Finite Group Databasebeta

← #32026147

Subgroups — #32026147

Order 18176.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986042C211yesmaximal, characteristic, complemented
2C2×#17986023C211yesmaximal, characteristic, complemented
3C2×#17986000C211yesmaximal, characteristic
4C2×#17986000C211yesmaximal, characteristic
5C2×#17985897C211yesmaximal, characteristic, complemented
6#22589735C212yesmaximal, complemented
7#22589728C212yesmaximal, complemented
8#22589721C212yesmaximal, complemented
9#22589714C212yesmaximal, complemented
10#22588061C211yesmaximal, characteristic, complemented
11C71×#2297C211yesmaximal, characteristic
12C22×#11765849C2211yescharacteristic
13C22×#11765849C2211yescharacteristic
14C22×C4⋊Dic71C2211yescharacteristic
15C22×C4⋊Dic71C2211yescharacteristic
16C22×Dic71⋊C4—21no—
17#17986042—24no—
18C2×#14556061C2212yes—
19C2×#14556059C2212yes—
20#17986023—24no—
21C2×C4⋊(C71⋊C8)C2212yes—
22C2×C4⋊(C71⋊C8)C2212yes—
23#17986000—24no—
24#17986000—24no—
25C2×#14556052C2212yes—
26C2×#14556052C2212yes—
27C2×#14556049C2212yes—
28C2×#14556049C2212yes—
29#17985897—21no—
30#17985897—21no—
31#17985897C2212yes—
32#17985897C2212yes—
33C2×C71⋊#95—21no—
34#17985761C2212yes—
35#17985761C2212yes—
36C22×#41×C71C2211yescharacteristic
37C2×#117×C71C2211yescharacteristic
38C22×C42×C71C2211yescharacteristic
39#391×C71C2211yescharacteristic
40#391×C71C2211yescharacteristic
41#390×C71C2211yescharacteristic
42#390×C71C2211yescharacteristic
43#386×C71—21no—
44#386×C71—21no—
45C4×C22⋊C4×C71—21no—
46C4×C22⋊C4×C71—21no—
47C23×Dic71—21no—
48C23×Dic71—21no—
49C2×#11765849—22no—
50C2×#11765849—22no—
51C2×#11765849—22no—
52C2×#11765849—22no—
53C2×#11765849—22no—
54C2×#11765849D414yes—
55C22×C71⋊C8C2312yes—
56C22×C71⋊C8C2312yes—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—22no—
59C2×C4⋊Dic71—21no—
60C2×C4⋊Dic71—22no—
61C2×C4⋊Dic71—22no—
62C2×C4⋊Dic71D414yes—
63C2×C4⋊Dic71C2312yes—
64C2×C4⋊Dic71C2312yes—
65C2×Dic71⋊C4—22no—
66C2×Dic71⋊C4—22no—
67C2×Dic71⋊C4—44no—
68#14556061—24no—
69#14556059—24no—
70C4⋊(C71⋊C8)—24no—
71C4⋊(C71⋊C8)—24no—
72#14556052—24no—
73#14556052—24no—
74#14556049—24no—
75#14556049—24no—
76C71⋊#95—22no—
77C71⋊#95—22no—
78C2×#41×C71—22no—
79C2×#41×C71—22no—
80C2×#41×C71—22no—
81C2×#41×C71—22no—
82C2×#41×C71—22no—
83C2×#41×C71Q814yes—
84C22×Q8×C71C2311yescharacteristic
85C22×D4×C71—21no—
86C22×D4×C71C2311yescharacteristic
87C23×C4×C71—21no—
88C23×C4×C71D411yescharacteristic
89C23×C4×C71D411yescharacteristic
90C23×C4×C71C2311yescharacteristic
91#117×C71—22no—
92#117×C71—22no—
93C2×C4⋊C4×C71C2311yescharacteristic
94C2×C4⋊C4×C71C2311yescharacteristic
95C2×C22⋊C4×C71—21no—
96C2×C22⋊C4×C71—21no—
97C2×C22⋊C4×C71—21no—
98C2×C22⋊C4×C71—21no—
99C2×C22⋊C4×C71—21no—
100C2×C42×C71—22no—
101C2×C42×C71—22no—
102C2×C42×C71—21no—
103C2×C42×C71D411yescharacteristic
104C2×C42×C71D411yescharacteristic
105C2×C42×C71C2311yescharacteristic
106C2×C42×C71C2311yescharacteristic
107#95×C71—21no—
108#95×C71—21no—
109#95×C71—21no—
110#95×C71—21no—
111C22×Dic71—24no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—42no—
115C22×Dic71—42no—
116C22×Dic71—42no—
117C22×Dic71—42no—
