Finite Group Databasebeta

← #13380645

Subgroups — #13380645

Order 1792.

2170 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#6505C211yesmaximal, characteristic, complemented
3C2×#90482C211yesmaximal, characteristic
4C2×#89997C211yesmaximal, characteristic, complemented
5C2×#89979C211yesmaximal, characteristic, complemented
6C2×#89738C211yesmaximal, characteristic, complemented
7#11729454C214yesmaximal, complemented
8#11729453C214yesmaximal, complemented
9#11729312C211yesmaximal, characteristic, complemented
10C23×C7×Q8C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C22×#1157C2211yescharacteristic
13C2×C7×C22⋊Q8—24no—
14C2×C7×C22⋊Q8—22no—
15C2×C7×C22⋊Q8—24no—
16C2×C7×C22⋊Q8C2214yes—
17C22×C7×C4⋊C4—21no—
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4—21no—
20C2×#6516—21no—
21C22×#1144C2212yescomplemented
22C2×#6505—24no—
23C2×#6505—24no—
24C2×#6505—24no—
25C22×#1143C2212yescomplemented
26#90482—24no—
27#90482—24no—
28C2×#6493C2212yes—
29C2×#6492—21no—
30C2×C4×C22⋊D7—22no—
31C2×#6489C2211yescharacteristic
32C2×#6487C2212yes—
33C22×#1136C2211yescharacteristic
34C22×#1136C2211yescharacteristic
35#89997—24no—
36C2×#6409C2214yescomplemented
37#89979—24no—
38C2×C22⋊(C7⋊C8)C2214yes—
39C2×#6384C2212yescomplemented
40#89738—24no—
41C2×#6383C2212yescomplemented
42#89324C2214yescomplemented
43#23392—71nomaximal, Sylow
44C25×C7D411yescharacteristic
45C22×C7×Q8—22no—
46C22×C7×Q8—21no—
47C22×C7×Q8—22no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8—22no—
51C22×C7×Q8D414yescomplemented
52C22×C7×Q8C2312yes—
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7—22no—
56C23×C4×C7—22no—
57C23×C4×C7—22no—
58C23×C4×C7—22no—
59C23×C4×C7—22no—
60C23×C4×C7—22no—
61C23×C4×C7D412yes—
62C23×C4×C7D411yescharacteristic
63C23×C4×C7C2312yes—
64C23×C4×C7C2311yescharacteristic
65C22×C22⋊D7—22no—
66C22×C22⋊D7—21no—
67C2×#1157—22no—
68C2×#1157—22no—
69C2×#1157—22no—
70C2×#1157—22no—
71C2×#1157—22no—
72C2×#1157D414yes—
73C22×D28C2311yescharacteristic
74C22×C4×D7—21no—
75C22×Dic14C2311yescharacteristic
76C7×C22⋊Q8—216no—
77C7×C22⋊Q8—28no—
78C7×C22⋊Q8—28no—
79C2×C7×C4⋊C4—28no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4D414yescomplemented
84C2×C7×C4⋊C4C2312yes—
85C2×C7×C22⋊C4—28no—
86C2×C7×C22⋊C4—24no—
87#6516—28no—
88C2×C22⋊Dic7—22no—
89C2×C22⋊Dic7—21no—
90C2×#1144—24no—
91C2×#1144—24no—
92C2×#1144—24no—
93#6505—232no—
94C2×#1143—24no—
95C2×#1143—24no—
96C2×#1143—24no—
97#6493—24no—
98#6493—24no—
99#6492—28no—
100C4×C22⋊D7—216no—
101C2×(C2×D7)⋊C4—22no—
102#6489—22no—
103#6489D414yes—
104C2×C4⋊Dic7C2312yes—
105#6487—24no—
106#6487—24no—
107C2×Dic7⋊C4—22no—
108C2×C4×Dic7—21no—
109C2×#1136—22no—
110C2×#1136—22no—
111C2×#1136—22no—
112C2×#1136—22no—
113C2×#1136—22no—
114C2×#1136D414yescomplemented
115C22×C7⋊C8C2312yes—
116C22×C7⋊C8C2312yes—
117#6409—28no—
118C22⋊(C7⋊C8)—28no—
119#6384—24no—
120#6383—24no—
121C22×#137—71no—
122C2×#525—71no—
123C22×C22⋊Q8D711yescharacteristic, complemented
124C2×#417—71no—
125C2×#419—71no—
126C2×#407—71no—
127#2001—74no—
128#2000—74no—
129#1736—71no—
130C24×C7—22no—
131C24×C7—24no—
132C24×C7—22no—
133C24×C7—22no—
134C24×C7—42no—
135C24×C7C2×D412yes—
136C24×C7C2×D411yescharacteristic
137C23×D7—21no—
138C2×C7×Q8—24no—
139C2×C7×Q8—22no—
140C2×C7×Q8—28no—
141C2×C7×Q8—28no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×Q8—22no—
