Finite Group Databasebeta

← #32038118

Subgroups — #32038118

Order 18176.

960 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986204C211yesmaximal, characteristic, complemented
2C2×#17986056C211yesmaximal, characteristic, complemented
3#22596692C211yesmaximal, characteristic, complemented
4#22589543C212yesmaximal, complemented
5#22589343C212yesmaximal
6#22589263C211yesmaximal, characteristic, complemented
7#22589220C212yesmaximal, complemented
8#22589098C211yesmaximal, characteristic, complemented
9#22589029C212yesmaximal, complemented
10#22588340C211yesmaximal, characteristic, complemented
11C4×#121×C71C211yesmaximal, characteristic
12C22×#11765853C2212yes—
13C2×#14556128C2211yescharacteristic
14C2×#14556107C2212yes—
15#17986204—24no—
16#17986056—24no—
17C2×#14556061C2212yes—
18C2×C4⋊(C71⋊C8)C2211yescharacteristic
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×#14556051C2212yes—
21C2×#14556051C2212yes—
22C2×#14556051C2212yes—
23C2×#14556049C2212yes—
24C2×C22⋊(C71⋊C8)C2212yes—
25C2×#14556039C2212yes—
26#17985905C2212yes—
27#17985897—21no—
28#17985897—21no—
29#17985869C2211yescharacteristic
30#17985859—21no—
31#17985859—21no—
32#17985761C2212yes—
33#17985761C2212yes—
34C2×#121×C71C2211yescharacteristic
35C2×C4×D4×C71—21no—
36C2×C4×D4×C71C2211yescharacteristic
37C22×C42×C71C2211yescharacteristic
38#390×C71C2211yescharacteristic
39#389×C71C2211yescharacteristic
40#388×C71—21no—
41#386×C71—21no—
42#380×C71C2211yescharacteristic
43C4×C4⋊C4×C71C2211yescharacteristic
44C4×C22⋊C4×C71—21no—
45C22×Dic142C2312yes—
46C2×#11765853—24no—
47C2×#11765853—24no—
48C2×#11765853—42no—
49C22×C71⋊C8C2312yes—
50C22×C71⋊C8C2312yes—
51#14556128—22no—
52#14556128—22no—
53#14556107—42no—
54#14556107—42no—
55C2×C4⋊Dic71—21no—
56C2×C4⋊Dic71—21no—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—21no—
59C2×C4⋊Dic71C2312yes—
60C2×Dic71⋊C4—22no—
61C2×C22⋊Dic71—22no—
62#14556061—42no—
63C4⋊(C71⋊C8)—22no—
64C4⋊(C71⋊C8)D414yescomplemented
65#14556051—24no—
66#14556051—24no—
67#14556051—42no—
68#14556049—42no—
69C22⋊(C71⋊C8)—24no—
70#14556039—24no—
71C71⋊#95—22no—
72C71⋊#95—22no—
73C22×D4×C71—21no—
74C22×D4×C71C2311yescharacteristic
75C23×C4×C71—21no—
76C23×C4×C71D411yescharacteristic
77C23×C4×C71D411yescharacteristic
78C23×C4×C71C2311yescharacteristic
79#121×C71—22no—
80#121×C71—22no—
81C4×D4×C71—22no—
82C4×D4×C71—22no—
83C4×D4×C71—44no—
84C2×C4⋊C4×C71—21no—
85C2×C4⋊C4×C71C2311yescharacteristic
86C2×C4⋊C4×C71C2311yescharacteristic
87C2×C4⋊C4×C71C2311yescharacteristic
88C2×C22⋊C4×C71—21no—
89C2×C22⋊C4×C71—21no—
90C2×C22⋊C4×C71—21no—
91C2×C22⋊C4×C71—21no—
92C2×C22⋊C4×C71—21no—
93C2×C42×C71—22no—
94C2×C42×C71—22no—
95C2×C42×C71—21no—
96C2×C42×C71D411yescharacteristic
97C2×C42×C71D411yescharacteristic
98C2×C42×C71C2311yescharacteristic
99C2×C42×C71C2311yescharacteristic
100#95×C71—21no—
101#95×C71—21no—
102#95×C71—21no—
103C2×Dic142—22no—
104C2×Dic142—22no—
105C2×Dic142—22no—
106C2×Dic142—22no—
107C2×Dic142—22no—