118C22×Dic71—42no—
119#11765849—28no—
120#11765849—44no—
121C2×C71⋊C8—22no—
122C2×C71⋊C8—22no—
123C2×C71⋊C8—22no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8—22no—
126C2×C71⋊C8C2×D414yes—
127C4⋊Dic71—24no—
128C4⋊Dic71—22no—
129C4⋊Dic71—22no—
130C4⋊Dic71—22no—
131C4⋊Dic71—42no—
132C4⋊Dic71C2×D414yes—
133Dic71⋊C4—44no—
134Dic71⋊C4—44no—
135C24×C71—21no—
136C24×C71C2×D411yescharacteristic
137#41×C71—28no—
138#41×C71—28no—
139#41×C71—44no—
140#41×C71—44no—
141C2×Q8×C71—21no—
142C2×Q8×C71—21no—
143C2×Q8×C71—21no—
144C2×Q8×C71—21no—
145C2×Q8×C71—21no—
146C2×Q8×C71C2×Q812yes—
147C2×D4×C71—24no—
148C2×D4×C71—21no—
149C2×D4×C71—21no—
150C2×D4×C71—21no—
151C2×D4×C71—21no—
152C2×D4×C71—21no—
153C2×D4×C71—42no—
154C2×D4×C71—42no—
155C2×D4×C71C2×Q812yes—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—21no—
159C22×C4×C71—24no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—41no—
165C22×C4×C71—41no—
166C22×C4×C71—42no—
167C22×C4×C71—42no—
168C22×C4×C71C2×D412yes—
169C22×C4×C71C2×D412yes—
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×Q812yes—
176C22×C4×C71#4112yes—
177C22×C4×C71#4112yes—
178C22×C4×C71#4112yes—
179C22×C4×C71C2411yescharacteristic
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C22⋊C4×C71—42no—
185C22⋊C4×C71—42no—
186C22⋊C4×C71—42no—
187C22⋊C4×C71—42no—
188C22⋊C4×C71—42no—
189C42×C71—22no—
190C42×C71—22no—
191C42×C71—21no—
192C42×C71—21no—
193C42×C71—21no—
194C42×C71—21no—
195C42×C71—42no—
196C23×C71—21no—
197C23×C71—21no—
198C23×C71—21no—
199C23×C71—21no—
200C23×C71—22no—
201C23×C71—41no—
202C23×C71—41no—
203C23×C71—41no—
204C23×C71—41no—
205C23×C71—41no—
206C23×C71—41no—
207C23×C71#12112yes—
208C23×C71#12112yes—
209C23×C71C22⋊Q812yes—
210C23×C71C22×D411yescharacteristic
211Q8×C71—22no—
212Q8×C71—22no—
213Q8×C71—41no—
214Q8×C71—41no—
215D4×C71—22no—
216D4×C71—22no—
217D4×C71—48no—
218D4×C71—41no—
219D4×C71—41no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—22no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—44no—
232C2×C4×C71—44no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—48no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71#12112yes—
243C2×C4×C71#12112yes—
244C2×C4×C71#12114yes—
245C2×C4×C71C22⋊Q814yes—
246C2×C4×C71C22⋊Q814yes—
247C2×C4×C71C22⋊Q812yes—
248C2×C4×C71#12614yes—
249C2×C4×C71AGL(1,ℤ/8)12yes—
250C2×C4×C71#13712yes—
251C2×C4×C71C22×D411yescharacteristic
252C2×C4×C71C22×D411yescharacteristic
253C2×C4×C71C22×Q811yescharacteristic
254C2×C4×C71C2×#4111yescharacteristic
255C2×C4×C71C2×#4111yescharacteristic
256C2×C4×C71C2×#4111yescharacteristic
257C2×Dic71—24no—
258C2×Dic71—48no—
259C2×Dic71—42no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—42no—
264C2×Dic71—84no—
265C71⋊C8—24no—
266C71⋊C8—42no—
267C22×C71—22no—
268C22×C71—22no—
269C22×C71—22no—
270C22×C71—22no—
271C22×C71—22no—
272C22×C71—22no—
273C22×C71—42no—
274C22×C71—42no—
275C22×C71—42no—
276C22×C71—81no—
277C22×C71—81no—
278C22×C71—81no—
279C22×C71—81no—
280C22×C71#39614yes—
281C22×C71C2×#12111yescharacteristic
282C22×C71C2×#12111yescharacteristic
283C22×C71C2×C22⋊Q811yescharacteristic
284C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C71C2×#13711yescharacteristic