145C2×C7×Q8—22no—
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8—44no—
149C2×C7×Q8—44no—
150C2×C7×Q8C2×D414yes—
151C22×C4×C7—22no—
152C22×C4×C7—24no—
153C22×C4×C7—24no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—24no—
159C22×C4×C7—24no—
160C22×C4×C7—24no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—44no—
168C22×C4×C7—42no—
169C22×C4×C7—42no—
170C22×C4×C7—44no—
171C22×C4×C7—44no—
172C22×C4×C7—44no—
173C22×C4×C7—44no—
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D412yes—
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D412yes—
181C22×C4×C7#4112yes—
182C22×C4×C7C2411yescharacteristic
183C2×C22⋊D7—28no—
184C2×C22⋊D7—24no—
185C2×C22⋊D7—44no—
186C2×C22⋊D7—42no—
187C2×C22⋊D7—42no—
188C2×C22⋊D7—44no—
189C22×Dic7—22no—
190C22×Dic7—21no—
191#1157—28no—
192#1157—28no—
193#1157—44no—
194#1157—44no—
195C2×D28—21no—
196C2×D28—21no—
197C2×D28—21no—
198C2×D28—21no—
199C2×D28—21no—
200C2×D28C2×D412yes—
201C2×C4×D7—24no—
202C2×C4×D7—42no—
203C2×C4×D7—42no—
204C2×Dic14—21no—
205C2×Dic14—21no—
206C2×Dic14—21no—
207C2×Dic14—21no—
208C2×Dic14—21no—
209C2×Dic14C2×D412yes—
210C7×C4⋊C4—216no—
211C7×C4⋊C4—24no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4C2×D414yes—
214C7×C22⋊C4—216no—
215C22⋊Dic7—28no—
216C22⋊Dic7—24no—
217#1144—216no—
218#1143—216no—
219(C2×D7)⋊C4—28no—
220C4⋊Dic7—22no—
221C4⋊Dic7C2×D414yes—
222Dic7⋊C4—28no—
223C4×Dic7—24no—
224#1136—28no—
225#1136—44no—
226C2×C7⋊C8—22no—
227C2×C7⋊C8—22no—
228C2×C7⋊C8—22no—
229C2×C7⋊C8—22no—
230C2×C7⋊C8—22no—
231C2×C7⋊C8C2×D414yes—
232C22×#41—71no—
233C23×Q8C2×D711yescharacteristic
234C24×C4C2×D711yescharacteristic
235C2×#137—144no—
236C2×#137—144no—
237C2×#137—144no—
238C22×Q16—72no—
239C22×SD16—72no—
240C22×M16—71no—
241C22×M16—71no—
242#525—144no—
243#525—144no—
244C2×#123—141no—
245C2×C22⋊Q8—24no—
246C2×C22⋊Q8—22no—
247C2×C22⋊Q8—24no—
248C2×C22⋊Q8—72no—
249C2×C22⋊Q8C2×D714yes—
250C2×#121—72no—
251C2×#120—141no—
252C2×C4×D4—142no—
253C2×#117—71no—
254C22×C4⋊C4—21no—
255C22×C4⋊C4C2×D711yescharacteristic
256C22×C22⋊C4—21no—
257#419—144no—
258#417—144no—
259C2×#103—74no—
260C2×#103—72no—
261C2×#102—72no—
262#407—144no—
263C2×C22⋊C8—74no—
264#340—74no—
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—28no—
268C23×C7—22no—
269C23×C7—22no—
270C23×C7—22no—
271C23×C7—22no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—48no—
275C23×C7—48no—
276C23×C7—44no—
277C23×C7—44no—
278C23×C7#12012yes—
279C23×C7#12012yes—
280C23×C7#12112yes—
281C23×C7#13714yescomplemented
282C23×C7C22×D411yescharacteristic
283C22×D7—22no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—42no—
287C22×D7—42no—
288C7×Q8—24no—
289C7×Q8—24no—
290C7×Q8—42no—
291C7×Q8—42no—
292C7×Q8—48no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—24no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—48no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—44no—
302C2×C4×C7—44no—
303C2×C4×C7—48no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—48no—
309C2×C4×C7#12014yes—
310C2×C4×C7#12012yes—
311C2×C4×C7#12012yes—
312C2×C4×C7#12114yes—
313C2×C4×C7#12114yes—
314C2×C4×C7#12112yes—
315C2×C4×C7#12714yes—
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C22×D411yescharacteristic
318C2×C4×C7C22×D411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C22×D411yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C22⋊D7—416no—