108C2×Dic142—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111#11765853—48no—
112C2×C71⋊C8—22no—
113C2×C71⋊C8—22no—
114C2×C71⋊C8—22no—
115C2×C71⋊C8—22no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8C2×D414yes—
118C4⋊Dic71—22no—
119C4⋊Dic71—22no—
120C4⋊Dic71—42no—
121C4⋊Dic71—42no—
122C4⋊Dic71—42no—
123C4⋊Dic71—42no—
124Dic71⋊C4—44no—
125C22⋊Dic71—44no—
126C24×C71—21no—
127C24×C71C2×D411yescharacteristic
128C2×D4×C71—24no—
129C2×D4×C71—21no—
130C2×D4×C71—21no—
131C2×D4×C71—21no—
132C2×D4×C71—21no—
133C2×D4×C71—21no—
134C2×D4×C71—42no—
135C2×D4×C71—42no—
136C2×D4×C71#4112yes—
137C22×C4×C71—22no—
138C22×C4×C71—21no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—21no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
151C22×C4×C71—41no—
152C22×C4×C71—42no—
153C22×C4×C71—42no—
154C22×C4×C71—42no—
155C22×C4×C71C2×D411yescharacteristic
156C22×C4×C71C2×D411yescharacteristic
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D411yescharacteristic
159C22×C4×C71C2×D411yescharacteristic
160C22×C4×C71C2411yescharacteristic
161C4⋊C4×C71—21no—
162C4⋊C4×C71—21no—
163C4⋊C4×C71—21no—
164C4⋊C4×C71—21no—
165C4⋊C4×C71—21no—
166C4⋊C4×C71—42no—
167C4⋊C4×C71#4112yes—
168C22⋊C4×C71—24no—
169C22⋊C4×C71—42no—
170C22⋊C4×C71—42no—
171C22⋊C4×C71—42no—
172C22⋊C4×C71—42no—
173C42×C71—22no—
174C42×C71—22no—
175C42×C71—21no—
176C42×C71—21no—
177C42×C71—42no—
178C42×C71C2×D412yes—
179C42×C71#4112yes—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71—21no—
183C23×C71—21no—
184C23×C71—21no—
185C23×C71—21no—
186C23×C71—22no—
187C23×C71—41no—
188C23×C71—41no—
189C23×C71—41no—
190C23×C71—41no—
191C23×C71—41no—
192C23×C71—41no—
193C23×C71#13712yes—
194C23×C71C22×D411yescharacteristic
195D4×C71—22no—
196D4×C71—22no—
197D4×C71—48no—
198D4×C71—41no—
199D4×C71—41no—
200C2×C4×C71—21no—
201C2×C4×C71—21no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—22no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—44no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—41no—
218C2×C4×C71—41no—
219C2×C4×C71—44no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—84no—
227C2×C4×C71#12112yes—
228C2×C4×C71#12112yes—
229C2×C4×C71AGL(1,ℤ/8)12yes—
230C2×C4×C71#13712yes—
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×C4×C71H(5,𝔽2)11yescharacteristic
236C2×C4×C71H(5,𝔽2)11yescharacteristic
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243Dic142—42no—
244Dic142—42no—
245Dic142—42no—
246Dic142—42no—
247C71⋊C8—24no—
248C71⋊C8—42no—
249C22×C71—22no—
250C22×C71—22no—
251C22×C71—22no—
252C22×C71—41no—
253C22×C71—41no—
254C22×C71—41no—
255C22×C71—41no—
256C22×C71—42no—
257C22×C71—42no—
258C22×C71—42no—
259C22×C71—81no—
260C22×C71—81no—
261C22×C71—81no—
262C22×C71—81no—
263C22×C71C2×#12111yescharacteristic
264C22×C71#52511yescharacteristic
265C22×C71#53511yescharacteristic
266C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C71C2×#13711yescharacteristic
268C22×C71C2×#13711yescharacteristic
269C22×C71#57511yescharacteristic