286C22×C71#57511yescharacteristic
287C22×C71#57511yescharacteristic
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—42no—
291C4×C71—41no—
292C4×C71—41no—
293C4×C71—81no—
294C4×C71—81no—
295C4×C71#39614yes—
296C4×C71C2×#12111yescharacteristic
297C4×C71C2×C22⋊Q811yescharacteristic
298C4×C71C2×C22⋊Q811yescharacteristic
299C4×C71C2×#12611yescharacteristic
300Dic71—44no—
301Dic71—82no—
302#23617—711nomaximal, Sylow
303C2×C71—24no—
304C2×C71—82no—
305C2×C71C2×#39611yescharacteristic
306C2×C71#303611yescharacteristic
307C2×C71#303711yescharacteristic
308C2×C71#305511yescharacteristic
309C2×C71#305611yescharacteristic
310C2×C71#307911yescharacteristic
311C2×C71#308011yescharacteristic
312C2×#428—711no—
313C2×#423—711no—
314C2×#417—711no—
315C2×#417—711no—
316#2297D7111yescharacteristic
317C2×#389—711no—
318#2068—712no—
319#2067—712no—
320#2066—712no—
321#2065—712no—
322#1736—711no—
323C71#2361711yescharacteristic, complemented, Sylow
324C22×#41C2×D7111yescharacteristic
325C22×M16—711no—
326C22×M16—711no—
327C2×#117C2×D7111yescharacteristic
328C22×C4⋊C4—711no—
329C22×C4⋊C4—711no—
330C22×C4⋊C4—1421no—
331C22×C42C2×D7111yescharacteristic
332#428—1424no—
333C2×#107—712no—
334C2×#106—712no—
335#423—1424no—
336C2×C4⋊C8—712no—
337C2×C4⋊C8—712no—
338#417—1424no—
339#417—1424no—
340C2×#103—712no—
341C2×#103—712no—
342C2×#102—712no—
343C2×#102—712no—
344#391C2×D7111yescharacteristic
345#391C2×D7111yescharacteristic
346#390C2×D7111yescharacteristic
347#390C2×D7111yescharacteristic
348#389—712no—
349#389—712no—
350#389—1421no—
351#389—1421no—
352#386—21no—
353#386—21no—
354C4×C22⋊C4—21no—
355C4×C22⋊C4—21no—
356C2×#95—1421no—
357#340—712no—
358#340—712no—
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41Dic14214yes—
365C22×Q8C22×D7111yescharacteristic
366C22×D4—21no—
367C22×D4C22×D7111yescharacteristic
368C23×C4—21no—
369C23×C4—1421no—
370C23×C4—1421no—
371C23×C4C22⋊D7111yescharacteristic
372C23×C4C22⋊D7111yescharacteristic
373C23×C4C22×D7111yescharacteristic
374C2×M16—714no—
375C2×M16—1422no—
376C2×M16—1422no—
377C2×M16—1422no—
378C2×M16—1422no—
379C2×M16—1422no—
380C22×C8—712no—
381C22×C8—712no—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—2844no—
385C2×C4⋊C4—714no—
386C2×C4⋊C4—712no—
387C2×C4⋊C4—712no—
388C2×C4⋊C4—1421no—
389C2×C4⋊C4—1422no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4—1422no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42C22⋊D7111yescharacteristic
406C2×C42C22⋊D7111yescharacteristic
407C2×C42C22×D7111yescharacteristic
408C2×C42C22×D7111yescharacteristic
409#107—1424no—
410#106—1424no—
411C4⋊C8—1424no—
412C4⋊C8—1424no—
413#103—1424no—
414#103—1424no—
415#102—1424no—
416#102—1424no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—1422no—
422#95—1422no—
423C24—21no—
424C24C2×C22⋊D7111yescharacteristic
425#41—28no—
426#41—28no—
427#41—44no—
428#41—44no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C2×Dic14212yes—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C2×Dic14212yes—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—2842no—
457C22×C4—2842no—
458C22×C4—2842no—
459C22×C4—2842no—