323C22⋊D7—48no—
324C2×Dic7—24no—
325C2×Dic7—22no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330D28—22no—
331D28—22no—
332D28—41no—
333D28—41no—
334C4×D7—48no—
335Dic14—22no—
336Dic14—22no—
337Dic14—41no—
338Dic14—41no—
339C7⋊C8—24no—
340C7⋊C8—42no—
341C25C22⋊D711yescharacteristic
342C2×#41—74no—
343C2×#41—142no—
344C2×#41—142no—
345C2×#41—142no—
346C2×#41—142no—
347C2×#41—142no—
348C22×Q8—22no—
349C22×Q8—21no—
350C22×Q8—22no—
351C22×Q8—22no—
352C22×Q8—22no—
353C22×Q8—22no—
354C22×Q8—71no—
355C22×Q8C22⋊D714yescomplemented
356C22×Q8C22×D712yes—
357C22×D4—71no—
358C22×D4—142no—
359C22×D4—141no—
360C23×C4—21no—
361C23×C4—21no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—141no—
369C23×C4C22⋊D711yescharacteristic
370C23×C4C22⋊D712yes—
371C23×C4C22×D712yes—
372C23×C4C22×D711yescharacteristic
373#137—1432no—
374C2×Q16—144no—
375C2×Q16—144no—
376C2×Q16—144no—
377C2×SD16—144no—
378C2×SD16—144no—
379C2×SD16—144no—
380C2×M16—74no—
381C2×M16—142no—
382C2×M16—142no—
383C2×M16—142no—
384C2×M16—142no—
385C2×M16—142no—
386C22×C8—72no—
387C22×C8—72no—
388#123—148no—
389C22⋊Q8—216no—
390C22⋊Q8—28no—
391C22⋊Q8—28no—
392C22⋊Q8—144no—
393C22⋊Q8—144no—
394#121—144no—
395#121—144no—
396#120—148no—
397C4×D4—1416no—
398#117—74no—
399#117—142no—
400C2×C4⋊C4—28no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—72no—
405C2×C4⋊C4—142no—
406C2×C4⋊C4C22⋊D714yescomplemented
407C2×C4⋊C4C22×D712yes—
408C2×C22⋊C4—28no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—141no—
411C2×C22⋊C4—142no—
412C2×C22⋊C4—142no—
413C2×C42—141no—
414#103—148no—
415#103—144no—
416#102—144no—
417C22⋊C8—148no—
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—24no—
421C22×C7—24no—
422C22×C7—24no—
423C22×C7—24no—
424C22×C7—24no—
425C22×C7—416no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7#39214yes—
431C22×C7C2×#12011yescharacteristic
432C22×C7C2×#12011yescharacteristic
433C22×C7C2×#12111yescharacteristic
434C22×C7C2×#13712yes—
435C22×C7#57512yes—
436C2×D7—42no—
437C2×D7—44no—
438C2×D7—81no—
439C2×D7—81no—
440C2×D7—82no—
441C4×C7—42no—
442C4×C7—44no—
443C4×C7—44no—
444C4×C7#39214yes—
445C4×C7C2×#12011yescharacteristic
446C4×C7C2×#12111yescharacteristic
447C4×C7C2×#12111yescharacteristic
448C4×C7C2×#12711yescharacteristic
449Dic7—42no—
450Dic7—44no—
451C24—24no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—42no—
456C24—141no—
457C24C2×C22⋊D712yes—
458C24C2×C22⋊D711yescharacteristic
459#41—148no—
460#41—148no—
461#41—284no—
462#41—284no—
463C2×Q8—24no—
464C2×Q8—22no—
465C2×Q8—28no—
466C2×Q8—28no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—42no—
474C2×Q8—44no—
475C2×Q8—72no—
476C2×Q8—141no—
477C2×Q8—141no—
478C2×Q8—141no—
479C2×Q8—141no—
480C2×Q8—141no—
481C2×Q8C2×C22⋊D714yes—
482C2×D4—72no—
483C2×D4—148no—
484C2×D4—144no—
485C2×D4—141no—
486C2×D4—141no—
487C2×D4—141no—
488C2×D4—141no—
489C2×D4—141no—
490C2×D4—284no—
491C2×D4—282no—
492C2×D4—284no—
493C2×D4—282no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—44no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—142no—
518C22×C4—141no—
519C22×C4—144no—
520C22×C4—282no—
521C22×C4—282no—
522C22×C4D4×D712yes—
523C22×C4D4×D712yes—
524C22×C4D4×D712yes—
525C22×C4#115912yes—
526C22×C4C2×C22⋊D712yes—
527C22×C4C2×C22⋊D711yescharacteristic
528C22×C4C2×C22⋊D712yes—
529C22×C4C2×C22⋊D712yes—