270C4×C71—22no—
271C4×C71—42no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71—41no—
275C4×C71—81no—
276C4×C71—81no—
277C4×C71#46814yes—
278C4×C71#54611yescharacteristic
279C4×C71#55011yescharacteristic
280C4×C71#55311yescharacteristic
281C4×C71#55511yescharacteristic
282Dic71—82no—
283Dic71—82no—
284#24659—711nomaximal, Sylow
285C2×C71—42no—
286C2×C71—82no—
287C2×C71#241411yescharacteristic
288C2×C71C2×#46811yescharacteristic
289C2×C71#305611yescharacteristic
290C2×C71#311011yescharacteristic
291C2×C71#311611yescharacteristic
292C2×C71#321411yescharacteristic
293C2×C71#321811yescharacteristic
294C2×#461—711no—
295C2×#432—711no—
296#2857—711no—
297C4×#121D7111yescharacteristic
298#2036—712no—
299#2000—712no—
300#1981—711no—
301#1973—712no—
302#1947—711no—
303#1935—712no—
304#1802—711no—
305C71#2465911yescharacteristic, complemented, Sylow
306C22×SD16—712no—
307C2×C4⋊Q8—711no—
308C2×C22⋊Q8—712no—
309C2×#121C2×D7111yescharacteristic
310C2×C4×D4—21no—
311C2×C4×D4C2×D7111yescharacteristic
312C22×C42C2×D7111yescharacteristic
313#461—1424no—
314#432—1424no—
315C2×#107—712no—
316C2×C4⋊C8—711no—
317C2×C4⋊C8—711no—
318C2×#103—712no—
319C2×#103—712no—
320C2×#103—712no—
321C2×#102—712no—
322C2×C22⋊C8—712no—
323C2×#97—712no—
324#391—712no—
325#390C2×D7111yescharacteristic
326#389—1421no—
327#389—1421no—
328#389C2×D7111yescharacteristic
329#388—21no—
330#386—21no—
331#384—711no—
332#380—1421no—
333#380—1421no—
334#380C2×D7111yescharacteristic
335C4×C4⋊C4C2×D7111yescharacteristic
336C4×C22⋊C4—21no—
337#340—712no—
338#340—712no—
339C22×Q8—712no—
340C22×D4—21no—
341C22×D4C22×D7111yescharacteristic
342C23×C4—21no—
343C23×C4C22⋊D7111yescharacteristic
344C23×C4C22⋊D7111yescharacteristic
345C23×C4C22×D7111yescharacteristic
346C2×SD16—2842no—
347C2×SD16—1424no—
348C2×SD16—1424no—
349C22×C8—712no—
350C22×C8—712no—
351C4⋊Q8—1422no—
352C4⋊Q8—1422no—
353C22⋊Q8—2842no—
354C22⋊Q8—2842no—
355#121—22no—
356#121—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—44no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—712no—
362C2×C4⋊C4—1422no—
363C2×C4⋊C4—1421no—
364C2×C4⋊C4—1421no—
365C2×C4⋊C4—1421no—
366C2×C4⋊C4—1421no—
367C2×C4⋊C4C22×D7111yescharacteristic
368C2×C4⋊C4C22×D7111yescharacteristic
369C2×C4⋊C4C22×D7111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1422no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42C22⋊D7111yescharacteristic
380C2×C42C22⋊D7111yescharacteristic
381C2×C42C22×D7111yescharacteristic
382C2×C42C22×D7111yescharacteristic
383#107—2842no—
384C4⋊C8—714no—
385C4⋊C8—1422no—
386#103—2842no—
387#103—1424no—
388#103—1424no—
389#102—2842no—
390C22⋊C8—1424no—
391#97—1424no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—1422no—
396#95—1422no—
397C24—21no—
398C24C2×C22⋊D7111yescharacteristic
399C2×Q8—1422no—
400C2×Q8—1422no—
401C2×Q8—1422no—
402C2×Q8—1422no—
403C2×Q8—1422no—
404C2×Q8—1422no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4#1176586512yes—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—1422no—
