no—
514C42—716no—
515C42—148no—
516C42—288no—
517C2×C7—44no—
518C2×C7#228011yescharacteristic
519C2×C7#229711yescharacteristic
520C2×C7#231111yescharacteristic
521C2×C7#234311yescharacteristic
522C2×C7#234311yescharacteristic
523C2×C7#234711yescharacteristic
524C2×C7#234711yescharacteristic
525C2×C7#259812yes—
526C2×C7#259912yes—
527C2×C7#260112yes—
528C2×C7#260112yes—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23#644014yes—
552C23C22×C4×D711yescharacteristic
553C23C2×#115712yes—
554C23C2×D4×D711yescharacteristic
555C23C2×D4×D711yescharacteristic
556C23C2×#115912yes—
557C23C2×#115912yes—
558C23#654811yescharacteristic
559C23#655111yescharacteristic
560C23D7×#4112yes—
561C23#655311yescharacteristic
562C23#655411yescharacteristic
563C2×C4—232no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—74no—
587C2×C4—74no—
588C2×C4—142no—
589C2×C4—142no—
590C2×C4—142no—
591C2×C4—142no—
592C2×C4—144no—
593C2×C4—144no—
594C2×C4—142no—
595C2×C4—142no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—142no—
601C2×C4—142no—
602C2×C4—142no—
603C2×C4—142no—
604C2×C4—144no—
605C2×C4—144no—
606C2×C4—144no—
607C2×C4—282no—
608C2×C4—282no—
609C2×C4—282no—
610C2×C4—282no—
611C2×C4—282no—
612C2×C4—282no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4—282no—
618C2×C4—282no—
619C2×C4—282no—
620C2×C4—282no—
621C2×C4—282no—
622C2×C4—282no—
623C2×C4—282no—
624C2×C4#644014yes—
625C2×C4D7×C22⋊C414yes—
626C7#1393811yescharacteristic, complemented, Sylow
627C22—24no—
628C22—24no—
629C22—24no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22C4×#115711yescharacteristic
639C22C2×#644011yescharacteristic
640C22C2×#644011yescharacteristic
641C22C2×D7×C22⋊C411yescharacteristic
642C22#9018612yes—
643C22#9019511yescharacteristic
644C22#9020412yes—
645C22#9021311yescharacteristic
646C22#9021411yescharacteristic
647C22#9023312yes—
648C22#9023812yes—
649C22#9023912yes—
650C22#9025612yes—
651C22#9027712yes—
652C22#9027712yes—
653C22#9029811yescharacteristic
654C22#9029911yescharacteristic
655C22#9030512yes—
656C22#9031911yescharacteristic
657C22#9032011yescharacteristic
658C22#9033212yes—
659C22#9034112yes—
660C22#9034912yes—
661C4—24no—
662C4—24no—
663C4—44no—
664C4—82no—
665C4—144no—
666C4—144no—
667C4—282no—
668C4—282no—
669C4—284no—
670C4—562no—
671C2—44no—
672C2#1173078011yescharacteristic
673C2#1173090511yescharacteristic
674C2#1173100811yescharacteristic
675C2#1173126211yescharacteristic
676C2#1173132711yescharacteristic
677C2#1173144811yescharacteristic
678C2#1173150411yescharacteristic
679C2#1173194212yes—
680C2#1173194312yes—
681C2#1173222012yes—
682C2#1173222212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C22⋊#95C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC22⋊#95D7
hypocenterC7#13938
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#89423C2semidirect
3. C2×#89421C2semidirect
8. #11727165C2semidirect
9. #11727161C2semidirect
10. #11727156C2semidirect
11. #11727156C2semidirect
19. C2×#6489C4semidirect
199. C4×Dic7C22⋊C4semidirect
626. C7#13938semidirect

Maximal subgroups

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

subgroupindex
C22×#64892
C2×#894232
C2×#894212
C7×C22⋊#952
#117298812
#117293012
#117292992
#117271652
#117271612
#117271562
#139387