530C22×C4C23×D711yescharacteristic
531Q16—1416no—
532SD16—1416no—
533M16—148no—
534M16—284no—
535C2×C8—74no—
536C2×C8—142no—
537C2×C8—142no—
538C2×C8—142no—
539C2×C8—142no—
540C2×C8—142no—
541C4⋊C4—216no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—74no—
545C4⋊C4—148no—
546C4⋊C4—142no—
547C4⋊C4C2×C22⋊D714yes—
548C22⋊C4—216no—
549C22⋊C4—148no—
550C22⋊C4—148no—
551C22⋊C4—144no—
552C42—144no—
553C2×C7—24no—
554C2×C7—44no—
555C2×C7C2×#39211yescharacteristic
556C2×C7#300411yescharacteristic
557C2×C7#300511yescharacteristic
558C2×C7#300712yes—
559C2×C7#305612yes—
560D7—82no—
561C23—22no—
562C23—22no—
563C23—28no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—48no—
571C23—48no—
572C23—44no—
573C23—44no—
574C23—142no—
575C23—281no—
576C23—282no—
577C23—282no—
578C23—281no—
579C23#650012yes—
580C23#650212yes—
581C23#650514yescomplemented
582C23#651612yes—
583C23C22×C22⋊D711yescharacteristic
584Q8—24no—
585Q8—24no—
586Q8—42no—
587Q8—42no—
588Q8—48no—
589Q8—142no—
590Q8—142no—
591Q8—281no—
592Q8—281no—
593D4—142no—
594D4—142no—
595D4—2816no—
596D4—288no—
597D4—281no—
598D4—281no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—144no—
616C2×C4—142no—
617C2×C4—281no—
618C2×C4—282no—
619C2×C4—282no—
620C2×C4—281no—
621C2×C4—288no—
622C2×C4#650014yes—
623C2×C4#650012yes—
624C2×C4#650114yes—
625C2×C4#650214yes—
626C2×C4#650212yes—
627C2×C4#650314yes—
628C2×C4#651612yes—
629C2×C4C2×D4×D711yescharacteristic
630C2×C4C2×D4×D711yescharacteristic
631C2×C4C2×D4×D711yescharacteristic
632C2×C4C2×#115911yescharacteristic
633C2×C4C22×C22⋊D711yescharacteristic
634C2×C4C22×C22⋊D711yescharacteristic
635C8—144no—
636C8—282no—
637C7#2339211yescharacteristic, complemented, Sylow
638C22—24no—
639C22—24no—
640C22—22no—
641C22—22no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—284no—
651C22—282no—
652C22—562no—
653C22—561no—
654C22—561no—
655C22#8999514yes—
656C22C2×#650011yescharacteristic
657C22C2×#650211yescharacteristic
658C22C2×#650512yes—
659C22#9051312yes—
660C22C2×#651611yescharacteristic
661C4—42no—
662C4—44no—
663C4—44no—
664C4—284no—
665C4—282no—
666C4#8999514yes—
667C4C2×#650011yescharacteristic
668C4C2×#650111yescharacteristic
669C4C2×#650211yescharacteristic
670C4C2×#650311yescharacteristic
671C2—24no—
672C2—44no—
673C2—562no—
674C2#1174017712yes—
675C2#1174018012yes—
676C2C2×#8999511yescharacteristic
677C2#1174150411yescharacteristic
678C2#1174152611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×C22⋊Q8C2
socleC23×C7C22×D4
prime radicalC2×#90482C2
hypercenterC22×C22⋊Q8D7
hypocenterC7#23392
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊Q8C2semidirect
2. C22×#6505C2semidirect
4. C2×#89997C2semidirect
5. C2×#89979C2semidirect
6. C2×#89738C2semidirect
7. #11729454C2semidirect
8. #11729453C2semidirect
9. #11729312C2semidirect
21. C22×#1144C22semidirect
25. C22×#1143C22semidirect
36. C2×#6409C22semidirect
39. C2×#6384C22semidirect
41. C2×#6383C22semidirect
42. #89324C22semidirect
51. C22×C7×Q8D4semidirect
83. C2×C7×C4⋊C4D4semidirect
114. C2×#1136D4semidirect
123. C22×C22⋊Q8D7semidirect
281. C23×C7#137semidirect
355. C22×Q8C22⋊D7semidirect
406. C2×C4⋊C4C22⋊D7semidirect
581. C23#6505semidirect
637. C7#23392semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C22×#65052
C2×#904822
C2×#899972
C2×#899792
C2×#897382
#117294542
#117294532
#117293122
#233927