460C22×C4—2842no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1424no—
464C22×C4D4×D7112yes—
465C22×C4D4×D7112yes—
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4C2×C22⋊D7111yescharacteristic
468C22×C4C2×C22⋊D7111yescharacteristic
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C2×Dic14212yes—
472C22×C4#1176586412yes—
473C22×C4#1176586412yes—
474C22×C4#1176586512yes—
475C22×C4C23×D7111yescharacteristic
476M16—2844no—
477M16—1428no—
478C2×C8—714no—
479C2×C8—1422no—
480C2×C8—1422no—
481C2×C8—1422no—
482C2×C8—1422no—
483C2×C8—1422no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—2844no—
489C4⋊C4—2842no—
490C4⋊C4—2844no—
491C4⋊C4—714no—
492C4⋊C4—1422no—
493C4⋊C4—1422no—
494C4⋊C4—1422no—
495C4⋊C4—1424no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—42no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#1455609912yes—
520C23#1455610012yes—
521C23#1455610712yes—
522C23C22×C22⋊D7111yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527D4—22no—
528D4—22no—
529D4—48no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—48no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—2848no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—2842no—
559C2×C4—2842no—
560C2×C4—5684no—
561C2×C4—1424no—
562C2×C4#1455609912yes—
563C2×C4#1455610012yes—
564C2×C4#1455610114yes—
565C2×C4#1455610514yes—
566C2×C4#1455610514yes—
567C2×C4#1455610712yes—
568C2×C4#1455612114yes—
569C2×C4#1455615312yes—
570C2×C4#1455615812yes—
571C2×C4C2×D4×D7111yescharacteristic
572C2×C4C2×D4×D7111yescharacteristic
573C2×C4C22×Dic14211yescharacteristic
574C2×C4C2×#1176586411yescharacteristic
575C2×C4C2×#1176586411yescharacteristic
576C2×C4C2×#1176586511yescharacteristic
577C8—2842no—
578C8—1424no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22#1798592214yes—
593C22C2×#1455609911yescharacteristic
594C22C2×#1455610011yescharacteristic
595C22C2×#1455610711yescharacteristic
596C22C2×#1455615311yescharacteristic
597C22C2×#1455615811yescharacteristic
598C22#1798679711yescharacteristic
599C22#1798679711yescharacteristic
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—2844no—
608C4—5682no—
609C4#1798592214yes—
610C4C2×#1455610111yescharacteristic
611C4C2×#1455610511yescharacteristic
612C4C2×#1455610511yescharacteristic
613C4C2×#1455612111yescharacteristic
614C2—24no—
615C2—82no—
616C2C2×#1798592211yescharacteristic
617C2#2259780011yescharacteristic
618C2#2259780811yescharacteristic
619C2#2259796111yescharacteristic
620C2#2259797211yescharacteristic
621C2#2259815511yescharacteristic
622C2#2259816211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2297C2
socleC23×C71C22×D4
prime radicalC22×#41×C71C22
hypercenter#2297D71
hypocenterC71#23617
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. C2×#17986042C2semidirect
2. C2×#17986023C2semidirect
5. C2×#17985897C2semidirect
6. #22589735C2semidirect
7. #22589728C2semidirect
8. #22589721C2semidirect
9. #22589714C2semidirect
10. #22588061C2semidirect
323. C71#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#179860422
C2×#179860232
C2×#179860002
C2×#179858972
#225897352
#225897282
#225897212
#225897142
#225880612
C71×#22972
#2361771