433C22×C4—1422no—
434C22×C4C2×C22⋊D7111yescharacteristic
435C22×C4C2×C22⋊D7111yescharacteristic
436C22×C4C2×C22⋊D7111yescharacteristic
437C22×C4C2×C22⋊D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4C23×D7111yescharacteristic
440SD16—2848no—
441C2×C8—714no—
442C2×C8—1422no—
443C2×C8—1422no—
444C2×C8—1422no—
445C2×C8—1422no—
446C2×C8—1422no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—2844no—
454C4⋊C4—2842no—
455C4⋊C4—2842no—
456C4⋊C4—2842no—
457C4⋊C4—2842no—
458C4⋊C4—1422no—
459C4⋊C4—1422no—
460C4⋊C4#1176586512yes—
461C22⋊C4—24no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—2844no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—22no—
471C42—42no—
472C42D4×D7112yes—
473C42#1176586412yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23#1455615812yes—
488C23C22×C22⋊D7111yescharacteristic
489Q8—2842no—
490Q8—2842no—
491Q8—2842no—
492Q8—2842no—
493D4—22no—
494D4—22no—
495D4—48no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
525C2×C4—2842no—
526C2×C4—2842no—
527C2×C4—2842no—
528C2×C4—2842no—
529C2×C4—2842no—
530C2×C4—2842no—
531C2×C4#1455610012yes—
532C2×C4#1455610012yes—
533C2×C4#1455615412yes—
534C2×C4#1455615812yes—
535C2×C4C2×D4×D7111yescharacteristic
536C2×C4C2×#1176586411yescharacteristic
537C2×C4C2×#1176586511yescharacteristic
538C2×C4C2×#1176586511yescharacteristic
539C2×C4#1455617411yescharacteristic
540C2×C4#1455617411yescharacteristic
541C8—2842no—
542C8—1424no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1455610011yescharacteristic
558C22#1798651011yescharacteristic
559C22#1798656311yescharacteristic
560C22C2×#1455615411yescharacteristic
561C22C2×#1455615811yescharacteristic
562C22C2×#1455615811yescharacteristic
563C22#1798679711yescharacteristic
564C4—22no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—5682no—
572C4—5682no—
573C4#1798624314yes—
574C4#1798664511yescharacteristic
575C4#1798667011yescharacteristic
576C4#1798669511yescharacteristic
577C4#1798671411yescharacteristic
578C2—42no—
579C2—82no—
580C2#2259175811yescharacteristic
581C2C2×#1798624311yescharacteristic
582C2#2259797211yescharacteristic
583C2#2259843911yescharacteristic
584C2#2259850211yescharacteristic
585C2#2259933111yescharacteristic
586C2#2259936411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC4×#121×C71C2
socleC23×C71C22×D4
prime radicalC2×#121×C71C22
hypercenterC4×#121D71
hypocenterC71#24659
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#17986204C2semidirect
2. C2×#17986056C2semidirect
3. #22596692C2semidirect
4. #22589543C2semidirect
6. #22589263C2semidirect
7. #22589220C2semidirect
8. #22589098C2semidirect
9. #22589029C2semidirect
10. #22588340C2semidirect
64. C4⋊(C71⋊C8)D4semidirect
305. C71#24659semidirect

Maximal subgroups

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

subgroupindex
C2×#179862042
C2×#179860562
#225966922
#225895432
#225893432
#225892632
#225892202
#225890982
#225890292
#225883402
C4×#121×C712
#2465971